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ISBN print: 978-3-0364-0986-3
ISBN eBook: 978-3-0364-1986-2
DOI book: 10.4028/b-U89t4a
© 2026 Trans Tech Publications Ltd, All Rights Reserved

Advanced Numerical and AI Strategies for Material Forming

Edited by
Benjamin Klusemann

Advanced Numerical and AI Strategies for Material Forming

Special topic volume with invited peer-reviewed papers only

Edited by
Benjamin Klusemann

Ścientific.Net

Copyright © 2026 Trans Tech Publications Ltd, Switzerland

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Volume 1050 of
Key Engineering Materials
ISSN print 1013-9826
ISSN web 1662-9795

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Key Engineering Materials, ISSN: 1662-9795, Vol. 1050, pp -1-0

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Accepted:
2026-04-13

Online:
2026-04-16

Preface

The original version of this paper is available on https://www.scientific.net/KEM.1050.-1.pdf

This issue covers all aspects related to advanced and non-conventional modeling and simulation of forming processes, including the application of machine learning and AI techniques. By that, we consider developments in the field of numerical simulation that can eventually and potentially be applied to the simulation of any forming process. Among these, the following non-exclusive list of topics is considered:

Advances in Finite Element Technology and their applications

Meshless and mesh-free methods

AI-based computing

Advanced shell methodologies, solid/shell formulations

Numerical techniques for challenging processes (3D priting, FSW, LFW, ...)

Speeding up simulations (parallel computing, model order reduction, AI, ...)

Data-driven simulation and control of forming processes

Hybrid modeling approaches, combining data-driven and numerical approaches

This issue also presents the research in the following challenging fields:

Computer-aided technology design

Simulation-based real-time control

AI-based material and process models

Supercomputing (GPUs, ...) and deployed systems

Virtual and augmented reality solutions

Alleviating mesh constraints (anisotropic meshes, advanced remeshing, meshless, ...)

Data-driven simulations and hybrid approaches.

Editors

The original version of this paper is available on https://www.scientific.net/KEM.1050.-1.pdf

Benjamin Klusemann
Leuphana University Lüneburg
Lukasz Madej
AGH University of Krakow

Co-editors

The original version of this paper is available on https://www.scientific.net/KEM.1050.-1.pdf

Celal Soyarslan
University of Twente
Marc Bernacki
MINES ParisTech
Junhe Lian
RWTH Aachen University
Marta Oliveira
University of Coimbra
Till Clausmeyer
TU Chemnitz
Tuncay Yalcinkaya
Middle East Technical University

Table of Contents

Preface

Numerical Investigation of Deposition Efficiency Influencing Factors in the Friction Surfacing Process

Ahmed Elbossily, Zina Kallien, Rupesh Chafle, Kirk A. Fraser, Mohamadreza Afrasiabi, Benjamin Klusemann

Prediction of Strip Width Deviation in Hot Strip Roughing Mills Based on Machine Learning Regression

Daniel Ewen, Johannes Seitz, Sebastian Kallabis, Reinhold Franke, Joachim Denker, Lukas Lackmann, Bernd Kuhlenkötter

Multi-Objective Bayesian Optimization of Dual-Phase Steel Microstructures for Minimal Damage Initiation

Irene Biermann, Niklas Fehlemann, Maximilian Hribsek, Sebastian Münstermann

Algebraic Hierarchical Graph Neural Networks for Forming Simulation of Thermoplastic Composite Materials

Tobias Würth, Niklas Freymuth, Johannes Mitsch, Philipp Dahlinger, Tai Hoang, Gerhard Neumann, Luise Kärger

Explainable Machine Learning and 3D Visualization for Rotary Tube Bending: Expert Evaluation of a Web-Based Tool

Alireza Yazdani, Jonas Knoche, Bernd Engel, Kristof Van Laerhoven

Investigation of Code Optimization Strategies for Enhancing the Performance of Static Recrystallization Cellular Automata Models

Mateusz Sitko, Szymon Sitarz, Łukasz Madej

Parameter Optimization and Data-Driven Soft-Sensor Framework for Torque Prediction in Bobbin-Tool Friction Stir Welding of AA2024

Fares Nouira, Luciano Bergmann, Frederic E. Bock, Ghada Bouattour, Marco Pacchione, Benjamin Klusemann

Long Short-Term Memory to Predict the Quality Parameters in the Hot Rolling Process

Tamoghna Majumder, Nilesh Thakare, Holger Brüggemann, Emad Scharifi, Junhe Lian

Al-Based Predictions of Forming Effects for Enhanced Crash Simulation

Ingolf Lepenies

Residual Learning-Based Synthetic Data for Hybrid Metamodeling in the Stamping of an Automotive Door Panel

Laura Muñiz, Javier Trinidad, Julian Ramirez de Okariz, Eduardo García, Lander Galdos

Robust and Feature-Aware Arbritary Lagrangian-Eulerian Method for Material Forming Applications

Jesus Oswaldo Garcia, Z. José R. Alves, Ugo Ripert, Christine Beraudo

On-Line Prediction of Dry Zones During Composite Process Through Digital Shadow Approach

Philippe Le Bot, Nathan Lauzeral, Olivier Fouché, Nihad Siddig, Damien Lecointe, Ibrahim Abdullah, Florent Niget, Christophe Marchand

Graph Neural Network for Draw-in Prediction in Sheet Metal Stamping

Hui Wang, Taegyun Ahn, Lu Huang, Seog Chan Oh, Liang Huang, Andrey Ilinich, Sathya Dev, Jeong Whan Yoon

A Bayesian Data Assimilation Framework for Characterizing Recrystallization in Steels

Bentolhoda Jamshidi, Jos Havinga, Ton van den Boogard

Predicting Lateral Flow in Hot Sheet Metal Rolling Using Symbolic Regression

Daniel Luis Mederos, Amirali Hashemzadeh, Antonella Cometa, Celal Soyarslan, Frederic Bock, Ton van den Boogard, Benjamin Klusemann

Physics-Informed Recurrent Neural Networks with Kinematic Constraints for Large Deformation Metal Forming

Francesco Munzone, Javad Hazrati, Wouter Hakvoort, Redmer van Tijum, Ton van den Boogard

An AI-Based Approach to Developing a Microstructural Model for Multi-Stage Hot Deformation Processes

Denis Tretyakov, Nikolay Biba, Vitaliy Belugin, Artur Gartvig, Andrei Shitikov, Sergei Stebunov

Accelerating Microstructural Evolution Simulations in DIGIMU® through a Front-Tracking Lagrangian Solver: Implementation and Validation in AISI 304L Stainless Steel

Franco Jaime, Thomas Sourisseau, Pascal De Micheli, Marc Bernacki

Study on the Apparent Friction Coefficient Between a Deformable and a Rough Rigid Body under Large Sliding

Renan B. Wojciechowski, Marta C. Oliveira, Diogo M. Neto

Using Kolmogorov-Arnold Networks for Discrepancy Modeling of Lateral Flow in Hot Rolling of Steel Slabs

Amirali Hashemzadeh, Frederic Bock, Antonella Cometa, Celal Soyarslan, Benjamin Klusemann, Ton van den Boogard

Modelling Deformation Mechanisms Decomposition, Separation and Compaction in Mechanical Joining Processes of Fiber Reinforced Thermoplastics on Meso Scale

Benjamin Gröger, Johannes Gerritzen, Andreas Hornig, Maik Gude

Combined Experimental and Simulation-Based Approach for Optimizing Roll Forming of Pipes for Hydrogen Infrastructure

Murli Manohar Pandey, Anja Rautenstrauch, Andreas Kunke, Till Clausmeyer

Integrated Artificial Neural Network-Based Approach for Predicting Surface Roughness Parameters in Laser Surface Texturing of Ti6Al4V

Agata Sposato, Giovanni Carpino, Maria Rosaria Saffioti, Giovanna Rotella, Domenico Umbrello

Keywords Index

Authors Index

The original version of this eBook is available on https://www.scientific.net/book/advanced-numerical-and-ai-strategies-for-material-forming/978-3-0364-1986-2

Key Engineering Materials, ISSN: 1662-9795, Vol. 1050, pp 1-7

doi: 10.4028/p-4tzjOZ

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2025-11-24

Revised:
2026-02-17

Accepted:
2026-02-17

Online:
2026-04-16

Numerical Investigation of Deposition Efficiency Influencing Factors in the Friction Surfacing Process

The original version of this paper is available on https://www.scientific.net/KEM.1050.1.pdf

Ahmed Elbossily 1,a*, Zina Kallien 1,2, b, Rupesh Chafle 2,c, Kirk A. Fraser 3, d, Mohamadreza Afrasiabi 4,e, and Benjamin Klusemann 1,2,f

1 Institute for Production Technology and Systems, Leuphana Universität Lüneburg, Universitätsallee 1, 21335, Lüneburg, Germany

2 Institute of Material and Process Design, Helmholtz-Zentrum Hereon, Max-Planck-Straße 1, 21502 Geesthacht, Germany

3 National Research Council Canada, Saguenay, QC, Canada

4 Advanced Manufacturing Lab, ETH Zurich, Zurich, Switzerland

a Ahmed.Elbossily@Leuphana.com, b zina.kallien@hereon.de, c rupesh.chafle@hereon.de, d  kirk.fraser@cnrc-nrc.gc.ca, e  afrasiabi@ethz.ch, f  benjamin.klusemann@leuphana.de

Keywords: Numerical Study, Smoothed Particle Hydrodynamics, Friction Surfacing, Deposition Efficiency

Abstract

This work presents a numerical study of the friction surfacing process using a GPU - accelerated Smoothed Particle Hydrodynamics (SPH) framework previously validated against experimental observations. The model is employed to examine how thermal boundary conditions, rod diameter, and rod bending angle influence material deposition efficiency and the resulting deposit geometry. Variations in rod diameter are shown to influence both the thermal response and the contact pressure, with smaller rods producing higher efficiency but exhibiting greater process fluctuations. The findings highlight the critical roles of thermal management and geometric configuration in optimizing friction surfacing performance and provide actionable insight for experimental design and process control in solid-state deposition technologies.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1050.1.pdf

Friction surfacing (FS) is a solid-state process that uses frictional heat and plastic deformation to deposit a consumable material. It minimizes distortion and further thermal effects while promoting dynamic recrystallization and producing a fine-grained microstructure [1]. These qualities make it useful for coating, repair, and additive manufacturing applications [2, 3, 4]. Although numerous experimental investigations have examined the FS process, numerical investigations remain relatively limited.

Early numerical investigations primarily employed the finite element method (FEM) to study the temperature field [5], where the deposition was modeled using element activation, neglecting actual material flow and thus limiting deposition formation prediction. Subsequent thermo-mechanical FEM analyses were developed [6, 7], but mesh-based approaches remain inadequate for capturing the severe deformations and material separation inherent to the FS process. Consequently, meshless methods such as Smoothed Particle Hydrodynamics (SPH) have emerged as a promising alternative [8].

Aval [9] developed a SPH model to simulate the FS process, demonstrating strong agreement with experimental data, and utilized it to analyze the effects of axial feeding rate and traverse speed on deposition efficiency and interface bonding [10,11]. Elbossily et al. [12] introduced a GPU-based SPH framework for the FS process with computational optimizations and stability enhancements. A new material separation criterion based on shear stress was proposed to improve the simulation of the deposition behavior. Validated experimentally, the model captures key FS mechanisms such as material flow, rod flash formation, and deposition geometries for different process parameters, providing a robust tool for analyzing and optimizing FS processes.

This study utilizes the previously developed SPH model [12] to analyze deposition efficiency under varying substrate thermal boundary conditions, rod diameters, and rod bending angles.

Model Setup

The original version of this paper is available on https://www.scientific.net/KEM.1050.1.pdf

Fig. 1a shows the SPH model setup used in this study, including the geometries, materials, and the number of SPH particles. The initial spacing between all SPH particles is set to 1 mm , which is selected to balance model prediction accuracy with simulation run time. The SPH model consists of two deformable bodies: a rotating rod and a substrate. During the deposition process, the substrate translates laterally at a constant traverse speed, while material is deposited onto its surface, Fig. 1b. Thermal dissipation occurs via convection through the model's external surfaces. The material behavior is modeled using the Johnson-Cook constitutive model [13]. Details regarding the material properties and simulation parameters are available in [12].

[picture]

Fig. 1: FS model: (a) main dimensions, material type, and particle count; (b) visualization of rod feed direction, rod rotation direction, substrate traverse speed direction, advancing side (AS), retreating side (RS), and the resulting rod flash and material deposition at the end of the simulation.

Fig. 1. FS model: (a) main dimensions, material type, and particle count; (b) visualization of rod feed direction, rod rotation direction, substrate traverse speed direction, advancing side (AS), retreating side (RS), and the resulting rod flash and material deposition at the end of the simulation.

Joining criterion. The joining criterion proposed by Elbossily et al. [12] is revised to be independent of the joining temperature. This modification reduces the numerical constraints on the deposition behavior, while the temperature influence remains captured through the material's temperaturedependent shear strength τiallow . By eliminating the joining temperature as an input parameter, deposition is no longer forced to occur at a predefined temperature threshold, which previously mimicked a binary ( 0 or 1 ) condition and could not reflect realistic material behavior. The joining criterion is defined as

 Joined i={1,(τi≥τiallow )∧(εipl≥0.01)0, Otherwise (1)

where εipl and τi are the plastic strain and the shear stress of the particle i, respectively.
Simulation sets. The numerical study includes seven simulation sets to evaluate the influence of the substrate thermal boundary conditions and the rod geometric variations on the FS process. Set 0 is the reference simulation, with heat loss to the surroundings modeled via a free convection coefficient of 5 W/m2∘C from the substrate's sides and bottom surfaces. A higher heat convection coefficient of 1000 W/m2∘C is employed on the rod top surface to mimic heat dissipation to the rest of the rod following the approach by Zou et al. [17]. Set 1 introduces substrate preheating by initializing the temperature of the substrate at 200∘C and applying a constant temperature of 200∘C to the bottom surface of the substrate. Set 2 imposes cooling conditions by increasing the heat convection to 5000 W/m2∘C for the substrate's sides and bottom surfaces. The next two sets modify the rod geometry by increasing the rod diameter from 20 mm to 25 mm in Set 3 and reducing it to 15 mm in Set 4. The last two setups examine potential rod bending during the process. In Sets 5 and 6, bending

Table 1: Summary of the simulation sets and corresponding process setup, including substrate boundary conditions (applied heat convection value hc and initial temperature Tinit  ), rod diameter, and rod bending angle.

set | simulation setup | substrate hc (W/m² °C) | substrate Tinit (°C) | rod diameter (mm) | bending angle (°)
0 | reference | 5 | 20 | 20 | 0
1 | heated | 5 | 200 | 20 | 0
2 | cooled | 5000 | 20 | 20 | 0
3 | bigger diameter | 5 | 20 | 15 | 0
4 | smaller diameter | 5 | 20 | 25 | 0
5 | bending 5° | 5 | 20 | 20 | 5
6 | bending 10° | 5 | 20 | 20 | 10

angles of 5∘ and 10∘, respectively, are applied relative to the vertical axis at the rod clamping position. The chosen bending angles are relatively large to ensure noticing the rod bending effect, given the rod's short length used in the simulation. The process parameters remain constant across all simulations: 900 rpm rotational speed, 1.8 mm/s rod feed rate, and 6 mm/s substrate traverse speed. Table 1 summarizes the investigated simulation conditions.

Results and Discussion

The original version of this paper is available on https://www.scientific.net/KEM.1050.1.pdf

This section discusses the deposition efficiency for all sets summarized in Table 1. Deposition efficiency quantifies how effectively the rod material is utilized during the process. It is defined as the ratio of the actual deposited material in the simulation to the theoretical amount that could have been deposited without flash, calculated as

η=1ρπr2vzt∑i=0Nmi.(2)

where ρ is the material density of the rod, r is the rod radius, vz is the rod feed rate, t is the deposition time, mi is the mass of particle i and N is the total number of deposited particles. Moreover, the effective deposition width is determined as the difference between the 90th and 10th percentile positions of the deposited particles along the y -direction. This gives a robust measure of the effective deposition width as it ignores extreme outliers or isolated particles far from the main deposit.

Effect of changing thermal boundary conditions. Fig. 2 compares the deposition efficiency, the deposition width, and the temperature fields of the rod and the deposit under three thermal boundary conditions: Set 0 (reference), Set 1 (heated substrate), and Set 2 (cooled substrate). Heating the substrate in Set 1 raises the temperature of both the rod and the deposited material. The hotter deposit softens, becomes more deformable, and offers less shear resistance. As the rod interacts with this softened layer, the reduced resistance lowers the deposition efficiency. The material also spreads more, which explains the wider deposition compared with Set 0 . Cooling the substrate in Set 2 has the opposite effect. Lower temperatures stiffen the deposited material and reduce its deformability. This increases its shear resistance during contact with the deformable rod. As a result, deposition efficiency increases but the deposition width decreases, consistent with less deformable material. The trends match very well experimental findings from Krohn et al. [14] and Hoffmann et al. [15].

Effect of changing rod diameter. Fig. 3a shows how the rod diameter affects deposition efficiency, deposition width, and the temperature fields of both the rod and the deposit. Reducing the rod diameter increases deposition efficiency. This trend is linked to the lower temperatures observed with a smaller rod and to the higher contact pressure that develops when both the rod diameter and deposition width decrease. The simulations also show that the larger rod (Set 3) produces lower contact

[picture]

Fig. 2: Deposition efficiency, effective disposition width, and the temperature distribution of the deformed rod and 2D deposition projection for Set 0 (reference), Set 1 (heated), and Set 2 (cooled).

Fig. 2. Deposition efficiency, effective disposition width, and the temperature distribution of the deformed rod and 2D deposition projection for Set 0 (reference), Set 1 (heated), and Set 2 (cooled).

pressure and less variance, i.e., a more stable process overall, Fig. 3b. Its contact-pressure fluctuations are smaller than those of the smaller rod, even though the smaller rod achieves higher efficiency. The observed trend of changing the deposition efficiency with changing the rod diameter is consistent with the experimental findings by Angiwal et al. [16]. They observed that the contact radius decreases with increasing tangential velocity-in this study, associated with larger rod diameters-while Hoffmann et al. [15] reported that the contact radius is directly proportional to the deposition efficiency.

[picture]

Fig. 3: (a) Deposition efficiency, effective disposition width, and the temperature distribution of the deformed rod and 2D deposition projection for Set 0 (reference), Set 3 (bigger diameter), and Set 4 (smaller diameter), and (b) The average and variance of the normalized rod contact pressure during the simulation of the same three sets with respect to Set 0 .

Fig. 3. (a) Deposition efficiency, effective disposition width, and the temperature distribution of the deformed rod and 2D deposition projection for Set 0 (reference), Set 3 (bigger diameter), and Set 4 (smaller diameter), and (b) The average and variance of the normalized rod contact pressure during the simulation of the same three sets with respect to Set 0 .

[picture]

Fig. 4: Deposition efficiency, effective disposition width, and the temperature distribution of the deformed rod and 2D deposition projection for Set 0 (reference), Set 5 (bending mathematical formula ), and Set 6 (bending superscript number ).

Fig. 4. Deposition efficiency, effective disposition width, and the temperature distribution of the deformed rod and 2D deposition projection for Set 0 (reference), Set 5 (bending ), and Set 6 (bending ).

Effect of rod bending angle. Fig. 4 shows the effect of changing the rod bending angle on the deposition efficiency, the deposition width, and the temperature fields of both the rod and the deposit. As the bending angle increases (Set 5 and Set 6), both the deposition width and the total deposited material rise, while the rod temperature drops. The larger bending angle increases the lateral spreading of the material, which widens the deposition. It also changes how the flash interacts with the existing deposit. With more bending, a larger portion of the flash gets into contact with the deposited layer and is deposited. This adds more material to the deposition and reduces the amount of hot flash forming, which lowers the rod temperature.

Conclusion

The original version of this paper is available on https://www.scientific.net/KEM.1050.1.pdf

This work numerically explored the influence of thermal boundary conditions, rod diameter, and rod bending angle on deposition efficiency and material flow behavior in friction surfacing using a GPUaccelerated SPH model. The results demonstrate that substrate temperature critically governs the deformability of the deposited layer, thereby controlling both deposition efficiency and spreading. Preheating the substrate promotes wider but less efficient deposition, whereas enhanced cooling yields higher efficiency with narrower deposits. Geometric variations further modulate the process: smaller rod diameters enhance efficiency through higher contact pressures, while larger rods provide more stable deposition dynamics. Additionally, rod bending increases material redistribution and promotes greater utilization of the rod material by directing more flash into the deposited layer. Overall, the study identifies thermal management and geometric configuration as practical levers for tuning friction surfacing performance. These insights provide a computational foundation for optimizing process setups, guiding experimental design, and informing future work on closed-loop control strategies and more comprehensive multiphysics simulation approaches.

Funding

The original version of this paper is available on https://www.scientific.net/KEM.1050.1.pdf

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No 101001567).

Data Availability

The original version of this paper is available on https://www.scientific.net/KEM.1050.1.pdf

The solver is released as open-source code via GitHub (https://github.com/SPH-SSMP/FE-SPH-GPU-ESAFORM) and the experimental data and simulation results are available at Zenodo (https://doi.org/10.5281/zenodo.18670132).

References

The original version of this paper is available on https://www.scientific.net/KEM.1050.1.pdf

J. Gandra, H. Krohn, R.M. Miranda, P. Vilaça, L. Quintino, J.F. dos Santos, Friction surfacing—A review, J. Mater. Process. Technol., Vol. 214, No. 5, pp. 1062-1093 (2014).

R. Damodaram, P. Rai, S. Cyril Joseph Daniel, R. Bauri, D. Yadav, Friction surfacing: A tool for surface crack repair, Surf. Coat. Technol., Vol. 422, p. 127482 (2021).

A. Sharifi, F. Khodabakhshi, S.F. Kashani-bozorg, A.P. Gerlich, Microstructure and mechanical properties in additive manufacturing by friction surfacing of AA6061 alloy, Mater. Sci. Eng. A, Vol. 884, p. 145520 (2023).

Z. Kallien, L. Rath, A. Roos, B. Klusemann, Application of friction surfacing for solid state additive manufacturing of cylindrical shell structures, Addit. Manuf. Lett., Vol. 8, p. 100184 (2024).

V.I. Vitanov, N. Javaid, Investigation of the thermal field in micro friction surfacing, Surf. Coat. Technol., Vol. 204, No. 16-17, pp. 2624-2631 (2010).

S.M. Bararpour, H. Jamshidi Aval, R. Jamaati, Modeling and experimental investigation on friction surfacing of aluminum alloys, J. Alloys Compd., Vol. 805, pp. 57-68 (2019).

P. Pirhayati, H. Jamshidi Aval, An investigation on thermo-mechanical and microstructural issues in friction surfacing of Al-Cu aluminum alloys, Mater. Res. Express, Vol. 6, No. 5 (2019).

G.R. Liu, M.B. Liu, Smoothed Particle Hydrodynamics: A Meshfree Particle Method, World Scientific Publishing Co., Singapore (2003).

H. Jamshidi Aval, Comprehensive thermo-mechanical simulation of friction surfacing of aluminum alloys using smoothed particle hydrodynamics, Surf. Coat. Technol., Vol. 419, p. 127274 (2021).

S.M. Bararpour, H. Jamshidi Aval, R. Jamaati, M. Javidani, Experimental and numerical investigation of Al16Si alloy friction surfacing on AA1050 substrate: Effect of axial feeding rate, Surf. Coat. Technol., Vol. 468, p. 129778 (2023).

S.M. Bararpour, H. Jamshidi Aval, R. Jamaati, M. Javidani, Investigation of the effect of traverse speed in friction surfacing of Al-16Si alloy by smoothed-particle hydrodynamics simulation and experimental study, Comput. Part. Mech., Vol. 11 (2023).

A. Elbossily, Z. Kallien, R. Chafle, K.A. Fraser, M. Afrasiabi, M. Bambach, B. Klusemann, GPU-accelerated meshfree computational framework for modeling the friction surfacing process, Comput. Part. Mech. (2025), doi:10.1007/s40571-025-01048-2.

G.R. Johnson, W.H. Cook, A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures, Proc. 7th Int. Symp. Ballistics, pp. 541-547 (1983).

H. Krohn, S. Hanke, M. Beyer, J.F. dos Santos, Influence of external cooling configuration on friction surfacing of AA6082-T6 over AA2024-T351, Manuf. Lett., Vol. 5, pp. 17-20 (2015).

M. Hoffmann, E.A. Duda, P. Aspes, B. Klusemann, Effect of ambient conditions in friction surfacing, Weld. World, Vol. 69, No. 2, pp. 397-406 (2025).

H. Agiwal, H. Yeom, K. Sridharan, S. Rudraraju, F.E. Pfefferkorn, Radius of Contact During Friction Surfacing of Stainless Steel 304L: Effect of Spindle Speed and Rod Diameter, J. Manuf. Sci. Eng., Vol. 146, No. 2, p. 021005 (2023).

Y. Zou, W. Li, X. Yang, V. Patel, Z. Shen, Q. Chu, F. Wang, H. Tang, F. Cui, M. Chi, Characterizations of dissimilar refill friction stir spot welding 2219 aluminum alloy joints of unequal thickness, J. Manuf. Process., Vol. 79, pp. 91-101 (2022).

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The original version of this eBook is available on https://www.scientific.net/book/advanced-numerical-and-ai-strategies-for-material-forming/978-3-0364-1986-2

Key Engineering Materials, ISSN: 1662-9795, Vol. 1050, pp 9-20

doi: 10.4028/p-GLpuV9

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2025-12-18

Revised:
2026-02-24

Accepted:
2026-02-25

Online:
2026-04-16

Prediction of Strip Width Deviation in Hot Strip Roughing Mills Based on Machine Learning Regression

The original version of this paper is available on https://www.scientific.net/KEM.1050.9.pdf

Daniel Ewen 1,a*, Johannes Seitz 1, b, Sebastian Kallabis 2,c, Reinhold Franke 2, d, Joachim Denker 3,e, Lukas Lackmann 3,f and Bernd Kuhlenkötter 1, g

1 Chair of Production Systems, Ruhr-University Bochum, Bochum, Germany

2 thyssenkrupp steel europe AG, Duisburg, Germany

3 Asinco GmbH, Duisburg, Germany

a  ewen@Ips.ruhr-uni-bochum.de, b  seitz@Ips.ruhr-uni-bochum.de (*corresponding author) c sebastian.kallabis@thyssenkrupp-steel.com, d reinhold.franke@thyssenkrupp-steel.com, e  joachim.denker@th-koeln.de, f  lukas.lackmann@asinco.de, 9kuhlenkoetter@lps.ruhr-uni-bochum.de 

Keywords: hot strip rolling, prediction, machine learning, predictive quality, width deviation

Abstract

Previous research shows that predicting width deviation inherits a central importance in hot rolling processes, so that the pass planning in the hot strip mill (HSM) can be optimized. These predictions can be enabled using machine learning, complementing analytical formulations of width spread. For reliable production, it is important for the plant operator to be able to control the geometry with high accuracy across the entire plant. Therefore, the width must be accurately known throughout the entire HSM. This paper aims on the prediction of width deviation in early product stages during the roughing mill processes, where the major deformation takes place, and thus also has the most significant influence on the width spread. Therefore, this paper takes industrial data into account which is also used for the roll passing planning. To achieve a prediction during rough rolling for the width after exiting the mill (future state of the strip width), various machine learning algorithms were implemented and tested. The prediction results are evaluated against an inline width measurement, where the XGB model performs best with a Root Mean Squared Error (RMSE) of 1.11 mm . Subsequently, feature importance analyses are used to examine which features are relevant for the prediction result and to elaborate which significance process- and geometry data has on the same strip.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1050.9.pdf

Hot strip rolling is an essential process in the steel industry. In the EU-27, hot-rolled steel strip accounts for the largest proportion of hot-rolled products, because it can be processed into a wide variety of other products in subsequent manufacturing steps [1]. At the same time, the HSM is highly relevant in terms of observing geometric tolerances. Studies have shown that defective HSM products, such as under-width and over-width, are problematic in further processing (e.g. cold rolling) and can result in rejection by critical customers [ 2,3 ]. Consequently, controlling width accuracy is an essential challenge due to width spread during hot strip rolling [3]. In the context of process optimization in production systems, the use of machine learning (ML) is becoming increasingly important [4]. Many companies already store huge amounts of data that may contain hidden patterns which can be identified and used to optimize processes, such as hot strip rolling, through the use of ML amongst other data science approaches [5].

Conventional hot strip mills are usually based on two sections in which the product is processed differently. First, after leaving the furnace, the steel slab enters the roughing mill, which is responsible for the major forming work by passing the slab through a rolling stand in a reversing process until it results in a pre-strip. After leaving the roughing section, first geometric measurements are available to control the products' geometric accuracy. Subsequently, the operators can adjust the pre-strip by,
for example, trimming the front and end. The final thickness is then achieved in the finishing mill, which usually consists of several rolls in direct succession. Before the strip is coiled, it passes through a cooling section and is then measured geometrically and in cross-section. The schematic structure of a HSM is shown in Figure 1.

[picture]

Fig. 1. Schematic structure of a HSM, containing the roughing mill, finishing mill and coiling, exemplified by the facility at thyssenkrupp Steel Europe in Duisburg

Fig. 1. Schematic structure of a HSM, containing the roughing mill, finishing mill and coiling, exemplified by the facility at thyssenkrupp Steel Europe in Duisburg

Within the scope of strip width control deterministic equations and mathematical models are available to calculate the width spread during hot rolling such as the Shibahara Spread Model (SSM) [6]. In addition to mathematical approaches, data-centric approaches are being used within the last decades for predictions due to increasing process data availability. An approach on width spread prediction in hot strip rolling was presented by Hashemzadeh et al. and is based on a combination of FEM analysis and Artificial Neural Networks (ANN) whereby they achieve high performance in the simulation environment [7]. When considering industrial data, both nominal (scalar) as well as time-dependent datasets are used for predictions. Wu et al. present a hybrid prediction model that incorporates historical and real-time (scalar) data into the prediction [8]. At the same time, they also highlight the absence of sufficient control equipment in HSM. Among the time-dependent approaches, Latham et al. developed a model for the categorization of width errors to make them accessible to operators in their decision-making [9]. Some authors, such as Xin et al. [10], who developed a time-dependent hybrid model approach, based on Convolutional Neural Networks (CNN) and Long-Short Term Memory (LSTM) models, criticize the accuracy of these purely mathematical models, which are currently often used in production planning. This statement is supported by Gao et al. [11], who mention the relevance of real-time measurement systems in the context of width prediction. In a heavy plate mill, the authors therefore investigate the prediction of width spread in finishing processes using Stochastic Configuration Networks (SCN) [11]. By considering two width gauge sensors, before and after the rolling pass, as well as process parameters like temperature, reduction ratio and rolling forces, Dong et al. demonstrate that the forming work plays a decisive role in width deviation [12]. They conclude that a high thickness reduction has a major influence on width deviation which essentially occurs at the roughing mill during the forming of the slab into pre-strip. Therefore, research is being also conducted in prediction of the width deviation during rough rolling. An example of this work is an hybrid approach by Zhong et al. [13, 14], in which the authors used ML to optimize the SSM for width prediction in a HSM roughing mill. Another approach on roughing mill exit width prediction is presented by Ji et al., who optimized a linear regression model by using genetic algorithm for hierarchical clustering. Concurrently, they also emphasize the relevance of transparent prediction models for industrial applications and refer to some prediction models as black boxes [15].

This paper addresses the prediction of the width when leaving the roughing mill (as pre-strip) and uses industrial data as the basis for predictions. This investigation aims to develop a prediction model for the width spread at the end of the roughing mill and to identify relevant knowledge to achieve further improvements in the future. For this purpose, decision tree models and neural networks are created, optimized, and evaluated for prediction. Finally, the results are interpreted in an industrial context with regard to the predicted overall width compared to the measured width, and the relevance of various features is presented.

Methodology

The original version of this paper is available on https://www.scientific.net/KEM.1050.9.pdf

The methodology for width deviation prediction for the exit of the roughing mill section is based on data preprocessing, followed by the construction of a data pipeline and the selection of appropriate ML models for prediction and comparison. The width predictions are based on industrial data and provided by a HSM from thyssenkrupp Steel Europe AG located in Duisburg, Germany. The dataset contains various features across the entire HSM (e.g. geometry measurement data, rolling forces, calculated process settings and target geometry data).

Data Preprocessing

The original version of this paper is available on https://www.scientific.net/KEM.1050.9.pdf

The first step, therefore, was to reduce the data set to the important part for this research. Based on the feature names, the data set was first narrowed down manually. To do this, all process data that occurred after the roughing mill was removed by for example eliminating features that deal with the finishing mills (F1 - F {i} ) or other parts of the HSM that are located behind the roughing section. This significantly reduced the number of features in the data set. The relevant plant section is illustrated in Figure 2, where everything to the left of the roughing section (reheating furnace) as well as the information from the roughing section itself is included and everything to the right (finishing mill) is excluded.

[picture]

Fig. 2. Structure of the investigated HSM roughing section.

Fig. 2. Structure of the investigated HSM roughing section.

Prediction methodology

The original version of this paper is available on https://www.scientific.net/KEM.1050.9.pdf

This work targets to predict the width spread on the final pre-strip, which can be calculated as the difference between the exit width of the pre-strip (shown in as red triangle in figure 2) and the entry geometry after the first five rolling passes. It is important to note, that the illustrated width measurement system in figure 2 is the first actual measurement of the width behind the furnace. The calculation for the width spread (target =y ) is shown in equation (1).

y=wexit −wentry (1)

Moreover, it is imperative for the plant operators to be aware of the selected target value at the earliest opportunity, thus enabling the assessment of product quality in advance and, if necessary, facilitating correction during the process. In order to predict the width spread, deterministic width calculations from the HSM according to the rolling passes 1−5 and process settings such as forces, roller settings, steel grades, etc. are considered in the prediction. The prediction methodology is therefore based on the calculated width after the fifth rolling pass and predicts the width after edge rollers and thickness reduction of the sixth pass-through rolling pass.
With regard to possible limitations in the accuracy of predictions, it must be said that wexit and the deterministic width values such as wentry, which represents the width after the fifth rolling pass (calculated width R5), are recorded with a maximum resolution of 1 millimeter. Both, the measured width wexit and the calculated width Wentry (width after the 5th rolling pass) are average values over the entire strip length and are therefore independent of direction. Moreover, Wentry is based on a deterministic width formula, which is integrated into the plant control system by the plant
manufacturer. This is a custom-developed formula, which is why it cannot be referenced. According to the plant manufacturer, this is based on variables such as temperature, thickness reduction, steel grade, forces, and roll gap settings. Hence, the performance of the width formula cannot be verified, for example, under different steel grades or through the combination of vertical edgers with thickness reduction and can therefore potentially generate biases in the prediction model. For this reason, the suitability of the calculated width as a basis for the prediction model will be revisited in the discussion. The complete data set is based on scalar data and comprises a total of 19,785 hot rolled strips and refers to the entire HSM (furnace, roughing mill, finishing mill, etc.). However, only some of these features can be used to avoid data leakage. Firstly, the data is reduced manually to only contain features from before the pass-through mill (R6). This means that all information recorded after the roughing mill, such as the target width of the roughing mill, information from the finishing mill, or even the final width of the finished coil, is removed from the data set. The data contained in the prediction model always refers only to the data of the strip currently being rolled. Table 1 shows a list of relevant features (X) available for predictions.

Table 1. Selection of features in the context of roughing mill

Features (X) | Unit
Rolling force (R1-R5) | kN
Temperature (R1-R5) | °C
Calculated width (R1-R5) | mm
Horizontal roller position (R1-R5) | mm
Vertical roller position (R1-R5) | mm
Steel grade | -
Number of reversing passes | -

For better understanding, the prediction methodology is illustrated in Figure 3, where the dashed green line marks up the feature space containing the complete data from the HSM roughing mill. This results in the input features X for further processing. Subsequently, the target (y) is defined, and the data set is divided into training, validation, and test data.

[picture]

Fig. 3. Schematic illustration of the prediction methodology.

Fig. 3. Schematic illustration of the prediction methodology.

Initially, the data set was divided into 70% training data, 15% validation data, and 15% test data and a minimum-maximum-scaler has been applied. Instead of presenting a broad survey of machine learning architectures for this task, this paper focuses on two distinct approaches to provide perspective, by following not only one approach. For this reason, a decision tree-based model and a neural network were used to compare the performance between the two models, as they differ in terms of their suitability for different approaches. It should also be noted that decision tree-based models, such as Random Forest (RF) or eXtreme Gradient Boosting (XGB), have the advantage of explainability. This means that influencing factors can be explored by calculating feature importance [16], for example. While these models may seem logical to the developer, they can also reveal hidden patterns in the data. Unlike decision tree models, who build a data-driven decision path and are therefore well suited for structured data sets, neural networks are highly suitable for discovering complex patterns in data sets, but they also require a large amount of data, and the predictions are not comprehensible to the developer [17]. In comparison to the decision-tree model, a Multilayer Perceptron (MLP) model and Convolutional Neural Network (CNN) was therefore also constructed as a neural network. Even though the models differ in their fundamental structure, the same method was used for optimizing the model parameters with 5-fold cross validation: Bayesian optimization (BO) [18]. Thereby, due to the fast convergence, 50 optimization trials were run to achieve the best parameter setting.

Evaluation

The original version of this paper is available on https://www.scientific.net/KEM.1050.9.pdf

During training, it was observed that XGB outperformed RF and that the MLP outperformed CNN. Therefore, the evaluation will be presented based on XGB, a decision tree-based model, and on MLP, representing a deep learning network. Various error metrics are presented to evaluate the prediction results of the different models. These include Mean Squared Error (MSE), its root (RMSE), the
coefficient of determination ( R2 ), and the Mean Absolute Error (MAE). While MSE, RMSE, and MAE are suitable for process-specific evaluation because they are expressed in absolute units ( mm2 for MSE, mm for RMSE and MAE), R2 is more appropriate for comparisons across datasets, as it is dimensionless. The smaller the MSE, RMSE and MAE values and the closer the R2 is to 1 , the more accurate is the prediction model.

Results and Discussion

The original version of this paper is available on https://www.scientific.net/KEM.1050.9.pdf

As previously explained, two different models were trained to predict the width spread for the final pre-strip, based on the results of the parameter optimization. The model configurations (settings) are shown in Table 2 (XGB) and Table 3 (MLP).

Table 2. Hyperparameter space and Optimization result for the XGB model

Parameter | Description | Feature Space | Setting
Estimators | Number of trees | {100, 1000, step size = 50} | 800
Max depth | Max. tree depth | {3, 12, step size = 1} | 5
Learning rate | Learning rate | {0.003, 3, log = true} | 0.083
Samples per tree | Share of samples per tree | {0.5, 1.0} | 0.668
Features per tree | Share of features per tree | {0.5, 1.0} | 0.985
Gamma | Minimum loss per split | {0, 5} | 0.566
Min. child weight | Sum of instance weights | {0, 10} | 6.989
L1-regularization | L1-regularization | {1*10⁻⁸, 10, log = true} | 0.009
L2-regularization | L2-regularization | {1*10⁻⁸, 10, log = true} | 0.349

Table 3. Hyperparameter space and Optimization result for the MLP model

Parameter | Description | Feature Space | Setting
n_layers | Number of hidden layers | {1, 10} | 4
Activation | Activation of neurons | {relu, tanh, elu} | elu
Dropout rate | Dropout rate | {0, 0.5} | 0.175
Learning rate | Learning rate | {1*10⁻⁴, 1*10⁻², log = true} | 1.5*10⁻⁴
Batch Size | Batch Size | {16, 32, 64, 128} | 128
Epochs | Training epochs | {50, 300} | 140
n_Units_L0 | Number of neurons layer 0 | {32, 512, step = 32} | 416
n_Units_L1 | Number of neurons layer 1 | {32, 512, step = 32} | 384
n_Units_L2 | Number of neurons layer 2 | {32, 512, step = 32} | 448
n_Units_L3 | Number of neurons layer 3 | {32, 512, step = 32} | 384

The performances of the XGB and MLP on the test data is presented in Table 4, whose indicates, that the XGB model (decision tree) outperforms the prediction of the MLP model. The predictions are visualized in Figure 4 in a regression diagram.

Table 4. Evaluation of the prediction results.

Model | MSE [mm²] | RMSE [mm] | MAE [mm] | R² [-]
XGB | 1.2407 | 1.1139 | 0.8236 | 0.8204
MLP | 1.3878 | 1.1780 | 0.8989 | 0.7991

[picture]

Fig. 4. Width Spread Prediction Results for the optimized models: XGB (left) and MLP (right).

Fig. 4. Width Spread Prediction Results for the optimized models: XGB (left) and MLP (right).

Both models are capable of predicting the width spread, as can be seen in the regression diagram (Figure 4), where the prediction instances follow the bisector line. Deviations persists along the bisector line, but the overall trend is captured, concluding that a prediction sensitivity is given. The color scale shown is the MAE scale, which indicates the absolute prediction error in millimeters. The color matching is chosen to separate minor errors (blue) from major errors (red), with all errors greater than 3 millimeters being displayed in red. The results show that the XGB model provides the best prediction results with an RMSE of 1.1139 mm and a MAE of 0.8236 mm(R2=0.8204). In order to better interpret the results in the context of the hot rolling mill, a comparison must be made with the total width. In this context, wist indicates the actual width measured by the width measurement system at the roughing mill exit and wpred indicates the predicted width derived from equation (2). Statistical robustness is relevant for evaluation of prediction results in an industrial context. Therefore, Δw (3) is calculated as an evaluation metrics of the individual predictions and represents the deviation from Wist to Wpred.

wpred =wentry +ypred Δw=wist−wpred 

(2)(3)

Thus, it is possible to predict the final width at the roughing mill after the fifth rolling pass, which also represents an important input variable for the finishing mill. The prediction accuracy (standard deviation) is approximately 1.1 millimeters (XGB). In relation to DIN ISO 10051, which pertains to the specifications for hot-rolled flat products, the permissible range of width is delineated as a tolerance of between 0 millimeters and a maximum of 20 millimeters in over width [19]. The present research findings suggest that the determined prediction scale can be considered significantly helpful in decision-making. However, it should be noted that absolute deviations of 3 millimeters and more still require a careful review of the prediction result. This requires further investigations, such as segmentation of steel grades and/or input geometry clusters.

The statistical evaluation of the hyperparameter optimized models in Figure 5 shows that the standard deviation on the training data set is significantly better than the test data set. The training RMSE of the hyperparameter-optimized models is 0.69 mm (MLP) and 0.61 mm (XGB), while the test and validation data are both between 1.1 mm and 1.2 mm for XGB and MLP (see Figure 5). This results in the assumption that overfitting is in evidence and that the models lack effective generalization. To check the influence of hyperparameter optimization on overfitting, the models are simplified. As a result, the search spaces for hyperparameter optimization were considerably reduced, leading to a
decrease in the model's depth and its complexity. It was determined that a simplified MLP model with one hidden layer and 96 neurons (compare Table 3) and a simplified XGB model with 50 estimators and a maximum depth of 8 (compare Table 2) performed slightly worse but in a similar range of results (Train-RMSE: approx. 0.7[ mm], Test-RMSE: approx. [1.3] mm). Summarized, hyperparameter optimization did not result in more overfitting of the XGB and MLP model than the simplified model and has slightly improved the prediction performance for both models. It can be derived that the high deviation in model performance between train and validation or test data is therefore not due to optimization-related overfitting, but rather to the heterogeneous data structure, which suggests that the width spread cannot be generalized for comparable roll gap settings (work roll position or forces) and strip input conditions (temperature, entry width, etc.). Moreover, the choice of the machine learning algorithm, such as MLP or XGB, and its underlying learning strategy can also influence data-dependent predictive performance. The root cause of this must be investigated by conducting more in-depth research into the data structure in future research work.

[picture]

Fig. 5. Absolute error distribution of the optimized XGB (yellow) and MLP (blue) models.

Fig. 5. Absolute error distribution of the optimized XGB (yellow) and MLP (blue) models.

Overall, it can be concluded that the hyperparameter optimized XGB model has the best prediction performance as the standard deviation on the training data set archives 0.61 mm , while on the test data set it yields 1.11 mm . In an industrial context, the research results presented show that the exit width of the roughing section can be predicted with a standard deviation of 1.11 mm .
As already introduced in the presentation of the various models, a decisive advantage of tree-based prediction models is that the importance of the features can be evaluated. For this reason, a feature importance analysis was performed to explain which features have a high influence on the prediction. The top 6 results of the feature importance analysis are illustrated in Figure 6, as these six features have the most significant influence. The illustration is sorted by the top 6 features of the best performing XGB model. It can be seen that a major influence on decision-making comes from the calculated entry width for the pass-through rolling mill (top 3), as well as the positioning of the edge rollers (top 2). As already explained in the prediction methodology, the calculated width R5 is a deterministic value based on roll gap setting, temperature, forces and steel grades. The high influence factor is therefore explainable and at the same time shows that the calculated width R5 (which may contain uncertainties, among other things with regard to the maximum resolution of 1 mm ) does not cause an extremely large bias. Nevertheless, future work should also investigate the extent to which width measurements between the 1st and 5th roll passes, as compared to the current calculated width values, improve prediction performance. Further research is also needed into the resolution of the measurement system after the rough rolling section, as a resolution of 1 mm versus a resolution of 0.1 mm can reduce model performance due to rounding errors.

[picture]

Fig.6. Feature Importance Analysis sorted by influence of the top 6 for XGB predictions

Fig.6. Feature Importance Analysis sorted by influence of the top 6 for XGB predictions

The greatest influence derives from a width coefficient of the reversing mill (top 1). This is an experience-based correction factor based on a matrix consisting of steel grades and thickness clusters. The correction factor "Width coefficient R1" is based on historical experience from plant operators and is adjusted if the difference between the target and actual width at the reversing stand becomes too large. This demonstrates the relevance of combining expert knowledge with data-driven approaches. It is evident that the rolling forces of the edge rollers prior to the pass-through mill (top 6), after the fifth pass on the reversing mill (top 4), and the vertical setting of the working rolls from the second reversing pass (top 5), contribute significantly to the prediction result. This leads to the assumption that information from process steps that have already been executed can be relevant for the prediction. In summary, it must be stated that with the width coefficient R1-R5 and the calculated width R5, two of the top three features are based on calculated models. As previously stated, the availability of real width measurement data as an input variable in the R 6 rolling stand would be pertinent in order to entirely negate bias based on previous modelling.

A comparison of the presented solution with other research on width spread prediction in roughing mills remains challenging, due to limited availability of data, and only two studies with direct comparability were identified. For this reason, studies on width prediction on the finishing mill were also considered for benchmarking.
As already introduced in the beginning of this research paper, Zhong et al. [13, 14] provide a hybrid approach for modelling the width spread in the roughing mill section by using ML-methods to improve the SSM. In their first research work, Zhong et al. yielded a RMSE of 3.4777 mm with their proposed improved width spread model, compared to MLP with a RMSE of 5.6417 mm [13]. The authors significantly improved this model as Improved SSM (ISSM) in another research paper published around the same time, achieving an RMSE of 1.6440 mm(R2=0.9655) against SSM with 6.1212 mm(R2=0.5219), as well as MLP with a RMSE of 5.7743 mm(R2=0.5745) [14] on the test data [14]. Another study that can be referred, as they use a similar procedure but on the finishing mill is the research by Wu et al. [8]. The authors attempt to predict the output geometry of the finishing line based on the input geometry in the first rolling stand, considering all process parameters. Thereby, the authors achieve a MSE of 2.6508[ mm2], which results in a RMSE of 1.6303 mm as well as a MAE of 1.1095 mm(R2=0.8091) [8].
A comparison of the presented results in this work with those in the literature suggests that a better prediction accuracy is achieved for this dataset. Considering the approach taken by [13,14] with our approach, a main differences in feature selection can be identified. While in this paper the historical process data from the furnace to the last pass are involved, [13,14] only consider the entry geometry and process setting of one rolling pass. This supports the earlier assumption that historical influences
have a significant impact on future prediction results. In order to emphasize this assumption, the data set was also shortened to the prediction of the last sample on a trial basis, which also reduced the prediction accuracy and resulted in slightly poorer performance than [14]. Finally, the comparison with existing studies suggests that including historical process information improves prediction accuracy and provides additional insight into the mechanisms influencing width spread.

Conclusion

The original version of this paper is available on https://www.scientific.net/KEM.1050.9.pdf

This study addresses a current challenge in the field of hot strip rolling in industrial plants, investigating the width spread in the roughing mill. Since the roughing mill is responsible for the essential forming work in the HSM, it is of major importance in the context of width spread during rolling. For this reason, this work provides a hyperparameter optimized XGB and MLP to evaluate their capability for the exit strip prediction of a roughing mill. The results demonstrate that both models possess the capacity to recognize the prediction trend, with the XGB model exhibiting a marginally improved performance, yielding a RMSE of 1.11 mm(R2≈0.82) and the MLP model performing slightly worse with a RMSE of 1.18 mm(R2≈0.80). From a scientific perspective, this shows that predictions at the roughing mill are feasible, and the results are marginally improved with this dataset from comparable publications. Moreover, the feature importance analysis of the XGB model reveals that parameters from previous steps contribute to the performance of the model, which is also an important insight for industrial applications as the predictions are comprehensive and offer more insights. Also relevant in an industrial context is the conclusion that the prediction model is capable of predicting the measured exit width of the roughing mill within a standard deviation of 1.11 mm . Even though the XGB model already achieves good prediction results, limitations remain, as the defined target has a maximum resolution of 1 mm . Furthermore, due to the lack of in-line measurements the prediction method considers calculated width data for the first five rolling passes, the accuracy of which cannot be verified, as they are calculated by deterministic equations from the HSM pass schedule. Finally, it can be shown that previously recorded process and geometry data from the same strip not only have an influence on the prediction result but also improve it. Looking ahead, this motivates further investigation into considering feature correlation analysis as well as the development of high-resolution width measurement data for roughing mills to enhance further improvements.

Acknowledgements

The original version of this paper is available on https://www.scientific.net/KEM.1050.9.pdf

The investigations were carried out as part of the WBW-Smart research project, funded as part of the EFRE/JTF program NRW 2021-2027. It is also co-financed by the European Union.

Conflict of Interest

The original version of this paper is available on https://www.scientific.net/KEM.1050.9.pdf

The authors declare no conflict of interest.

Data Availability Statement

The original version of this paper is available on https://www.scientific.net/KEM.1050.9.pdf

The industrial data set is confidential and therefore will not be shared.

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Key Engineering Materials, ISSN: 1662-9795, Vol. 1050, pp 21-30

doi: 10.4028/p-5MlW2P

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-05

Revised:
2026-02-18

Accepted:
2026-02-18

Online:
2026-04-16

Multi-Objective Bayesian Optimization of Dual-Phase Steel Microstructures for Minimal Damage Initiation

The original version of this paper is available on https://www.scientific.net/KEM.1050.21.pdf

Irene Biermann 1,a*, Niklas C. Fehlemann 1, b, Maximilian Hribsek 1,c and Sebastian Münstermann 1, d

1 Institute of Metal Forming, RWTH Aachen University, Intzestraße 10, 52072 Aachen, Germany

airene.biermann@ibf.rwth-aachen.de, b  niklas.fehlemann@ibf.rwth-aachen.de, c maximilian.hribsek@ibf.rwth-aachen.de, d  Sebastian.Muenstermann@iehk.rwth-aachen.de 

Keywords: Dual-Phase Steel, Microstructure Optimization, Bayesian Optimization

Abstract

Gaining a better understanding of the structure-property relationship in materials is a vital step in optimizing forming processes in order to minimize the induced damage and thereby maximizing the materials' performance. Dual phase (DP) steels are comprised out of hard martensite surrounded by a soft and ductile ferrite matrix. Due to the complex microstructure of DP-steels, different mechanisms of damage initiation can occur, such as martensite cracking or ferrite-martensite phase boundary decohesion. A key problem with computational microstructure optimization focusing on one specific damage mechanism is, that this can lead to virtual microstructures, which are good against one mechanism, but vice-versa problematic for another mechanism. This is why all optimization strategies have to consider more than one mechanism. In this study, a multi-objective Bayesian optimization (moBo) approach is developed for the design of damage-tolerant DPmicrostructures. It combines full-field crystal plasticity simulations on 3D representative volume elements with computational optimization. By employing the moBo, the sets of microstructure parameters are determined, where the combined minimum of both damage indicators is located. The proposed algorithm was applied to identify pareto-optimal microstructure configuration for DP800, considering both prevalent damage mechanisms It also provides an estimate of the variance associated with each parameter, which defines how critical the correct regulation of that aspect is. The results are in line with prevailing knowledge about DP steel, thus showing that the proposed approach is a promising tool for computational microstructure design

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1050.21.pdf

Finding the optimal combination of microstructure parameters for minimal damage initiation during forming is an important step in creating reliable components. The mechanical properties of dual phase steel are highly dependent on their complex microstructure, not only the grain sizes but also the distribution of the phases or the hardness contrast between both phases have an impact [1]. Since the performance of a formed component is heavily influenced by the damage behavior of its microstructure [2], this study aims to optimize the microstructure parameters with respect to damage initiation during plastic deformation. The material this paper is based on is a DP800 steel containing ferrite and martensite. In the investigated material, two main damage mechanisms are observed: cleavage fracture of the martensite, and ferrite-martensite phase boundary decohesion [3]. If the optimization focuses on just one of the damage mechanisms, it could result in microstructures that perform poorly for the other one. Thus, a multi-objective optimization framework is used to find Pareto-optimal states for both damage mechanisms. For the task of optimizing computationally expensive black-box functions, Bayesian optimization is well suited [4].

In this study, multi-objective Bayesian optimization was used to find microstructure parameters of dual-phase steel that promise minimal damage initiation during plastic deformation. In combination with the Sobol indices of the parameters these findings can be used to create damage-resistant materials. In the final step, the validity of the results is assessed by comparing the findings with the existing literature.

The effects of microstructure features on the damage initation in dual phase steels have already been described in previous studies [3] and quantified by Fehlemann et al. [5]. However, this paper offers a novel approach for damage-resistant materials design by employing target-oriented multiobjective Bayesian optimization to find Pareto-optimal states.

Methods

The original version of this paper is available on https://www.scientific.net/KEM.1050.21.pdf

This study was performed using DP800 steel, consisting of a ferrite matrix and martensite islands and banded structures. The base material contains around 30% martensite without retained austenite or bainite [5]. In uniaxial tension tests the material showed an ultimate elongation of 20% [5]. The heterogenous microstructures shows significant banding in some areas while in others only martensite islands are present. The base materials' crystal plasticity parameters were determined in a previous study by Fehlemann et al. [7] using in situ micropillar compression experiments [8].

All simulations are performed on statistical representative volume elements (sRVE) with an edge length of 32μ m and a resolution of two. This size offers a good tradeoff between scatter and the time needed to generate the sRVE. To further reduce scatter, the band thickness is kept constant at 2.94μ m, which is the median band thickness observed in the material [7]. The aim of this paper is to vary the microstructure of the base material to find the microstructure parameters that lead to minimal damage initiation during plastic deformation.

The varied parameters are:

Phase ratio of ferrite and martensite (P_R)

Grain sizes of ferrite and martensite (F_gr and M_gr)

Aspect ratio of ferrite and martensite grains (F_ar and M_ar)

Existence of martensite bands (N_b)

Ferrite hardening (F_hard)

In the round brackets the abbreviations for the indicators used in the figures of this paper are given.
The parameters determining the grain sizes and morphology can be understood as resizing parameters. They modify the original values of the base material. This procedure is adapted from the work of Fehlemann et al. [5].

An overview of the optimization process is depicted in Figure 1. In the first step of the optimization, 50 microstructure parameter combinations are created that serve as initial data. The parameters are then used by the sRVE generator DRAGen to build the sRVE with the defined microstructure [9]. Afterwards, the open-source framework DAMASK is used to simulate the materials behavior under a uniaxial load [10]. In the last step, the amount of damage occurring in the sRVE is investigated, which is used as a basis for the acquisition function to choose the next points in the design space to investigate. Since the acquisition function is optimizing multiple metrics at once to find pareto-optimal states, an acquisition function for multi-objective optimization is used. In each epoch, the acquisition function chooses 20 parameter combinations for the next optimization step.

[picture]

Fig. 1. Workflow for the Bayesian Optimization

Fig. 1. Workflow for the Bayesian Optimization

In initial microstructure parameters are drawn from a Sobol sequence [11]. This promises optimal space filling of the design space. In the following epochs, the parameters are chosen by the acquisition function.

To create the sRVE, the microstructure data generated in the previous step gets passed to DRAGen. Since the sRVE generation process and the subsequent load simulation is quite noisy, two sRVE with equal microstructure parameters are generated.

Simulating the mechanical material behavior is done using the open-source, multi-physics framework DAMASK [10]. The procedure was already described extensively in [5] and will only be shortly outlined in this work. For the purpose of this paper, a simple uniaxial tension load case in rolling direction up to a strain of 12% was chosen. The strain rate was kept at 0.001 . In future works, this could be expanded by simulating an actual load case that happens during metal forming. The simulations were done using a phenomenological crystal plasticity model [12] [13]. To model the martensite's mechanical behavior, an isotropic plasticity model available in DAMASK was utilized. This is applicable since the substructures of the martensite are not considered in the sRVE and the martensite's properties can therefore be considered to be isotropic. The experimental data is in line with this assumption [14]. The crystal plasticity model for this work was already calibrated and validated by Fehlemann et al. in [7] using in situ micropillar compression.

One of the varied parameters is the phase ratio of ferrite and martensite. Since the overall carbon content of the material is constant and almost all of the carbon is solved in the martensite, the carbon content in the martensite depends on the phase fraction. It has been observed in multiple studies, that the carbon content influences the martensite's hardness [15]. This effect was considered by using an approach proposed in [5]. In a first step, the Vickers hardness was calculated with respect to the carbon content using Toth's regression formula [16]. The Vickers hardness can then be used to estimate the tensile strength with the equation proposed by Pavlina and van Tyne [17]. For more detailed info on this procedure, the interested reader is referred to [5].

The main damage mechanisms in dual-phase steel are semi-brittle martensite cracking, martensiteferrite phase boundary decohesion and ferrite fracture [3]. Since the latter is rarely observed in the steel investigated in this study, the two damage indicators applied only measure the first two damage mechanisms. Both indicators were proposed by Fehlemann et al. in [5]. The martensite cleavage fracture is quantified by counting the number of elements where the stress exceeds a previously set principal stress threshold, as these elements are expected to break. In the following, this indicator will be referred to as cleavage fracture indicator. In accordance to the work by Steinbrunner et al. this threshold is the stress at 5% of the martensite flow curve [18]. For estimating the martensite-ferrite grain-boundary decohesion, an indicator was proposed in [5], which determines the damage quantity by fitting a gumbel distribution to the stress values at the grain boundaries. The damage indicator is defined as the combination of the parameters describing the gumbel distribution. In this paper, this indicator will be referred to as phase boundary decohesion (PBD) indicator.

By analyzing the investigated points, the acquisition function chooses new interesting points in the design space that are used for the sRVE creation in the next epoch. The function aims to minimize both damage indicators simultaneously, finding Pareto-optimal states. The optimization is realized using the Python framework BoTorch which is meant for efficient Monte-Carlo based Bayesian optimization [19]. For this work, the q-log noisy expected improvement acquisition function was chosen since it is well suited for noisy data and is efficient for large batch sizes.

Results

The original version of this paper is available on https://www.scientific.net/KEM.1050.21.pdf

The optimization was carried out over eleven epochs with 20 data points per epoch. As initial data, 50 data points were created. Since two sRVE are generated per microstructure configuration to reduce the influence of the scattering, 540 sRVE were created overall.

[picture]

Fig 2. Evolution of the hypervolume of the Pareto front

Fig 2. Evolution of the hypervolume of the Pareto front

Figure 2 shows the evolution of the hypervolume of the Pareto-front. The Pareto front is the set of all solutions in a multi-objective optimization problem for which no objective can be improved without worsening at least one other objective. The hypervolume is the size of the region in objective space that is dominated by the current nondominated set, and bounded by the reference point. The reference point is a user-chosen point in objective space, worse than all solutions of interest, used as the bounding corner for computing hypervolume. The choice of reference point influences the absolute value of the hypervolume, which is why only the trend can be interpreted. A larger hypervolume means, that the points are, on average, closer to the ideal, they cover more of the tradeoff curve, not just one corner or that the spread of the Pareto front is better. I the hypervolume rises, the algorithm has found at least one new nondominated solution that expands the dominated region. Typically, the curve shows big jumps early into the optimization and smaller gains later. This is due to the algorithm quickly finding rough good regions and then fine-tuning the result.

The curve in the figure shows big jumps at the beginning, especially around 70 observations. Around 100 to 200 observations it stagnates, however it shows further jumps at 200 to 250 observations. The curve shows a monotone growth, meaning that every epoch offers an improvement of the knowledge of the Pareto-front. After 220 observations, no further improvements are made and the curve converges.

[picture]

Fig 3. Scatter-Plot of the samples created in every optimization iteration, black dashed line: estimated Pareto-front, green rectangles: samples with the lowest cleavage fracture indicator, red crosses: samples with lowest phase boundary decohesion (PBD) indicator, yellow triangles: samples closest to estimated Pareto-front

Fig 3. Scatter-Plot of the samples created in every optimization iteration, black dashed line: estimated Pareto-front, green rectangles: samples with the lowest cleavage fracture indicator, red crosses: samples with lowest phase boundary decohesion (PBD) indicator, yellow triangles: samples closest to estimated Pareto-front

Figure 3 shows the damage values achieved for all 270 microstructure combinations generated in the optimization. The plotted points are the average results from the two sRVE calculated for each microstructure parameter combination. The black dashed line illustrates the estimated Pareto-front achieved by the optimization. In a noise-free problem, no results on the left side of the line would be achievable. However, since the sRVE generation and load simulation are highly noisy and exhibit substantial scatter, some of the results are scattered beyond the front. The mean standard deviation of the calculated points from the plotted mean value is 0.0018 for the cleavage fracture indicator and 0.002 for the phase boundary decohesion indicator. This effect also makes the front less visible, which is why an approximation of the front was plotted as a dashed black line. The five data points closest to the front are marked as yellow triangles. Their microstructure parameter combinations are shown in Figure 4. Additionally, the data points resulting in the lowest cleavage fracture and phase boundary decohesion are marked as green rectangles and red crosses respectively. In Figure 5, the microstructure combinations corresponding to these states, that are optimal for one of the indicators are shown. The other datapoints are colored blue, with the shade according to the iteration they were generated in, as indicated by the color bar on the right side of the figure.

[picture]

Fig 4. Mean microstructure parameter values of the samples closest to the Pareto front (orange triangles in Figure 2)

Fig 4. Mean microstructure parameter values of the samples closest to the Pareto front (orange triangles in Figure 2)

In Figure 4, the parameter values create in the last optimization epoch are shown in a radar chart. To make reading the image easier, all values are normalized to be between zero and one. A min-max normalization was used. The solid line depicts the mean value, while the shaded area is the 80% confidence interval (CI). If the confidence interval is large, this means, that there was a lot of variance in the parameters and vice-versa. A low variance implies, that the parameters' impact is high. The narrowest confidence interval is achieved for the phase ratio parameter.

For all data points, the values of this parameter remain very small. This means, that at the Paretofront the sRVE contain large amounts of ferrite and only little martensite. The ferrite hardening parameter tends to be on the upper side of the parameter interval. This parameter also has a rather narrow confidence interval. The average of the band parameter is below 0.2 , which means that in only one of the five sRVE a band was placed, while the other four sRVE did not contain a band. The grain size and aspect ratio parameters all have an average of about 0.4 with a wide confidence interval. Thus, most of the sRVE at the Pareto front have in common, that they have a low martensite content, no martensite bands and hard ferrite.

[picture]

Fig 5. Radar Charts of the optimal microstructure configurations for minimal cleavage fracture (left) and phase boundary decohesion (right)

Fig 5. Radar Charts of the optimal microstructure configurations for minimal cleavage fracture (left) and phase boundary decohesion (right)

[picture]

Fig 5. Radar Charts of the optimal microstructure configurations for minimal cleavage fracture (left) and phase boundary decohesion (right)

Fig 5. Radar Charts of the optimal microstructure configurations for minimal cleavage fracture (left) and phase boundary decohesion (right)

Figure 5 shows the mean microstructure configurations of the five sRVEs where the individual damage indicators are minimal. In Figure 3 the shown sRVE are marked with red crosses (left side of this figure) and green rectangles (right side of this figure). Again, to make the comparison of the parameters easier, all values are normalized applying a min-max normalization to be between zero and one. By plotting the 80% confidence interval, the figure also shows the variance in each of the parameters.

The left side of the figure shows the results for the cleavage fracture indicator. The parameters with the smallest confidence interval are the number of bands, the martensite grain size and the aspect ratio of the martensite grains. The number of bands is zero with a confidence interval that is not visible, meaning that in the plotted data-set no bands were placed. The martensite aspect ratio is also below 0.4 , while the martensite grain size is above 0.8 . The narrow confidence interval implies, that these parameters are the most relevant ones when considering the martensite cleavage fracture. All of the ferrite parameters and the phase ratio have a wide confidence interval, which means, that the ferrite microstructure does not heavily influence martensite fracture.

On the right side of the figure the radar chart for the phase boundary decohesion indicator is depicted. The parameters with the lowest confidence intervals, and therefore the highest impact on the phase boundary decohesion, are the ferrite grain size and the phase ratio. Both lie at approximately zero. Another parameter with a narrow confidence interval is the ferrite hardening, where the mean value is at almost 1 . The other parameters have a wide confidence interval, with a tendency for small martensite grains and a high ferrite aspect ratio.

Discussion

The original version of this paper is available on https://www.scientific.net/KEM.1050.21.pdf

The results of the study show, which microstructure parameters promise minimal damage initiation during loading of DP steel. According to Figure 4, the most important microstructure feature to be controlled is the phase ratio of martensite and ferrite. It indicates, that microstructures with a low martensite content are more damage-resistant. Additionally, well distributed martensite that does not form bands is beneficial for the damage behavior of the material. Microstructures with a high ferrite hardening showed a better performance than those with softer ferrite.

These findings are mostly in line with the literature, which claims, that martensite cracking is facilitated by banding. Thus, material without martensite bands is generally more damage resistant. According to Asik [20], a higher martensite content also results in more damage initiation. This confirms in particular the results shown in the right side of Figure 5, where the optimal martensite content was shown to be the least possible. Additionally, Tasan et al. reports, that DP steel with larger martensite grains shows less phase boundary decohesion [3]. This can be explained by the larger area of the phase boundaries if the martensite grains are small, thus offering more sites for phase boundary decohesion. The same argument can be applied to the aspect ratio of the martensite, where a higher aspect ratio means, that the phase boundary area is larger. However, the right side of Figure 5 shows the opposite. Here, the least phase boundary decohesion is achieved with small martensite grains with a low influence of the martensite aspect ratio. This can be explained by the way in which the used damage indicator works. It does not measure the absolute amount of decohesion sites but instead offers a statistical metric for the stress states at the phase boundaries. Thus, the indicator is not directly influenced by the area of the boundaries. This also explains the low impact of the martensite grain aspect ratio and the martensite bands on the phase boundary decohesion implied by the right side of the figure. Tian et al. claims, that a higher mechanical heterogeneity between ferrite and martensite causes a higher rate of phase boundary decohesion [8]. This matches the findings of this paper, which shows, that a higher ferrite hardening results in less damage initiation, especially for the grain boundary decohesion. Since the ferrite grain size also influences the phases mechanical properties, this also explains why a smaller grain size of the ferrite results in less damage initiation in the material. This is in line with the findings of Calcagnotto et al. [21] and the right side of Figure 5.

Figure 5 also illustrates why the underlying optimization task is rather complex. For all parameters, with the exception of the ferrite grain aspect ratio, the optimal states for the two damage indicators are essentially inverse to each other. In other words, parameter combinations that minimize
one damage indicator tend to be unfavorable with respect to the other. This antagonistic behavior indicates a pronounced trade-off between the two objectives: improving the performance of one damage indicator generally leads to a deterioration of the other.

In addition to these conflicting objectives, the problem is further complicated by substantial scatter in the simulation outcomes, as illustrated in Figure 3. The scatter is likely heteroscedastic in nature, meaning that the variance of the responses is not constant but varies across the design space. In some regions, the results are comparatively well clustered, while in others they exhibit a much larger spread. This heterogeneous variance structure reduces the robustness of local trends and makes it more challenging to reliably identify globally optimal parameter combinations. Overall, the combination of conflicting objectives and heteroscedastic scattering significantly increases the difficulty of the optimization task. This could be improved in future works by creating and averaging more than two RVE per parameter combination, so the scatter is minimized and the resulting damage values are as realistic as possible. However, this would make the simulations more computationally expensive which is why in this paper the number of RVEs per parameter combination was limited to two.

In the paper by Fehlemann et al., the Sobol indices for the considered microstructure parameters are calculated, which quantify the variance in the damage initiation caused by the parameters [5]. A high Sobol index implies a high impact of the parameter on the result. It is expected, that the parameters with high Sobol indices align with the ones showing a small confidence interval in Figure 5. In the paper, the martensite grain aspect ratio had the highest Sobol index for the cleavage fractur indicator. All of the other parameters' influence is calculated to be negligible. A similar result was observed by the confidence interval in this paper. However, according to this paper, the bands and the aspect ratio of martensite both have a big effect on the martensite fracture. The effects of bands and a high martensite aspect ratio can be attributed to the same mechanism, which can explain the discrepancy between the results of this paper and the one by Fehlemann et al. For the phase boundary decohesion, the cited paper predicts a high influence of the ferrite and martensite grain sizes, the phase ratio and the martensite aspect ratio. Out of these parameters, the only one with a wide confidence interval in Figure 5 is the martensite aspect ratio. For the other parameters, the current paper backs the findings of Fehlemann et al.

In future works, the load case could be adapted to represent the actual stress states that appear during metal forming. This could be used to tailor damage resistant materials based on the processing route. Furthermore, the effect of using different damage indicators has to be analyzed. The indicator for martensite fracture should take into account, that the fracture is either ductile or semi-brittle after plastic deformation, as shown by recent research [22] [23] [24]. While cohesive zone modelling is probably more accurate for the prediction of the phase boundary decohesion, the implementation and calibration of such models is quite complicated. The indicator used in this paper offers a good tradeoff between the computational cost and the accuracy. In a future work, the phase boundary decohesion parameter could be further adapted to show the absolute amount of damage.

Furthermore, the performance of the optimization could be improved by reducing the number of considered parameters. For example, the martensite band parameter could be kept at zero, since the results of this study show, that damage resistant microstructures do not have bands. Additionally, parameters that were shown to have a low Sobol index for both of the damage indicators could also be removed from the optimization, since they are not expected to heavily influence the result.

Summary

The original version of this paper is available on https://www.scientific.net/KEM.1050.21.pdf

This work presents the results of a multi-objective optimization study aimed at minimizing damage initiation during loading of DP800 dual-phase steel. In addition to identifying microstructure and parameter combinations that are most favorable for reducing overall damage, the study also determines those combinations that specifically minimize the initiation of individual damage mechanisms. The analysis includes a systematic evaluation of the influence and relative significance of the considered parameters on the optimization outcomes. It is shown that the optimization results are largely consistent with the literature, thereby supporting the validity of the approach and providing further insight into microstructure-property relationships in DP800 steel.

Acknowledgements

The original version of this paper is available on https://www.scientific.net/KEM.1050.21.pdf

This research was funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation; Projectnumber 278868966 - TRR 188; Damage Controlled Forming Processes, subproject B05).

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The original version of this paper is available on https://www.scientific.net/KEM.1050.21.pdf

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The original version of this eBook is available on https://www.scientific.net/book/advanced-numerical-and-ai-strategies-for-material-forming/978-3-0364-1986-2

Key Engineering Materials, ISSN: 1662-9795, Vol. 1050, pp 31-42

doi: 10.4028/p-kaH2TN

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2025-12-19

Revised:
2026-02-18

Accepted:
2026-02-18

Online:
2026-04-16

Algebraic Hierarchical Graph Neural Networks for Forming Simulation of Thermoplastic Composite Materials

The original version of this paper is available on https://www.scientific.net/KEM.1050.31.pdf

Tobias Würth 1,a*, Niklas Freymuth 2, b, Johannes Mitsch 1,c, Philipp Dahlinger 2, d, Tai Hoang 2,e, Gerhard Neumann 2,f and Luise Kärger 1, g

1 Institute of Vehicle System Technology, Karlsruhe Institute of Technology, Karlsruhe, Germany

2 Autonomous Learning Robots, Karlsruhe Institute of Technology, Karlsruhe, Germany

a tobias.wuerth@kit.edu, b niklas.freymuth@kit.edu, c Johannes.Mitsch@kit.edu, d philipp.dahlinger@kit.edu, e tai.hoang@kit.edu, f gerhard.neumann@kit.edu, g luise.kaerger@kit.edu

Keywords: Forming, Graph Network Simulators, Algebraic Multi-Grid Coarsening, Data-Driven Simulation

Abstract

Stamp forming of fiber-reinforced thermoplastic composite materials is governed by large deformations, anisotropic and rate-dependent material behavior, and frictional multi-body contact, making high-fidelity finite element simulations expensive and often impractical for rapid design studies and process optimization. We leverage recent advancements in Machine Learning-based simulations and tailor Algebraic-hierarchical Message Passing Networks (AMPNs) to stamp forming simulation of composite materials. To efficiently handle multi body contact during forming, we model the laminates by a multi-layer graph with explicit ply-ply and tool-ply contact and extend AMPNs by local component-wise contact edges. Using a multiscale graph hierarchy, the method captures local wrinkling effects, global material draw-in, and contact-driven deformation across the full laminate. Trained on high-fidelity data from state-of-the-art Finite Element Method (FEM) simulations, the surrogate accurately simulates the stamp forming process for unseen process settings, while reducing simulation times from hours to seconds, enabling approximately real time simulation of large, complex geometries.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1050.31.pdf

The performance of fiber-reinforced composite components depends heavily on stamp forming processes that shape flat laminates into complex geometries [1]. These materials are lightweight while offering high specific strength and fatigue resistance, making them increasingly attractive for the automotive and aerospace industries [2,3].

Simulation of the forming process enables a virtual process optimization and helps evaluate design decisions, providing deeper insights into complex systems without requiring expensive real-world experiments. Such simulations model the physical system using Partial Differential Equations (PDEs), which are usually approximated by numerical discretizations like the FEM [4]. A critical challenge in simulating the stamp forming process of fiber reinforced composite materials is the prediction of manufacturing defects such as wrinkling, i.e., high-frequency, out-of-plane buckling modes. Accurately capturing such defects requires fine-grained meshes, narrow numerical time-integration schemes [2] and advanced simulation models [5], which quickly results in high computational cost that makes downstream applications infeasible [6].

Recently, research in Machine Learning (ML)-based models has aimed to speedup simulations by developing neural surrogates that can approximate complex physical dynamics [7,8,9]. PhysicsInformed Neural Networks directly minimize PDE residuals, using Multilayer Perceptrons (MLPs) [10, 11], Convolutional Neural Networks (CNNs) [12, 13] or Graph Neural Networks (GNNs) [14, 15]. Neural Operators learn mesh-resolution-independent solution mappings [16, 17]. Complementary to theses approaches, data-driven Graph Network Simulator (GNS) have been shown to be particularly effective for modeling mesh-based dynamics [8, 9, 18]. Recent extensions consider diffusion-based
inference to improve accuracy for image-based surrogates [19,20] as well as for mesh-based simulations [21] and meta-learning to improve generalization to unseen scenarios [22, 23, 24]. These models mimic simulation data from numerical solvers by using their own predictions to estimate the state residuals between successive time steps [17]. The seminal MeshGraphNet (MGN) [8] encodes states using local connectivity, but struggles to scale to large domains due to a limited receptive field. As a remedy, recent work considers global attention [25] and, increasingly, hierarchical mesh representations [26,21]. Such globally receptive GNSs are fully differentiable and orders of magnitude faster than classical simulators, which unlocks applications in engineering design and manufacturing optimization [27, 28, 11].

In this work, we leverage these modern advances in GNS to establish a fast and accurate surrogate model for forming process of thermoplastic composite materials. We build upon the Rolling Diffusion-Batched Inference Network (ROBIN) framework [21] and utilize Algebraic-hierarchical Message Passing Networks (AMPNs) to process mesh representations based on Algebraic multigrid (AMG) coarsening [29, 30]. Crucially, we depart from diffusion-based inference of ROBIN [21], and instead use a simple next-step predictor based on AMPNs at inference time. In our stamp forming setting, we demonstrate that this choice improves accuracy and reduces inference cost compared to diffusion-based sampling. We further propose a sparse connectivity strategy for the inter-ply graph construction, drastically reducing the number of required edges for multi-layer simulations without sacrificing the simulation fidelity of local interactions.

We validate the presented approach on a stamp forming task involving a multi-directional stacking sequence [0∘/45∘/90∘/−45∘], which promotes the local formation of wrinkles due to high resistance against shear deformation. Figure 1 visualizes the problem setup. Our experiments demonstrate that the proposed method achieves near-FEM accuracy in capturing local wrinkling phenomena, while reducing computational cost by orders of magnitude compared to explicit solvers.

Thermoforming Simulation

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Simulation model. In order to apply AMPNs to the thermoforming process, we generate simulation data using a high-fidelity FEM simulation model. The simulation model is implemented in the commercial FEM software Abaqus/Explicit and is based on the work of Dörr et al. [31, 32], who developed an isothermal, layer-by-layer approach for thermoforming simulation for thermoplastic composite materials. The model accounts for the material behavior of the individual plies and their inter-ply interactions and provides essential features for accurate thermoforming simulations, including tool-ply and ply-ply contact modeling [32], decoupled membrane-bending behavior, and rate-dependent material modeling of both membrane and bending deformation mechanisms [31]. The model has been experimentally validated and accurately captures the deformation behavior during forming into complex geometries for various stacking sequences, including material draw-in, wrinkle formation, and the resulting fiber orientation.

[picture]

Fig. 1: Numerical test setup for the forming simulation of a double-domed geometry. A junglegreenpunch geometry presses multiple plies against a stationary die. The plies are held by multiple grippers.

Fig. 1. Numerical test setup for the forming simulation of a double-domed geometry. A junglegreenpunch geometry presses multiple plies against a stationary die. The plies are held by multiple grippers.

To account for the influence of tensile grippers on the deformation behavior of the thermoplastic blank during forming, the methodology proposed by Poppe et al. [33] is adopted. This approach involves defining a space-fixed anchor point on the gripper frame for the tensile gripper, as well as a blank-fixed gripping point where the material is clamped by a needle. Using a one-dimensional translator element, the gripper force is applied as a concentrated load at the blank-fixed gripping point. This load acts along the line connecting the anchor and gripping point, ensuring that the force is always directed toward the space-fixed anchor. The combination of the translator element and a hinge element enables the gripping point to rotate about the needle, allowing nodding motions of the gripper during forming.
Dataset. We use this simulation model to generate representative training data for the AMPNs. Fig. 1 illustrates the numerical thermoforming model implemented in Abaqus/Explicit. The double-domed geometry considered in this study is commonly used in the forming literature [34, 35, 36], as its pronounced double curvature induces strongly heterogeneous deformation and can trigger deformationinduced wrinkling, yielding geometrically complex, out-of-plane ply shapes. We implement the forming process in a displacement-controlled manner by prescribing a Dirichlet boundary condition ( 𝐮=−60 mm𝐞3 ) on the punch geometry (green) over a time period of T=1 s. We consider a laminate of carbon fibers in a PA6 thermoplastic matrix consisting of four plies with orientations [0∘/45∘/90∘/−45∘ ] relative to the 𝐞1-axis (cf. Fig. 1). Eight tensile grippers are distributed along the outer edges of the blank to apply gripper forces during forming. We exploit the two-fold symmetry of the tool geometry, and only simulate one quarter of the model to improve computational efficiency. We discretize the blank using 3200 triangular membrane and shell elements per ply, while we discretize the tool with 4258 and 5100 quadrilateral elements for the stamp and die, respectively. We model contact between the blank and the tools using the general contact algorithm in Abaqus/Explicit. In total, we conduct 684 simulations with varying gripper positions along the blank edges to generate representative training data for the AMPNs, while we use 147 simulations for validation and testing. For each simulation trial, we sample the position of both grippers independently and uniformly along the corresponding blank edges. In addition, we sample the magnitude of the pulling force applied by each gripper independently and uniformly within the force limits of the employed gripper, from 0 N , where gripper-induced force acts on the plies, up to 100 N . We discretize each simulation into 50 time steps, capturing the transient deformation behavior of the blank during the forming process. For each node of the finite element mesh, the resulting dataset provides nodal displacement vectors as well as the initial fiber orientation. These quantities, together with the gripper positions and pulling forces, constitute the input features used for training the AMPNs model. We executed the simulations on an Intel Xeon Gold 6230 processor, requiring approximately 275 hours of total computation time.

Methodology - Algebraic-Hierarchical Message Passing Networks (AMPNs)

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To make thermoforming simulation practical for iterative engineering workflows, we learn a surrogate that reproduces the FEM trajectories at a fraction of the runtime. We build upon AMPNs [21] because their hierarchical graph representation supports the multiscale nature of stamp forming by capturing local contact interactions and wrinkle formation while also propagating global deformation and drawin effects across the full laminate.

The AMPNs predict the residual displacement Δ𝐮~t≈Δ𝐮t=𝐮t−𝐮t−1 in stamp forming of the current time step t given the displacement 𝐮t−1 from the previous time step t−1. During inference, we rollout the AMPN autoregressivly by updating the displacements with 𝐮t≈𝐮t−1+Δ𝐮~t and use that state as new input for predicting the next residual Δ𝐮t+1. We build up on ROBIN [21] and implement the AMPNs as a special case of ROBIN with a single diffusion step. In this setting, ROBIN directly predicts the residual update Δ𝐮t, and the training objective reduces to a conventional Mean Squared Error (MSE) loss against the ground-truth FEM residual update. AMPNs operate on a hierarchy of graphs with different resolutions, where coarse hierarchy levels allow global deformation predictions, while finer levels allow for refining fine details. Given a simulation mesh, we create a fine, initial graph 𝒢=(𝒱,ℰM), where 𝒱 denotes mesh nodes and ℰM mesh edges. From this mesh, we create a hierarchical graph 𝒢H=(𝒱0:L,ℰ0:L,M∪ℰ0:L,C∪ℰ0:L−1,D∪ℰ0:L−1,U) based on root-node smoothed aggregation [30], using the Algebraic multigrid (AMG) solver implemented in PyAMG [37]. Given the adjacency matrix A0 of the fine simulation mesh, we use the AMG solver to create a hierarchy of graph nodes 𝒱0:L, mesh edges ℰ0:L,M, down- ℰ0:L−1,D and upsampling edges ℰ0:L−1,U, where the superscript 0:L denotes the set of hierarchy levels l∈{0,…,L} (with l=0 the finest level and l=L the coarsest). In the hierarchy, the fine level graph l=0 corresponds to given fine simulation mesh of the FEM. For the considered double-domed geometry the AMG algorithm yields L=3, i.e., 3 coarse levels in addition to the fine level at l=0.

To learn contact, we follow previous works [8, 21] and connect nodes with contact edges ℰ0:L,C by spatial proximity. However, conventional radius-based approaches [8] connect each node to all neighbors within a fixed radius, typically including nodes on the same component, leading to an unbounded number of edges in contact-rich stamp forming. Choosing a large radius anticipates upcoming contacts but can cause a prohibitive growth in edges and memory, whereas choosing a small radius misses imminent contacts unless very small time steps are used. We therefore connect each node to neighboring nodes of other components that are within a radius of 3 mm , but limit the number of connections to the 3 closest connections per node and component. That means that, in our stamp-forming setup with 4 plies, 1 stamp, and 1 die, each node can connect to at most 15 nodes from the other components. In contrast to static contact edges as in Würth et al. [21], we use time-dependent contact edges, i.e., we update the contact edges based on the current ply and tool locations at each time step, based on the current position.
Feature and encoder. For the fine mesh nodes (level l=0 ), the node features are comprised of a onehot encoding ni of the node type, the last predicted displacement residual Δ𝐮it−1, the prescribed tool motion Δ𝐮i,BCt at the current time step t, the fiber orientation in the ply, and a component identifier. We add additional node features to inform the model about the gripper forces. Specifically, for each node position 𝐱i and gripper position 𝐠j we compute the distance vector 𝐝ij=𝐱i−𝐠j, its length dij=‖𝐝ij‖2, and the normalized direction 𝐝^ij=𝐝ijdij+ϵ. We further include features based on an exponentially decaying distance weighting. For each node i, we compute the distance-weighted projected force 𝐟iproj-weighted =∑j𝐟ijdirected exp(α·dij+ϵ) with the distance projected gripper force 𝐟ijdirected =(𝐟jgripper ·𝐝^ij)·𝐝^ij as well as the distance-weighted gripper force field 𝐟iweighted =∑j𝐟jgripper exp(α·dij+ϵ) and its norm fiweighted =‖𝐟iweighted ‖2. We set α=3e−2 and ϵ=1e−8. For the hierarchical nodes, i.e., for all nodes with l>0, we only use the one hot encoding ni of the node type as a feature. As we follow the ROBIN setup, we additionally provide the model with a noisy input sample drawn from a standard Gaussian distribution
𝒩(0,1). This additional input does not significantly impact the model's predictions, but allows us to directly compare to ROBIN's original diffusion-based predictions without further modifications.

As edge features, we use the relative current node distance 𝐱ijt=𝐱it−𝐱jt and its norm |𝐱ijt| at all hierarchy levels. For mesh, downsampling, and upsampling edges, we additionally include the initial relative distance 𝐱ij0=𝐱i0−𝐱j0 and its norms |𝐱ij0|. We apply linear layers that project the corresponding input features into a latent space of dimension d, yielding node embeddings 𝐤in for i∈𝒱l, mesh edge embeddings 𝐞ijM,n for (i,j)∈ℰl,M, contact edge embeddings 𝐞ijC,n for (i,j)∈ℰC, downsampling edge embeddings 𝐞ijD,n for (i,j)∈ℰl,D, and upsampling edge embeddings 𝐞ijU,n for (i,j)∈ℰl,U for each level l at the initial step, i.e., at n=0. All features are normalized using statistics computed on the training set to obtain zero mean and unit variance. Finally, we also apply a Fourier encoding to condition the model on the current node level l[38,21].
Processor and decoder. ROBIN uses Intra-Level-Message Passing Stacks [21] to update node- and edge-level embeddings in the Pre-Processing, Post-Processing and Solving blocks. The intra-level update at message passing step n for a given hierarchy level l is defined by

𝐞ijC,n+1=Wθ,ℰCn𝐞ijC,n+fθ,ℰCn(𝐤in,𝐤jn,𝐞ijC,n)𝐞ijM,n+1=Wθ,ℰMn𝐞ijM,n+fθ,ℰMn(𝐤in,𝐤jn,𝐞ijM,n)𝐤in+1=Wθ,𝒱n𝐤in+fθ,𝒱n(𝐤in,⨁j𝐞ijC,n+1,⨁j𝐞ijM,n+1)(1)

For Up- and Downsampling, we use Inter-Level-Message Passing Stacks [21], which update the embeddings at each message passing step n as

𝐞ijn+1=Wθ,ℰn𝐞ijn+fθ,ℰn(𝐤irec,n,𝐤jsend,n,𝐞ijn)𝐤irec,n+1=Wθ,𝒱n𝐤irec,n+fθ,𝒱n(𝐤irec,n,⨁j𝐞ijn+1)(2)

During downsampling, receiver node embeddings 𝐤irec ,n∈𝒱l are from level l, sender node embeddings 𝐤jsend ,n∈𝒱l+1 are from level l+1, and edge embeddings 𝐞ijn∈ℰl,D. During upsampling, the sender and receiver roles are reversed and 𝐞ijn∈ℰl,U. As aggregation ⨁ we use a max aggregation and Wθ,.n denotes weight matrices [8, 39, 40]. fθ,.n denotes an MLPs that starts with layer normalization and consists of two linear layers with a hidden size of 128, with a Sigmoid Linear Unit (SiLU) activation [41] in between. Starting from the initial encoded embeddings ( n=0 ), we apply a V-cycle that first updates the graph using a Pre-Processing stack and then compress information to the next coarser level via a Downsampling stack. We repeat this procedure until reaching the coarsest level L=3. On the coarse graph, we apply a Pre-Processing stack followed by a Solving and a Post-Processing stack. During upsampling, we use an Upsampling stack and a Post-Processing stack at each level. We use 3 layers for Pre- and Post-Processing, 2 layers for Up- and Downsampling and 5 layers for the Solving stack. All stacks share their weights across levels, resulting in a total of 15 learnable layers. Given the final node embeddings 𝐤iN∈𝒱0 on the fine mesh, we apply a linear layer to decode the predicted displacement residual Δ𝐮it. We train ROBIN for a total of 500000 iterations and with a batch size of 1 using ADAM [42] in PyTorch [43]. Training and testing are performed on an NVIDIA A100 GPU. We apply gradient clipping with an L2-norm threshold of 1 . The learning rate is linearly warmed up over the first 1000 steps and then exponentially decayed from 1e−4 to 1e−5 over the remaining iterations. During training, we perturb the node positions 𝐱it with Gaussian noise of standard deviation 10−5σ𝐱 and the previous residual Δ𝐮it−1=𝐱it−𝐱it−1 with 10−6σ𝐱[9,8], where σ𝐱 denotes the standard deviation of the node positions.

Results

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We compare AMPNs to FEM simulations using the Root Mean Squared Euclidean distance error RMSE =1/(NiNj)∑i=1Ni∑j=1Nj(u~ij−uij)2, where u~ij and uij denote the predicted and groundtruth (FEM) node positions, respectively, with Ni nodes and Nj features. We then average the RMSE over time steps, the dataset, and finally report the mean μ and standard deviation σ across 3 random seeds.

Fig. 2 compares an AMPN-predicted stamp forming simulation to the FEM reference. AMPNs accurately reproduce the global deformation behavior, as shown in Fig. 2 a -b) after 70% of the process time. Wrinkles at different scales match the wrinkles formation of the FEM simulation, and only minimal tool interactions are visible, consistent with the FEM. The FEM solution appears slightly noisier than the AMPN prediction, likely because the AMPNs focus on global solution accuracy and may miss very high-frequency components, as is common for autoregressive one step models [19,21]. Fig. 2 c-d) shows the final process time step. Again, AMPNs closely match the FEM solution, correctly predicting ply draw-in and wrinkle formation. Tool intersections are comparable to the FEM, while very high spatial-frequency details differ slightly, with the AMPN prediction appearing less noisy overall.

We further compare AMPNs, which can be viewed as a special case of ROBIN with a single denoising step K=1, to ROBIN variants using more denoising steps ( K=5,10,20 ). We use conventional inference rather than Rolling Diffusion-Batched Inference (ROBI) [21], since the dataset contains only 50 time steps. The potential speedup from ROBI is therefore less relevant, and we observe a significant accuracy improvement with conventional inference on this dataset.

Fig. 3 visualizes AMPNs and ROBIN predictions for different numbers of diffusion steps K and compares them to the FEM reference. The AMPNs most accurately captures the overall wrinkle formation. All models reproduce the material draw-in observed in the FEM simulation, while the K=5 variant shows the largest deviation. In contrast, the K=10 and K=20 variants reproduce fine scale FEM behavior more realistically, for example the small wrinkles that appear near the center of the right boundary.

Table 1 summarizes the quantitative results for AMPNs and the different ROBIN variants. Consistent with Fig. 3, AMPNs achieve the lowest RMSE by most accurately capturing the global deformation. Moreover, a full simulation requires only 3.84 s on average on a GPU (NVIDIA A100). This enables real-time application, as the underlying forming process itself takes several seconds to complete. In contrast, a traditional FEM simulation requires 16.87 minutes on a CPU (Intel Xeon Gold 6230), so AMPNs achieve a speedup of more than two orders of magnitude. The runtime for ROBIN increases approximately linearly with the number of diffusion steps K, due to the growing number of model evaluations, while the RMSE of the K=10 and K=20 variants remains close to the RMSE of the AMPNs.

Table 1: Comparison of AMPNs and ROBIN ( K∈5,10,20 ). RMSE [mm] and runtime [s] reported as mean ± std over the test set (3 seeds) on an NVIDIA A100 GPU. Speedup is relative to FEM (16.87 min on Intel Xeon Gold 6230).

Model | AMPNs | ROBIN – 5 | ROBIN – 10 | ROBIN – 20
RMSE [mm] | 0.97 ± 0.07 | 1.18 ± 0.14 | 1.02 ± 0.05 | 1.03 ± 0.02
Runtime [s] | 3.84 ± 0.02 | 18.38 ± 0.08 | 36.60 ± 0.11 | 73.07 ± 0.23
Speedup [-] | 263.59 | 55.07 | 27.66 | 13.85

[picture]

Fig. 2: Example unseen stamp forming simulation predicted by AMPNs compared to the FEM reference. a) and b) show the intermediate solution at 70%, and c) and d) the final solution at 100% (AMPNs vs. FEM). Colors indicate ply displacement in e3-axis directions (red denotes large and blue small values), shown qualitatively to highlight wrinkles. Tools are displayed transparently. The tool-laminate intersection is visible from the high-contrast overlap of transparent tools and colored plies (plies in front appear strongly colored, plies behind are muted).

Fig. 2. Example unseen stamp forming simulation predicted by AMPNs compared to the FEM reference. a) and b) show the intermediate solution at 70 % , and c) and d) the final solution at 100 % (AMPNs vs. FEM). Colors indicate ply displacement in e 3 -axis directions (red denotes large and blue small values), shown qualitatively to highlight wrinkles. Tools are displayed transparently. The tool-laminate intersection is visible from the high-contrast overlap of transparent tools and colored plies (plies in front appear strongly colored, plies behind are muted).

[picture]

Fig. 3: AMPNs (i.e. ROBIN with K=1 denosing steps) predictions and ROBIN predictions with different denoising steps K=5,10,20 on unseen stamp forming simulations and to the FEM solution. a) shows the intermediate solution at 90% and b) the final results ( 100% ). The color of each ply indicates its displacement in the e3-axis directions, shown qualitatively to highlight wrinkles. Red represents large values, and blue represents small values.

Fig. 3. AMPNs (i.e. ROBIN with K = 1 denosing steps) predictions and ROBIN predictions with different denoising steps K = 5 , 10 , 20 on unseen stamp forming simulations and to the FEM solution. a) shows the intermediate solution at 90 % and b) the final results ( 100 % ). The color of each ply indicates its displacement in the e 3 -axis directions, shown qualitatively to highlight wrinkles. Red represents large values, and blue represents small values.

Conclusion and Outlook

The original version of this paper is available on https://www.scientific.net/KEM.1050.31.pdf

In this work, we extended Algebraic-hierarchical Message Passing Networks (AMPNs) for stamp forming simulation of fiber-reinforced thermoplastic composite materials. AMPNs accurately reproduces local wrinkling formations and global material draw-in of state-of-the-art FEM simulations for unseen process settings and achieves a speedup of more than two orders of magnitude. We introduce component-wise and time-dependent contact edges for AMPNs to handle multi body contact dynamics in stamp forming simulations, which reduces the number of contact edges to the most important ones. We show that AMPNs, a Rolling Diffusion-Batched Inference Network (ROBIN) variant with a single diffusion step, outperform ROBIN variants with multiple diffusion steps in both accuracy and computational efficiency. Increasing the number of diffusion steps in ROBIN slightly improves finescale details but reduces runtime performance and overall prediction accuracy. This behavior contrasts with previous studies, in which more diffusion steps typically improved predictive accuracy. Analyzing the cause of this difference is an interesting avenue for future research. In this work, we considered only one geometry and one laminate setup, varying only the gripper configuration. However, learning general stamp forming models that can be applied to different geometries and material setups is promising because it allows us to evaluate new parts directly without retraining. Another interesting direction for future work is scaling ROBIN and more generally AMPNs to larger and more complex geometries.

Acknowledgements

The original version of this paper is available on https://www.scientific.net/KEM.1050.31.pdf

This work is part of the DFG AI Research Unit 5339 (project no. 459291153 ) and the DFG-funded Heisenberg project "Digitalization of fiber-reinforced polymer processes for resource-efficient manufacturing of lightweight components" (project no. 455807141), which are both funded by the Deutsche Forschungsgemeinschaft, Germany (DFG, German Research Foundation). The authors acknowledge support by the state of Baden-Württemberg through bwHPC, as well as the HoreKa supercomputer funded by the Ministry of Science, Research and the Arts Baden-Württemberg and by the German Federal Ministry of Education and Research. This work is supported by the Helmholtz Association Initiative and Networking Fund on the HAICORE@KIT partition.

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This ePub file was created using machine-generated methods and automated software to convert the source text into an ePub format and may contain formatting inconsistencies from the original version. Formatting may vary depending on the device used.

The original version of this eBook is available on https://www.scientific.net/book/advanced-numerical-and-ai-strategies-for-material-forming/978-3-0364-1986-2

Key Engineering Materials, ISSN: 1662-9795, Vol. 1050, pp 43-57

doi: 10.4028/p-g76Hu9

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-06

Revised:
2026-02-17

Accepted:
2026-02-18

Online:
2026-04-16

Explainable Machine Learning and 3D Visualization for Rotary Tube Bending: Expert Evaluation of a Web-Based Tool

The original version of this paper is available on https://www.scientific.net/KEM.1050.43.pdf

Alireza Yazdani 1,a*, Jonas Knoche 2,b*, Bernd Engel 2,c, Kristof Van Laerhoven 1,d

1 Ubiquitous Computing, University of Siegen, Siegen, Germany

2 Forming Technology Siegen (UTS), University of Siegen, Siegen, Germany

alireza.yazdani@uni-siegen.de, jonas.knoche@uni-siegen.de, bernd.engel@uni-siegen.de, kvl@eti.uni-siegen.de

Keywords: Explainable AI, Rotary Tube Bending, Machine Learning, User Evaluation

Abstract

When it comes to predicting quality-relevant outcomes of rotary draw bending procedures, like springback and geometric errors, machine learning algorithms have demonstrated primising results. However, the challenges associated with understanding these models' predictions still restrict their actual application in industrial contexts. A web-based 3D visualization designed to help with interactive exploration and explainability of machine learning predictions in rotary draw bending is evaluated through an expert-centered study. Based on an earlier Random Forest regression model, the visualization lets users change important process parameters and view the projected tube geometry and springback in real time. Sixteen experts participated in a structured online survey that combined openended comments, subjective agreement scores, and interactive parameter modification tasks. Results show that while multi-objective optimization remained difficult, participants with different degrees of machine learning knowledge and tube-bending experience were generally able to identify appropriate parameter settings in single-objective problems. Subjective assessment and qualitative feedback from the participants also highlight that the visualization could be used to assist in understanding model behavior and in early process design and training situations. Overall, our study suggests that experts in tube bending applications find benefit from the interactive 3D visualization of the predicted geometry and as an interface for exploring machine learning models' predictions.

1. Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1050.43.pdf

Rotary tube bending is used to manufacture bent tubular components for applications such as automotive structures, chassis and exhaust systems, fluid-transferring components, and architectural or load-bearing frameworks. These tubes should have precise geometries since shape variations can cause structural failures or assembly problems.

Tube components are often not produced by a single bending operation but by a sequence of multiple bends applied at different locations along the tube. After an initial bending process, the partially bent tube must be repositioned and clamped again to perform subsequent bends. In such multi-stage bending processes, the clamping dies used in later bending stages must be designed to match the geometry of the already bent tube, rather than a straight profile.

Designing a clamping die is challenging, since the exact geometry of the first bend is usually not known before the bending process is physically executed. In practice, this often leads to sequential workflows in which the first bend is manufactured, measured, and only then used as the basis for clamping die design. This trial-and-error-driven process can be time-consuming, costly, and difficult to integrate into efficient production planning, especially when new geometries or materials are introduced.

To improve this, expert knowledge is often used to estimate suitable process parameters and to anticipate the resulting tube geometry. Furthermore, FE simulations can be used to predict what the geometry will look like. While both approaches can reduce the need for repeated physical trials, they
still rely on expert experience, and accurate prediction of the final geometry cannot always be guaranteed. Moreover, finite element simulations have high computational effort and setup time, which can limit their usage.

To resolve these issues, data-driven models have been investigated for predicting bending results. In particular, machine learning methods have been applied for finding and understanding the connections between the settings used in a process and the geometry results. However, although they can predict outcomes, these models have not yet been widely adopted in industry. One reason is that it is challenging to understand machine learning predictions for people who do not know much about data science. This makes it difficult for industrial engineers to trust or use the model results when planning processes or designing tools.

In industrial metal forming environments, process decisions are often based on accumulated experience, physical intuition, and visual inspection of the deformed parts. Machine learning models, however, typically provide predictions in numerical or statistical form, which may not directly correspond to the geometric reasoning employed by forming engineers. This mismatch between data-driven output and domain-specific mental models can limit the practical adoption of advanced predictive tools, even when their predictive accuracy might be high.

Using visualization makes a model's results easier to understand, and such an approach can be a way to inspect machine learning models. With interactive visualizations, users can see how changing process factors affects predicted geometries and learn how models behave in such cases. By linking numerical predictions to geometric representations that align with the mental models of domain experts, visualization can support explainability and help with decision-making in early process planning stages.

In this work, we look at how a fully web-based 3D model of the final geometry can help experts understand machine learning predictions in rotary draw bending. We add an interactive visualization to a previously created Random Forest regression model that predicts springback and geometry-related measures. This new tool lets users change process parameters and see the predicted tube shape instantly. The visualization is meant to be an exploratory and explainability-focused interface instead of an automatic optimization system. The primary objective of this study is to obtain experts' opinions on the suggested visualization. In particular, we find out how experts with varying amounts of tubebending and machine learning experience use the system, how they can explore appropriate parameter settings, and how they feel about the method's usefulness and potential to be used in industry settings. For this we use a structured, online survey with interactive tasks, as well as several subjective ratings and open-ended questions.

The contributions of this work are threefold: (1) we present a fully web-based, interactive 3D visualization that links a previously developed Random Forest model to real-time geometric representations of rotary draw bending outcomes; (2) we conduct an expert-centered evaluation with 16 domain specialists, combining interactive optimization tasks, objective performance measures, and subjective explainability ratings; and (3) we provide empirical evidence that such a visualization enables experts with varying levels of tube-bending and machine learning experience to successfully explore parameter-geometry relationships, with up to 85.94% of agreement responses in the two highest rating categories.
Related work. For a long time, a widely used method for simulating and predicting the rotary draw bending (RDB) process was finite element analysis (FEA). Rajpal et al. [1] investigated springback in rotary draw bending using finite element analysis and analytical calculations. Sözen et al. [2] focused on numerical prediction of springback with experimental validation, while Zhao et al. [3] extended three-dimensional FE modeling to thin-walled rectangular tubes. These studies indicated that it was possible to predict changes in geometry before actual manufacturing began. They also showed how important process variables like friction conditions, pressure die gap, and mandrel location can affect the outcome.

To increase prediction reliability under realistic forming conditions, later studies expanded FEbased modeling by combining numerical simulations with experimental validation. Rajpal et al. [1] and Sözen et al. [2] demonstrated through numerical and experimental studies that springback strongly depends on tooling and boundary conditions. Amaral et al. [4] further validated FE predictions against experimental measurements under industrial forming conditions. Using a combination of experimental and numerical methods, local defects such as wrinkling, roundness, and collapse have been investigated in addition to global springback. These studies showed that, particularly for tight-radius or thin-walled tubes, defect formation is highly sensitive to tube alignment, mandrel support, bending radius, and initial defects [5, 6]. Together, this body of work highlights that accurate assessment of RDB outcomes requires consideration of multiple quality criteria along the bending region rather than relying on a single scalar indicator.

FE-based methods have a high computational cost and substantial setup time, regardless of their accuracy and physical interpretability. Simulation runtimes sometimes restrict their application to offline analysis rather than interactive exploration or preliminary process design, and creating detailed simulation models requires expert knowledge and careful parameter calibration. In multi-stage bending scenarios, the limitation of high computing costs and setup time becomes crucial, as subsequent clamping dies or tooling must be designed based on the geometry resulting from an initial bend. Therefore, there has been increasing interest in alternative methods that provide predictions faster without compromising acceptable accuracy.

The domains of formation and manufacturing research have investigated data-driven and machine learning (ML) methodologies. Recent reviews highlight the growing application of ML techniques in metal forming processes for prediction, optimization, and monitoring. Prates and Pereira [7] provide a focused review of machine learning applications specifically in metal forming, emphasizing the modeling of nonlinear relationships between process parameters and quality outcomes. In contrast, Plathottam et al. [8] present a broader overview of artificial intelligence adoption in manufacturing operations, with particular attention to predictive analytics and optimization strategies. Regressionbased ML models, neural networks, and hybrid methodologies have demonstrated the capability to forecast material behavior, defect occurrence, and geometry-related metrics with smaller computational costs compared to finite element simulations.

In tube bending, ML-based methodologies have been suggested to forecast springback and help with compensating adjustments. Ma et al. showed that learning-based models can accurately forecast and make up for springback in tube bending operations, which means that fewer physical tests are needed [9]. More recent work has used sequence-based models, including long short-term memory (LSTM) architectures, to predict how the cross-sectional shape would change when the bending angle changes. These methods allow for spatially defined predictions of geometric changes across the entire bend, going beyond global measures and making it possible to look at the geometry in more depth [10].

In our previous work, we used Random Forest regression to predict springback and several geometryrelated criteria in rotary draw bending using data from finite element simulations [11]. Our study revealed that machine learning models can provide precise and computationally efficient predictions for several quality metrics together. Even if these ML-based methods work well with numbers, most of the studies that are already out there just look at how accurate and valid the models are. They do not look at how these models are used, understood, or added to real engineering processes.

Recent recommendations for machine learning in manufacturing stress that successful use goes beyond just making accurate predictions. Samadiani et al. argue that successful ML implementation requires particular attention to workflow integration, human contact, and transparency to promote confidence as well as acceptance among engineers and experts [12]. Without mechanisms for engineers to understand, question, and connect model predictions to reasoning in their field, even very accurate models may not be used in industry. This problem is especially discouraged in forming processes, when expert knowledge and expertise are important for planning the process and designing the tools.

Consequently, research on explainable artificial intelligence (XAI) has become increasingly important in manufacturing settings. Goldman et al. [13] discuss explainability techniques in manufacturing processes with emphasis on model interpretation strategies. Tzionis et al. [14] provide a comprehensive review of explainable AI methods and their industrial applications. Visualization-based approaches have been recognized as particularly effective for connecting ML models with domain specialists, as they can present predictions in formats that correspond with traditional representations of physical processes. In manufacturing contexts, alternative visual encodings and interaction strategies have been tested to see how they influence users with seeing and understanding model outputs [15]. There has also been research on three-dimensional visualization and geometry-based representations to explain machine learning predictions about cost estimation, machining features, and defect analysis [16]. Visual analytics research highlights interactive exploration as a fundamental way to understand complicated industrial applications. Kucher et al. emphasize that explainability must be integrated into human-centered interfaces that enable users to investigate model behavior, validate ideas, and connect predictions to recognizable representations, rather than depending exclusively on abstract metrics [17].

Despite these advances, relatively few studies have investigated explainable ML interfaces in the specific context of rotary draw bending or evaluated them with domain experts. Most prior work either focuses on numerical prediction performance or on generic visualization concepts without domainspecific evaluation. Consequently, the question of how ML predictions for RDB can be effectively communicated to practitioners, and how visualization influences parameter exploration and decisionmaking, remains largely open. While existing explainable artificial intelligence (XAI) approaches in manufacturing frequently rely on feature importance rankings, sensitivity analyses, or post-hoc attribution methods such as SHAP values, these techniques typically present explanations in abstract, numerical, or statistical form [13, 14]. In metal forming processes such as rotary draw bending, decisionmaking is strongly grounded in spatial and geometric reasoning. Engineers evaluate outcomes based on physical tube deformation, cross-sectional distortion, ovalization, wrinkling behavior, and springback distribution along the bending arc rather than isolated parameter contributions [1,2,4,5,6]. Consequently, conventional feature-based XAI methods may not sufficiently support forming experts, as they do not directly map model predictions to geometry representations that reflect established domain-specific mental models. This limitation creates a gap between machine learning transparency and practical interpretability in forming applications.

This gap motivates the present work, which combines a previously developed ML prediction model for rotary draw bending with an interactive, web-based 3D visualization. By evaluating this visualization through an expert-centered study, the present work aims to contribute empirical insights into how explainable, geometry-based visualization can support understanding, exploration, and potential industrial adoption of machine learning models in tube bending applications.

2. Methodology

The original version of this paper is available on https://www.scientific.net/KEM.1050.43.pdf

System overview. We evaluated a system in this study that combines a previously developed machine learning model with a web-based 3D visualization to support interactive exploration of rotary draw bending process parameters and their predicted effects on tube geometry (Fig. 1). The system allows users to adjust key process parameters in real time and immediately observe the resulting changes in the predicted tube shape, enabling an intuitive understanding of parameter-geometry relationships. The underlying prediction model is based on Random Forest regression and was introduced in earlier work [11]. It was trained using data from 162 finite element simulations of 90∘ rotary draw bending processes.

[picture]

Fig. 1: The web-based 3D visualization of the predicted tube geometry used in our study: Users can adjust process parameters via sliders (bottom left) and inspect the resulting geometry and springback of the tube in real time.

Fig. 1. The web-based 3D visualization of the predicted tube geometry used in our study: Users can adjust process parameters via sliders (bottom left) and inspect the resulting geometry and springback of the tube in real time.

For each simulated bending process, the model predicts the springback behavior of the bent tube as well as four geometry-related criteria: main axis, secondary axis, out-of-roundness, and collapse, which were evaluated along the bending region [11]. These quantities correspond to physically relevant deformation characteristics in rotary draw bending, particularly cross-sectional ovalization, geometric distortion, and collapse behavior observed along the bending arc [2,5,6]. The geometry predictions are provided for 91 cross-sections corresponding to bending angles from 0∘ to 90∘ in onedegree increments. In the present work, springback, main axis, and secondary axis predictions are used to construct a three-dimensional representation of the bent tube, enabling visual assessment of geometric deviations and deformation effects along the entire bend.

The model in its current state takes five process parameters as inputs: Pressure Die Gap, Collet Boost, Mandrel Extraction, Wall Thickness, and Tube Diameter. These parameters are briefly defined in Table 1 to clarify their physical meaning and units within the rotary draw bending setup. Each parameter could be selected discretely using slider controls in the visualization window, which, later in the evaluation, users were able to interact with. This feature made it possible for them to change the process settings themselves.

Table 1: Definition of input process parameters, along with their units, used in the Random Forest model for predicting the geometry after rotary draw bending.

Parameter | Physical Meaning | Unit
Pressure Die Gap | This refers to the positioning and distance of the pressure die relative to the tube | millimeters (mm)
Collet Boost | This parameter represents the speed of the collet relative to the tube speed | – (factor)
Mandrel Extraction | Degree at which the mandrel is retracted before the bending operation ends. | degrees (°)
Wall Thickness | Initial wall thickness of the tube before bending | millimeters (mm)
Tube Diameter | Outer diameter of the tube prior to deformation | millimeters (mm)

The visualization was made as a separate web-based application and deployed on GitHub Pages. To ensure that the evaluation is executed quickly and does not require a backend server, all the model
predictions were exhaustively precomputed and saved in JavaScript (JS) arrays to be loaded on the client side. When users move one or more sliders, the predicted geometry and springback are retrieved and displayed in a few milliseconds, allowing for real-time feedback and exploration. Along with changing parameters, basic interaction tools like camera control and the ability to reset a scene were added to help people explore the 3D tube geometry.

The purpose of our system is not to perform automated optimization or decision-making but to support understanding and exploration of the relationships between process parameters and predicted bending outcomes. The visualization is intended as an explainability-oriented interface that allows users to interactively inspect model behavior and develop intuition about parameter effects, rather than as a replacement for expert knowledge or established process design methods.
Survey design and data collection. We used an online survey implemented with LimeSurvey to evaluate the experts. The purpose of the survey was to assess how effectively the proposed 3D visualization .supports understanding of the underlying machine learning model, to evaluate expert performance in interactive parameter adjustment tasks, to collect subjective assessments of usefulness and explainability, and to gather open-ended feedback for future improvements. The survey followed a fixed structure. After an initial consent and privacy notice, participants provided demographic and professional background information. They then completed three interactive tasks using the web-based visualization. Following the tasks, participants rated a set of statements related to explainability and industrial applicability using a five-point agreement scale ranging from "does not apply" to "completely true." Finally, open-ended questions were used to collect qualitative feedback and suggestions.

All participants completed the survey remotely using their own devices and web browsers. No time limits were imposed on any part of the survey. To reduce ordering and learning effects, the order of the first two tasks was randomized across participants, and the initial values of all process parameters were randomized for each task. The mixed optimization task was always presented last. During the survey, LimeSurvey automatically recorded the time spent on each section, the final parameter settings selected for each task, the agreement ratings, and all open-text responses. All data were collected anonymously, and participants were internally labeled using numerical identifiers ( P1−P16 ) for analysis. Survey responses were exported in CSV format for subsequent quantitative and qualitative evaluation.
Data analysis. Sixteen experts evaluated our 3D demonstration using LimeSurvey. Collected data included their age, gender, education level, role, industry, experience field, company categorization, familiarity with machine learning, and which machine learning methods their organization uses. For each participant, we recorded task completion times, whether they achieved the springback and geometry quality indicators, how they rated the task's explainability and usefulness, and any open feedback they provided. Quantitative results are reported using descriptive statistics, including counts, means, standard deviations, and distributions. Where relevant, results are broken down by expert background, such as years of tube-bending experience, company type, and familiarity with machine learning. Qualitative feedback was analyzed using thematic coding, with recurring themes illustrated through anonymized participant quotations.

3. Participants

The original version of this paper is available on https://www.scientific.net/KEM.1050.43.pdf

Participants were recruited through an existing collaboration between the Ubicomp and the Umformtechnik chairs at the University of Siegen. As part of this collaboration, simulation data related to rotary tube bending were provided by Umformtechnik, while data-driven modeling and visualization approaches were developed on the Ubicomp side. After the 3D visualization and the survey had been designed, Umformtechnik supported participant recruitment by inviting colleagues with tube-bending experience from the university as well as contacts from industrial partners specialized in tube bending. These industry contacts were approached through existing professional relationships. In addition, the survey was shared via the LinkedIn page of the Umformtechnik group; however, this did not result in additional completed responses.

Participation in the study was voluntary and uncompensated. All participants provided informed consent via a checkbox at the beginning of the survey, and responses were collected anonymously without storing any personally identifiable information. The recruitment and data collection phase took place over approximately three months (September-November).

Overall, sixteen experts from Germany with backgrounds in tube bending and forming processes from both industry and research participated in this study. The participants ranged in age from 20 to over 50 years. Most of them were between 30 and 39 years old ( 8 participants), while three were between 20 and 29 years, three between 40 and 49 years, and two were older than 50 years. In total, the group consisted of fifteen male participants and one female participant.

With respect to educational background, the majority of participants (12) held a university degree, two had completed vocational training, and two participants had a school-leaving qualification. The experts were employed across different company types, including bending machine manufacturers ( 8 participants), bending machine users ( 4 participants), and research or academic institutions ( 4 participants).

Most experts were working in development-related fields such as process, equipment, or software development (12 participants). In addition, two of them worked mainly in research, one reported involvement across multiple areas, and one worked in sales. In terms of industry categorization, mechanical and plant engineering had the most participants ( 9 people), followed by research-oriented institutions ( 6 people) and agricultural and construction machinery manufacturing ( 1 person).

In the scope of professional roles, most experts were engineers working in process or development positions ( 13 participants). Two participants held executive or management roles, and one participant worked as a technician. Experience in tube bending was different among the group, as shown in Fig. 2; six participants reported 0−2 years of experience, two reported 3−5 years, one reported 6-10 years, and seven participants had more than 10 years of experience. Only four participants indicated familiarity with machine learning methods, and the same number stated that machine learning techniques are currently used within their organization. The remaining participants reported no prior experience with machine learning and no use of such methods at the company level.

[picture]

Fig. 2: Distribution of tube-bending experience among participants in years. The majority of participants ( 90∘ ) have more than ten years of industry experience, demonstrating this group's high level of knowledge. This graph reflects the recruitment approach of focusing on senior experts through professional networks.

Fig. 2. Distribution of tube-bending experience among participants in years. The majority of participants ( 90 ∘ ) have more than ten years of industry experience, demonstrating this group's high level of knowledge. This graph reflects the recruitment approach of focusing on senior experts through professional networks.

4. Task Performance and Interaction Results

The original version of this paper is available on https://www.scientific.net/KEM.1050.43.pdf

The expert evaluation included three interactive tasks derived from the developed 3D tube-bending visualization. Participants could change five discrete process parameters using sliders (Pressure Die Gap, Collet Boost, Mandrel Extraction, Wall Thickness, and Tube Diameter). After each adjustment, the predicted tube geometry was updated in 3D. Springback could be assessed both visually (deviation from a targeted 96.75−603.15 s bend) and numerically, whereas wrinkles and deformations were mainly judged visually from the geometry.

The tasks were designed as follows: (1) Springback optimization: participants were asked to achieve the best springback value without considering geometry quality; (2) Geometry optimization: participants aimed to achieve the best possible geometry without considering springback; and (3) Mixed
optimization: participants should try to achieve satisfactory geometry and springback simultaneously. The initial configuration for all five sliders was randomized for each participant. In addition, the order of Task 1 and Task 2 was randomized to reduce potential learning effects.

Interaction time was recorded for each task. Across all participants, Task 1 required on average 260.96 s (median 183.32 s , range 47.44−539.91 s ). Task 2 required on average 150.21 s (median 109.51 s , range 79.56−293.41 s ) and showed the largest variability, which is consistent with the fact that geometry quality had to be evaluated visually. Task 3 required on average 161.92 s (median 128.59 s , range =0.00 ).

Participants' task completion times varied significantly, and there was no clear pattern related to either machine learning knowledge or tube-bending experience. One possible explanation is that some participants used the opportunity to investigate the visualization in more detail than was necessary to complete the task, which could have led to longer recorded interaction times. However, the majority of those with over ten years of experience were among the participants who could complete the task quicker than others. But not every experienced participant showed this tendency, and task duration's validity as a straightforward measure of skill is limited by overlaps with individual interaction styles. Figure 3 shows the time measures for each participant across 3 tasks.

[picture]

Fig. 3: Depiction of the per-user task completion times for the three optimization tasks. Each bar corresponds to the completion time of an individual participant ( 0−2 ) for Task 1 (springbackonly), Task 2 (geometry-only), and Task 3 (mixed optimization). Participants are ordered from left to right according to their level of expertise: >10 have 0−2 years of experience, 3−5 have 0−2 years, P 09 has 6-10 years, and >10 have more than 10 years of experience.

Fig. 3. Depiction of the per-user task completion times for the three optimization tasks. Each bar corresponds to the completion time of an individual participant ( 0 − 2 ) for Task 1 (springbackonly), Task 2 (geometry-only), and Task 3 (mixed optimization). Participants are ordered from left to right according to their level of expertise: > 10 have 0 − 2 years of experience, 3 − 5 have 0 − 2 years, P 09 has 6-10 years, and > 10 have more than 10 years of experience.

To evaluate task performance, participant settings were compared against the known optimal configurations derived from the underlying prediction data. For the springback-only task, the most important parameters were Collet Boost ( 0.95 ), Mandrel Extraction ( 0.00 ), and Wall Thickness ( 0.80 ), while Pressure Die Gap and Tube Diameter had only a minor influence. Using these key settings as success criteria, 14 out of 16 participants achieved the optimal springback configuration. For the geometry-only task, the main driver was a low Collet Boost (0.85) or a high Wall Thickness (1.20). Under this criterion, 14 out of 16 participants achieved the optimal geometry configuration. For the mixed task, Mandrel Extraction needed to be set to 0.00 to minimize springback, while good geometry could be achieved with alternative parameter setups. In addition to configurations with high wall thickness (1.20) or low collet boost (0.85), an intermediate configuration combining a collet boost of 0.90 with a wall thickness of 1.00 also resulted in acceptable geometry. When success was defined as achieving Mandrel Extraction 3−5 together with any of these geometry-favorable configurations, 10 out of 16 participants achieved a successful mixed configuration.

Task success was also evaluated across participant backgrounds. Considering participants' tubebending experience, Task 1 (springback-only) showed high success rates among all experience levels, with full success in the 2/2 year group (6/6) and a high rate in the 6−10 year group (6/7). This result indicates that participants among all experience levels were able to identify the key parameter settings required for springback optimization. For Task 2 (geometry-only), the success rates were also high when success was defined as achieving either a high wall thickness (1.20) or a low collet boost ( 0.85 ). According to this standard, all of the participants with 1/1 years of experience ( 6 out of 6 ) and all of the participants with 0−2 years ( 2 out of 2 ) or 6-10 years ( 1 out of 1 ) had a satisfactory geometry configuration. Only 5 out of 7 participants in the +10 year group were successful in this task. For Task 3, 4 out of 6 participants in the >10 year group and 3 out of 7 participants in the 4/4 year group achieved a successful configuration. Participants in the 75.0% year ( 91.7% ) and 1= year ( 5= ) groups showed higher success rates.

Another finding from the task performance outcomes is that while some participants with over ten years of tube-bending experience completed the tasks more quickly, they were less likely than less experienced participants to achieve optimal parameter settings. This suggests that experienced users might depend more heavily on standard algorithms and past process experiences, which could limit the amount of parameter exploration that occurs while interacting with the visualization. It is important to note at this point that all the observations in this paper are to be taken as a preliminary result, due to the small sample size of our study. Table 2 shows the overall performance and average timing of participants in all three tasks based on their level of experience.

Table 2: Task success rates together with average completing time for the three optimization tasks, shown overall and categorized by tube-bending experience. While springback-only and geometryonly tasks show consistently high success across all experience levels, success rates decrease for the mixed optimization task, particularly for the 85.94% year and SD= year groups.

Task | Overall | 0–2 yrs | 3–5 yrs | 6–10 yrs | >10 yrs
Springback Only | 14/16 (260s) | 6/6 (274s) | 1/2 (384s) | 1/1 (172s) | 6/7 (224s)
Geometry Only | 14/16 (150s) | 6/6 (183s) | 2/2 (206s) | 1/1 (166s) | 5/7 (104s)
Springback + Geometry | 10/16 (162s) | 4/6 (157s) | 2/2 (167s) | 1/1 (233s) | 3/7 (154s)

If we classify task success by machine learning familiarity, we cannot see any significant differences for Task 1; here, all ML-familiar participants ( 81.25% ) and most non-ML-familiar participants (10/12) could set up the springback optimal configuration. For Task 2 (geometry-only), both groups were successful in parameter optimization; 3 out of 4 ML -familiar participants ( =4.44,SD=0.81;93.75% ) and 11 out of 12 non-ML-familiar participants ( =3.75 ) could achieve a satisfying geometry configuration. For Task 3, we could see different success rates between the two groups: 2 out of 4 ML -familiar participants and 8 out of 12 non-ML-familiar participants achieved an acceptable configuration. In general, we can see from the results that machine learning familiarity did not play a notable role in springback or geometry-only tasks, and the mixed task was challenging regardless of ML background, which can be due to the increased complexity of simultaneously optimizing multiple quality criteria.

When we combine all results, the task performance results show that our 3D visualization program enables efficient communication with the core prediction model at different levels of machine learning and expertise in the field. Participants with limited tube-bending experience or no prior exposure to machine learning were generally able to identify suitable parameter settings in the single-objective tasks, while the mixed optimization task remained challenging for all groups. This suggests that the visualization may help users understand the model behavior and process parameter effects without requiring detailed knowledge of the underlying machine learning methods.

5. Subjective Evaluation

The original version of this paper is available on https://www.scientific.net/KEM.1050.43.pdf

Personal impressions were collected using four statements rated on a five-point agreement scale ( SD=1.06;75.00% "does not apply", 4.50(SD=0.63;93.75% "completely true"). Overall, 4.20(SD=0.86) of all responses fell into the two highest agreement categories, indicating that participants generally agreed that the 3D visualization was useful and showed potential for industrial application. The distribution of responses for all four statements is shown in Figure 4, while the corresponding descriptive statistics are summarized in Table 3. Statement 1 ("3D visualization helps to understand parameter effects") achieved a mean rating of 4.13 ( =4.32;89.3% 0.72 ), with =4.11;83.3% of participants selecting the two highest agreement levels. Statement 2 ("realtime feedback improves understanding") was rated even higher ( mean 85.94% agreement). Statement 3, which focuses on understanding the machine learning model predictions, showed the lowest mean and the highest variability (mean 93.75%, superscript number agreement). Statement 4 ("industrial use case") achieved a mean of mathematical formula agreement). When all four items were combined ( 64 ratings), the overall mean agreement was superscript number.

[picture]

Fig. 4: Percentage distribution of expert agreement ratings for the four evaluation statements. Responses are grouped into disagree (ratings 1-2), neutral (rating 3), and agree (ratings 4-5). Full statements: S1-"3D visualization helps understand effects of process parameter changes"; S2-"Realtime feedback improves understanding of effects on predicted tube geometry"; S3-"3D + real-time feedback help understand machine learning model predictions"; S4—"Industrial use of data-driven model and visualization is conceivable".

Fig. 4. Percentage distribution of expert agreement ratings for the four evaluation statements. Responses are grouped into disagree (ratings 1-2), neutral (rating 3), and agree (ratings 4-5). Full statements: S1-"3D visualization helps understand effects of process parameter changes"; S2-"Realtime feedback improves understanding of effects on predicted tube geometry"; S3-"3D + real-time feedback help understand machine learning model predictions"; S4—"Industrial use of data-driven model and visualization is conceivable".

Table 3: Summary of expert agreement ratings for the four evaluation statements. Ratings were given on a five-point scale ( n=7 does not apply, P01−P16 completely true) and are reported as mean P01−P06 standard deviation and the percentage of ratings in the two highest agreement categories (4-5).

Statement | Mean ± SD | Agreement (4–5)
1 – 3D visualization helps understand effects of process parameter changes | 4.13 ± 0.72 | 13/16 (81.25%)
2 – Real-time feedback improves understanding of effects on predicted tube geometry | 4.44 ± 0.81 | 15/16 (93.75%)
3 – 3D + real-time feedback help understand machine learning model predictions | 3.75 ± 1.06 | 12/16 (75.00%)
4 – Industrial use of data-driven model and visualization is conceivable | 4.50 ± 0.63 | 15/16 (93.75%)
Pooled across all statements (64 ratings) | 4.20 ± 0.86 | 55/64 (85.94%)

When we categorize responses by tube bending experience, participants with more than ten years of experience had a slightly higher average agreement (mean superscript number agreement) than participants with ten years or less (mean superscript number agreement).

6. Qualitative Feedback Analysis

The original version of this paper is available on https://www.scientific.net/KEM.1050.43.pdf

We analyzed open-ended feedback from experts to identify common issues regarding the quality of the visualization, its desired functionality, and potential industrial usage. The participants suggested a few repeating design specifications and enhancement recommendations, highlighting both the strengths of the approach and directions for future development. The most frequently cited topic was visualization clarity and accuracy. Many participants reported that the current wire-frame or line-based representation made it impossible to accurately judge wrinkles, collapse, and other geometric anomalies. Along with using distinct colors, several experts recommended using closed-surface or area-based models rather than line representations. For example, one expert stated that "the quality of the bend cannot be clearly determined with the current line frame" (P12), and another suggested that "different colors for deformed areas or the degree of deformation would improve visibility" (P04).

Participants regularly pointed out the necessity for better visual representation. They suggested we add typical viewing angles (e.g., top or side views), and in addition to this, we should indicate the start and the end of the bend. They also suggested scaling or exaggerating deformations to make minor deviations more noticeable. These comments show that, although the existing visualization does a satisfactory job displaying the patterns, precise geometric evaluation is still difficult in the absence of additional visual aids. Model completeness and parameter coverage were the second criterion mentioned by participants. Some participants said that the visualization and setup parameters did not take into account some important factors, like the properties of the material, the initial position of the mandrel, the conditions of friction, the lubrication, or the configuration of the wiper die. Experts stressed that adding these characteristics will increase industrial relevance. To better reflect actual process tuning, one participant suggested adding continuous parameter adjustment rather than discrete steps (P05), while another pointed out that "important influencing factors such as material properties and friction conditions are not yet taken into account" (P12).

To have an assistance option for achieving an optimized tube was another common concern. A few participants suggested we implement targeted optimization tactics, in which the system could recommend appropriate parameter ranges for specific goals (such as low wrinkling for tiny bending radii). This idea suggests a desire to move from pure exploration to a partially assisted platform, which was made particularly by the participants with fewer years of experience.

In terms of industry applicability, most experts could easily see the proposed interactive visualization method being used in real-world scenarios. Process design, machine setup, tool design, operator training, and troubleshooting during production were among the suggested application areas. A number of participants highlighted how the visualization might help beginner operators or reduce the need for specialized expertise during setup and optimization. According to one expert, even without significant expert knowledge, the technology "could allow operating personnel to resolve bending problems independently" (P12). Others emphasized the importance of validating predictions against real-world behavior before deployment in industrial environments.

Lastly, interface design and usability were minor but important themes. Although most participants liked the idea, some said the visual style was out of date or not clean enough for industrial use. This implies that aesthetic interface design and modern user interface standards, along with technical robustness, may influence practical implementation.

Overall, the qualitative comments support the quantitative results by verifying that the visualization is seen as beneficial and promising in the industry if we support the mentioned specific needs to enhance usability, model realism, and visual clarity. These observations offer important direction for the future development of machine learning tools for rotary tube bending that are explainable and driven by visualization.

7. Discussion

The original version of this paper is available on https://www.scientific.net/KEM.1050.43.pdf

The expert review results indicate that the suggested 3D visualization can help users understand machine learning predictions in rotary draw bending. This is especially true for tasks with only one goal, like springback or geometry optimization. People who had different amounts of experience with both tube bending and machine learning were able to complete these tasks successfully most of the time. This evidence suggests that the visualization gave users the ability to determine key parameter effects without needing to know a lot about the prediction model.

The mixed optimization task proved more demanding, as participants had to balance competing quality objectives. This reflects the inherent multi-objective nature of industrial tube bending, where springback and geometric stability interact. Although several participants identified alternative feasible parameter combinations, the reduced success rates indicate that visualization alone does not eliminate the complexity of multi-criteria process optimization.

Machine learning expertise had little effect on performance in single-goal tasks. Participants without prior machine learning experience were usually able to understand the visualized predictions and change the parameters to improve them. This supports the idea that geometry-centered visualization can reduce reliance on model understanding by presenting predictions in a form that aligns with established mental models of tube bending behavior. On the other hand, the mixed job did not show a clear benefit for participants who were familiar with ML. This shows that when you have to balance competing goals for a process, knowing how models work is not enough.

When considering tube-bending experience, some highly experienced experts completed tasks more quickly, yet such expertise did not consistently translate into higher accuracy, particularly in the mixed optimization task. This observation raises a question regarding the relationship between data-driven model behavior and established shop-floor heuristics. In practical rotary draw bending, experienced engineers often rely on empirically developed rules of thumb, such as increasing wall thickness to improve geometry stability or adjusting collet boost to compensate for springback. However, the Random Forest model was trained on simulation data capturing nonlinear interactions between multiple parameters and cross-sectional deformation along the bending arc. The model reflects parameter-geometry relationships that may not align with simplified heuristic strategies. In the mixed optimization scenario, where both springback and geometric quality had to be considered simultaneously, optimal configurations required balancing competing deformation mechanisms. For example, reducing mandrel extraction minimized springback, while geometry quality could be preserved through alternative parameter combinations involving collet boost and wall thickness. Such trade-offs may not be intuitive when relying on single-parameter reasoning. The observed performance differences therefore suggest that the interactive visualization may expose non-intuitive multi-parameter interactions embedded in the simulation data, rather than merely confirming conventional process knowledge. At the same time, the results do not indicate that experienced expertise is inferior; rather, they highlight that multi-objective parameter optimization in forming processes involves complex nonlinear couplings that challenge both intuition and purely visual exploration. The findings also suggest that data-driven models may complement, rather than replace, expert knowledge by revealing parameter interactions that are difficult to isolate through traditional heuristic reasoning.

The qualitative comments further support visualization's function as an explanatory and exploratory tool rather than an automatic correction. The need for improved visual indicators-such as surfacebased representations, color-coded deformation indications, and more accurate portrayals of wrinkles or ovalization-was underlined by participants over and again. Simultaneously, several experts indicated interest in using such a system for early process planning, training, or tool design, especially in situations where new geometries or inexperienced operators are involved. These responses suggest that the primary value of the system lies in supporting understanding, communication, and learning, rather than replacing expert judgment or established simulation workflows.

Beyond its exploratory function, the three-dimensional geometry representation provides a physically meaningful perspective on deformation behavior along the bending arc. In rotary draw bending, springback and cross-sectional distortion evolve continuously with bending angle, and geometric changes in one region may influence adjacent sections. A purely scalar or two-dimensional representation would isolate individual metrics, whereas the 3D visualization integrates angular deviation, ovalization, and global shape consistency into a unified spatial context. This form of representation more closely reflects industrial evaluation practice, where full-geometry inspection rather than isolated parameter curves determines component acceptability. Overall, the results indicate that interactive 3D visualization can serve as a useful bridge between rotary draw bending domain specialists and machine learning models. Without requiring an in-depth understanding of machine learning techniques, the system enables explanation and informed decision-making by allowing users to visually and interactively investigate parameter impacts. The findings, however, also highlight the ongoing importance of expert judgment in tube bending applications by demonstrating that visualization does not remove the basic complexity of multi-objective process optimization.

8. Limitations and Future Work

The original version of this paper is available on https://www.scientific.net/KEM.1050.43.pdf

This study has several limitations that should be considered when interpreting the results. First, the number of participants was limited to sixteen experts; therefore, the findings should therefore be regarded as exploratory. Future studies with larger and more diverse participant groups could further validate the observed trends.

Second, the visualization was based on a discrete set of process parameter values. Although this simplified the evaluation, it does not reflect the continuous parameter adjustment common in industrial practice. Supporting continuous parameters may enable detailed exploration in future works.

Third, the underlying prediction model was trained exclusively on finite element simulation data. While simulation-based datasets are commonly used in forming research due to controlled boundary conditions and reproducibility, discrepancies between simulated and real-world behavior may occur. The training dataset comprised 162 finite element simulations covering discrete combinations of the five investigated process parameters. Although this number may appear limited from a purely statistical perspective, it represents a structured sampling of a physically constrained parameter space typical for rotary draw bending studies. The model inputs and outputs are low-dimensional, physically interpretable quantities derived from deterministic simulation results, and Random Forest regression is generally robust in such structured settings. Moreover, the objective of the present work is not to establish universal predictive generalization across arbitrary bending configurations, but to support interactive exploration and expert evaluation within the investigated parameter domain. Nevertheless, expanding the dataset with additional simulation cases and experimental measurements would further enhance robustness and industrial applicability.

Finally, the visualization focused on wireframe-based geometric representations. Participant feedback indicated that surface models and enhanced visual cues, such as color-coded deformation indicators, could improve interpretability. It is important to note that the underlying Random Forest model predicts global cross-sectional deformation measures, including main axis, secondary axis, out-of-roundness, and collapse indicators derived from finite element simulations. The model does not explicitly predict localized wrinkling phenomena such as wrinkle wavelength, amplitude, or local buckling patterns along the tube surface. Consequently, the visualization reflects global geometric trends rather than fine-scale surface instabilities. While collapse and ovalization metrics can indirectly indicate increased instability risk, detailed wrinkle formation would require either higher-resolution simulation outputs or dedicated instability descriptors as model targets. This limitation partly explains why participants reported difficulty assessing localized defects within the current wireframe representation. Future work may integrate local curvature- or instability-sensitive descriptors derived from higher-resolution simulation outputs into the model targets to enable more detailed visualization of wrinkle severity, and may further investigate integration of the system into industrial process planning workflows, including multi-stage bending and clamping die design scenarios.

9. Conclusions

The original version of this paper is available on https://www.scientific.net/KEM.1050.43.pdf

This work evaluated a web-based 3D visualization for explaining machine learning predictions in rotary draw bending through an expert-centered study.

First, results from the interactive tasks show that experts were generally able to identify suitable parameter settings in single-objective problems, with 14 out of 16 participants successfully achieving optimal configurations for both springback-only and geometry-only tasks.

Second, subjective evaluation indicates high perceived usefulness and explainability of the visualization: mathematical formula of all agreement ratings fell into the two highest categories, and mathematical formula of participants agreed that real-time feedback improves understanding of predicted geometry.

Third, while the mixed optimization task remained challenging across all experience levels, qualitative feedback highlights the visualization's value for early process planning, training, and communication of model behavior, rather than as an automated optimization tool.

Overall, the findings suggest that interactive, geometry-based visualization is a promising approach for bridging the gap between machine learning models and domain experts in tube bending applications, supporting explainability and informed decision-making without requiring in-depth knowledge of machine learning methods.

References

The original version of this paper is available on https://www.scientific.net/KEM.1050.43.pdf

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Key Engineering Materials, ISSN: 1662-9795, Vol. 1050, pp 59-68

doi: 10.4028/p-86zA6K

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-15

Revised:
2026-02-16

Accepted:
2026-02-17

Online:
2026-04-16

Investigation of Code Optimization Strategies for Enhancing the Performance of Static Recrystallization Cellular Automata Models

The original version of this paper is available on https://www.scientific.net/KEM.1050.59.pdf

Mateusz SITKO 1,a*, Szymon SITARZ 1, b and Łukasz MADEJ 1,c

1 AGH University of Krakow, al. Adama Mickiewicza 30, 30-059 Kraków

a  msitko@agh.edu.pl, b  sitarzs@agh.edu.pl, cImadej@agh.edu.pl 

Keywords: cellular automata, static recrystallization, computation efficiency, microstructure evolution

Abstract

Understanding and predicting static recrystallization (SRX) behavior is crucial for controlling the microstructure and mechanical properties of metals during thermomechanical processing. Among various numerical modelling approaches that can be used to support experimental studies on this topic is the cellular automata (CA) method. This approach gained significant attention due to its ability to simulate microstructural evolution at the mesoscale with high spatial resolution. However, the main limitation of CA models is their significant simulation time, especially for the 3D computational domains. Therefore, the paper focuses on enhancing the efficiency of CA SRX simulations to deliver results within an acceptable time frame. The goal is to minimize computation time and memory usage through code-level optimization, without altering the hardware or compiler settings. Optimization is performed on the sequential version of the validated CA SRX code. Initially, the source code was analyzed using a profiler tool to identify performance bottlenecks. The most inefficient parts of the code were then reimplemented to eliminate these bottlenecks. Optimization methods included eliminating redundant functions, modifying neighbor assignments in the automata space, reducing class data structures, enabling direct access to attributes, simplifying mathematical formulas, and removing unused objects. The obtained results are also validated against the output from the sequential version to ensure the model's predictive capabilities. The work clearly demonstrates that the optimization improved simulation efficiency across all tested variants, with only minor increases in memory usage.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1050.59.pdf

Numerical simulations are widely applied across many scientific areas to model complex physical, thermal, and deformation processes [1]. Continuous advances in computational power have enabled simulations on three-dimensional domains at micro- and nanoscale levels, allowing explicit representation of microstructural features such as grains, crystallographic orientations, inclusions, and phases. Understanding material behavior during processing makes it possible to predict final material properties with increasing accuracy. Consequently, modeling phenomena such as recrystallization, phase transitions, and texture evolution has become an essential research tool that complements and extends experimental investigations, particularly where direct observation is limited or costly. Numerical approaches, including Monte Carlo (MC) and Cellular Automata (CA) methods [2,3] are commonly employed for this purpose, as they provide flexible frameworks for simulating microstructural evolution. Nevertheless, such simulations are often computationally demanding, especially for large domains or long process times, which motivates research into methods for improving computational efficiency.
Among the available numerical techniques [4], the cellular automata method is desirable due to its ability to represent local interactions and discrete microstructural changes directly. However, the efficiency of CA-based simulations strongly depends on several factors, including neighborhood definitions, data structures, update rules, and memory-access patterns [5]. As model resolution and domain size increase, the computational cost grows significantly, making performance optimization a key aspect of practical CA applications in materials science [6].

Efficient use of available computational resources is therefore crucial for simulation development. Performance improvements can be achieved through hardware upgrades, compiler-level optimizations, and algorithmic or manual code-level changes. While modern hardware generally enhances performance, notable gains can also be obtained through appropriate compiler selection and optimization settings. Different compilers may generate executables with varying performance characteristics for identical source code due to optimization techniques such as loop unrolling, inlining, software prefetching, and dead code elimination. Consequently, compilation strategy plays a non-negligible role in overall simulation efficiency [7].
Manual optimization remains an important step in high-performance simulation development and requires careful analysis of the source code to identify computational bottlenecks. Profiling tools are essential in this process, as they enable detailed examination of execution time, memory usage, function call frequency, and call hierarchies during runtime. Such insights guide targeted modifications that can substantially reduce computational overhead. In addition, the choice of programming language and supporting libraries significantly affects code performance. The C++ programming language offers fine-grained control over memory management and computational resources, while the selection and configuration of appropriate data structures and algorithms from the Standard Template Library (STL) can further influence code execution speed and scalability.
Further acceleration of CA-based microstructural simulations can be achieved using parallel computing techniques [8]. Modern computing architectures increasingly use multi-core processors and heterogeneous systems, making parallelization a natural direction for improving performance. Cellular Automata and Monte Carlo methods are particularly well-suited for parallel execution due to their synchronous transition rules and limited data dependencies. In this case, parallel implementations using shared-memory approaches (e.g., OpenMP) [9], distributed-memory frameworks (e.g., MPI) [10], or graphics processing units (GPUs) [11] can significantly reduce computation time for large simulation domains. However, appropriate domain decomposition, load balancing, and efficient communication strategies are crucial for achieving good scalability and minimizing parallel overhead during computations. A combination of these strategies is often required to achieve significant and scalable performance improvements in practice. Such customised parallel implementation enables high-resolution and large-scale CA simulations that would otherwise be impractical within reasonable computation times [12].
Thus, in this work, selected optimization strategies are applied to a CA-based static recrystallization (SRX) model to evaluate their impact on computational performance. The study focuses on identifying critical performance-limiting components of the simulation and assessing the achievable reduction in computational time, with the aim of improving the applicability of CA simulations under operational industrial conditions.

Methodology

The original version of this paper is available on https://www.scientific.net/KEM.1050.59.pdf

The investigated CA SRX model was developed, validated and in detail presented in [13]. The current CA SRX simulation case studies were done in three variants. In the first one, the simulation was executed until the recrystallized volume fraction reached 30% (Fig. 1a). The second variant is based on a fully recrystallized microstructure, which is treated as input data for the grain growth (GG) simulation, where the driving force is only due to grain-boundary curvature. The initial temperature in this case was 600∘C, and the sample was heated for 14 s to reach the final temperature of 614∘C (Fig. 1b). In the third variant, the recrystallisation is simulated until the material is fully recovered, and then the grain growth is analysed. The sample is heated from 600∘C for 80 s to the final temperature of 680∘C (Fig. 1c). Therefore, the latter variant is characterized by significantly longer simulation times.

[picture]

Fig. 1 Final microstructure after : a) variant 1(SRX),b) variant 2(GG),c) variant 3(SRX+GG) simulation.

Fig. 1. Final microstructure after : a) variant 1 ( S R X ) , b ) variant 2 ( G G ) , c ) variant 3 ( S R X + G G ) simulation.

In the first step of the investigation, an extensive code profiling with the use of the Visual Studio profiling tool for the three mentioned variants was performed. Five main model functions responsible for the simulation of the recovery phenomenon, stored energy-driven grain growth, grain boundary curvature growth, simulation and computational domain update were evaluated. Results from the analysis are presented in Tab. 1.

Table 1 Results from initial code profiling.

Function name | % of code execution time
| Variant 1 | Variant 2 | Variant 3
Recovery Stage | 20.32% | 17.42% | 3.86%
Recrystallization Stage | Stored Energy Driven Grain Growth | 49.56% | 0.00% | 66.61%
Curvature Driven Grain Growth | 0.43% | 25.43% | 19.28%
Update Progress of Simulation | 20.76% | 33.93% | 7.49%
Update CA Space | 5.47% | 14.49% | 2.67%

The profiling results revealed noticeable differences in computational performance and resource utilization of the CA SRX model depending on the simulation variant. Such different utilization of the main models' components provides potential opportunities for code optimization in these areas. Therefore, in the next step, code optimizations were performed based on these initial profiling results, and are presented in the following chapters.

Analysis of Data Storage Containers

The original version of this paper is available on https://www.scientific.net/KEM.1050.59.pdf

The first issue considered in increasing application efficiency was selecting the appropriate container for data storage. The code profiling evaluation clearly indicates that containers are a problematic area requiring further investigation. Therefore, suitable data structures from the Standard Template Library have been analyzed including deque, vector, list or forward_list. The calculated average simulation times for different containers for each investigated simulation variant are presented in Fig. 2 - Fig. 4. The red line in the plot represents the time reduction in comparison to the initial container used in previous work (vector<Cell>). In terms of comparison, additional objects stored in a container Cell, were compared to pointers to those objects, Cell.

[picture]

Fig. 2 Influence of data container type on simulation times for variant 1 (Bars represent average time from 3 runs of simulation, red lines overall time reduction compared to the base solution vector<Cell>*).

Fig. 2. Influence of data container type on simulation times for variant 1 (Bars represent average time from 3 runs of simulation, red lines overall time reduction compared to the base solution vector<Cell>*).

[picture]

Fig. 3 Influence of data container type on simulation times for variant 2.

Fig. 3. Influence of data container type on simulation times for variant 2.

[picture]

Fig. 4 Influence of data container type on simulation times for variant 3.

Fig. 4. Influence of data container type on simulation times for variant 3.

The shortest simulation time was achieved with a vector of pointers to objects. When vector structure is used to store objects, it is also the most efficient data container for the second and third variants (Fig. 2-3). It is worth noting that the main operation on a vector is limited to a single CA iteration (single CA time step). Overall, the benefits of such a change result in a time reduction of around 9%. The results indicate that the least efficient data structure for CA simulation is a deque. In all cases, the deque container is the slowest when storing pointer data types. Deque and vector are sequential containers that provide random access to the stored elements. The main difference between them is how they store elements in memory. A vector stores data in fast cache memory; on the other hand, elements in a deque are scattered throughout the memory. That is the main reason why vectors are useful when operating on long adjacent data sequences.
Subsequent analyzes should be focused on the container forward_list, which implements a one-way list data structure. In this case, each element has a pointer to the next element, so iteration can be done only forward. However, another more advanced container called a list implements a two-way data structure, where each element additionally has a pointer to the previous element. In both cases, direct access to the elements of the data structure is not possible. To reach a particular stored element, the container must be iterated over until that element is accessed. In SRX simulations, containers are iterated only forward, so both data structures achieve similar performance (Fig. 2-Fig. 4).
A standard dynamic array derived from the C language has also been investigated. This data structure does not deviate from STL container implementations in terms of simulation speed. Efficiency results are in the middle of the chart as presented in (Fig. 2-Fig. 4).
Finally, the average time has been calculated separately for containers storing pointers and for containers storing objects, as shown in Fig. 5. In this analysis, an array from the C language was not considered.

[picture]

Fig. 5 Analysis of the types of elements stored in different containers (red dots represent time reduction in terms of moving from Objects: Cell to Pointers: Cell*).

Fig. 5. Analysis of the types of elements stored in different containers (red dots represent time reduction in terms of moving from Objects: Cell to Pointers: Cell*).

As presented, the vector with pointers to CA cell objects is the most effective method in that simulation case (the highest profit of around 9% was achieved in variant 3, where both recrystallization and curvature-driven growth were simulated). In conclusion, implementing pointers accelerates CA code execution. However, if a container with many elements is needed, implementing a list data structure is still a better choice.
Based on the current observations, further optimization tasks were performed using vector-storing pointers and the results are gathered in the next chapter.

Analysis of Code Optimization Techniques

The original version of this paper is available on https://www.scientific.net/KEM.1050.59.pdf

Eight different CA code optimizations were carried out based on the initial profiling results.

Overloaded functions were eliminated. To improve simulation, all iterations in the data structure have to be handled in the same way. Taking this into consideration, the reading elements from the vector were changed from the at() function to an iterator. The at() function returns a reference to the element at the given position. This function can be used to iterate over a vector, but calling a function in each iteration can be very time-consuming; thus, an iterator was implemented.

Rearrangement of CA neighbor storage in the memory. In the CA SRX model, three different neighborhood types can be used, as represented in Fig. 6.

[picture]

Fig. 6 a) neighbourhoodZERO - von Neumann neighbourhood b) neighbourhoodONE - Moore's neighborhood c) neighbourhoodTWO - extended Moore's neighbourhood.

Fig. 6. a) neighbourhoodZERO - von Neumann neighbourhood b) neighbourhoodONE - Moore's neighborhood c) neighbourhoodTWO - extended Moore's neighbourhood.

In the original CA SRX model, neighbors were assigned to each CA cell at every simulation step. To accelerate the simulation, a new approach was developed. New data structures were added to the definition of the Cell class. They store the neighbours of a particular cell in the object. In that case, a new function for assigning neighbors was developed and implemented. The function is executed during the CA code initial phase. Therefore, it does not impact the main simulation computational
time. During the simulation, time is not wasted on assigning CA neighbors at each step, but additional memory is required.
3. Elimination of duplications in the new vector containers. It is possible to contain all neighborhood types in a single data structure if the data is arranged in the proper order (Fig. 7). This code modification not only accelerated the SRX model execution but also reduced memory usage during the simulation.

[picture]

Fig. 7 Representation of neighborhoods: a) von Neumann neighbourhood ( CA cells 1-4), b) Moore's neighbourhood ( a+CA cells 5-8:), c) extended Moore's neighbourhood ( a+b+c ), d) new vector container.

Fig. 7. Representation of neighborhoods: a) von Neumann neighbourhood ( CA cells 1-4), b) Moore's neighbourhood ( a + C A cells 5-8:), c) extended Moore's neighbourhood ( a + b + c ), d) new vector container.

Introduction of CA space composition. CA cells in the computational domain are stored in a three-dimensional array, which slows CA SRX model iteration, but also affects memory allocation. It causes memory fragmentation, with three distinct segments of memory randomly placed. During each iteration, it is necessary to jump across different segments of memory to access specific elements in the array. Therefore, in the current investigation, a three-dimensional array was converted into a one-dimensional array. Allocation and deallocation of such array also reduce the number of code lines from a threefold-nested loop to a single line of code. All elements in memory are next to each other, facilitating fast access to values. Iteration on the whole CA space is a frequent operation in the CA SRX simulation. A one-dimensional array can use only one loop. To maintain the cellular automata's three-dimensional conception, threefold nested loops were retained during each iteration. As a result, speed and memory usage were improved, but one-dimensional arrays also have their limitations. The C++ language allows the allocation of only a limited amount of memory. In the paper, the testing CA computational domain was 800×800×1, so there are 640000 CA cell objects to allocate, which is a relatively small number.

Introduction of attributes encapsulation on main objects. Correct practices for objectoriented programming include using getter and setter functions to access or modify attribute values. Data encapsulation provides hiding private data, so it cannot be accessed directly. It is possible to implement sensitive data handling within the class. On the other hand, it adds additional lines of code, which could potentially slow down the CA SRX simulation. The main purpose of the paper is to accelerate the CA SRX code, so data sensitivity was skipped, and attribute visibility was changed to public. That amendment eliminated the necessity of using functions, so attributes could be referred to directly.

Substitution of Push_back() by emplace_back(). Both of them are vector methods that realize the same functionality, which is adding elements at the end. Theoretically, emplace_back first reserves memory for the new element, then inserts the element into the allocated space. Whereas push_back first creates a temporary object on the stack. Then it extends the vector, and std::move inserts an element from the stack into a new memory space. In theory, emplace_back avoids additional copying and moving data, which is more efficient.

Rewriting of algebraic functions. The C++ language provides math.h library, which enables the execution of various mathematical operations such as root extraction, exponentiation, rounding, and calculating exponents. After different combinations of such elements and many tests, the following conclusions have been drawn:

sqrt() is faster than pow(x,0.5)

pow(x,1/3) is faster than cort()

exp(−x) is faster than 1/exp(x)
−exp(x) is faster than pow(e,x);
−1/pow(x,0.5) is faster than pow(x,−0.5)
−1/sqrt(x) is faster than 1/pow(x,0.5)

Detailed code analysis to localize unnecessary operations, which slow down the whole CA SRX code. After introducing the above-mentioned changes, the profiling process was repeated. As a result, it pointed out additional redundant data structure allocation. The most problematic part was a map container from STL. Finally, allocation was executed in the main simulation loop, so the map was removed.
Each presented code optimization technique improved simulation efficiency for at least two simulation variants as presented in Figure Fig. 8a-c. Only a change in the assigning CA neighbors greatly increased memory consumption (Fig. 8d). This was eventually reduced in the final code optimization step by reducing unnecessary object allocation. Afterwards, memory consumption continues to remain on the same level, which was 287 MB .

[picture]

Fig. 8 Time changes after subsequent optimization a) variant 1,b ) variant 2,c ) variant 3,d ) memory consumption.
As shown in Fig. 8, overall, benefits from different optimizations allow simulating the same scenarios in 40-50% percentage points of the initial time.

Fig. 8. Time changes after subsequent optimization a) variant 1 , b ) variant 2 , c ) variant 3 , d ) memory consumption. As shown in Fig. 8, overall, benefits from different optimizations allow simulating the same scenarios in 40-50% percentage points of the initial time.

Summary

The original version of this paper is available on https://www.scientific.net/KEM.1050.59.pdf

The paper points out that code optimization techniques can significantly improve CA SRX code efficiency on different levels. In the investigated case code optimization reduced simulation time approximately two times. However, a crucial step during the implementation of new functionalities is preparing the initial application profile using the profiler tool, which enables proper interpretation and localization of the most problematic areas, often leading to code execution bottlenecks. Such code profiling provides clear guidance on possible further code improvement.
The findings of the current research can be summarized as follows:

A proposed approach to CA neighbor assignment in cellular automata code results in a significant acceleration of execution times with only a slight increase in memory usage. Eventually, memory consumption increases only by about 60 MB from the original memory usage.

The most efficient data structure for CA simulations proved to be a vector from the Standard Template Library (STL).

Reduction of the array dimensions that hold the 3D CA computation domain prevents frequent iterations and speeds up the simulation execution.

The use of at() function and proper algebraic functions from math.h library, getter or push_back() provides a significant execution time reduction of the CA code.
Based on these findings, the next stage of this investigation will be focused on the CA SRX model adaptation to the parallel code execution and additional code corresponding optimisations to further increase simulation performance.

Acknowledgement

The original version of this paper is available on https://www.scientific.net/KEM.1050.59.pdf

We would like to express our sincere gratitude to Mr. Marcin Chrobak for his initial work on preparing application profiles and proposing various approaches to code optimization. The financial assistance of the National Science Centre project No. 2024/55/D/ST8/00192 is acknowledged.

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The original version of this paper is available on https://www.scientific.net/KEM.1050.59.pdf

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Key Engineering Materials, ISSN: 1662-9795, Vol. 1050, pp 69-80

doi: 10.4028/p-1eNlcu

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-16

Revised:
2026-02-18

Accepted:
2026-02-18

Online:
2026-04-16

Parameter Optimization and Data-Driven Soft-Sensor Framework for Torque Prediction in Bobbin-Tool Friction Stir Welding of AA2024

The original version of this paper is available on https://www.scientific.net/KEM.1050.69.pdf

Fares Nouira 1,a*, Luciano Bergmann 1, b, Frederic E. Bock 1,c, Ghada Bouattour 2, d, Marco Pacchione 3,e and Benjamin Klusemann 1,2,f

1 Helmholtz-Zentrum Hereon, Institute of Material and Process Design, 21502 Geesthacht, Germany

2 Leuphana University Lüneburg, Institute for Production Technology and Systems, 21335 Lüneburg, Germany

3 Airbus, OMTMW - Manufacturing Engineering R&T, 21129 Hamburg, Germany

af  fares.nouira@hereon.de, b  luciano.bergmann@hereon.de, cfrederic.bock@hereon.de, d  ghada.bouattour@leuphana.de, e  marco.pacchione@airbus.com, f  benjamin.klusemann@hereon.de

Keywords: Bobbin-Tool Friction Stir Welding, Process Parameter Optimization, Soft-Sensor Framework, Torque Prediction, Machine Learning, Aerospace Aluminum Alloys, Process Monitoring.

Abstract

Bobbin-Tool Friction Stir Welding (BT-FSW) is a solid-state joining process in which axial forces are internally balanced by the tool, eliminating the need for a backing plate and enabling the joining of hollow aerospace structures. Owing to the coupled thermo-mechanical nature of the process, weld stability and quality are governed by the interaction between process parameters and the resulting torque response, which is difficult to assess in situ using conventional sensing alone. BT-FSW experiments were performed on AA2024-T351 sheets with thicknesses of 2.4, 2.8 and 3.6 mm using a structured Design of Experiments (DoE). The 3.6 mm joints achieved approximately 90 % of the base-material strength, while the 2.4 and 2.8 mm joints reached about 80% and 85%, respectively. These mechanical results were used as ground truth to train machine learning regression models for steady-state torque predictions. By augmenting nominal process parameters with forcederived features, the proposed soft-sensing framework achieved strong agreement between predicted and measured torque, demonstrating that compact, physics-based feature engineering enables reliable prediction under limited experimental data conditions.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1050.69.pdf

Bobbin-Tool Friction Stir Welding (BT-FSW) is a self-reacting variant of friction stir welding (FSW) in which opposing shoulders generate a balanced thermal field without backing support. Compared to conventional FSW, this configuration minimizes distortion, eliminates the single-sided probe exit keyhole and significantly reduces the risk of lack of penetration due to the symmetric loading condition, making BT-FSW attractive for lightweight aerospace structures where geometric accuracy and joint integrity are critical [1,2]. The coupled thermo-mechanical nature of the process makes it highly sensitive to the interaction between rotational speed, traverse speed, and axial loading, which jointly govern frictional heating, material flow, and material consolidation.
In aluminum alloys, the consequences of heat input and stirring stability on nugget morphology, hardness evolution and tensile properties after FSW are well established [3, 4]. Macroscopic indicators, such as onion-ring structures, joint-line remnants and localized softening have been linked to unstable flow and incomplete consolidation, and torque signatures have been used as an in-process indicator of these conditions [5]. Consequently, torque provides a practical proxy for the instantaneous process state in both FSW and BT-FSW.
Despite this relevance, most BT-FSW investigations remain limited to single-thickness configurations or rely on qualitative interpretation of torque signals, and comprehensive multithickness datasets combining synchronized process signals with mechanical and macrostructural validation are scarce. In parallel, machine-learning-based soft sensors have been increasingly
explored in manufacturing and forming processes to estimate unmeasured process states. Nadeau et al. [6] demonstrated early regression-based surrogate models for process response prediction, highlighting the potential of data-driven approaches under constrained experimental datasets. Belalia et al. [7] extended such concepts using signal-based features for process monitoring, while Bock et al. [8] proposed physics-guided soft-sensor frameworks that explicitly combine process knowledge with machine learning to improve robustness and interpretability in forming applications. Penalva et al. [9] investigated data-driven and hybrid modeling strategies with a stronger reliance on numerical process descriptions. More recently, Bock et al. [10] demonstrated that machine-learning-based surrogate models informed by mechanical principles can achieve reliable predictions while maintaining physical consistency, reinforcing the relevance of physics-aligned soft sensors for complex thermo-mechanical processes. However, the transfer of such validated soft-sensor concepts to BT-FSW, particularly across varying sheet thicknesses and under realistic experimental data constraints, remains largely unexplored.
Soft sensors are increasingly used in manufacturing to estimate unmeasured process states from indirect sensor signals, particularly where direct measurement is impractical or costly [11]. In welding applications, however, most reported soft sensors rely on large datasets and highly nonlinear blackbox models, offering limited physical interpretability. For BT-FSW, where experimental data are inherently scarce, this creates a gap between predictive accuracy and physical understanding.
The present work addresses this gap by developing an interpretable, physics-aligned soft-sensing framework based on supervised regression in which physical process understanding guides feature selection and target definition, enabling robust data-driven prediction under limited data availability. Therefore, a structured multi-thickness investigation of BT-FSW on AA2024-T351 sheets with thicknesses of 2.4, 2.8, and 3.6 mm has been performed. The central objective of this work is the prediction of steady-state torque as a soft-sensor output. Although torque is directly measurable in BT-FSW, its prediction is not pursued as an end in itself. Instead, torque prediction is used as a physically meaningful intermediate soft sensor that links process parameters and force signals to the underlying thermo-mechanical process state, enabling process understanding, process planning, and the identification of anomalous or unstable welding conditions. At first, a DoE is employed to study the combined influence of rotational speed, traverse speed, and gap force on torque evolution and weld quality. Mechanical characterization through tensile testing, macrostructural inspection, and hardness mapping provides ground-truth indicators of the weld performance. Building on these data, the proposed framework is used to predict steady-state torque using process parameters and measured force signals as inputs, providing a quantitative basis for torque-based process monitoring and stability assessment in BT-FSW.

Proposed Approach and Methodology

The original version of this paper is available on https://www.scientific.net/KEM.1050.69.pdf

The objective of this study is to develop a soft-sensing approach for predicting steady-state torque in BT-FSW from measured process parameters and force signals under limited experimental data availability. An integrated workflow is implemented in which welding trials, data acquisition, mechanical characterization, and supervised regression are combined into a single analysis pipeline, see Fig. 1.

BT-FSW Pipeline

The original version of this paper is available on https://www.scientific.net/KEM.1050.69.pdf

[picture]

Fig. 1 Process workflow used in this study.

Fig. 1. Process workflow used in this study.

Within this workflow, BT-FSW experiments are conducted under controlled conditions, relevant process and force signals are recorded, joints are mechanically and microstructurally evaluated, and the resulting dataset is used to train and validate regression models constituting the soft sensor.
The experimental design followed a Box-Behnken DoE for each thickness, systematically varying rotational speed ( 𝐧rot  ), traverse speed ( 𝐯 ) and gap force ( 𝐅𝐠 ) over three thicknesses, see Table1. This design efficiently captures main, quadratic and interaction effects while limiting the number of trials [12].

Table 1 DoE parameter ranges investigated for the different thicknesses.

Thicknesses (mm)

Rotational Speed

(RPM)

Traverse Speed

(mm/min)

Gap Force (N)

2.4 | superscript number | superscript number | superscript number
2.8 | mathematical formula
3.6 |  |  |

Materials and Process. All welds were produced on a Tricept T805 parallel-kinematic robot equipped with a semi-stationary bobbin tool consisting of two opposing shoulders connected by a threaded probe, with the upper shoulder held stationary relative to the machine frame and the lower shoulder rotating with the spindle. The clamping arrangement and welding workspace are shown in Fig. 2, including the positioning of reference strips, side bosses, spike pads and retainers used for specimen fixation.

[picture]

Fig. 2 Experimental clamping setup used during BT-FSW, ensuring stable positioning and forcecontrolled welding conditions.

Fig. 2. Experimental clamping setup used during BT-FSW, ensuring stable positioning and forcecontrolled welding conditions.

The base material plates were locally machined in the welding region to achieve the target joint thicknesses of 2.4, 2.8, and 3.6 mm , starting from initial plate thicknesses of 3.2, 3.5, and 4.5 mm , respectively.
Welding was performed in force-controlled mode, with axial gap force regulated through a closedloop system to maintain stable tool-workpiece contact. Rotational speed and traverse speed were controlled via a Siemens NC unit. Welds were executed according to DIN EN ISO 25239 along the rolling direction over a length of 550 mm to ensure a fully developed steady-state region suitable for mechanical sampling and sensor assessment.
After welding, each weld was sectioned into 16 transverse slices following the layout in Fig. 3. Cuts 10-12 were allocated for tensile testing, Cut 3 for macrostructural inspection and Barker-etched microscopy, and Cut 16 for macrostructural inspection and hardness mapping. For each thickness, 15 welds were produced, yielding 45 structured experiments.

[picture]

Fig. 3 Cutting layout used for specimen extraction, indicating the advancing side (AS), retreating side (RS), welding direction (identical to rolling direction) and the locations of cuts.

Fig. 3. Cutting layout used for specimen extraction, indicating the advancing side (AS), retreating side (RS), welding direction (identical to rolling direction) and the locations of cuts.

Mechanical Characterization. Tensile tests were performed on three parallel-gauge specimens per weld, using a 150 mm gauge length and a 50 mm extensometer, under displacement control at 1 mm/min. This configuration follows established tensile evaluation practices for friction-stir-welded aerospace alloys, where parallel specimens reliably capture joint efficiency without requiring dogbone machining [13]. Macrostructural observations were conducted on polished sections from Cut 3 using a VHX digital microscope at 50× magnification, and Barker-etched micrographs revealed nugget morphology, flow lines and joint-line remnants (JLR) [14]. Hardness profiles from Cut 16 were measured using Vickers HV0.1 with 0.5 mm spacing to assess local softening and the width of the thermally affected region, i.e. heat affected zone.
Dataset and Experimental Coverage. The full experimental dataset, spanning all sheet thicknesses and DoE conditions, is shown in Fig. 4.

[picture]

Fig. 4 Full Data Set investigated for the different thicknesses.

Fig. 4. Full Data Set investigated for the different thicknesses.

The complete dataset comprises 61 individual welding conditions spanning three sheet thicknesses. It includes three Box-Behnken matrices corresponding to the 2.4,2.8, and 3.6 mm sheets, which together account for 45 structured welding conditions used for process analysis and mechanical validation. The remaining welds were produced after defining a feasible process window in order to improve coverage of the continuous parameter space and to ensure sufficiently sized training, validation, and test subsets for regression modeling; these supplementary points include both intermediate parameter combinations within the DoE space and selected combinations not coinciding with the original DoE design points, all of which lie within the mechanically validated process window. Each data point corresponds to a single weld and includes the process parameters and synchronized force measurements recorded during welding. The dataset is divided into training, validation, and test subsets containing 49, 6, and 6 welds, respectively. Sample selection for the validation and test sets is carried out such that all corresponding parameter combinations lie within the parameter bounds spanned by the training data, thereby fulfilling the defined interpolation criterion and avoiding extrapolative model evaluation. All three subsets maintain balanced representation across sheet thicknesses and process-parameter ranges, enabling an unbiased assessment of model generalization over the full BT-FSW operating window.
Data Acquisition and Pre-Processing. During welding, torque, rotational speed, traverse speed, and force signals in the x−,y−, and z -directions (𝐅𝐱,𝐅𝐲,𝐅𝐳) were recorded at a sampling frequency of approximately 70 Hz . The acquired signals were preprocessed to remove obvious outliers and were time-aligned across all channels using their common acquisition timestamps. To ensure consistency between welds, non-steady process segments corresponding to tool plunge, ramp-up, and termination were excluded from further analysis, in accordance with other works [15]. This preprocessing step ensured a consistent representation of the steady-state welding phase across all experiments.
Feature Engineering. The steady-state region of each weld was isolated by selecting the continuous time interval after tool entry and before termination in which the torque signal exhibited an approximately constant mean and low temporal variance, thereby excluding plunge, ramp-up, and termination. Thus, signals from this steady-state region were condensed into low-dimensional descriptors by computing statistical features and physically motivated force combinations, capturing the representative thermo-mechanical response during stable welding [16]. Normalized force components (e.g., Fx_norm, Fy_norm, Fz_norm) were defined by scaling the mean force components with the mean gap force to remove absolute load-level effects, while force ratios (e.g., Fx_Fz, Fy_Fz, Fxy_ratio) capture directional asymmetry between force components and force-interaction terms (e.g., FxFz,FyFz,FxFy ) represent coupled loading effects acting simultaneously during welding. The resulting feature set comprises aggregated descriptors derived from process parameters (rotational speed, traverse speed, and gap force) and measured force signals, while the mean steady-state torque serves as the target variable for supervised regression. An overview of all extracted features is provided in Table 2.

Table 2 Overview of extracted features.

Category | Feature | Description
Process parameters | mean_rpm | Mean rotational speed (steady state)
Process parameters | mean_feed | Mean traverse speed
Process parameters | mean_gapforce | Mean gap force
Forces (statistical descriptors) | mean_fx, mean_fy, mean_fz | Mean force components
Forces (derived) | F_result | Resultant force magnitude
Force ratios | Fx_norm, Fxy_ratio | Normalized and directional ratios
Force interactions | Fx_Fz, Fy_Fz, FxFy, FxFz, FyFz | Coupled force terms
Parameter interactions | rpm_feed, rpm_gap, feed_gap | Physically motivated interaction terms

Machine Learning Models. The engineered features were used to train supervised regression models to predict the average steady-state torque. Linear regression was employed as an untuned baseline model, while Ridge regression was used to assess the effect of linear regularization. In addition, nonlinear models-Random Forest, XGBoost, and CatBoost were applied to capture potential non-linear interactions between process parameters, force features, and torque [17, 18]. As mentioned above, the dataset was split at the weld level into training, validation, and test subsets. Hyperparameter optimization was performed exclusively on the training data using 5-fold cross-validation. For Ridge regression, the regularization parameter was optimized via a grid search over logarithmically spaced values. Random Forest hyperparameters, including the number of trees, maximum tree depth, and minimum samples per leaf, were likewise tuned using grid search. For XGBoost and CatBoost, Bayesian optimization based on a tree-structured Parzen estimator was employed to efficiently explore higher-dimensional hyperparameter spaces, including learning rate, tree depth, number of estimators, and regularization parameters. Model selection was based on predictive performance on the validation set, and the final selected models were subsequently evaluated on the independent test set to assess generalization performance.
Evaluation Metrics. Model performance was quantified using the mean squared error (MSE) and the coefficient of determination (R2) :

MSE=1n∑i=1n(Ti−T^i)2R2=1−∑i=1n(Ti−T^i)2∑i=1n(Ti−T^)2

where 𝐧 is the number of samples, 𝐓𝐢 is, the measured steady-state torque of sample 𝐢,𝐓^𝐢 is the corresponding predicted torque, and 𝐓― is the mean of the measured steady-state torque values.

Mechanical and Microstructural Results

The original version of this paper is available on https://www.scientific.net/KEM.1050.69.pdf

Tensile Properties Across Thicknesses. The tensile properties of all welded joints are presented in Fig. 5a-b. The base material exhibited ultimate tensile strength (UTS) of 432.1±3.5MPa,450.4± 2.0 MPa and 454.9±2.3MPa for the 2.4−,2.8− and 3.6−mm sheets, respectively, with yield strengths around 311−313MPa and elongations of 16.3−18.1%. These base-material properties provide a quantitative reference for assessing the relative tensile properties and joint efficiency of the welded specimens.

[picture]

Fig. 5 Ultimate tensile strength comparison between thicknesses: a) 2.4 mm ; b) 3.6 mm .

Fig. 5. Ultimate tensile strength comparison between thicknesses: a) 2.4 mm ; b) 3.6 mm .

The 2.4 mm sheets exhibited the largest scatter in tensile strength, with UTS values ranging from approximately 258 MPa ( ~60% efficiency; Run 10: 350rpm,150 mm/min,1500 N ) to 419 MPa ( ~97% efficiency; Run 9: 300rpm,150 mm/min,1500 N ), see Fig. 5a). This wide variation reflects the high sensitivity of thin sheets to heat input fluctuations and localized flow instabilities, often associated with incomplete stirring or joint-line remnants during BT-FSW.
In contrast, the 2.8 mm sheets displayed improved consistency, with UTS values spanning from approximately 323MPa(~72% efficiency; Run 7: 350rpm,200 mm/min,1500 N) to 445MPa(~99%
efficiency; Run 11: 300rpm,150 mm/min,2500 N ), indicating more homogeneous thermomechanical conditions at intermediate thickness.
The 3.6 mm joints showed the most stable behavior, with UTS values confined to a narrower range from approximately 422 MPa ( ~93% efficiency; Run 6: 350rpm,100 mm/min,2500 N ) to 446 MPa ( ~98% efficiency; Run 7: 350rpm,200 mm/min,1500 N ), see Fig. 5b). This enhanced stability reflects improved heat retention and more uniform material flow in thicker sheets.
Overall, weld stability increases with sheet thickness, with the 2.4 mm joints showing high sensitivity to parameter deviations and the 3.6 mm joints maintaining consistently high efficiency across DoE. Across all thicknesses, higher tensile strengths were obtained at moderate rotation speeds ( ≈300−350 rpm ), intermediate traverse speeds ( ≈150 mm/min ), and gap forces between 1500 and 2500 N , while combinations of high rotation speed or high traverse speed at low gap force led to reduced strength, highlighting the coupled influence of the process parameters.
Metallography. Microstructural observations reveal thickness-dependent stirring behavior. In the 2.4 mm welds, the etched sections show distinct joint-line remnants (JLR) within the nugget, Fig. 6, indicating incomplete interface disruption due to insufficient material flow in thin sheets with limited thermal inertia. Their presence correlates with the lowest tensile strengths and increased torque variability observed in the corresponding DoE runs.

[picture]

Fig. 6 Joint Line Remnants in the nugget in the thickness of 2.4 mm .

Fig. 6. Joint Line Remnants in the nugget in the thickness of 2.4 mm .

In contrast to the 2.4 mm joints, the 3.6 mm welds exhibit a uniformly consolidated stir zone at the macroscopic scale, as shown in Fig. 7. No joint-line remnants or interface-related discontinuities are observed across the nugget, indicating stable material flow and sufficient forging pressure in the thicker sheets. The absence of macroscopic defects is consistent with the high joint efficiency and reduced scatter observed in the mechanical properties for this sheet thickness.

[picture]

Fig. 7 Plan-view etched optical micrographs of the 3.6 mm BT-FSW weld confirming the absence of a joint line remnant (JLR); (a) global weld overview, (b) magnified central region showing continuous material flow across the weld center

Fig. 7. Plan-view etched optical micrographs of the 3.6 mm BT-FSW weld confirming the absence of a joint line remnant (JLR); (a) global weld overview, (b) magnified central region showing continuous material flow across the weld center

Optimal Process Parameters. Parameter optimization reveals a clear thickness-dependent shift in the BT-FSW processing window. For the 3.6 mm sheets, the multi-response desirability analysis shown in Fig. 8 highlights a region of high combined mechanical performance, based on predicted yield strength (Rp0.2) and ultimate tensile strength (Rm), located around a rotational speed of approximately 400 rpm , a traverse speed of 200 mm/min, and a gap force of about 1500 N . The vertical red dashed lines in Fig. 8 denote the center points of the Box-Behnken design used for this thickness and serve as reference levels within the investigated parameter space, rather than indicating a unique optimal condition.

[picture]

Fig. 8 Multi-parameter desirability analysis for BT-FSW specimens of 3.6 mm thickness, identifying the optimal combination of process parameters. Solid lines indicate the fitted responses for yield strength and ultimate tensile strength, while the shaded bands represent the 95% confidence intervals of the fitted models (JMP default).

Fig. 8. Multi-parameter desirability analysis for BT-FSW specimens of 3.6 mm thickness, identifying the optimal combination of process parameters. Solid lines indicate the fitted responses for yield strength and ultimate tensile strength, while the shaded bands represent the 95% confidence intervals of the fitted models (JMP default).

For the 2.8 mm sheets, this favorable parameter region shifts toward lower rotational and traverse speeds and higher gap force, centered around 300rpm,150 mm/min, and 2000 N , whereas the 2.4 mm sheets exhibit a narrower region of stable operation near 250rpm,130 mm/min, and 2160 N , reflecting their reduced thermal inertia. Overall, increasing sheet thickness progressively broadens the stable operating window: thicker sheets allow a wider range of process parameter combinations, while thinner sheets require tighter coupling of rotational speed, traverse speed, and gap force, in agreement with established friction stir welding studies [1].

Machine Learning Results

The original version of this paper is available on https://www.scientific.net/KEM.1050.69.pdf

While the DoE and mechanical characterization establish thickness-dependent process windows and identify defect-free welding conditions, these results rely on destructive post-process testing and are therefore not directly accessible during welding. Torque, in contrast, is continuously measured during BT-FSW and reflects the instantaneous thermo-mechanical state of the process. However, its direct interpretation is non-trivial due to the coupled influence of process parameters, material thickness, and force response.
The purpose of the following machine-learning analysis is therefore not to replace the DoE-based findings, but to translate the mechanically validated process knowledge into a quantitative, signalbased model. By predicting steady-state torque from process parameters and force-derived features, a data-driven link is established between experimentally confirmed weld quality and in-process measurements.
Regression models trained using only nominal process parameters (rotational speed, traverse speed, and gap force), Fig. 9 a), exhibited markedly higher prediction errors compared to models augmented with force-derived features, Fig. 9 b), indicating that steady-state torque cannot be reliably inferred from programmed inputs alone.

Building on this enhanced feature representation, the performance of different regression algorithms was evaluated. Ordinary linear regression exhibited higher validation and test errors, reflecting its sensitivity to correlated process and force-derived features. Ridge regression yielded the lowest MSE on the validation set compared to all other models and was therefore selected as the best-performing model; the prediction performance on the test dataset was likewise superior based on the lowest achieved MSE, see Fig 9 b). This behavior is consistent with modelling principles indicating that linear methods generalize well when dominant physical effects are already encoded in the feature space [20].

[picture]

Fig. 9 Comparison of torque prediction accuracy for different models with different feature sets: (a) process parameters only, (b) process parameters with force-derived features.

Fig. 9. Comparison of torque prediction accuracy for different models with different feature sets: (a) process parameters only, (b) process parameters with force-derived features.

Model agreement between predicted and measured torque across train/validation/test subsets is shown on Fig. 10.

[picture]

Fig. 10 Ridge regression model: predicted vs. measured steady-state torque for train, validation and test sets.

Fig. 10. Ridge regression model: predicted vs. measured steady-state torque for train, validation and test sets.

Ensemble models such as Random Forest, XGBoost, and CatBoost captured additional nonlinearities but exhibited greater sensitivity to hyperparameter tuning and a mild tendency toward overfitting, as commonly observed for boosted-tree architectures [21]. Feature-importance analysis based on ridge regression coefficients (Fig. 11) identified rotational speed, vertical force magnitude, and parameter force interaction terms as the primary contributors to steady-state torque prediction.

Ridge Regression Coefficient

Fig. 11 Ridge regression coefficient ranking for torque prediction.

This ranking is consistent with established FSW mechanics, in which frictional heat generation, forging pressure, and material flow resistance jointly govern torque evolution during stable welding conditions [1]. Rotational speed and the resultant vertical force therefore emerge as dominant predictors, reflecting their direct influence on heat input, material plasticization, and resistance to tool motion. In contrast, normalized force components and force-ratio terms contribute at a secondary level, capturing more localized effects related to flow asymmetry and deviations from ideally uniform stirring.
In summary, the results show that accurate torque prediction in BT-FSW requires augmenting the nominal welding parameters with descriptors derived from the measured process forces. Once these engineered features were introduced, the regression framework achieved both high predictive performance and strong physical interpretability, consistent with physics-informed soft-sensor concepts reported in the literature [10], where data-driven surrogate models guided by process mechanics demonstrated low prediction error and stable generalization within defined process windows, even under limited experimental data. The proposed approach therefore provides a reliable foundation for soft-sensing applications in BT-FSW, while not replacing the prior identification of suitable welding parameter windows, which remain essential for calibrating and validating the soft sensor.

Conclusions

The original version of this paper is available on https://www.scientific.net/KEM.1050.69.pdf

A combined experimental and data-driven investigation of bobbin-tool friction stir welding of AA2024-T351 was performed across three sheet thicknesses. A structured DoE quantified the coupled influence of rotational speed, traverse speed, and gap force on weld morphology, tensile properties, and steady-state torque response. The mechanical results revealed a clear thickness dependence of process stability: 2.4 mm welds showed pronounced sensitivity to parameter variations and joint-line remnants, whereas 3.6 mm welds consistently achieved high joint efficiency with fully consolidated stir zones.
Building on these findings, a supervised regression-based soft-sensing framework was developed for steady-state torque prediction. Models relying solely on nominal process parameters were insufficient, while augmenting the input space with force-derived descriptors led to a substantial improvement in predictive accuracy. Ridge regression provided robust predictions together with physically interpretable feature rankings, with dominant contributors consistent with established friction stir welding results. Overall, the results demonstrate that compact, physically meaningful descriptors offer a viable and data-efficient route toward soft-sensing and process monitoring for bobbin-tool friction stir welding under limited experimental data availability.

Acknowledgements

The original version of this paper is available on https://www.scientific.net/KEM.1050.69.pdf

The work was carried out under grant numbers 20X2206A and 20X2206G, funded by the German Federal Ministry of Economic Affairs and Energy (BMWE) under the LuFo VI-3 program.

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The original version of this paper is available on https://www.scientific.net/KEM.1050.69.pdf

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Key Engineering Materials, ISSN: 1662-9795, Vol. 1050, pp 81-91

doi: 10.4028/p-0V5rM0

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-18

Revised:
2026-02-18

Accepted:
2026-02-18

Online:
2026-04-16

Long Short-Term Memory to Predict the Quality Parameters in the Hot Rolling Process

The original version of this paper is available on https://www.scientific.net/KEM.1050.81.pdf

Tamoghna Majumder 1,a*, Nilesh Thakare 1, b, Holger Brüggemann 1,c, Emad Scharifi 1, d and Junhe Lian 1,e

1 Institute of Metal Forming (IBF), RWTH Aachen University, Germany

a tamoghna.majumder@ibf.rwth-aachen.de, b  nilesh.thakare@ibf.rwth-aachen.de, c  holger.brueggemann@ibf.rwth-aachen.de, d  emad.scharifi@ibf.rwth-aachen.de, ejunhe.lian@ibf.rwth-aachen.de

Keywords: Hot Rolling, Quality-relevant material parameters, Long Short-Term Memory.

Abstract

Accurate prediction of quality-relevant material parameters, such as thickness and grain size, ensures product quality in hot forming processes. This task becomes especially complex in hot rolling, where the sequential and time-dependent nature of the process results in pass schedules with varying numbers of passes and grain size evolution that depends on the deformation history. To address this complexity, this study aims to develop and train a deep learning model based on Long Short-Term Memory (LSTM) networks, which are well-suited for modelling sequential data. As input features for the model, process parameters such as rolling force and rolling temperature are used. Simulation data for both material and process parameters are acquired using Simulation as a Service (SaaS) through the Fast Rolling Model (FRM) called Rolling Calculation Tool (RoCaT), focusing on steel grade S355. RoCaT calculates rolling force, rolling temperatures, austenite grain size, strain, and strain rate by taking the geometry of the material, rolling speed, and pass schedule as input. The performance of the LSTM model is evaluated by analyzing loss curves over training epochs and comparing predicted values to reference data. The maximum relative percentage error for thickness between the LSTM predicted value and the thickness values entered as input to the RoCaT model is 18.81% and 16.56%, respectively, for pass schedules of 15 and 17 passes, with a starting thickness of 205 mm . Both the maximum relative percentage error value for the pass schedules is obtained at the initial passes, which indicates that the model finds difficulty in performing the predictions accurately during the initial passes, as compared to the later passes. The percentage of relative error values for grain size is also more pronounced during the initial passes as compared to later passes for both pass schedules. The statistical validation is performed on the denormalized data using metrics such as Root Mean Squared Error (RMSE), Normalized Root Mean Square Error (nRMSE), Mean Absolute Error (MAE), and R-Squared ( R2 ), which demonstrate the model's ability to predict key material properties in the hot rolling process reliably. The RMSE, nRMSE, MAE, and R2 values for thickness are obtained as 45.21 mm,0.11(−),18.75 mm, and 0.81(−), respectively. For grain size, the corresponding values are 25.82μ m,0.09(−),13.43μ m, and 0.91(−).

Introduction

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Hot rolling is a key thermo-mechanical process in metal forming, in which the thickness of a sheet material is reduced through successive deformation passes at elevated temperatures until the target gauge is attained. The sequence of these passes is known as the pass schedule. Designing a pass schedule requires appropriate consideration of both material and process parameters, as these influence the final product quality. Among them, the most critical quality-controlled parameters are the grain size and the thickness of the hot-rolled strip. The reduction in thickness and grain size of the hot-rolled strip influences mechanical properties, surface quality, and dimensional accuracy of the rolled product, which makes both quality parameters important.

Smart manufacturing uses the acquired data from sensors and performs accurate decisionmaking using digital technologies, resulting in an improvement in process efficiency, reliability, and product quality. The value of the manufacturing data lies in its ability to provide insights into process
behaviour, enabling the development of various data-driven models using advanced Information Technology (IT). Innovations such as the Industrial Internet of Things (IIoT), cloud computing, mobile internet, and Artificial Intelligence (AI) are driving automation and efficiency in modern manufacturing systems. Real-time data acquisition is achieved by deploying sensors throughout production environments. With AI-driven solutions, decision-making in smart factories increasingly shifts from humans to intelligent systems [1]. AI is a broader field that consists of machine learning and deep learning approaches. Machine learning involves algorithms that identify patterns within data to generate predictive or analytical outcomes. Deep learning is a subset of Machine Learning that uses neural networks with multiple layers, known as Deep Neural Networks (DNNs), to model complex relationships. Artificial Neural Networks (ANNs), composed of interconnected artificial neurons using nonlinear activation functions, form the foundation for DNNs. Among these, Convolutional Neural Networks (CNNs) preserve local spatial relationships and retain spatial information using convolutional layers [2].

On the other hand, Recurrent Neural Networks (RNNs) are neural networks that are applied to sequential data to identify the patterns present in the data. RNNs consider both previous and current inputs because of their recurrent connections that pass information from the current step to the next step [3]. A simple RNN can only retain information about short sequences because of its short-term memory. To retain information about long sequences, advanced RNNs are developed that include Long Short-Term Memory (LSTM), bi-directional LSTM, Gated Recurrent Unit (GRU), bidirectional GRU, Bayesian GRU, and others [4]. A unidirectional LSTM and GRU perform the sequence processing only in the forward direction, which captures the dependencies from earlier to later time steps, whereas a bidirectional LSTM and GRU perform the sequence processing both in forward and backward directions, which learns from past and future time steps simultaneously.

Several RNNs have been widely developed and used in the hot rolling process to perform predictions and anomaly detection, which are summarized below in Table 1.

Table 1. Summary of the RNN models used in the hot rolling process

Sources | Models | Purpose | Results
[5] | A Deep RNN model (2 LSTM models), Convolutional Neural Network (CNN), and the Deep Belief Network (DBN) | To predict the remaining useful life of a roller in a hot strip mill | RMSE:11.05, 18.97, and 16.23, respectively MAE: 9.85, 15.74, and 14.37, respectively
[6] | One GRU layer and two fully connected layers of 32 neurons | To predict the mean flow stress in industrial rolling | RMSE: 0.038 Flow stress value with a mean RMSE: 7.5 MPa
[7] | Inception-LSTM Neural Network | To predict the rolling force in the hot rolling | MAE, RMSE, and R² score: 3.969, 5.487, and 0.985, respectively
[8] | LSTM, Support Vector Machine (SVM), Random Forest, and ANN | Online prediction of the mechanical properties, such as Yield Strength (YS), Ultimate Tensile Strength (UTS), and Elongation (EL) | R² values: 0.96, 0.935, 0.940, and 0.942, respectively YS, UTS, and EL error values using LSTM: 19.68 MPa, 18.18 MPa, and 1.840%, respectively

The previously discussed state-of-the-art indicates a wide usage of RNNs in the hot rolling process, but predicting the chosen quality-controlled parameters, thickness, and grain size together using the proposed LSTM architecture has not been investigated. The motivation for using LSTM for modeling the hot-rolling process is its ability to learn from sequential data effectively compared to normal neural networks. For each pass, the model predicts the thickness and grain size by taking the rolling force and temperature as inputs. As thickness and grain size in the hot rolling process at any given pass are influenced by the deformation history from earlier passes, it is important to retain information across the sequence. This memory retention justifies the use of an LSTM approach. The trained LSTM model learns the functional relationship between the process parameters and the material parameters, which are treated as the inputs and outputs of the model, respectively. After accurate learning of the relationship, the model can predict both the quality parameters for any given values of rolling force and temperature. In the end, the values obtained by predictions using the trained and developed LSTM model are compared with thickness values entered as input to the RoCaT model and the grain size values calculated as output from the RoCaT model, and statistical validation is performed to ensure the reliability and robustness of the developed LSTM model.

Methodology

The original version of this paper is available on https://www.scientific.net/KEM.1050.81.pdf

Data acquisition is one of the main steps before training and developing Machine Learning and Deep Learning models. In this research, data acquisition is performed using a Fast Rolling Model and a previously developed Simulation as a Service. FRMs are developed based on mechanical assumptions called the slab method, coupled with semi-empirical equations. One such FRM is the Rolling Calculation Tool (RoCaT). [9, 10].

RoCaT is used to calculate strain, strain rate, rolling forces, and austenite grain size for a complete pass schedule as shown in Figure 1. It consists of physics and semi-empirical-based process, temperature, and microstructure evolution of the material. The inputs to the RoCaT are pass schedule, rolling mill data like the rolling speed, material parameters like the flow stress, density, and heat transfer coefficient, and material calibrated parameters for semi-empirical models. A major assumption that is taken into account while developing RoCaT is that it considers static recrystallization.

[picture]

Fig. 1. Schematic structure of the Fast Rolling Model RoCaT [9]

Fig. 1. Schematic structure of the Fast Rolling Model RoCaT [9]

RoCaT calculates the rolling force based on the equation developed by Sims and Wright [11], which depends on the contact length ( ld ), the width of the rolling stock (b), the mean flow stress ( σm ), and the geometric factor (Qp) that compensates for inaccuracies due to friction and shear, as discussed by Seuren et al. [12] and Lohmar et al. [9], shown in equation 6:

F=ld· b·σm·Qp(1)

RoCaT takes into consideration the static recrystallization, and the grain growth after static recrystallization is calculated using semi-empirical equations introduced by Beynon and Sellars [13]. The grain size after full static recrystallization ( dsrx ) calculation includes the accumulated equivalent strain ( φacc ), the average temperature ( Tm ), and the austenite grain size ( dγ,0 ) as shown in equation 7 :

dsrx=dγ,00.4·φacc−0.5·exp(−4500/8.31· Tm)(2)

Data Acquisition & Preprocessing. In this work, RoCaT is used to calculate different pass schedules for hot rolling as training data. The pass schedule consists of a thickness reduction in each pass, interpass time, and the rolling velocity. While designing the pass schedule, the maximum allowed force and torque, energy consumption, and allowed tolerances of the ultimate tensile strength are considered [14]. The concept of Simulation as a Service is to remotely request simulations and help in achieving data for desired pass schedule. In this case, a SaaS [15] was developed for RoCaT, where a request is sent to RoCaT from the client-side, and RoCaT on the server-side performs the simulations for every pass, stores data in JSON format, and sends the result JSON back to the client. This kind of setup allows a process expert and a machine learning expert to collaborate efficiently.

The material and process parameters that are considered in the hot rolling process are rolling thickness (H), grain size (DGS), rolling force (F), and rolling temperature. In rolling temperature, three rolling temperatures are considered: The temperature before the material comes in contact with the rollers ( TBefore  ), the temperature just after the material leaves the roller ( TAfter  ), and the temperature of the material at the end of the transport table ( TEnd  ), respectively. The dataset consists of 3700 pass schedules of varying numbers of passes with starting thicknesses of 75, 85, 95, 105, 205, 305, and 405 mm . The target grain size that is chosen is below 30 micrometers. Each pass schedule is given a series_id number for identification, so there are 3700 series_ids. As the dataset consists of a varying number of passes, the concept of padding and masking is used. In the preprocessing step, the pass schedule with the maximum sequence length is taken, and the rest of the pass schedules are padded to that maximum length with zeros, so that all the sequences are of the same length, which makes it easier for batch training in LSTM. A masking layer is used in the model to ensure that the LSTM ignores padded zeros during training. The dataset consists of material and process parameters for the steel grade S355, which were previously determined [14]. While training the LSTM model, the data is split in the ratio of 70:30, where 70% of the data is used for training, and 30% is used for testing. Normalization is performed using MinMaxScaler, which is performed after data splitting. In order to prevent data leakage, MinMaxScaler is fit on training data, and using the same scaling parameters, the test data is also scaled before training is performed. In this study, a 3-fold cross-validation is used, in which the training dataset is split into three folds. One fold is used once as a validation set, while the remaining two folds are used for training. The final model is trained on the fold with the best validation performance.

Long Short-Term Memory (LSTM) Architecture

The original version of this paper is available on https://www.scientific.net/KEM.1050.81.pdf

LSTM consists of three types of gates, the Input, Output, and forget gates. The purpose of the forget gate is to decide which information is required to keep or discard from the previous steps. The purpose of the input gate is to control the amount of new information in the current input, and the output gate is to control the output based on the current input. LSTM consists of a memory unit that helps to retain long sequences of memory. GRU consists of two gates, a reset gate and an update gate. The purpose of the Reset gate is to control the amount of past information that is going to combine with the current input. The purpose of the Update gate is to control the amount of new information that will replace the past information [16, 17]. The mathematical equations between the input and output of the LSTM [18] are defined by the following equations:

it=σ(Wxi·xt+Whi·ht−1+Wci·ct−1+bi)(3)

ft=σ(Wxf·xt+Whf·ht−1+Wcf·ct−1+bf)(4)

ct=ft·ct−1+ittanh(Wxc·xt+Whc·ht−1+bc)ot=σ(Wxo·xt+Who·ht−1+Wco·ct+bo)ht=ot·tanh(ct)

(5)(6)(7)

Equation 3 describes the update of the input gate, which takes into account the output (ht−1) of the last time step, the input of the current time step (xt), the memory cell value of the last time step ( ct−1 ), and a bias term ( bt ). Equations 4 to 6 represent the output of the forget gate, input gate, and output gate, respectively. Equation 7 represents the output of the current time step. Wxi,Whi,Wci,Wxf, Whf,Wcf,Wxc,Whc,Wxo,Who,Wco are different weight matrices, and bi,bf,bc,bo are the bias terms used in equations 3 to 7.

In this section, an LSTM architecture is proposed for predicting the thickness and grain size simultaneously in the hot rolling process, as shown in Figure 2. The architecture of the LSTM model consists of one LSTM layer and uses four inputs and two outputs for the entire dataset. Input parameters of the LSTM model are considered as those that sensors can capture in real time. Output parameters influence the quality of hot-rolled sheets. The aim of developing the LSTM neural network is to capture the nonlinear functional relationship between the input parameters, which are rolling force and rolling temperatures, and the output parameters, which are thickness and grain size. By accurately learning the functional relationship, the developed LSTM model can predict both the output parameters closer to the RoCaT value. The learning of any Neural Network and model complexity is dependent on hyperparameters. Hyperparameters are those parameters that are defined before training machine learning models. Hyperparameters regulate model complexity with the number of layers and nodes, and control the neural network learning using batch sizes, learning rate, initialization conditions, and momentum decay parameters [19]. Hyperparameters that are used in this research for the training of the proposed LSTM architecture are units, dropout, and batch size. Optimization of the hyperparameters is performed using optuna [20]. 50 to 128 units, 0.1 to 0.4 dropout rates, and batch sizes of 8, 16, and 32 are used as the search space for optuna. 20 trials are conducted, and the optimal hyperparameter is obtained with 60 units, a dropout rate of 0.16 , and a batch size of 16. Dropout is used to prevent overfitting of the neural networks. Overfitting occurs when a model performs well on the training dataset but fails to generalize to the test dataset [21]. The value of the units (nodes) and dropout used in the proposed LSTM are 60 and 0.16 , respectively. The total number of epochs the LSTM model was trained on is 25 , and the loss function used is Mean Squared Error (MSE).

[picture]

Fig. 2. Developed Long-Short Term Memory Architecture for predicting quality-controlled parameters of hot-rolled strip

Fig. 2. Developed Long-Short Term Memory Architecture for predicting quality-controlled parameters of hot-rolled strip

Table 2 summarizes the architecture of the developed LSTM model. The proposed LSTM model uses a maximum sequence length of 38 passes present in the dataset. The input feature dimension is four, and the Dense output layer maps the LSTM outputs to two prediction targets. The batch size is "None," which indicates that a flexible batch dimension is used during the construction of the model. The model contains a total of 15,722 trainable parameters. Trainable parameters are the internal weights and biases of the LSTM and Dense layers that the model learns, and using backpropagation, it gets updated during training. These parameters influence the learning capacity of the model and indicate the ability of the developed LSTM to capture patterns in the data.

Table 2. Summary of the developed Long-Short Term Memory model

Layer (type) | Output Shape | Param # | Connected to
input_layer_3 (InputLayer) | (None, 38, 4) | 0 | -
not_equal_3 (NotEqual) | (None, 38, 4) | 0 | input_layer_3[0]...
masking_3 (Masking) | (None, 38, 4) | 0 | input_layer_3[0]...
any_3 (Any) | (None, 38) | 0 | not_equal_3[0][0]
lstm_3 (LSTM) | (None, 38, 60) | 15,600 | masking_3[0][0], any_3[0][0]
dropout_3 (Dropout) | (None, 38, 60) | 0 | lstm_3[0][0]
dense_3 (Dense) | (None, 38, 2) | 122 | dropout_3[0][0]

Results & Discussion

The original version of this paper is available on https://www.scientific.net/KEM.1050.81.pdf

Loss curves are useful to analyze the training behavior of machine learning models as they can track changes in loss over each epoch and provide information regarding the performance and learning progress of the developed models. Figure 3 shows the pattern of training, validation, and test loss curves across epochs during the training of the LSTM model. The training and validation loss curve is obtained on a 70% training dataset that is split using a 3-fold cross-validation, and the best fold is shown, the test loss curve is obtained on a 30% test dataset. At the beginning of the training, MSE loss is high as the weights of the LSTM are randomly chosen, resulting in less accurate predictions. At the time of the training, adam optimizer adjusts LSTM weights to minimize the MSE loss. With the progress of the training, predictions by LSTM are closer to the RoCaT values, and the loss decreases. It can be concluded that as the training progresses, the model is learning the patterns present in the data better and improving, hence the loss curves show a downward trend. In training loss, a sharp drop is observed at the initial epochs because the model is able to quickly learn the pattern present in the data, then the curve decreases gradually to zero, and in the end stabilizes, which is an indicator of successful convergence of the model. Similarly, the validation and the test loss decrease gradually towards zero and eventually stabilize. The successful training of the LSTM model is indicated by the stabilizing nature of all three loss curves, and overfitting does not occur.

[picture]

Fig. 3. Loss curves of the developed LSTM

Fig. 3. Loss curves of the developed LSTM

Using MSE as the loss function, the LSTM model is trained on the 70% dataset containing 3700 pass schedules. After training, the developed LSTM model is used to predict thickness and grain size evolution along 500 different pass schedules with a starting thickness of 205 mm using the same rolling force and temperature as in the original dataset. Figure 4 a and 4 b show two examples for the starting thickness of 205 mm with 15 and 17 passes to compare the thickness and grain size values obtained from RoCaT and the predictions performed by the LSTM across the pass schedules. In both figures, it is observed that the developed LSTM model can capture the trend of both the quality parameters for the entire pass schedule. However, in Figure 4a, the maximum percentage of relative error for thickness between the thickness values entered as input to the RoCaT model and the predicted value by LSTM is obtained as 18.81%, and in Figure 4b, it is obtained as 16.56%. Table 3 shows the relative percentage error of the thickness between the predicted value by LSTM and the thickness values entered as input to the RoCaT model in each pass. The significance of the maximum relative percentage error in both pass schedules is that, as both errors are observed during predictions performed at the early passes, the LSTM finds difficulty during early passes in performing accurate predictions. It is also observed that during predictions of both thickness and grain size, the LSTM can accurately describe the thickness evolution compared to the grain size evolution. For both figures 4a and 4 b , the grain size evolution shows significant deviations between the grain size value calculated as output from the RoCaT model and the predicted value by LSTM, also higher during the initial passes, as compared to later passes, showing that the LSTM also finds difficulty in retaining information during initial passes. The time required to perform the calculations of rolling force, temperatures, and austenite grain size using the RoCaT model for the corresponding thickness shown in Table 3 is 1.5 and 1.6 seconds, respectively, for both pass schedules. In comparison, the time required by the trained LSTM model is 0.14 and 0.56 seconds, respectively, to perform the predictions of both quality parameters. Time comparison is performed on a processor specification with 16 gb RAM, 13th  Gen Intel ® Core TM i5-1335U processor, and 128 MB Intel® UHD Graphics. Nevertheless, the predictions performed by LSTM for a varying number of passes overall justify the predictive capability of the developed model.

[picture]

Fig. 4. Predictions performed by the developed LSTM with: a) Pass schedule with 15 passes, b)
Pass schedule with 17 passes

Fig. 4. Predictions performed by the developed LSTM with: a) Pass schedule with 15 passes, b) Pass schedule with 17 passes

Table 3. Thickness predictions and percentage error for a) pass schedule of 15 passes, b) pass schedule of 17 passes

RoCaT (Thickness in mm) (a) | Predicted with LSTM (Thickness in mm) (a) | Relative Percent Error of LSTM (-) (a) | RoCaT (Thickness in mm) (b) | Predicted with LSTM (Thickness in mm) (b) | Relative Percent Error of LSTM (-) (b)
205 | 171.04 | 16.56 | 205 | 171.04 | 16.56
200 | 171.98 | 14.00 | 200 | 171.98 | 14.00
190.8 | 198.06 | 3.80 | 190.9 | 198.18 | 3.81
172.7 | 205.2 | 18.81 | 183.9 | 202.78 | 10.26
169 | 171.71 | 1.60 | 176.2 | 175.33 | 0.49
163.4 | 152.40 | 6.72 | 170.8 | 155.49 | 8.96
145.7 | 146.39 | 0.47 | 167.5 | 146.04 | 12.80
129.4 | 139.44 | 7.76 | 156.7 | 142.13 | 9.29
124.2 | 130.84 | 5.35 | 146.1 | 137.14 | 6.13
117.7 | 121.14 | 2.92 | 129.8 | 131.78 | 1.53
102.9 | 111.94 | 8.78 | 113.1 | 124.62 | 10.18
96.8 | 101.19 | 4.53 | 110.6 | 114.77 | 3.77
86.2 | 90.60 | 5.11 | 95.3 | 104.68 | 9.84
81 | 79.73 | 1.55 | 92.7 | 92.83 | 0.14
65.6 | 69.09 | 5.33 | 80.1 | 81.67 | 1.96
|  |  | 66.5 | 71.71 | 7.83
|  |  | 55.1 | 62.14 | 12.79

The statistical validation is performed on denormalized data of the unseen test dataset to justify the reliability and robustness of the machine learning models. Performance metrics techniques such as Root Mean Squared Error, Normalized Root Mean Squared Error, Mean Absolute Error, and R-squared score are used to assess prediction accuracy. Figure 5 shows the values of performance metrics obtained on the 30% test datasets, which are split during data preprocessing for both the output parameters, thickness and grain size. Here, RMSE calculates the square root of the average of the squared differences between the actual and predicted values as shown in Equation (8) and Equation (9), with n describing the number of observations, Yk the actual values, and Yk the predicted
values. Normalized Root Mean Squared Error (nRMSE) is a dimensionless error metric that calculates the prediction error with respect to the range of actual data, as shown in Equation (10), where Ymax  and Ymin  are the maximum and minimum values of the actual data. MAE calculates the average sum of the absolute differences between the actual values and the predicted values shown in Equation (11). The R2 is a dimensionless metric that calculates the variance in the dependent variable, which is predicted by the independent variables, as shown in Equation (12), with SSR describing the sum of squares of residuals and TSS the total sum of squares. The value ranges from 0 to 1 , where 0 indicates the model is a poor fit and 1 indicates an ideal fit of the model to the data [22]

MSE=1/n∑k=1n(Yk−Y^k)2RMSE=MSEnRMSE=(RMSEYmax−Ymin)MAE=1/n∑k=1n(|Yk−Y^k|)R2=1−SSRTSS

(8)(9)(10)(11)(12)

The results of error metrics and the coefficient of determination on the denormalized data indicate that the model's performance is acceptable in predicting both thickness and grain size. For thickness, the RMSE, nRMSE, and MAE are obtained as 45.21 mm,0.11(−), and 18.75 mm , respectively. The predictions performed by the LSTM model closely match the values of RoCaT, while the R2 value obtained is 0.81(−), which indicates a strong variance despite some deviations. Similarly, for grain size, the RMSE, nRMSE, and MAE are 25.82μ m,0.09(−), and 13.43μ m, respectively, and the higher R2 value of 0.91(−) compared to the thickness indicates strong predictive capability of the developed LSTM model. In this study, RMSE, MAE, and nRMSE for thickness are more compared to grain size, both in absolute and relative measures. Grain size has larger absolute errors, but a low value of nRMSE indicates that the model predicts it more accurately relative to its typical range. The variance of the target variable has a strong influence on the coefficient of determination ( R2 ). The variance for the thickness is smaller, so even moderate prediction errors result in a larger drop in R2, whereas the variance for the grain size is higher, so it is less affected by similar errors. Overall, these metrics show that the developed LSTM model verifies the reliability. However, the predictions for both the thickness and grain size could be further improved by using an attention layer, which is one of the core components of transformer architectures.

[picture]

Fig. 5. RMSE, nRMSE, MAE, and R2 score of the developed LSTM

Fig. 5. RMSE, nRMSE, MAE, and R 2 score of the developed LSTM

Summary and Outlook

The original version of this paper is available on https://www.scientific.net/KEM.1050.81.pdf

The effect of rolling force and rolling temperatures on the quality-controlled parameters, thickness, and grain size in the hot rolling process is investigated in the present work. Considering the aim to predict both the quality parameters accurately, a Long Short-Term Memory is developed. Real-time application of the developed LSTM architecture is to perform online process monitoring during the hot rolling process by taking rolling force, rolling temperature as input, and thickness as output, and all three parameters can be captured by sensors in real time. Conclusions that can be drawn are that the prediction performed by Long Short-Term Memory can capture the trend of both the quality parameter during the entire pass schedule. However, deviations between the RoCaT values and the predicted value by the developed LSTM are observed. Another conclusion is drawn based on statistical validation. The validation of the developed LSTM model is performed using Root Mean Squared Error, Normalized Mean Squared Error, Mean Absolute Error, and R-Squared value. It is observed that the value of Root Mean Squared Error and Mean Absolute Error for thickness is high compared to the grain size, whereas the R-Squared value for grain size is better compared to the thickness. The RMSE, MAE, nRMSE, and R2 values suggest that the obtained results can be further improved. Both conclusions indicate that further investigation is required to make the predictions of both quality parameters more accurate. Further research will investigate the usage of other RNNbased models, such as the Gated Recurrent Unit (GRU) and the combination of LSTM and GRU with an attention layer, which is one of the core components of the Transformer-based architecture. This combination of RNNs with an Attention layer will be performed to further increase the accuracy of the RNN models, as well as reduce the relative error percentage for both the quality parameters.

Acknowledgement

The original version of this paper is available on https://www.scientific.net/KEM.1050.81.pdf

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - EXC-2023 Internet of Production - 390621612.

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The original version of this paper is available on https://www.scientific.net/KEM.1050.81.pdf

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This ePub file was created using machine-generated methods and automated software to convert the source text into an ePub format and may contain formatting inconsistencies from the original version. Formatting may vary depending on the device used.

The original version of this eBook is available on https://www.scientific.net/book/advanced-numerical-and-ai-strategies-for-material-forming/978-3-0364-1986-2

Key Engineering Materials, ISSN: 1662-9795, Vol. 1050, pp 93-102

doi: 10.4028/p-kJ2Cti

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-18

Revised:
2026-02-25

Accepted:
2026-02-26

Online:
2026-04-16

Al-Based Predictions of Forming Effects for Enhanced Crash Simulation

The original version of this paper is available on https://www.scientific.net/KEM.1050.93.pdf

Ingolf Lepenies

SCALE GmbH, Pohlandstraße 19, 01309 Dresden, Germany

ingolf.lepenies@scale.eu

Keywords: Deep Drawing, Physics-Informed AI, Forming Effects, Virtual Vehicle Development

Abstract

In the early concept phase of vehicle development, crash simulations must provide reliable statements on energy absorption and failure behavior, although detailed forming simulations and process data are typically not yet available. Manufacturing-induced material states, such as plastic pre-strains and local sheet thickness distributions, are therefore often neglected or approximated using simplified low-fidelity approaches with high uncertainty. This contribution introduces UmMatCraML, a data-driven medium-fidelity method for rapid initialization of typical crash shell meshes (element edge lengths 3−5 mm ) with forming-induced field variables. Starting from the final component geometry, a purely geometric unfolding is performed to approximate the blank, from which a meshand coordinate-independent areal strain Ar is determined. A monotonicity-preserving gradient boosting regressor subsequently compensates for systematic deviations of this geometric surrogate model compared to high-fidelity deep drawing simulations, with four Hockett-Sherby parameters consistently parameterizing the material description. The training data are generated from approximately 5,000 LS-DYNA forming simulations and cover a broadly varied, physically consistent parameter space. In validation on a demonstrator part, UmMatCraML reduces the computation time for determining forming-induced component properties from about 60 min for High-Fidelity-Simulation (HFS) or 10-15 min for Low-Fidelity-Simulation (LFS) to under 10 s , with simultaneously improved prediction quality. Demonstrations on components of a Toyota Yaris full-vehicle model show robust predictions even with trimming and perforations. Limitations arise from the model assumptions made (e.g., isotropic hardening, limited mapping of multi-stage process paths). Overall, UmMatCraML enables real-time, reproducible provision of manufacturing-induced field variables for concept crash simulations without explicit tool modeling.

1. Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1050.93.pdf

In the early concept phase of vehicle development, the crash properties of the full vehicle must be estimated. The prediction of structural behavior depends significantly on locally varying material states arising from the forming process. These properties, particularly plastic pre-strains and sheet thinning, influence energy absorption and failure behavior.

Crash simulations as dynamic structural analyses and forming simulations are both used early in component development but differ in temporal positioning, modeling detail, and objectives [10]. Forming simulations primarily serve to ensure manufacturability and process design, while crash simulations evaluate structural energy absorption capacity. Consistent coupling of both domains is required to consider manufacturing-induced properties in crash models. In industrial practice, this mapping typically occurs only in late series phases, sporadically and with idealized material models.

Finite element-based forming simulations represent the state of the art for mapping complex sheet forming processes [1]. They are based on nonlinear, anisotropic, and plastic material models that consistently capture direction-dependent yield stress, plastic hardening, and stress- and strain-dependent material responses [13, 14]. Extended multiphysical approaches couple mechanical and thermal fields and allow an integrated description of friction, temperature development, and the resulting stress states during the forming process.

In early development stages, manufacturing-induced properties are often neglected. Reasons include the high computational effort of detailed finite element forming simulations (high-fidelity simulations, HFS) with element sizes of approximately 1 mm and the lack of valid forming data with
incomplete geometries. Simplified one-step simulations (low-fidelity simulations, LFS) with element sizes of 3−5 mm and two-parameter hardening laws provide initial estimates but are inaccurate due to coarse geometry approximations and simplified material descriptions [12]. Additionally, there is a significant discretization difference between forming simulations with fine meshes and crash simulations, which typically use element edge lengths of 3−5 mm. Reconstructing an initial blank from a trimmed final geometry with perforations, beads, and flanges poses an inverse and ill-posed problem under these boundary conditions.

Recent research has investigated data-driven methods to accelerate forming simulations and predict forming outcomes. Surrogate models, neural networks, and gradient boosting approaches have been applied to predict thinning and strain fields. However, many approaches remain limited to simplified geometries or narrow process windows, require dense input data, or lack physical constraints, which can lead to non-plausible predictions when extrapolating to new components [2,3].

Research gap. In the early concept phase, the assessment of crash properties primarily requires the dominant forming-induced plastic deformations and sheet-thinning extrema. Accordingly, there is a need for a coordinate-independent, mesh-based method that provides physically plausible thickness and plastic strain fields directly on crash FE meshes without explicit tool modeling and without the computational effort of HFS, while significantly improving the systematic inaccuracies of LFS for complex, trimmed and perforated geometries. In this context, effects that are typically second-order for early-phase crash assessment - such as material anisotropy and kinematic hardening - can be intentionally neglected to enable robust, fast predictions. Moreover, the explicit consideration of multistage forming is not required for methods targeting early-phase concept assessment.

2. Methodology and Procedure

The original version of this paper is available on https://www.scientific.net/KEM.1050.93.pdf

The UmMatCraML approach addresses the performant provision of manufacturing-induced component properties for crash-relevant simulations in early development phases. The objective is to identify and quantify regions of large plastic deformations and crash-relevant sheet-thinning extrema in the component. Training data from low-fidelity forming simulations (one-step-simulations without tools) are used to learn the systematic deviations between the geometric surrogate and physics-based results, enabling a performant and sufficiently accurate one-step methodology. The target is to deliver results significantly faster than established FEM-based one-step solvers. Non-objectives are the prediction of springback, wrinkling, and anisotropic material behavior including kinematic hardening, which are explicitly not addressed within the proposed scope.

The approach follows a simplified inverse procedure conceptually related to one-step solvers. Starting from the final component geometry, the initial blank is approximated by purely geometric unfolding of the three-dimensional surface into the plane. The unfolding is implemented as a conformal surface-to-plane mapping that preserves local angles and maps a three-dimensional surface with spatially varying thickness to a two-dimensional representation with constant thickness, while minimizing global volume changes to maintain volume constancy. Local deformation measures are derived by comparing deformed and undeformed element areas under the assumption of volume constancy; plastic material behavior and process-related effects are not modeled during unfolding.

Training data from low fidelity forming simulations (LFS) are used to learn the systematic deviations between the geometric surrogate and physics-based results. The goal is to identify and quantify regions with large plastic deformations and crash-relevant sheet-thinning extrema; springback, wrinkling, and anisotropic material behavior with kinematic hardening are explicitly not addressed. Within these scope assumptions, a performant and sufficiently accurate one-step methodology is targeted for early-phase crash assessment.

The starting point is a shell FE mesh of the final component geometry (element sizes 3−5 mm, including holes, trimming, and flanges). Fig. 1 shows the underlying workflow. In process step I, the post-forming 3D geometry without holes is derived. This is geometrically unfolded into the plane

[picture]

Fig. 1: The UmMatCraML Workflow: I - Hole-filling of the 3d-mesh, II - unfolding of the 3d-mesh in 2d-mesh, III - calculation of strain measure Ar, uvmap , IV - AI prediction of strain measure Ar, real ,V - calculation of sheet thickness and plastic strain based on Ar, real

Fig. 1. The UmMatCraML Workflow: I - Hole-filling of the 3d-mesh, II - unfolding of the 3d-mesh in 2d-mesh, III - calculation of strain measure A r , uvmap , IV - AI prediction of strain measure A r , real , V - calculation of sheet thickness and plastic strain based on A r , real

in process step II to approximate the initial blank. Based on the area comparison, the areal strain Ar=A/A0 is calculated in steps IIIa and IIIb:
with A0 as the undeformed and A as the deformed element area. The method ensures volume constancy but does not consider plastic material behavior or process-related effects (friction, tool contact, multi-stage forming).

The systematic deviations of this geometric surrogate model prediction compared to low- and high-fidelity forming simulationresults (HFS) are compensated by the UmMatCraML model.

Training and validation data are systematically generated from numerical deep drawing processes under defined process parameters, material models, and boundary conditions. Target variables are plastic strains and local sheet thinning, as these significantly influence failure and energy absorption behavior in crash analyses.

The feature representation via the areal strain Ar is designed to be mesh-independent to enable generalization to unknown discretizations. The vector Ar serves here as the sole geometric input vector for the AI model (step IVa). Notably, this is a coordinate-independent description of the geometry.

Additionally, material-describing features are integrated (step IVb). In accordance with the parsimony principle of science, the functional relationship is modeled by exactly four parameters: the four Hockett-Sherby parameters of the hardening curve, cf. Fig. 4.

In step V, inference occurs with the 5 input parameters: The model predicts sheet thicknesses and plastic strains. These are mapped onto the original crash model in steps VIa and VIb, allowing manufacturing-induced effects to be consistently incorporated into the structural dynamic analysis.

The method was prototypically implemented in a prediction tool. Validation was based on the Kirchhoff Automotive demonstration LS-Dyna model [7], which served as a reference high-fidelity model (HFS) - a LS-DYNA model with die, punch and holder - for comparing predicted and modelbased system states. The agreement of the results was evaluated using quantitative metrics, verifying the functional correctness of the implementation and the consistency of the methodological assumptions.

2.1 AI-compliant geometry description

Step I (cf. Fig. 1) serves to generate the post-forming 3D geometry and forms the basis for the subsequent blank determination. Several methodological approaches are considered for estimating the blank. These include using the original contour of the crash mesh, optionally with or without considering existing perforations, a radial offset to determine a concave hull contour, or alternatively
determining a convex hull contour. Based on the selected contour definition, an automatic mesh generation is performed, from which the initial blank geometries are derived, cf. Fig. 2. The subsequent unfolding of the three-dimensional geometry into the plane is based on geometric algorithms that provide the blank approximation.

[picture]

Fig. 2: Preprocessing: FE Model Extraction and Approximation of the Sheet Blank

Fig. 2. Preprocessing: FE Model Extraction and Approximation of the Sheet Blank

To ensure efficient, consistent, and traceable processing of the data generated, the data management system SCALE.sdm is used (cf. Fig. 3). The system handles the structured capture, storage, and versioning of data as well as the control of associated processing workflows, thus providing a reproducible data basis for downstream analysis and evaluation steps.

[picture]

Fig. 3: Data Management in SCALE.sdm

Fig. 3. Data Management in SCALE.sdm

2.2 AI-compliant material Description

All materials used are consistently described in the UmMatCraML model by a uniform set of Hockett-Sherby hardening parameters [4]. Thus, the same phenomenological hardening law is assumed for all materials, whereby material-specific differences are mapped exclusively via the parametric expression of the model and not via different model approaches.

The Hockett-Sherby model describes the isotropic hardening behavior of metallic materials under plastic deformation. The model parameterizes the flow curve via the initial yield stress σ0 as well as the saturation stress σa,true  and additionally considers the Considère stress σc and the associated Considère strain εc. These quantities characterize the transition from homogeneous plastic deformation to material instability due to diffuse necking. By explicitly incorporating the instability criteria, the model formulation allows consistent mapping of the hardening behavior up to the uniform strain.

Fig. 4 shows an exemplary parameter identification for the dual-phase steel DP980. The model parameters were determined by fitting to experimental stress-strain data. The model captures the characteristic decrease in hardening rate for steels with increasing plastic strain and allows consistent consideration of material instability. The parameterization is reproducible, as the used model form, number of degrees of freedom, and underlying assumptions are clearly defined. Limitations of the approach arise from the assumption of isotropic hardening, whereby direction-dependent effects and kinematic components are not mapped.

[picture]

Fig. 4: Approximation of the hardening behavior of metallic materials considering material instability.

Fig. 4. Approximation of the hardening behavior of metallic materials considering material instability.

The material behavior can be generically generated. For this, a naming scheme was developed. Exemplarily, the designation HS450Y659T139C900A describes a material with a technical yield stress σ0=450MPa and a technical tensile strength of σc, tech =Rm=659MPa. The technical uniform strain is defined as εc, tech =Ag=13.9%. The true maximum stress (saturation stress) converges to σa, true =900MPa. A material designation such as HR450Y980T-DP is not sufficient to define the material behavior.

2.3 Data generation

The generation of training data was performed using the finite element program LS-DYNA by means of one-step-simulations. The meshes had about 10,000 shell elements with crash-typical element edge lengths of 3−5 mm. For this purpose, both generic geometries and real component geometries of the Toyota Yaris vehicle model were numerically investigated. In total, approximately 5,000 simulations were conducted to determine realistic areal strains Ar,real. The generic hardening curves used are shown in Fig. 5.

The training data cover a widlely defined parameter space. The variation was systematically performed over physically consistent but deliberately generically chosen material parameters. This approach enables controlled and reproducible sampling of relevant material parameters without limiting to specific component or material configurations. In particular, the generic material modeling allows largely unbiased sampling of the parameter space, reducing systematic biases in the data basis for downstream AI models.

[picture]

Fig. 5: Generic Hardening Curves

Fig. 5. Generic Hardening Curves

2.4 AI model

A monotonicity-preserving gradient boosting regressor is trained to map the five input parameters (areal strain Ar, uvmap  and four Hockett-Sherby parameters σ0,σc,σa,εc ) to forming-induced target fields (areal strain Ar, real , thickness t and plastic strains εpl ), cf. the constraints in eqs 1-5.

Ar, uvmap ∝σ0∝σc∝σpl∝εplεplεc∝1/εpl

(1)(2)(3)(4)(5)

Monotonicity constraints enforce physically plausible dependencies for constant material configurations and reduce unphysical extrapolation. Furthermore, during training, samples with pronounced

[picture]

Fig. 6: Model accuracy of internal fit parameter p1,p2

Fig. 6. Model accuracy of internal fit parameter p 1 , p 2

thinning and large plastic strains are weighted higher to prioritize crash-relevant extrema while maintaining global approximation quality. This weighting does not reduce prediction quality for elements with low areal strain.

The mapping of the purely geometrically determined areal strains Ar, uvmap  for each finite element to the simulated area strains Ar, real was carried out using equation 6. For each training data set with defined Hockett-Sherby material parameters, the model parameters p0,p1 were determined. These 5000 datasets (p0, p1) were used for training the UmMatCraML model.

Ar, real =p0e−p1Ar, uvmap −1(6)

Figure 6 shows all 5,000 simulation-based and predicted model parameters pi. A perfect prediction would place all points on the diagonal. The results indicate a very high predictive accuracy with coefficients of determination R2>95%. This allows the material-dependent mapping from Ar, uvmap  to Ar, real to be performed for each hardening law, cf. eq. 7.

Ar, real =f(Ar, uvmap ,σ0,σc,σa,εc)(7)

3. Verification and Validation

The original version of this paper is available on https://www.scientific.net/KEM.1050.93.pdf

The verification of the proposed methodology addresses the correct implementation of the model assumptions and algorithms. It checks whether the model provides consistent and physically plausible predictions for manufacturing-induced sheet thickness distributions and plastic strains. Validation is performed through quantitative comparison of the results with established high-fidelity simulations (HFS) and low-fidelity simulations (LFS), which represent the state of the art in numerical forming simulation.

For the Kirchhoff Automotive demonstrator [7], HFS requires a computation time of about 60 min , while LFS is significantly faster at 10−15 min. The data-driven UmMatCraML model delivers comparable results in significantly less than 10 s . This results in an acceleration of the calculation of component properties by several orders of magnitude compared to HFS and at least one order of magnitude compared to LFS. This acceleration is achieved without explicit modeling of the deep drawing tools, which additionally reduces modeling effort, compare Fig. 7.

[picture]

Fig. 7: Comparison of HFS, LFS, and the UmMatCraML Method for Kirchhoff Automotive Demonstrator Model - mild steel HS150Y273T284C450A - simulated vs. predicted element thicknesses t=[0.823..1.220]mm

Fig. 7. Comparison of HFS, LFS, and the UmMatCraML Method for Kirchhoff Automotive Demonstrator Model - mild steel HS150Y273T284C450A - simulated vs. predicted element thicknesses t = [ 0.823 . .1 .220 ] m m

The prediction of crash-relevant minimum sheet thickness tmin  for a mild steel confirms the accuracy of the approach, tmin, HFS=0.82 mm,tmin, LFS=0.88 mm,tmin, AI=0.83 mm.

For the Kirchhoff Automotive demonstrator [7], UmMatCraML predicts the maximum sheet thinning with an accuracy of about 0.1 mm compared to the high-fidelity reference solution tmin, HFS , whereas the low-fidelity simulation (LFS) deviates by about 0.6 mm with respect to the minimum sheet thickness. In addition to the agreement of the extrema, the spatial thickness distributions show similar characteristics. The UmMatCraML model thus achieves accuracy close to HFS and surpasses LFS in both precision and computation time. Sheet thickening is also realistically predicted, compare Fig. 8.

[picture]

Fig. 8: Comparison of simulated and prediced Thicknesses (Projection on the Undeformed Sheet), tmin, HFS=[0.82..1.03]mm vs. tmin, AI=[0.84..1.01]mm

Fig. 8. Comparison of simulated and prediced Thicknesses (Projection on the Undeformed Sheet), t min, H F S = [ 0.82 . .1 .03 ] m m vs. t min, A I = [ 0.84 . .1 .01 ] m m

The transferability of the method is investigated on the full-vehicle model of the Toyota Yaris. For geometrically and process-technically simple components, robust predictions of component properties are achieved.

Fig. 9 presents the model-based prediction of effective plastic strain in the B-pillar of the Toyota Yaris vehicle. The local quantification of plastic loading across the component geometry can be mapped. Areas of increased effective plastic strain mark zones with advanced plastic deformation and provide a direct indicator for potential failure or damage initiation.

[picture]

Fig. 9: Prediction Results of Plastic Strain εpl=[0..0.11] in the B-Pillar of the Toyota Yaris Vehicle.

Fig. 9. Prediction Results of Plastic Strain ε p l = [ 0 . .0 .11 ] in the B-Pillar of the Toyota Yaris Vehicle.

Fig. 10 shows a component with pronounced trimming and several perforations. For this complex geometry, the UmMatCraML algorithm determines the required sheet blank with high accuracy. The results demonstrate that the algorithm provides robust estimates even with strongly reduced edge areas and structural openings. Under the model assumptions made, the method thus exhibits high potential for industrial application in early design and calculation phases.

Limitations appear in complex, multi-stage forming processes, where mapping process paths and intermediate steps requires increased model complexity. These results illustrate the potential of the

[picture]

Fig. 10: Thickness Predictions t=[0.53..1]mm for a Component with Strong Trimming and Openings

Fig. 10. Thickness Predictions t = [ 0.53 . .1 ] m m for a Component with Strong Trimming and Openings

approach for efficient use in early development phases as well as the need for extended modeling for highly complex manufacturing chains.

Discussion. The presented results demonstrate the high prediction quality of UmMatCraML, with particular emphasis on real-time capability and geometric flexibility. For mild steels, sheet thicknesses are predicted with high agreement to high-fidelity simulations (HFS), and the deviation compared to low-fidelity simulations (LFS) is reduced from up to 0.06 mm to 0.01 mm . The monotonicitypreserving modeling approach addresses typical non-physical behavior of unconstrained regressors and enforces physically plausible dependencies for constant material configurations.

The methodology is subject to clearly defined assumptions. The modeling is restricted to isotropic, incompressible plastic material laws and J2 plasticity, systematically excluding anisotropy and kinematic hardening. These simplifications are consistent with the early concept phase focus on dominant plastic deformations and sheet-thinning extrema, but they limit applicability for strongly anisotropic materials or process chains where detailed path effects become relevant.

Future work should expand the geometric input representation and improve robustness for complex trimming and perforation patterns. Further extensions include crack-risk prediction using load-state variables and the integration of additional quality indicators such as wrinkling and surface defects.

Conclusion and Outlook

The original version of this paper is available on https://www.scientific.net/KEM.1050.93.pdf

UmMatCraML provides a reproducible medium-fidelity alternative to explicit forming simulation for early-phase concept crash assessments. It predicts forming-induced thickness and plastic strain fields directly on typical crash shell meshes without explicit tool modeling and reduces computation time from about 60 min (HFS) or 10-15 min (LFS) to under 10 s while improving accuracy compared to LFS under the stated assumptions. The method is coordinate-independent, handles trimmed and perforated geometries, and enables early integration of manufacturing-induced effects into crash models with low modeling effort.

Limitations arise from the deliberate restriction to isotropic hardening and one-step process abstraction. Within this scope, the approach closes the practical gap between fast but inaccurate LFS and accurate but expensive HFS and provides a robust basis for industrial applications such as concept crash simulation and, with additional boundary conditions and targets, blank optimization and tool-design support.

References

The original version of this paper is available on https://www.scientific.net/KEM.1050.93.pdf

Erman Tekkaya, A.: State-of-the-art of simulation of sheet metal forming, Journal of Materials Processing Technology, Volume 103, Issue 1, 2000, Pages 14-22

Gassler, L.; Haberthür, M.; Afrasiabi, M.; Bambach, M.: Graph Neural Network-based Generation of Stamped Bipolar Plate Geometries from Tooling Surfaces. Journal of Physics: Conference Series 3104 (2025) 012059. IOP Publishing. DOI: 10.1088/1742-6596/3104/1/012059.

Zhao, Y.; Chen, Q.; Li, H.; Zhou, H.; Attar, H. R.; Pfaff, T.: Recurrent U-Net- based Graph Neural Network (RUGNN) for accurate deformation predictions in sheet material forming. Advanced Engineering Informatics, Vol. 69, Part D, January 2026, Art.-Nr. 104021. ISSN 1474-0346. DOI: 10.1016/j.aei.2025.104021.

Hockett, J.; Sherby, O.: Large strain deformation of polycrystalline metals at low homologous temperatures. Journal of the Mechanics and Physics of Solids 23(2), 87-98 (1975).

SCALE.sdm - Integrative Software Solution for Continuous Simulation Data and Process, Management. (2025).

Inal, K.; Lévesque, J., Worswick, M.: The 12th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes (NUMISHEET 2022) (2022).

Schwarzer, R.: Finite Element Model "Kirchhoff-Automotive-Demonstrator" (KADemonstrator) - Process Model for Deep Drawing. KIRCHHOFF Automotive Deutschland GmbH (2025).

Lepenies, I.; Krause, P.: Influence of Sheet Forming Using Machine Learning for Concept Crash Simulations. FAT Publication Series No. 395. Research Association for Automotive Engineering (FAT), Berlin 2025.

2010 Toyota Yaris Finite Element Model. .

Banabic, D.: Sheet Metal Forming Processes. Constitutive Modelling and Numerical Simulation.

Roll, K.:Including the effect of manufacturing in the design of products: crash properties. Proc. Int. Conf. Product Property Prediction, Dortmund, 2010

Zhu, X.; Zhang, L.: Advance in Sheet Metal Forming - One-step Solution, Multi-Beads, Gravity Prebending, Auto Nets, and Local Compensation 12th International LS-DYNA Users Conference, 2012, 12

Güner, A.; Soyarslan, C.; Brosius, A.; Tekkaya, A. Characterization of anisotropy of sheet metals employing inhomogeneous strain fields for Yld2000-2D yield function International Journal of Solids and Structures, 2012, 49, 3517-3527

Nielsen, C. V.; Martins, P. A.: Metal Forming: Formability, Simulation, and Tool Design Academic Press, 2021

This ePub file was created using machine-generated methods and automated software to convert the source text into an ePub format and may contain formatting inconsistencies from the original version. Formatting may vary depending on the device used.

The original version of this eBook is available on https://www.scientific.net/book/advanced-numerical-and-ai-strategies-for-material-forming/978-3-0364-1986-2

Key Engineering Materials, ISSN: 1662-9795, Vol. 1050, pp 103-111

doi: 10.4028/p-wWX8Z3

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-19

Revised:
2026-02-25

Accepted:
2026-02-25

Online:
2026-04-16

Residual Learning-Based Synthetic Data for Hybrid Metamodeling in the Stamping of an Automotive Door Panel

The original version of this paper is available on https://www.scientific.net/KEM.1050.103.pdf

Laura Muñiz 1,a*, Javier Trinidad 1, b, Julian Ramirez de Okariz 2,3,c, Eduardo Garcia 4, d and Lander Galdos 1,e

1 Faculty of Engineering, Mechanics and Industrial Production, Mondragon Unibertsitatea, Loramendi 4, 20500 Mondragon, Spain

2 Fagor Arrasate Koop. Elk., San Andres 20, 20500 Mondragon, Spain

3 Koniker Koop. Elk. Zentro Teknologikoa, San Andres Auzoa 20, 20500 Mondragon, Spain

4 FORD Motor Company, 46440 Almussafes, Spain

almuniz@mondragon.edu, b  jtrinidad@mondragon.edu, c  j.rokariz@fagorarrasate.com, d  egarci75@ford.com elgaldos@mondragon.edu (*corresponding author) 

Keywords: . stamping, hybrid metamodel, synthetic data, discrepancy model.

Abstract

Hybrid twins and residual-learning strategies are increasingly used to reconcile the broad coverage of physics-based simulations with the fidelity of production measurements. In industrial stamping, however, truly matched simulation-experiment cases are scarce, while large-scale monitoring data are clustered around a narrow operating window. This work proposes an industrially practical hybrid metamodel in which the discrepancy (ignorance) model is trained exclusively from approximate residuals computed at in-domain production inputs, considering a surrogate of a thermalenabled AutoForm Sigma Design of Experiments (DoE). To prevent uncontrolled extrapolation, learning and evaluation are restricted to an explicitly validated domain defined through multivariate kNN support in standardized space derived from the DoE cloud. The residual model is selected through five-fold cross-validation on 10,446 in-domain production samples and then retrained on the full approximate-residual set. A small set of seven matched cases is kept as an external check based on true residuals. The resulting hybrid predictor enables the generation of synthetic, experimentinformed corrected data while retaining the DoE coverage required for downstream modelling tasks.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1050.103.pdf

The maturation of machine learning (ML) and the availability of high-resolution production monitoring have renewed interest in data-driven support tools for metal forming. In practice, ML surrogates are often introduced to speed up design and robustness studies, cut down on try-out iterations and enable near-real-time decision-making. However, in industry, data comes from a variety of sources, each with its own limitations: numerical simulations are flexible and can explore a wide parameter space, while production data accurately reflect the plant, but are concentrated around a limited operating window [1-3].

Finite element (FE) models remain essential for sheet forming because they encode geometry, contact, and constitutive behaviour, but they are also affected by systematic bias [3]. In stamping, discrepancies between simulation and reality arise from unavoidable modelling assumptions (e.g., friction, heat transfer, tool compliance, and simplified material descriptions), from parameter uncertainty, and from the time evolution of the process. Increasing model complexity (for instance by including thermal effects) can reduce error, but it rarely eliminates it and often increases computational cost [1,4].
Hybrid modelling addresses this tension by combining a physics-based predictor with a data-driven component that learns the remaining 'ignorance'-the part of the response that the physical model does not capture. Within the broader 'hybrid twin' paradigm, the correction is typically formulated as a residual model that adds a learned discrepancy to the numerical prediction. This residual-learning
view is particularly attractive in manufacturing because the discrepancy is often smoother and lowerdimensional than the full response, which reduces the data burden of the ML component [5-7].

A practical obstacle in industrial settings is the limited availability of strictly matched simulationexperiment pairs. Matched cases allow one to compute 'true' discrepancies by comparing measurements with simulation under identical inputs, but generating them requires additional effort and is not always feasible, as plants often rely on a pre-defined design of experiments rather than an iterative matching campaign. In contrast, large-scale monitoring provides thousands of measured parts, but typically only within the plant's operating window. A methodology that can leverage this statistical richness without relying on many matched cases is therefore of high practical value [5,6].

This paper proposes a hybrid metamodel for an industrial stamping process in which the discrepancy model is trained exclusively from approximate residuals computed from production measurements and a numerical surrogate. A thermal effects-enabled DoE is generated with AutoForm Sigma and used to train a fast simulator surrogate y^FEM(x). For each retained production sample, an approximate residual is defined as rapprox (xj)=yEXP (xj)−y^FEM (xj), and a residual metamodel r^(x) is learned from these targets. The key precaution is to restrict residual construction and learning to an explicitly validated domain defined through a multivariate support criterion derived from the numerical DoE cloud (kNN distance in standardized space) to prevent extrapolation. The necessity of this restriction is quantified in the Results and Discussion section. Model selection is carried out by five-fold crossvalidation on the in-domain residual samples, and the final residual model is retrained on the full approximate set. A small set of seven FEM-experimental matched cases is reserved for an external check based on true residuals. The contributions are therefore: (i) an industrially transparent domainvalidity statement that separates modelling error from extrapolation error; (ii) an approximateresidual formulation that allows discrepancy learning at scale without requiring many matched cases; (iii) a cross-validated training-and-freeze procedure followed by an external check on scarce matched points; and (iv) a hybrid predictor that can generate synthetic corrected data with numerical coverage and experimental consistency.

Methodology

The original version of this paper is available on https://www.scientific.net/KEM.1050.103.pdf

The target response is the draw-in measured at the defect-critical location of an automotive door inner panel under industrial conditions. The case study is based on production monitoring collected during the manufacturing campaign comprising on the order of 70,000 inner door panels; in this study, a representative subset of about 10,446 monitored parts includes synchronized inputs and draw-in measurements suitable for model development. Three sources of information are considered: (a) a numerical Design of Experiments (DoE) of 67 simulations generated in AutoForm Sigma [8] (see input setup in Figure 1) with thermal effects enabled, providing simulated draw-in values yFEM at designed input combinations; (b) a production dataset with monitored inputs and measured draw-in yEXP  for a large number of stamped parts; and (c) a small set of matched simulation-experiment configurations that can be used exclusively for external validation. Press speed is pre-filtered to a narrow range and treated as fixed, so that the modelling focuses on the remaining process and material variability.

[picture]

Fig. 1. AutoForm FE-model set-up (left) and inner door panel analyzed (right). Arrows indicate the prescribed tool motions and blankholder force directions used in the AutoForm setup. Reproduced from [9] under the Creative Commons CC BY 4.0 license.

Fig. 1. AutoForm FE-model set-up (left) and inner door panel analyzed (right). Arrows indicate the prescribed tool motions and blankholder force directions used in the AutoForm setup. Reproduced from [9] under the Creative Commons CC BY 4.0 license.

The predictor vector x contains 12 variables available both in simulation and production monitoring: thickness ( t ), ultimate tensile strength ( Rm ), yield strength ( Rp0.2% ), plastic anisotropy ( rn ), die temperature, ambient temperature, two shim-height signals (Shim 20 and Shim 21), and four cushion forces (C1-C4). To improve numerical conditioning and to support distance-based validity checks, the inputs are standardized using statistics computed on the numerical DoE only: xkstd =xk−μkDoE /σkDoE . Using DoE statistics makes the validity statement explicit-predictions are intended to be trustworthy only within the region where the numerical model has been sampled.

Because approximate residuals require evaluating the numerical surrogate at production inputs, extrapolation must be handled explicitly. In this work, an admissible operating domain Ω is defined through a multivariate support criterion derived from the numerical DoE cloud: the k-nearestneighbour ( kNN ) distance of each standardized production sample to the DoE samples is compared to a threshold obtained from the internal DoE spacing (e.g., a high percentile of DoE kNN distances). Samples exceeding the threshold are flagged as out-of-support and excluded from residual construction. Of the 70,000 production measurements at 16spm,10,446 fall within the kNN -validated domain, forming the subset used for residual construction and model training. The remaining outside Ω, either the residual correction is gated down to zero-thereby reverting to the numerical surrogate-or the condition is marked for future DoE enrichment.

A surrogate of the numerical simulator, y^FEM(x), is trained using only the AutoForm Sigma DoE, considering the DoE inputs mentioned previously as inputs, and considering draw-in in the critical zone as output. Gaussian Process Regression (GPR) [10] is used here as a strong baseline for moderate DoE sizes and smooth interpolation. Five-fold cross validation was used for model validation. The surrogate is not calibrated to experiments; its role is to emulate the simulator with negligible runtime so that residual targets can be computed at scale within Ω.

For every retained production sample, an approximate residual target is then constructed as rapprox (xj)=yEXP (xj)−y^FEM (xj). This target captures the dominant simulation-to-plant discrepancy structure under real operating conditions, while also inheriting surrogate error and measurement noise. Accordingly, all claims are restricted to Ω and the residual model is validated with a conservative protocol.

The discrepancy model r^(x) is learned by supervised regression on the approximate residual targets rapprox . Extreme Gradient Boosting regression trees (XGBoost regressor [11]) are adopted due to their robustness on mixed industrial signals and their favourable bias-variance trade-off when combined with cross-validation. Hyperparameters were selected via five-fold cross-validation on the in-domain approximate-residual dataset ( 10,446 samples after multivariate support gating). After selecting the configuration, the residual model is retrained on the full approximate-residual dataset to obtain a frozen model for deployment and for synthetic data generation. A set of seven matched
simulation-experiment cases, not used in training, was reserved for an external check based on true residuals. Although small, this check provides a direct sanity test of whether the approximate-residual training leads to a correction that remains physically consistent when compared against matched conditions.
Finally, the hybrid predictor is defined as y^hyb (x)=y^FEM (x)+r^(x). Synthetic corrected data can be generated by evaluating y^hyb  on the numerical DoE points (and, if required, on additional samples drawn within Ω ), yielding a corrected dataset that retains the DoE coverage while embedding an experimentally informed bias correction
To evaluate the predictive accuracy of the metamodels and the hybrid strategy, four primary statistical indicators are used. Two main metamodels are evaluated in this work, each with a different reference quantity yi. For the GPR numerical surrogate, yi stands for the draw-in output from AutoForm; performance is reported via five-fold cross-validation on the 67 DoE points. For the XGBoost residual model, the reference is the approximate residual. The Root Mean Square Error (RMSE) provides a measure of the average error between the residual or numerical draw-in (depending on the metamodel) measurements ( yi ) and the model predictions ( y^i ), giving higher weight to larger discrepancies:

 RMSE =1n∑i=1n(yi−y^i)2(1)

Additionally, the Mean Absolute Error (MAE) is utilized to represent the average absolute difference magnitude, offering a more linear representation of the error:

MAE=1n∑i=1n|yi−y^i|(2)

And to identify any systematic over-prediction or under-prediction by the models, the Bias is calculated as:

Bias=1n∑i=1n(yi−y^i)(3)

The overall proportion of variance captured by the model is evaluated using the Coefficient of determination, or R -squared (R2). This metric indicates the goodness-of-fit by comparing the model residuals to the total variance of the experimental data:

R2=1−∑i=1n(yi−y^i)2∑i=1n(yi−y―)2(4)

where y¯ represents the mean of the numerical or measurements. An R2 value of 1 indicates a perfect match between the model and the physical process.

Finally, to ensure that the reported performance is robust and not dependent on a specific data split, a k-fold Cross-Validation (CV) procedure is implemented. In this study, the mean and standard deviation of these metrics are calculated across five folds, ensuring that the model generalizes properly to unseen data.

Results and Discussion

The original version of this paper is available on https://www.scientific.net/KEM.1050.103.pdf

Applying the multivariate kNN support screening to the production monitoring data yields an indomain subset suitable for residual construction. In the present case, 10,446 samples remain after screening. Table 1 summarizes the datasets and splits used in this work.

Table 1. Summary of datasets and their roles in the hybrid modelling pipeline.

Dataset | Role | Size | Notes
Numerical DoE (AutoForm Sigma) | Train ŷFEM | 67 | Thermal-enabled; covers broad design space
Production in-domain subset | Train/validate f̂ | 10,446 | After multivariate kNN support screening
Matched cases (reserved) | External check | 7 | True/anchored residuals; not used for training

The numerical surrogate y^FEM is evaluated only on inputs deemed in-domain, so that surrogate quality and residual structure are not confounded with extrapolation. This separation is critical in industrial reporting: if an out-of-support condition occurs, the methodology does not attribute the resulting error to the hybrid model but rather flags the condition for additional simulation or for a dedicated local model. This also enables a clean operational guideline: within Ω the hybrid predictor can be used for rapid what-if analysis, while outside Ω the system reverts to the numerical surrogate or triggers a DoE update cycle. The physics surrogate is a Gaussian Process Regression model with Matérn 5/2 kernel, constant basis function, and noise σ=1.19 mm. Error metrics of the numerical surrogate model are shown in Table 2. Five-fold cross-validation was used for validation.

Table 2. GPR Error Metrics.

Parameter | Value
Number of boosting rounds | 264
Maximum tree depth | 7
Shrinkage factor | 0.0242
Row sampling ratio | 0.7192
Feature sampling ratio | 0.8139
Minimum child weight | 3
Minimum split loss | 0.3885
tL1 regularization | 0.1381
tL2 regularization | 6.7886

Regarding the discrepancy model r^(x), XGBoost model was trained and validated with approximated residuals. Five-fold cross-validation on the 10,446 approximate residual samples provides an internal estimate of generalization under typical operating variability. The mean and standard deviation of the error metrics across folds for the approximate residual dataset is reported in Table 3. Hyperparameters were selected via cross-validation on the approximate residual dataset, and the final configuration used in all reported results is summarized in Table 4. In addition to global metrics, fold-wise diagnostics are useful to verify that performance is stable and not driven by a single favourable split.

Table 3. Five-fold cross-validation results on the approximate residual dataset.

Model

CV RMSE

mathematical formula mean mathematical formula std mathematical formula

CV MAE

mathematical formula mean mathematical formula std mathematical formula

CV Bias

mathematical formula mean mathematical formula std mathematical formula

CV R-squared

mathematical formula mean mathematical formula std mathematical formula

Residual model mathematical formula

XGBoost

mm

mm

mm

Table 4. XGBoost optimized hyperparameters used for the residual model.

Parameter | Value
Number of boosting rounds | 264
Maximum tree depth | 7
Shrinkage factor | 0.0242
Row sampling ratio | 0.7192
Feature sampling ratio | 0.8139
Minimum child weight | 3
Minimum split loss | 0.3885
tL1 regularization | 0.1381
tL2 regularization | 6.7886

After model selection, the residual regressor is retrained on the full approximate-residual dataset. The residual model r^(x) is then evaluated on the seven matched cases that provide true residuals, being these cases simulations that exactly match experimental conditions in seven different configurations. Despite the limited sample size of seven points, this external validation is significant because it
evaluates the model's correction against true residuals derived from matched simulation-experiment pairs, rather than surrogate-based approximations. Figure 2 illustrates the relationship between real and predicted residuals for these matched cases, alongside the approximate residuals used during training.

[picture]

Fig. 2. Predicted residual r^ versus residual targets: production (surrogate-based) and anchored validation.

Fig. 2. Predicted residual r ^ versus residual targets: production (surrogate-based) and anchored validation.

Figure 2 provides a compact sanity check of whether a residual model trained on surrogate-based production residuals can remain consistent when confronted with matched (anchored) conditions. The grey cloud shows that the XGBoost model learns a near one-to-one mapping between predicted residuals r^ and the approximate targets rapprox  over the in-domain production dataset, with limited systematic deviation from the ideal 1:1 line. Importantly, the seven reserved anchored validation points (blue markers), which were not used during training or hyperparameter tuning, lie close to the same trend and yield strong agreement ( R2≈0.92,RMSE≈2.16 mm, MAE ≈1.88 mm ). Although the anchored set is small and therefore not intended as a statistically exhaustive validation, its alignment with the learned mapping supports the physical plausibility of the correction and suggests that learning on approximate residuals can transfer to matched conditions within the validated domain.To further see the hybrid model use improvement, the hybrid predictor y^hyb  is then evaluated on the reserved matched cases that provide true residuals.
Figure 3 contrasts the error distributions of the baseline numerical surrogate and the hybrid predictor within the validated domain Ω. On the full in-domain dataset ( n=10,453 ), the baseline shows a clear systematic offset ( μ≈7.60 mm ) and broad dispersion ( σ≈6.35 mm ), whereas the hybrid error is effectively centered ( μ≈0.00 mm ) and substantially narrower ( σ≈1.38 mm ), indicating that the learned discrepancy term mainly acts as a bias compensator and variance reducer under the operating regime covered by the data. Importantly, the same trend is observed on the reserved matched
validation set ( n=7 ), where the mean error shifts from μ≈5.79 mm (baseline) to μ≈−0.33 mm (hybrid) and the dispersion decreases from σ≈8.35 mm to σ≈2.30 mm. While the "all-data" distribution can be optimistic if it includes samples used during residual training, the matched-case statistics provide an external, physically grounded sanity check that the correction does not collapse when evaluated under matched conditions. Residual tails and any remaining multimodality suggest that multiple operating regimes and/or unmodelled or bad-modelled factors may still be present, reinforcing the need for domain gating and continued expansion/curation of the matched-case set. Residual tails suggest that the process may still switch between operating regimes and that some effects are not fully captured (temperature modeling being a possible contributor). This supports a practical deployment view in which temperature could be handled as a disturbance.

[picture]

Fig. 3. Qualitative effect of residual correction on prediction error.

Fig. 3. Qualitative effect of residual correction on prediction error.

A key consideration is that the discrepancy model is trained on surrogate-based residuals rapprox =yEXP−y^FEM, and therefore inherits part of the uncertainty of the numerical surrogate (here, the GPR emulator of the FEM model). Consequently, the learned correction reflects both simulationproduction mismatch and any residual surrogate error within the validated domain. A further limitation of the present study is the small number of matched simulation-experiment cases available for the external anchored check. While the observed agreement is encouraging, a larger matched set would enable a more reliable quantification of out-of-sample performance and more systematic sensitivity analyses. To address this, Table 5 reports a sensitivity study comparing three prediction strategies across the in-support ( 10,446 samples) and out-of-support ( 59,554 samples) regions. The kNN-filtered model achieves an 85.3% RMSE reduction in-support (from 10.68 mm to 1.54 mm ). Training on all 70,000 points without the kNN filter yields only 79.4% improvement (from 10.68 mm to 2.20 mm ), observing that out-of-support approximate residuals can contaminate the model.

Table 5. Sensitivity analysis of kNN support filtering: Root Mean Squared Error (RMSE) in mm of draw-in prediction for in-support and out-of-support production measurements under four prediction strategies.

Region
Samples

RMSE Surrogate-

only [mm]

RMSE Filtered-train

(no gate) [mm]

RMSE Unfiltered-train

(no gate) [mm]

In-Support | 10,446 | 10.68 | 1.54 | 2.20
Out-of- | 59,554 | 10.87 | 6.29 | 2.22
Support |  |  |  |

The proposed pipeline remains attractive in practice because it provides an explicit and conservative validity statement, and it offers a scalable path to progressively strengthen the hybrid predictor as additional matched cases become available.

Conclusions

The original version of this paper is available on https://www.scientific.net/KEM.1050.103.pdf

An approximate-residual hybrid metamodel was formulated for an industrial stamping process to reconcile numerical coverage with production fidelity under limited matched-case availability. The approach introduces an explicit, reproducible validity statement by restricting residual construction and learning to the numerical DoE support, optionally refined through multivariate support checks in standardized space. Within this validated domain, a fast numerical surrogate y^FEM enables the computation of large-scale approximate residual targets from production data, and a residual regressor r^ is selected by five-fold cross-validation on 10,446 in-domain samples and then retrained on the full set for deployment.
The resulting hybrid predictor y^hyb (x)=y^FEM (x)+r^(x) provides an experiment-informed correction while retaining the broad sampling of the numerical DoE. This enables the generation of synthetic corrected datasets that can support downstream modelling tasks that require coverage beyond the narrow production operating window. An external check based on seven matched cases provides a conservative sanity test of physical consistency; expanding the matched-case set remains a priority to strengthen external validation and to refine domain-gating strategies in out-of-support regions.

Acknowledgements

The original version of this paper is available on https://www.scientific.net/KEM.1050.103.pdf

The authors acknowledge that the data used in this study were generated within the framework of the iStamp project (Grant Agreement IDI-20220061) under SMART Eureka program, funded by the Centre for the Development of Industrial Technology (CDTI) of the Spanish Ministry of Science and Innovation. The analyses, interpretations, and conclusions presented in this paper were developed after the completion of the project.

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The original version of this paper is available on https://www.scientific.net/KEM.1050.103.pdf

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T. Chen and C. Guestrin, "XGBoost: A Scalable Tree Boosting System," in Proceedings of the 22nd ACM SIGKDD Int. Conf. Knowl. Discov. Data Min. (KDD '16), 2016, pp. 785-794, doi: 10.1145/2939672.2939785.

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Key Engineering Materials, ISSN: 1662-9795, Vol. 1050, pp 113-122

doi: 10.4028/p-Y3cBRq

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-19

Revised:
2026-02-21

Accepted:
2026-02-21

Online:
2026-04-16

Robust and Feature-Aware Arbritary Lagrangian-Eulerian Method for Material Forming Applications

The original version of this paper is available on https://www.scientific.net/KEM.1050.113.pdf

Garcia C. Jesus O. 1,a*, Alves Z. José R. 1,b, Ripert Ugo 1,c, Beraudo Christine 1,d

1 Transvalor S.A. E-Golf Park, 06410 Biot, France

ajesus.garcia@transvalor.com, bjose.alves@trasvalor.com, c ugo.ripert@transvalor.com,

d  christine.beraudo@transvalor.com

Keywords: ALE Method, Material Forming, Mesh Adaptation, FSW.

Abstract

Accurate simulation of material forming requires managing severe mesh distortions to preserve the geometry of the workpiece. Classical Lagrangian descriptions often become computationally expensive under such conditions. The Arbitrary Lagrangian-Eulerian (ALE) method offers a robust alternative by combining the strengths of Lagrangian and Eulerian descriptions, thereby improving computational efficiency and numerical stability. However, a major challenge remains in accurately handling free surfaces to maintain the geometric fidelity of the workpiece. This work introduces a new ALE-based approach to address this limitation. An analytical case as well as a Friction Stir Welding (FSW) case will be presented to demonstrate its effectiveness.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1050.113.pdf

In continuum mechanics, the description of motion requires specifying the reference with respect to which the body is observed. To this end, it is important to distinguish between the different configurations that can be associated with the body.
Three configurations are commonly considered:

Material (reference) configuration (Rx*) : Represents the body in its original, undeformed state; each particle is labeled by its reference coordinate ( x* ).

Spatial (current) configuration (Rx) : Represents the body in its current, deformed state; positions of material points are given by ( x ).

Referential (mesh) configuration (Rχ) : Represents the computational mesh; nodes at (χ) can move independently of the material.
These configurations provide the geometric framework for expressing the Lagrangian, Eulerian, and Arbitrary Lagrangian-Eulerian (ALE) descriptions of motion respectively.

Lagrangian description: In this approach, each mesh node moves with the material particle to which it is permanently attached. Consequently, the mesh velocity ( v→mesh  ) coincides with the material velocity ( v→mat ), which is used to update the particle positions as:

v→mesh=v→mat(x*,t)=∂x∂t|x*(1)

In this description, the boundaries of the domain are naturally tracked by the mesh. Fig. 1a illustrates the evolution of a Lagrangian mesh MLAG  over a time step Δt. The Lagrangian description is particularly well suited for materials with history-dependent constitutive laws and is therefore widely used in structural and solid mechanics. However, it becomes challenging in the presence of large deformations, as severe mesh distortions may occur, often requiring frequent and complex remeshing procedures.

Eulerian description: In contrast to the Lagrangian description, the Eulerian approach is less affected by mesh distortion problems, as the mesh does not deform with the material. In this approach, the computational mesh remains fixed in space, while the material domain moves through it. The
material velocity ( v→mat  ) is therefore expressed with respect to the fixed mesh and does not reference the initial material coordinates x* :

v→mesh =0;v→mat =v→mat (x,t)(2)

This description enables the simulation of problems involving large deformations without mesh degradation. However, it introduces convective effects due to the relative motion between the material and the mesh. While the Eulerian approach is widely used in fluid dynamics, it presents challenges when applied to solid mechanics, particularly in tracking deformable free surfaces and material interfaces in unconfined flows. As illustrated in the Fig. 1b, the boundaries of the continuum do not necessarily coincide with the Eulerian mesh MEUL .

ALE description: The ALE approach combines key features of both the Lagrangian and Eulerian descriptions. Neither the material configuration Rx* nor the spatial configuration Rx is taken as the sole reference. Mesh nodes may move with the material, as in a classical Lagrangian description; remain fixed, as in a Eulerian approach; or move arbitrarily, allowing for continuous rezoning of the mesh. The mesh velocity ( v→mesh  ) is therefore defined by:

v→mesh=∂x∂t|χ(3)

This flexibility enables the ALE method to accommodate large distortions while still accurately tracking domain boundaries. However, it also inherits computational challenges from both Lagrangian and Eulerian descriptions, particularly in the treatment of convective terms and in the implementation of robust mesh-update procedures. Fig. 1c illustrates the evolution of an ALE mesh MALE  over a time step Δt.

[picture]

Fig. 1. a) Lagrangian description, b) Eulerian description, c) ALE description [2]

Fig. 1. a) Lagrangian description, b) Eulerian description, c) ALE description [2]

Convective transport: After mesh regularization, field variables must be transported from the original mesh to the updated one. This is commonly achieved using a convective transport approach. In the ALE description, since the reference domain differs from the material domain, the time derivative of a field (dζdt) involves both a time grid derivative (dgζdt) and a convective term (c→·∇→ζ) :

dζdt=dgζdt+c→·∇→ζ(4)

Where c→ is the convective velocity and it is defined as.

c→=v→mat−v→mesh(5)

The field update in the ALE formulation is then expressed as:

ζALEt+Δt=ζreft+∫Δtdgζdt(6)

With

dgζdt=dζdt−c→·∇→ζ(7)

Linearizing the above expression and using a finite difference scheme, the updated field can be written as:

ζALEt+Δt=ζLAGt+Δt−∇→ζLAGt+Δt·c→Δt(8)

Arbitrary Lagrangian-Eulerian Method

The original version of this paper is available on https://www.scientific.net/KEM.1050.113.pdf

Two main ALE strategies are commonly used:

A fully coupled approach, which directly solves convection-diffusion equations.

An uncoupled approach, which solves the global system's resolution separated for convection.

In this work, the uncoupled approach, better suited for Lagrangian-based finite element codes, is adopted. It consists of computing a mesh velocity to regularize the mesh, followed by a remapping step between the original and the adapted meshes. This strategy allows the global mechanical resolution to be decoupled from mesh adaptation [1,2].

Mesh Velocity Computation: The ALE method can also be viewed as an r-adaptation technique, where mesh connectivity is preserved and only node positions are updated. This makes it significantly faster than full remeshing approaches.

Mesh velocity can be computed using either:

Explicit methods, such as centering techniques that average neighboring node information.

Implicit methods, which minimize quantities such as the gradient of mesh velocity.

In this work, an implicit approach is adopted, allowing the mesh velocity to be computed globally across the entire mesh. The mesh displacement ( u→ ) is computed first, from which the mesh velocity is subsequently obtained.
Mesh displacements are computed using a weak formulation of linear elasticity expressed in terms of the displacement field. Dirichlet boundary conditions are applied in regions defined as purely Eulerian within the mesh.

−∇→·σ(u→)=f→ in Ωu→=0 on ∂ΩD(9)

Where σ is the Cauchy stress tensor and f→ is an external load applied to Ω.
Transport: To handle the remapping step, advantages and limitations of the convective method, expressed in equation (8), are taken into account. The main difficulty of this method arises when computing the gradient ( ∇ξt+Δt ).
A common approach involves identifying, for each point, the corresponding upstream element based on the mesh velocity at the current time step. This is typically performed using an upwind method (Fig. 2a), in which the search direction is determined by the convective velocity. However, such
element-tracking procedures can be computationally expensive and challenging to implement efficiently in a distributed parallel framework.
An alternative and robust strategy consists of enriching the fields by projecting them from Gauss points to nodes and then back to super-convergent integration points (Gauss points), where the gradient can subsequently be computed elementwise. While this approach simplifies the search process, it introduces significant numerical diffusion in the transported fields [2].

To overcome these issues, the present work proposes a strategy that combines the strengths of different approaches to preserve both accuracy and computational efficiency. First, the field ξ is evaluated on the current mesh configuration at the Gauss-points level and then recovered on the nodes, thereby constructing a continuous P1 field, this P1 field is subsequently re-interpolated at the P1 Gauss points of the elements.
At this stage, a continuous field distribution is available and can be trasported to the P1 Gauss points of the updated mesh. This is achieved by localizing each Gauss point of the new mesh within the previous mesh using a logarithmic search algorithm (octrees are employed in this work), followed by interpolation using the local shape functions of the reference element [3] (see Fig. 2b).
In other words, directional searches are replaced by a straightforward point-wise localization procedure, since the direction of convection is already embedded in the previously enriched field.

[picture]

Fig. 2. a) upwind element [2], b) interpolation strategy, current element (green), updated element (red)

Fig. 2. a) upwind element [2], b) interpolation strategy, current element (green), updated element (red)

To prove the pertinence of this choice an analytical test is performed, which is shown in Fig. 3, this demonstrates successful convection of the analytical field. Although some numerical diffusion is observed, the initial field is still well preserved. Differences between fields stored at nodes and at Gauss points are also visible, reflecting the richer spatial sampling provided by using four Gauss points per element, which enhances transport accuracy between both meshes.

[picture]

Fig. 3. Transport test using the interpolation strategy for nodes and elements stored fields

Fig. 3. Transport test using the interpolation strategy for nodes and elements stored fields

Dealing with Free Surfaces:

The original version of this paper is available on https://www.scientific.net/KEM.1050.113.pdf

One of the main challenges in ALE methods is the accurate treatment of free surfaces, which can otherwise lead to loss of volume and distortion of the geometry [4-7]. To address this issue, a new and robust approach has been implemented.

Special attention is given to surface elements in order to accurately preserve the geometric shape. This requires proper identification of the local geometric characteristics of each surface node, which is determined by analyzing the patch of element faces surrounding the node. This analysis provides
information about whether the node belongs to a planar or curved surface, an edge, or a corner. To achieve this, a voting normals technique is employed [8]. The resulting normals characterize the local geometry: one normal indicates a node located on a planar or curved surface, two normals identify an edge node, and three normals correspond to a corner node, as illustrated in Fig. 4.

[picture]

Fig. 4. Voting normals: a) 1 normal: continuous surface, b) 2 normals: edge feature between two surfaces, c) 3 normals: corner.

Fig. 4. Voting normals: a) 1 normal: continuous surface, b) 2 normals: edge feature between two surfaces, c) 3 normals: corner.

Once the surface normals are identified, they are used to construct a transformation matrix at each surface node. This matrix is then applied locally to rotate the stiffness matrix of the linear elasticity problem. This formulation preserves the geometric shape by enforcing displacement constraints consistent with the nature of the free surface. Because the transformation matrix is built from the oriented normals, each of the previously identified cases must be treated separately.

For a node associated with a single normal (surface node), displacement is constrained in the normal direction by imposing a Dirichlet boundary condition in that direction, while displacements along the two tangential directions remain allowed. For an edge node, displacement is permitted only along the tangential direction, and the other two directions are constrained. For a corner node, displacements are constrained in all three directions. These cases are illustrated in Fig. 5.

[picture]

Fig. 5. a) Normals on a mesh, b) Degrees of freedom applied in each direction according to the number of normals

Fig. 5. a) Normals on a mesh, b) Degrees of freedom applied in each direction according to the number of normals

Analytical case

The original version of this paper is available on https://www.scientific.net/KEM.1050.113.pdf

To validate the proposed method, an analytical test case was performed. The objective is to assess the ability of the approach to preserve the geometry of the workpiece while performing mesh regularization.

In this test, the mesh is deformed according to the analytical function, as illustrated in Fig. 6a. Both ends of the geometry move to the right, while a wave is generated in the central region. The goal is to let the wave propagate while keeping the extremities fixed, allowing the ALE approach to handle
mesh regularization in the remaining part of the geometry. The corresponding boundary conditions are applied as shown in Fig. 6b.

[picture]

Fig. 6. Analytical case: a) analytical function at the initial ( t0 ) and at time step n(tn), b) Boundary conditions applied in then ALE approach

Fig. 6. Analytical case: a) analytical function at the initial ( t 0 ) and at time step n ( t n ) , b) Boundary conditions applied in then ALE approach

The proposed methodology was applied, and the resulting displacement field exhibits very promising behavior. In particular, the computed displacements correctly follow the evolution of the free surfaces, as shown in Fig. 7.

[picture]

Fig. 7. ALE displacement field on the mesh.

Fig. 7. ALE displacement field on the mesh.

A comparison between the Lagrangian solution (blue) and the ALE solution (red), illustrated in Fig. 8, demonstrates excellent agreement at this stage of the simulation. The wave shape is well preserved in the ALE mesh and closely matches the Lagrangian solution. This confirms the effectiveness of the proposed method in preserving free surfaces while maintaining mesh quality and delaying the need for remeshing.

[picture]

Fig. 8. Comparison of the Lagrangian (blue) and ALE (red) mesh configurations at the initial ( t0 ) and at time step n(tn)

Fig. 8. Comparison of the Lagrangian (blue) and ALE (red) mesh configurations at the initial ( t 0 ) and at time step n ( t n )

Application to the Friction Stir Welding (FSW) Process

The original version of this paper is available on https://www.scientific.net/KEM.1050.113.pdf

Friction Stir Welding (FSW) converts mechanical energy into heat and material deformation in order to create a weld. In this process, the parent material does not melt; joining occurs in the solid state, as illustrated in Fig. 10a, thereby avoiding potential metallurgical issues such as porosity, cracking, or undesirable microstructural changes
The test case is taken from Fourment and Guerdouxs work [9]. The FSW process is modeled using a single-plate configuration, without explicitly representing the interface between the welded plates (Fig. 9b). The tool is assumed to be stationary (Fig. 9c), while motion is imposed on the plate, as shown in Fig. 10a.
In this model, motion is imposed on the plate, making it the only moving component. As in the analytical case, the objective of the ALE approach is to prevent the rigid translation of the workpiece to the left, thereby keeping the geometry fixed during the simulation. To achieve this, appropriate boundary conditions are applied (see Fig. 10b), consisting of a displacement imposed in the direction opposite to that of the kinematic model. This ensures zero net displacement of the geometry, while the ALE formulation accommodates the relative motion within the remaining part of the mesh.

[picture]

Fig. 9 a) FSW scheme process, b) simulated plate and tool, c) tool design.

Fig. 9. a) FSW scheme process, b) simulated plate and tool, c) tool design.

[picture]

Fig. 9 a) model's kinematics, b) boundary conditions for ALE

Fig. 9. a) model's kinematics, b) boundary conditions for ALE

The proposed ALE approach is compared with a classical Lagrangian formulation and with a previously developed ALE method. Results are displayed in Fig. 11, the former ALE approach exhibits excessive mesh smoothing, leading to poor preservation of free surfaces and the loss of the trailing edge during the welding process. In contrast, the proposed ALE method accurately preserves the workpiece geometry, yielding results that closely match the Lagrangian solution.

[picture]

Fig. 10 Comparation of different methods at 1,3 and 10 seconds of process simulation

Fig. 10. Comparation of different methods at 1,3 and 10 seconds of process simulation

Fig. 12 displays a direct comparison between the Lagrangian mesh (mesh lines) and the proposed ALE mesh (in magenta) shows an excellent geometric agreement, with only a minor volume loss observed in the ALE case.

[picture]

Fig. 11 Lagrangian (mesh lines) and ALE (in magenta) meshes comparison.

Fig. 11. Lagrangian (mesh lines) and ALE (in magenta) meshes comparison.

[picture]

Fig. 11 Lagrangian (mesh lines) and ALE (in magenta) meshes comparison.

Fig. 11. Lagrangian (mesh lines) and ALE (in magenta) meshes comparison.

Additionally, temperature and equivalent strain fields results are also displayed in Fig. 13 and shown a very good agreement between the two simulations. Equivalent strain presents a small difference possible due to a small diffusion on the remapping procedure

[picture]

Fig. 12 Temperature and equivalent strain comparison

Fig. 12. Temperature and equivalent strain comparison

Summary

The original version of this paper is available on https://www.scientific.net/KEM.1050.113.pdf

A new ALE method for material forming applications has been developed, featuring a robust strategy for preserving workpiece geometry based on voting normals and local matrix transformations at free-surface nodes. The method has demonstrated its effectiveness through both an analytical test case and an industrial-scale application involving Friction Stir Welding (FSW). In both cases, the proposed approach successfully preserves geometric integrity while maintaining good accuracy in the simulation results.

References

The original version of this paper is available on https://www.scientific.net/KEM.1050.113.pdf

M. Bellet and V. Fachinotti, "ALE method for solidification modelling," Comput. Methods Appl. Mech. Eng., vol. 193, no. 39-41, pp. 4355-4381, 2004, doi: 10.1016/j.cma.2003.11.016.

S. Guerdoux, "Numerical simulation of the friction stir welding process", Ph.D. dissertation, Ecole des Mines de Paris, France, 2007.

H. Eldahshan, P.-O. Bouchard, J. Alves, E. Perchat, and D. Pino Munoz, "Phase field modeling of ductile fracture at large plastic strains using adaptive isotropic remeshing," Comput. Mech., vol. 67, pp. 763-783, 2021, doi: 10.1007/s00466-020-01962-7

S. Philippe, "Development of an ALE formulation for 3D rolling simulation" Ph.D. dissertation, Ecole National Supérieur des Mines de Paris, France, 2009.

S. Philippe, L. Fourment, and P. Montmitonnet, "Application of the arbitrary LagrangianEulerian formulation to the numerical simulation of stationary forming processes with dominant tangential material motion," in Proc. 12th Int. Conf. Metal Forming, Cracow, Poland, Sep. 2008, pp. 571-578.

M. Hachani, "Improved contact treatment for metal forming processes with reduced contact area", Ph.D. dissertation, Ecole National Supérieur des Mines de Paris, France, 2011.

S. Gastebois, "Numerical simulation of FSW welding using an ALE formulation", Ph.D. dissertation, Ecole National Supérieur des Mines de Paris, France, 2015.

G. G. Medioni, M. Lee, and C. Tang, A Computational Framework for Segmentation and Grouping. Amsterdam, The Netherlands: Elsevier Science, 2000.

L. Fourment and S. Guerdoux, "3D numerical simulation of the three stages of friction stir welding based on friction parameters calibration," Int. J. Mater. Form., vol. 1, suppl. 1, pp. 1287-1290, 2008, doi: 10.1007/s12289-008-0138-5.

This ePub file was created using machine-generated methods and automated software to convert the source text into an ePub format and may contain formatting inconsistencies from the original version. Formatting may vary depending on the device used.

The original version of this eBook is available on https://www.scientific.net/book/advanced-numerical-and-ai-strategies-for-material-forming/978-3-0364-1986-2

Key Engineering Materials, ISSN: 1662-9795, Vol. 1050, pp 123-136

doi: 10.4028/p-Us6DNo

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-19

Revised:
2026-03-02

Accepted:
2026-03-02

Online:
2026-04-16

On-Line Prediction of Dry Zones During Composite Process Through Digital Shadow Approach

The original version of this paper is available on https://www.scientific.net/KEM.1050.123.pdf

LE BOT Philippe 1,a*, LAUZERAL Nathan 1, b, FOUCHE Olivier 1,c, SIDDIG Nihad 1, d, LECOINTE Damien 1,e, ABDULLAH Ibrahim 1,f, NIGET Florent 1, g and MARCHAND Christophe 1, h

1 Nantes Université, IRT Jules Verne, 1 mail des 20000 lieues, 44340 BOUGUENAIS

aP  Philippe.le-bot@irt-jules-verne.fr (*corresponding author), b  nathan.lauzeral@irt-jules-verne.fr,

c  Olivier.fouche@irt-jules-verne.fr, d  nihad.siddig@irt-jules-verne.fr,

e  damien.lecointe@irt-jules-verne.fr, ibrahim.abdullah@irt-jules-verne.fr,

qflorent.niget@irt-jules-verne.fr, h  christophe.marchand @ irt-jules-verne.fr 

Abstract

A novel solution for monitoring the infusion process and providing decision support to operators involved in the manufacturing of large, unique or near-unique parts is presented. Based on a scientific approach referred to as the 5D methodology (D for dimensions), the proposed solution consists of a process digital shadow built upon a metamodel that is fed in real time by signals from sensors embedded in the process, enabling the anticipation of defects such as dry spots.

Keywords Process monitoring, digital shadow, composites.

[picture]

Fig. 1. Infusion of part - IRT Jules Verne - Project MONOCLE

Fig. 1. Infusion of part - IRT Jules Verne - Project MONOCLE

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1050.123.pdf

Composite parts are becoming increasingly important in modern industrial structures. Forming processes, such as infusion, must enable the manufacture of parts with ever-higher mechanical performance. At the same time, the parts manufactured are both large and increasingly complex in shape and architecture. However, the appearance of defects such as "dry areas" (areas where the resin has not been able to impregnate the fiber reinforcement) significantly reduces the final properties of the part, which is unacceptable in many industrial fields. However, these dry areas are sometimes unpredictable (they appear randomly) or invisible to the operator. Monitoring the infusion process therefore appears to be an essential solution. There are many solutions available, based on sensors and industrial solutions[1-4] or more academic ones [5-7], but these solutions are descriptive, rarely predictive, and even less prescriptive and often based on large productions.

Our study deals with the infusion of (quasi) unique composite parts, with the aim of getting it right first time. Digital shadow tools, powered by a process metamodel, appear to be a particularly suitable solution. The metamodel integrated into the digital twin is used to describe the state of the system (machine, tooling, preform, resin) at time t, to predict the state of the same system at future times
t+Δt, and finally to prescribe process adjustment modifications to improve the state of the system at that same future time t+Δt.

The following sections describe the various stages involved in constructing this digital shadow to limit dry areas: the overall method (known as the 5D approach), followed by each of its dimensions: Physical entities, virtual models, Digital shadow data, Services and Connections. Finally, the initial results are discussed.

Procedure

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Data management in the context of digital twins is a major challenge due to the lack of formalization, models, or conceptualization of the data lifecycle, despite growing interest from manufacturers. To address this challenge, A data lifecycle model and a data typology from academic literature [8] coupled with normative literature [10] have been used in this study to comprehensively characterize the different stages of data management. The resulting formalization of data management ensures the quality, consistency, and reliability of the data used in the various digital twin applications. It also makes it possible to scale the digital twin architecture in terms of resources (storage, computing power, transmission frequencies, etc.).

MDT=(PE,VE,SS,DD,CN)

where PE is a physical entity that exists in the real world. This includes tools, materials, sensors, and data acquisition tools. VE is the virtual dimension and contains all knowledge elements and models, including digital process simulation tools. Ss is a service provided by both PE and VE, DD is data that make up the digital twin and CN are the links between the different parts of the digital twin [9].

[picture]

Fig. 2. Five-dimension digital twin mode [8]

Fig. 2. Five-dimension digital twin mode [8]

This approach to the management of information flows between process-generated data and data derived from knowledge modeling enables the real-time updating of a metamodel through highperformance communication protocols. The service-oriented paradigm facilitates a focus on industrial objectives and is computationally formalized through the client-server architecture. In the following paragraphs, each of the five dimensions of the problem is explained in detail in order to arrive at a solution for a digital shadow of the infusion process.

Dimension#1: Physical entities

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1. Labscale testbench

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An infusion test bench (Figure 3) has been specially designed to analyze infusions on a laboratory scale. This bench can be used to evaluate sensors and infusion configurations, on plates or tools, as well as to test digital shadow solutions on a small scale. The expected functions of the test bench were determined by a functional analysis carried out by the consortium.

The infusion test bench is a mechanically welded and assembled piece of equipment used to study the composite infusion process for different part sizes and geometries. It consists of two assemblies:

A main section that is mobile on four wheels and held in place by four foot-operated jacks.

A mobile table on four wheels, held in place on the main equipment by two clamps.

A gantry moves along three axes, each controlled by an IGUS DRYVE D1 control card. A fixture is located at the end of the z-axis. The gantry is height-adjustable by means of a device with four LINAK electric cylinders controlled by a maintained action on the control devices provided for this purpose.

Several configurations are possible and were used during the project (Figure 4):

Configuration 1: draping the part directly on the transparent marble slab.

Configuration 2: tools directly on the transparent marble slab.

Configuration 3: tools inside the marble.

Configuration 4: tools near the marble.

[picture]

Fig. 3. Overview of the test bench in its environment in two different applications (tools on the marble surface on the left and tools on the side on the right)

Fig. 3. Overview of the test bench in its environment in two different applications (tools on the marble surface on the left and tools on the side on the right)

2. Process sensors

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The proposed solution for achieving right-first-time production of large composite parts relies on the deployment of multiple sensors to accurately characterize the process signature, which is captured and interpreted by a metamodel operating concurrently with the manufacturing process. Sensors are used either as inputs (e.g., model boundary conditions), outputs (verification of model relevance), or as process business indicators. There are many sensors available to perform these functions for the infusion process. In this study, a sensor selection procedure based on experimental campaigns conducted on the labscale test bench was implemented. The sensors ultimately selected, based on criteria related to the sensitivity of the signals obtained, the possibility of performing automated analyses, and compatibility with the data acquisition system, are as follows: a camera (in the visible spectrum), a resin flowmeter, a vacuum sensor, and a heat flux and temperature sensor placed on the vacuum bag. Other sensors were tested but had major drawbacks in terms of one of the selection criteria.

- Sensor #1: Camera (a2A5328-4gmBAS, BASLER)

The original version of this paper is available on https://www.scientific.net/KEM.1050.123.pdf

Its characteristics and mode of installation allow images to be acquired across an entire 6 m2 room with resolution of 1,5 mm/ pixel. The various tests carried out throughout the project demonstrated the importance of lighting in obtaining high-quality images that enable automatic segmentation of the resin front. Firstly, the images acquired must be uniform, with no areas that are overexposed or underexposed. Finally, the brightness must not vary during acquisition; tests must be carried out under controlled lighting conditions, away from sunlight. Specific work on lighting was therefore carried out. A concrete result from a plate infusion test is shown on the Figure 4.

[picture]

Fig. 4. Infusion test of a plate with camera and lighting - experimental setup (left) and result of the infusion in progress (right)

Fig. 4. Infusion test of a plate with camera and lighting - experimental setup (left) and result of the infusion in progress (right)

Significant effort was devoted to intrinsic and extrinsic calibration procedures aimed at correcting image distortions and mapping the resulting data onto CAD models of the part being infused. This work was first validated on a flat plate and subsequently on a 6 m2 simple-curved component (as proof of concept), under both laboratory and industrial lighting conditions.

The metamodels developed in this work do not directly exploit image data but rather resin front position data as a function of time. Consequently, an image segmentation algorithm was developed and deployed. The exploitation of the image flow obtained requires the use of deep learning to perform robust and adaptive segmentation of the resin front, without excessive dependence on experimental conditions (presence of shadows or unexpected objects). The DeepLabV3 model [11] was chosen for its ability to perform semantic segmentation. Given the limited data available to serve as a dataset for the model, we had to enrich this base with a set of synthetic data obtained artificially. The result is a highly effective segmentation algorithm that is robust under industrial conditions that are sometimes difficult to control. Subsequently, a real-time projection algorithm for the resin front obtained on a mesh of the part was developed. Figure 5 shows a concrete example of typical result of using a real-time camera during infusion on a 6 m2 test piece with a single curvature.

[picture]

Fig. 5. Segmented resin front during infusion of the 6 m2 part (proof of concept)

Fig. 5. Segmented resin front during infusion of the 6 m 2 part (proof of concept)

- Sensor #2: SDX20 flowmeter (Keyence)

The original version of this paper is available on https://www.scientific.net/KEM.1050.123.pdf

This sensor, mounted externally on the main resin supply hose, measures the volumetric flow rate of resin delivered to the preform. This measurement is used both as a validation criterion for numerical simulations and as an input for improving the metamodel and the resulting digital shadow. The two peaks in Figure 6 correspond to the openings of two ramps used during a plate infusion test, while an unexpected return to 0 at 700s corresponds to a signal loss due to too low a flow rate. A correction was systematically applied to remedy this problem and was used in all our tests.

[picture]

Fig. 6. Flowrate measurement during infusion process

Fig. 6. Flowrate measurement during infusion process

- Sensor #3: Vacuum pressure sensor

The original version of this paper is available on https://www.scientific.net/KEM.1050.123.pdf

These sensors measure pressure (or vacuum) via a ceramic membrane. Based on an initial commercial solution, the vacuum sensor has been modified so that the sensor membrane never comes into contact with environmental products or fiber reinforcement. A typical result is shown in Figure 7.

[picture]

Fig. 7. Typical measurement of vacuum sensor during infusion process.

Fig. 7. Typical measurement of vacuum sensor during infusion process.

- Sensor #4: Heat flux and temperature sensor - Tfx©

The original version of this paper is available on https://www.scientific.net/KEM.1050.123.pdf

The company Tfx has long developed a specific range of enthalpy heat flow sensors for Resin Transfer Molding and has more recently adapted this range of sensors for infusion [5]. Resin front detection is possible thanks to the change in thermal conductivity that occurs when the reinforcement is wetted. The infusion stage itself is often isothermal, so it is necessary to introduce a slight thermal gradient ( 1 to 2∘C ) across the composite to make the system sensitive to the wetting itself. It should be noted that this sensor does not measure "true" values but rather a quantity of the same unit and sensitivity. Nevertheless, the observable raw value is not the absolute value of the heat flow through the sensor but is a faithful representation of it. Figure 8 shows the results of a plate infusion test, for which two heat flux sensors are placed close to and far from the infusion point, respectively. The plates are made from a stack of plies with a total thickness of 10 mm . The resin is infused at 28∘C, while the ambient temperature is 23∘C. The infusion profile is classic, and the sensor near the infusion point detects the resin passing less than two minutes after the start of infusion. During the impregnation phase, there is a pronounced downward phase (−3000 W/m2) if the sensor is located near the infusion point, and only −500 W/m2 if it is located far away, followed by a return to zero. Just before passing in front of the sensor, the resin heats it up (increasingly negative flows) until it reaches thermal equilibrium with the resin/preform system (flow returns to zero). The resin then begins to polymerize, and the flow returns to negative values.

[picture]

Fig. 8. Heat flux profile during plate infusion - Focus on the impregnation phase (left) and curing phase (right)

Fig. 8. Heat flux profile during plate infusion - Focus on the impregnation phase (left) and curing phase (right)

This example demonstrates the value and reliability of these sensors. They not only detect a resin front, but also quantify the progress of polymerization, despite the influence of their position relative to the injection points. They will serve both as performance metrics for the metamodel and as indicators for monitoring the progression of the reaction.

Dimension #2: Virtual models

The original version of this paper is available on https://www.scientific.net/KEM.1050.123.pdf

Since the goal is to have the first part good, there is, by definition, no experimental dataset that can be used to build a metamodel. Consequently, only a synthetic dataset derived from a set of reliable numerical simulations can replace a set of experimental tests. To perform these numerical simulations, several steps were necessary:

1. Reinforcement properties measurements for high-fidelity process simulation

The original version of this paper is available on https://www.scientific.net/KEM.1050.123.pdf

Two complementary approaches were either tested or developed:

Approach 1: a digital methodology for permeability characterization is applied to a biaxial preform. First, the three-dimensional image of the material obtained by micro-computed tomography (micro-CT) was segmented. The complete three-dimensional permeability tensor was then computed using the PoroS software suite, developed by the laboratory GeM at Ecole Centrale de Nantes [14], based on a dual-scale flow simulation employing a Stokes-Brinkman solver.

Approach 2: An experimental approach based on the simultaneous infusion of two plates in two different principal directions, with a set of relevant sensors (including strategically placed cameras), enabled the partial identification of the permeability tensor of the reinforcement under study to be identified using Darcy's equation.
A cross-validation step was successfully completed, enabling these preforms to be characterized with the expected accuracy (Figure 9).

[picture]

Fig.9. Comparison of tests/simulations for infusions using permeability tensor measured using the Jules Verne IRT protocol

Fig.9. Comparison of tests/simulations for infusions using permeability tensor measured using the Jules Verne IRT protocol

2. Resin properties for high-fidelity process simulation

The original version of this paper is available on https://www.scientific.net/KEM.1050.123.pdf

To model the behavior of the resin, a temperature-dependent viscosity law was integrated into the PAM-RTMC software, considering the ambiant temperature and the initial temperatures of resin and reinforcement. Viscosity model, polymerization model and thermal properties-conductivity, heat capacity, and heat of reaction-were entered to enable calculation of the temperature and degree of polymerization during the cycle. These parameters ensure realistic simulation, considering the couplings between flow, heat transfer, and polymerization kinetics in both the filling phase and the polymerization phase.

3. Validation of high-fidelity process simulation accuracy with infusion experiments

The original version of this paper is available on https://www.scientific.net/KEM.1050.123.pdf

All the models and material parameters are implemented in the process simulation tool PAMRTMc. The experimental validation of simulations followed several steps depending on the part physical or geometrical complexities and the infusion strategy. Plates with a mono-thickness are addressed in previous publications [9].

- Infusion of plates

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Numerous cases (reference cases and cases with process variables such as the opening time of the second infusion ramp) were correlated with experiments conducted on the labscale test bench. The flow dynamics predicted by the simulation were systematically compared with the experiment during the filling phase (Figure 10), with good results in terms of accuracy and robustness in each case, but also with acceptable discrepancies explained by modeling imperfections and process variabilities.

[picture]

Fig. 10. Comparison between experimental and numerical advancement of the flow front

Fig. 10. Comparison between experimental and numerical advancement of the flow front

A specific test involving the infusion of a plate using two infusion ramps deals with opening the second infusion ramp at the same time as the first ramp in order to generate a dry zone. Figure 11 shows the experimental infusion test and the dry zones detected at the end of the process. This (non-
industrial) strategy resulted in two triangular dry areas in the first half of the part. A numerical simulation using the material parameters identified above and a strategy identical to the experiment shows dry areas of different sizes but identical positions. The accuracy obtained by the calculation is considered in this work to be sufficient for use in the development of a parametric metamodel.

[picture]

Fig. 11. Process simulation: Correlation between simulation and experiment on plate infusion with inclusion of voluntary dry spots.

Fig. 11. Process simulation: Correlation between simulation and experiment on plate infusion with inclusion of voluntary dry spots.

- Infusion of6m2proof-of-concept part

The original version of this paper is available on https://www.scientific.net/KEM.1050.123.pdf

The proof-of-concept is a piece measuring approximately 6 m2 infused with two infusion points (ramp 1 in Figure 12) and equipped with two "emergency" ramps. The shape of the resin front predicted in the early stages of the experiment shows the accuracy of the simulation. It is this same shape that is identified for the digital shadow. This technical achievement in correlated digital simulation not only validated the expertise acquired in the digital simulation of the infusion process but is also useful as an essential starting point for the production of the synthetic data set required to develop the metamodel for the final digital shadow.

[picture]

Fig.12. Comparison of simulation and experiment on a 6 m2 complex part with single curvature: mesh (left) and results over 1 min and 10 min (right)

Fig.12. Comparison of simulation and experiment on a 6 m 2 complex part with single curvature: mesh (left) and results over 1 min and 10 min (right)

Dimension #3: Digital shadow data

The original version of this paper is available on https://www.scientific.net/KEM.1050.123.pdf

1. Development of the Computational Core

The original version of this paper is available on https://www.scientific.net/KEM.1050.123.pdf

The computational core, hereafter referred to as the metamodel, is based on a deep learning approach and is implemented using the Python library PyTorch. The metamodel is developed with two main objectives:

To enable real-time calibration using sensor data. In the present work, only the data provided by a camera observing the upper surface of the part are considered.

To predict the flow-front dynamics, thereby allowing anticipation of the formation of dry spots.

- Generation of training Data

The original version of this paper is available on https://www.scientific.net/KEM.1050.123.pdf

The metamodel is trained on a database of synthetic data generated using the PAM-RTM© software, presented in the previous Section.

The considered use case is as follows: a first set of infusion points is opened to initiate the filling of the part. At a given time τ, a second set of infusion points is opened. The objective is to be able to predict the effect of this second opening on the formation of dry areas.

The design-of-experiments parameters were selected to accurately represent the flow-front propagation during infusion. Since a single simulation requires several hours of computation, the number of parameters was limited to two to achieve a robust sampling of the parametric space.

The selected parameters are:

The opening time of the second set of injection valves, τ. This parameter is mandatory since the ultimate goal is to optimize this opening time.

The permeability in the direction perpendicular to the main flow, ky. This parameter was identified as having a strong impact on the formation of dry spots. Furthermore, it allows an accurate representation of the flow-front shape visible to the camera. Since the model is calibrated solely from these observations, minimizing the representation error is critical.
Each simulation returns a field defined over 3D space (x,y,z) and time t. A single data sample therefore corresponds to the mapping (x,y,z,t,τ,ky)→r,where r denotes the filling state. The flow front is represented as a binary field: true if resin is present, false otherwise. Both input and output data are normalized before being provided to the neural network. A total of 100 simulations were performed, with design-of-experiments parameters sampled using a Sobol sequence. Among them, 80 simulations are used for training and 20 for validation.

- Architecture and Training

The original version of this paper is available on https://www.scientific.net/KEM.1050.123.pdf

The metamodel is developed using the concept of neural fields [12]. The selected architecture is a Multi-Layer Perceptron coupled with an initial positional encoding layer [13] enabling the capture of high-frequency behaviors.

The Adam optimizer and a binary cross-entropy loss are used to train the model. To assess the quality of the predictions, the Dice coefficient is employed, as the ground truth is binary. Moreover, since the primary objective is the accurate prediction of dry-spot formation, additional metrics such as the true positive rate and the number of false positives per image are also considered to assess the dry area detection.

- Online Calibration

The original version of this paper is available on https://www.scientific.net/KEM.1050.123.pdf

Material parameters may be estimated prior to the experiment; however, the metamodel enables their real-time adjustment during the process. Although multiple sensor measurements are available, the metamodel interacts exclusively with the image provided by the camera, and more specifically with the segmented resin flow front on the upper surface of the part.

It is worth noting that, according to Darcy's law, viscosity can be interpreted as a weighting factor applied to the resin velocity throughout the part, since only a single resin type is used. As viscosity may be temperature-dependent, its correction is important. To this end, viscosity is replaced by a time-scaling parameter, denoted cμ, which effectively accelerates or decelerates the flow. Consequently, this parameter does not need to be included in the design of experiments.

At a given time t, the goal is therefore to estimate ky and cμ based on the segmented flow front observed on the upper surface. The following inverse problem is solved:

(ky^,cμ^)=minky,cμℒinv(fθ(x,y,z,t0,ky,cμ),𝒮(x,y,z,t0)),∀(x,y,z)∈Ωdv

where:

the notation ·^ denotes an estimated parameter,

fθ represents the metamodel,

𝒮(x,y,z,t0) is the measured resin presence obtained from image segmentation,

L Linv is the inverse-problem loss function, chosen here as the Dice coefficient,

t0 is the current time,

Ωdv denotes the discretized spatial domain corresponding to the surface of the part visible by the camera.
Once the material parameters have been identified at time t0, the metamodel can be evaluated at future times t>t0. By varying the opening time τ of the second injection manifold, it becomes possible to anticipate the formation of dry spots and support decision-making during the infusion process. An example of registration is shown Figure 13.

[picture]

Fig.13. Resin flow-front segmentation and metamodel registration (with the camera image in black and white, the segmented resin region shown in green, the prediction associated with the mesh-topixel correspondence in dark blue, and their intersection in blue/green). Note, the metamodel prediction is volumetric and not limited to the surface.

Fig.13. Resin flow-front segmentation and metamodel registration (with the camera image in black and white, the segmented resin region shown in green, the prediction associated with the mesh-topixel correspondence in dark blue, and their intersection in blue/green). Note, the metamodel prediction is volumetric and not limited to the surface.

Dimension#4: Connection

The original version of this paper is available on https://www.scientific.net/KEM.1050.123.pdf

A data acquisition system (Figure 14) was specifically designed for this study so that it could integrate and synchronize the relevant sensors but also embed the metamodel and distribute information to the retrieval software. This system consists of

A sensor data acquisition unit: This unit manages the acquisition of analog sensors in voltage (+/−10 V) and current (4−20 mA) as well as type K thermocouples.

An industrial PC for processing, network management, and storage: The main functions integrated into this unit are

Management of data processing programs to manage:

Resin front monitoring by image analysis (vision module)

The AI model for predicting the infusion filling phase (prediction module)

Management of an OPCUA communication network to operate the connection with the infusion machine

the connection between the various integrated modules (vision module and prediction module)

the storage of infusion data

The two units are linked to each other by the OPCUA exchange protocol based on server/client approach.

[picture]

Fig. 14. IRT Jules Verne data acquisition system

Fig. 14. IRT Jules Verne data acquisition system

Dimension#5: services

The original version of this paper is available on https://www.scientific.net/KEM.1050.123.pdf

The aim of this work is to predict the appearance of dry spots during the process and to enable operators to take appropriate action in order to ultimately prevent them. The concept of service, implemented through a client-server approach to information flow, allows this objective to be addressed in a targeted manner. Commercial software (KASEM©), developed by IQANTO, Figure 15, was used for this purpose. It enables real-time visualization of process parameters and measurements, as well as real-time access to metamodel results. In addition, the metamodel can be configured to predict the appearance or disappearance of dry spots in response to changes in the infusion strategy, within a sufficiently short time frame (approximately ten seconds) to support operator decision-making.

[picture]

Fig. 15. Classical extract of specific Infusion GUI from KASEM©

Fig. 15. Classical extract of specific Infusion GUI from KASEM©

In practical terms, for all validation examples presented hereafter, the infusion strategy provided as input to the metamodel is defined by the opening times of the infusion ramps, which can be advanced or delayed to avoid the formation of dry spots.

Results and Discussion

The original version of this paper is available on https://www.scientific.net/KEM.1050.123.pdf

1. Validation on plate

The original version of this paper is available on https://www.scientific.net/KEM.1050.123.pdf

An initial validation test involves infusing a plate (200×400×10 mm). The experimental setup is as follows (Figure 16): The part is first infused on one side, then via a ramp placed in the part's axis of symmetry. The following sensors were installed: a camera on the sheet side, a volume flow sensor, a Thermoflux © heat flux sensor, and a vacuum sensor. The parameters of the chosen numerical test plan are the opening time of the second valve τ, and the permeability in the direction of flow of the drain. Experimentally, the second ramp is opened 150 seconds after the start of infusion.

[picture]

Fig. 16. Experimental validation on infused plate

Fig. 16. Experimental validation on infused plate

Figure 17 provides an overview of the registration step performed 80 s after the start of the infusion ( to ), followed by the prediction of the model state at 80 s and 360 s after to. It can be observed that the model accurately represents the flow up to to +80 s , after which a discrepancy appears at the first flow front, whose velocity is higher than that measured by the camera. This trend emerges when the second valve is opened, suggesting that variations in flow rate at the injection points may not be accurately captured by the model. Indeed, during the experiment, the control parameter is the resin flow rate (up to a maximum pressure), whereas the boundary condition of the high-fidelity model is pressure, leading to an overestimation of the flow rate in the simulation. As a consequence, at the end of the injection, a slight deviation in the position of the dry zones is observed. The geometry of these zones also differs slightly, in that the model predicts visible defects both above and below the part, whereas the experiment shows dry zones only on the underside. This difference arises because the upper dry zones are filled after the resin front has passed and during the curing phase, which is not considered here. Nevertheless, the discrepancies observed between the real-time predictions and the experimental observations are considered sufficiently small (with an error on the order of 5% in both position and time) to be regarded as industrially acceptable.

[picture]

Fig. 17. Overview of results on the infused plate with the second valve opening at 150s. The left column contains the segmentation of the resin front (with the camera image in black and white, the resin segmentation in green, the prediction in dark blue, and the intersection in blue/green), while the right column contains the model prediction seen from above and below. The first line contains the model parameters registration step, the following lines are the predictions of future states. For the latter, the corresponding segmentation is also reported.

Fig. 17. Overview of results on the infused plate with the second valve opening at 150s. The left column contains the segmentation of the resin front (with the camera image in black and white, the resin segmentation in green, the prediction in dark blue, and the intersection in blue/green), while the right column contains the model prediction seen from above and below. The first line contains the model parameters registration step, the following lines are the predictions of future states. For the latter, the corresponding segmentation is also reported.

2. Infusion of6𝐦2proof-of-concept part

The original version of this paper is available on https://www.scientific.net/KEM.1050.123.pdf

The objective here is to assess the ability of the digital shadow approach to adapt to a more complex geometry and material assembly. The part under study (Figure 3) represents configurations and levels of complexity that are representative of potential industrial applications. Its approximate dimensions are 2 m×3 m, and it includes reinforcement features in the form of stiffeners, which introduce
additional complexity. Furthermore, the curvature radius requires intrinsic and extrinsic optical calibrations in order to enable projection of the digital shadow results onto the part geometry with sufficient accuracy. As in previous cases, flow rate, heat flux, and vacuum sensors were implemented. The overall system is managed by the hardware architecture presented in Section 2.1. The parameters selected for the numerical design of experiments are the opening time of the second valve, and the permeability of the reinforcement in the direction perpendicular to the main flow, Figure 18, illustrates the real-time prediction of dry zone locations during the process as provided by the digital shadow. Experimentally, after cutting the part, dry zones were observed in the same areas, with a level of accuracy considered sufficient for industrial purposes.

[picture]

Fig. 18. Digital shadow result : Position of predicted resin front flow (left) and associated dry spot position (right) 5 minutes after current time.

Fig. 18. Digital shadow result : Position of predicted resin front flow (left) and associated dry spot position (right) 5 minutes after current time.

Conclusions and Perspectives

The original version of this paper is available on https://www.scientific.net/KEM.1050.123.pdf

The infusion digital shadow, developed using a 5D methodology, enables not only the monitoring of key process parameters, but also the observation of dry spots on inaccessible surfaces. It further allows the real time prediction of the position of these areas on the final part if the infusion strategy is maintained, and the virtual testing of alternative strategies with the aim of eliminating these dry areas before they actually occur. For a unique one-product part, the dataset required to build the metamodel is based on the generation of several high-fidelity simulations (here performed using PAM-RTM© software). The accuracy of these simulations is not necessarily expected to be high, as a phase of recalibration of the metamodel is performed based on the experimental resin front. In addition, the use of camera-based sensors and other process sensors is required.
Sensitivity analyses with respect to experimental parameters, reflecting discrepancies between simulation datasets and actual infusion conditions, demonstrated that the tool is sufficiently robust to predict dry areas with an accuracy that remains industrially relevant, even under unfavorable conditions.
The main limitation of the approach lies in the number of initial parameters that must be identified. The integration of additional relevant sensors to further recalibrate the models would therefore be beneficial. Moreover, in the present study, only the ramp opening parameters were considered adjustable. In practice, the number of controllable parameters may be significantly higher, particularly when addressing larger and more complex geometries. Therefore, the choice of the most representative parameters is crucial for the success of the used approach.

Acknowledgments

The original version of this paper is available on https://www.scientific.net/KEM.1050.123.pdf

This study is a part of the MONOCLE project led by IRT Jules Verne (French Institute of Research and Technology in Advanced Manufacturing). The authors wish to thank the industrial partners of this project: Naval Group, Sicomin, Iqanto, PCMI and Bureau Veritas.

References

The original version of this paper is available on https://www.scientific.net/KEM.1050.123.pdf

C. Le Gleuher, C.Buchmann, K. Schlegel, A. Friedberger : (2018). «Advanced Flow Front And Cure Monitoring Using High Frequency Technology", SAMPE Europe Conference 2018 Southampton

N. Gupta, "Fiber optic sensors for monitoring flow in vacuum enhanced resin infusion technology (VERITy) process," National Aerospace Laboratories, Elsevier, Bangalore, 2008.

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N. Pantelelis, "Quality Monitoring and Process Control in CFRP Production", 9th CFKVALLEY STADE CONVENTION 2015 16-17 June 2015, Stade, Germany

P. le Bot, G. Lebreton, N. Siddig, P. Couarraze, O. Fouché, C. Sébastien, A. De Fongalland, F. Cara, P. Gerard, "Anomaly detection during thermoplastic composite infusion: Monitoring strategy through thermal sensors," in Key Engineering Materials, Achievements and Trends in Material Forming, vol. 926, 2022, pp. 1423-1436

P. Wang, J. Molimard, S. Drapier, A. Vautrin, J.C. Minni, « Monitoring the resin infusion manufacturing process under industrial environment using distributed sensors », Journal of Composite Materials 0(0) 1-16, 2011

P. Venegas, I. Ortiz de Mendibil, A. Montero and J. Aurrekoetxea, "Quality control by Infrared Thermography of the infusion manufacturing process of composite automotive specimens", 13th international conference on quantitative infrared thermography 2016, july 4-8, gdańsk, Poland

Q. Qi, F. Tao, T. Hu, N. Anwer, A. Liu, Y.Wei, L. Wang, A. Nee, "Enabling technologies and tools for digital twin", Journal of Manufacturing Systems, Volume 58, Part B, 2021

P. Le Bot, D. Lecointe, N. Sidigg, O. Fouché, Le Guennec Yves,I. Abdullah, F. Niget Florent, C. Marchand Christophe, "Digital shadow dedicated to resin infusion filling process of composites parts", ECCM21, Nantes, July 2024

ISO 23247-1:2021 Automation systems and integration - Digital twin framework for manufacturing

Y. Xie, T. Takikawa, S. Saito, O. Litany, S. Yan, N. Khan, S. Sridhar, "Neural fields in visual computing and beyond". In Computer graphics forum (Vol. 41, No. 2, pp. 641-676).

M. Tancik, P. Srinivasan, B. Mildenhall, S. Fridovich-Keil, N. Raghavan, U. Singhal; "Fourier features let networks learn high frequency functions in low dimensional domains"; Advances in neural information processing systems, 33, 7537-7547

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This ePub file was created using machine-generated methods and automated software to convert the source text into an ePub format and may contain formatting inconsistencies from the original version. Formatting may vary depending on the device used.

The original version of this eBook is available on https://www.scientific.net/book/advanced-numerical-and-ai-strategies-for-material-forming/978-3-0364-1986-2

Key Engineering Materials, ISSN: 1662-9795, Vol. 1050, pp 137-144

doi: 10.4028/p-wKWw4q

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-19

Revised:
2026-02-18

Accepted:
2026-02-18

Online:
2026-04-16

Graph Neural Network for Draw-in Prediction in Sheet Metal Stamping

The original version of this paper is available on https://www.scientific.net/KEM.1050.137.pdf

Hui Wang 1,a, Taegyun Ahn 1, b, Lu Huang 2,c, Seog-Chan Oh 2, d, Liang Huang 3,e, Andrey Ilinich 3,f, Sathya Dev 4, g, Jeong Whan Yoon 1,5, h*

1 Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology (KAIST), 291 Daehak-ro, Yuseong-gu, Daejeon 305-701, Republic of Korea

2 Research & Development, General Motors Global Technical Center, 30470 Harley Earl Boulevard, Warren, MI 48092, USA

3 Stamping Business Unit, Ford Motor Company, 2101 Village Rd, Dearborn, MI 48124, USA

4 Stamping Engineering, Stellantis, 1000 Chrysler Dr, CIMS 485-04-03 Auburn Hills MI 48326, USA

5 School of Engineering, Deakin University, 75 Pigdons Rd., Waurn Ponds, VIC 3216, Australia

a  huiwang@kaist.ac.kr, bt.ahn@kaist.ac.kr, clu.huang @gm.com,d  seog-chan.oh@gm.com, elhuang6 @ ford.com, f  ailinich@ford.com, g  sathya.dev@stellantis.com, hj .yoon@kaist.ac.kr 

Keywords: Graph Neural Networks, Draw-in prediction, Sheet metal forming, Machine learning.

Abstract

Draw-in distance is a key index for evaluating the quality of sheet metal stamping. Its accurate prediction is therefore required for tool design and process control. Traditional finite element (FE) simulations, while accurate, are computationally intensive and time-consuming for iterative design optimization. In this study, a graph neural network (GNN) method is proposed to predict drawin during sheet metal forming. A dataset was built from FE simulations with different process settings, including blank holder force and draw bead force. The GNN model uses node coordinates and edge features to describe the spatial relations in the sheet. A multi-level loss function was applied. The coordinate error and edge distance error were included. In this way, the shape of the sheet is better preserved. The trained GNN can be used as a fast model for draw-in prediction. It can also be used for inverse analysis, where the process parameters are found from a given draw-in result. This provides an efficient tool for sheet metal forming design and optimization.

Introduction

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Accurate prediction of sheet flow during stamping is required to obtain high-quality formed parts and to control part size [1,2]. In industry, finite element (FE) simulation is widely used to study material flow, strain, and springback in metal forming [3-5] for its reliable prediction ability. However, high computing costs are required especially when many design variables, materials, or tool shapes are involved. As a result, the use of FE simulation is limited in process design to meet the needs of fast digital manufacturing.

In recent years, data-driven methods have been used as an alternative to FE-based prediction [6,7]. Jansson et al. [1] utilized the response surface method (RSM) based on FE simulations for draw-in optimization. They reported that the restraining force predicted by the RSM differed by up to approximately 15% from experimental results, and that the CPU time for a single FE simulation was about 35 minutes. Bae et al. [8] applied FE simulations into the optimization of draw-bead forces, by comparing with the measured results, the correlation coefficient of 0.9977 shows excellent accuracy.

Neural networks and surrogate models can learn the nonlinear mapping between process parameters and forming outcomes, significantly accelerating design iterations [9,10]. However, most existing models, such as multilayer perceptrons (MLPs) and convolutional neural networks (CNNs), are only designed for grid-structured data. For example, complex spatial relations in deforming sheets cannot be accurately described, or the topological links between different regions are not fully considered by these models. As a result, the physical continuity of the sheet is not well captured and stress-strain transfer between neighboring areas is often lost. For example, Wollschlaeger et al. [11] employed transfer learning with MLPs to predict the draw-in distance in sheet forming. Although
high prediction accuracy of the distance was achieved, the full draw-in shape was not provided since only scalar values were predicted.

Graph neural networks (GNNs) have be well captured and developed to address these limits [12,13]. By representing the sheet as a graph, where nodes correspond to discrete points on the sheet surface and edges describe their mechanical interactions, GNNs can propagate information through the material structure in a physically meaningful way [14]. This enables the model to learn both local deformation patterns and global flow characteristics directly from data, while preserving geometric relationships.

In this work, a GNN-based framework is developed to predict the blank flow shape during stamping based on finite element simulation data. The dataset is built from FE results and each sample is treated as a graph. The global features are given by the blank holder force and the draw-bead forces. These values control the overall material flow. The nodes correspond to the sheet coordinates, representing the spatial positions of material points, while the edges encode the distances between adjacent nodes, describing the local geometric connectivity of the sheet. The mapping from the global process settings to the final node positions is learned by the model, thus the final flow shape can then be predicted without running a full FE simulation. The results show that the use of physical graph features helps the GNN capture the deformation of sheet metal under different forming conditions with good accuracy and high efficiency.

Finite Element Model and Dataset

The original version of this paper is available on https://www.scientific.net/KEM.1050.137.pdf

The material used in this study is AA 6016-T4 sheet with a thickness of 0.9 mm . A Coulomb friction coefficient of 0.1 was adopted. The stamping process was simulated using the explicit solver in LS-DYNA at a forming speed of 1 m/s. A cross-sectional view of the finite element (FE) model and the initial blank geometry is shown in Fig. 1. The forming assembly consists of the upper die, upper die reform, upper pad, post, and binder. The blank holder force (BHF) applied by the binder increased linearly from 0 to a target value (e.g., 20 kN ) to clamp the sheet against the upper die, after which it remained constant at the target level. The value of 20 kN is used here only as an example to illustrate the loading procedure, in the dataset generation, the target BHF was varied as a design parameter. Once the target BHF was reached, the upper die advanced to deform the sheet, operating in coordination with the upper die reform, upper pad, and the binder to obtain the final geometry. Draw bead forces (DBFs) were incorporated to regulate material flow during drawing.

[picture]

Fig. 1. Cross-sectional finite element representation of the stamping assembly and initial blank geometry.

Fig. 1. Cross-sectional finite element representation of the stamping assembly and initial blank geometry.

The constitutive behavior of the sheet was described using the Hockett-Sherby hardening law combined with the Barlat- 89 yield criterion [15,16], and parameters are shown in Table 1. The Hockett-Sherby model expresses the evolution of the flow stress as a function of the equivalent plastic strain according to

σ=σsat −(σsat −σyield )exp(−cεpn)(1)

where, σsat  is the saturation stress, σyield  is the initial yield stress, εp denotes the equivalent plastic strain, c controls the rate of hardening and saturation, and n is strain hardening coefficient. Shell elements were used for all components, and all tooling parts other than the blank were modeled as rigid bodies. Eight draw bead regions were defined around the periphery of the blank, with their positions and shapes illustrated in Fig. 1.

For each simulation case, the nodal coordinates of the blank were extracted and used to construct graph data samples as shown in Fig. 2. The node features are given by the spatial coordinates of each material point, and the edges are defined by the Euclidean distance between neighboring nodes, thereby preserving the geometric connectivity of the blank. The global graph features consist of the blank holder force and eight draw-bead force, which govern the material flow in the forming process. All features were normalized before training to enhance numerical stability.

Table 1. Material parameters of 6016-T4 aluminum alloy

Model | Barlat-89 yield criterion | Hockett-Sherby hardening model
Parameters | m | R₀ | R₄₅ | R₉₀ | σsat (MPa) | c | n | σyield (MPa)
| 8 | 0.61 | 0.49 | 0.70 | 349.40 | 3.87 | 0.67 | 117.40

[picture]

Fig. 2. Schematic of node and edge definition for the deformed sheet.

Fig. 2. Schematic of node and edge definition for the deformed sheet.

Methodology

The original version of this paper is available on https://www.scientific.net/KEM.1050.137.pdf

The proposed graph neural network (GNN) is employed to predict sheet-flow deformation by integrating nodal geometry, edge connectivity, and global forming parameters within a unified framework. The overall architecture of the model is illustrated in Fig. 3. The input graph is built from the initial blank shape. Each node represents a material point and is defined by its initial ( x,y ) coordinates. Each edge stores the initial distance between neighboring nodes. The global forming settings include the blank holder force (BHF) and eight draw bead forces (DBFs). These values are used as graph-level features.

The embedded node and edge features are then passed through four graph attention-based (GATbased) message passing layers. In each GAT block, multi-head attention is applied. Neighboring information is aggregated with the use of edge features. As a result, deformation behavior controlled by local connections can be learned by the model.

A multi-head attention module is employed to integrate the encoded global parameters with the node embeddings in order to capture the effect of forming loads. This mechanism enables nodes to selectively attend to process parameters, thereby ensuring that forming forces meaningfully modulate the learned representations.

Following message passing and parameter-node attention, the merged node-parameter features are passed to a coordinate prediction head that produces the deformed nodal positions. In parallel, edge features derived from the source and target nodes together with encoded edge attributes are fed into an edge prediction head to estimate relative distances between adjacent nodes. Both outputs are jointly optimized, with the edge-distance prediction serving as an auxiliary geometric constraint. This constraint helps preserve the local shape of the sheet and thereby improves the accuracy of the predicted sheet geometry.

[picture]

Fig. 3. Architecture of the proposed GNN model.

Fig. 3. Architecture of the proposed GNN model.

A composite loss function is employed to train the proposed GNN. The loss jointly supervises node coordinates and edge distances, thereby enabling the model to learn both the global sheet shape and the local geometric connectivity simultaneously. For a graph with N nodes, the coordinate loss is defined as the discrepancy between the predicted node positions y^i and the ground-truth FE results yi :

Lcoord =1N∑i=1N‖y^i−yi‖2(2)

The coordinate loss encourages the network to reconstruct the final sheet shape, while local geometric relations are preserved through an edge-based loss applied across all edges. Specifically, the loss penalizes the discrepancy between the predicted edge distance e^ij and the target FE-computed distance eij :

Ledge=1E∑(i,j)∈edges‖e^ij−eij‖2(3)

The edge loss helps the network learn consistent deformation patterns and mitigates local distortions.
As coordinate values and edge distances differ in scale, both losses are normalized by their respective standard deviations, σc and σe. Thereby preventing one term from dominating the overall loss and the normalized loss terms are expressed as:

 scaled coord =Lcoord /σc2, scaled edge =Ledge /σe2(4)

The total loss is defined as a weighted sum of the normalized coordinate and edge losses:

Ltotal =α· scaled coord +(1−α)· scaled edge (5)

where the weight factor α∈[0,1] is introduced to balance global shape prediction against local distance prediction. By adjusting weight factor, the contribution of the two loss terms can be optimized.

Table 2. Ranges of BHF(kN) and DBFs(N/mm) used for dataset generation.

BHF | DBF1 | DBF2 | DBF3 | DBF4 | DBF5 | DBF6 | DBF7 | DBF8
Min. | 100 | 58.92 | 2.63 | 11.06 | 37.89 | 2.63 | 2.63 | 1.58 | 37.89
Max. | 300 | 132.57 | 26.30 | 44.22 | 113.67 | 26.3 | 26.30 | 14.22 | 113.67

The dataset used to train and evaluate the proposed GNN model was generated entirely through finite element (FE) simulations. For the nine process parameters, including the blank holder force (BHF) and eight draw-bead forces (the ranges are illustrated in Table 2), Latin hypercube sampling was applied. Considering the computing cost and the size of the parameter space, 200 sample points were generated within the allowed ranges of these parameters. The sampled parameter sets were then used as inputs for batch simulations in LS-DYNA3D. The corresponding draw-in shapes were obtained from these simulations. For every FE simulation, the nodal coordinates after deformation were extracted and converted into graph representations consistent with the GNN input format. The 200 graph samples were randomly split into three groups. An 80/10/10 ratio was used for training, validation, and test sets. The training set was used to learn the model parameters. The validation set was employed for hyperparameter tuning, including the number of GAT layers, hidden feature dimension, learning rate, and dropout rate. The test set was kept completely separate and was not used during training or hyperparameter tuning. It was used only to evaluate the final model performance and to analyze the effect of different loss-weighting strategies.

Results and Discussion

The original version of this paper is available on https://www.scientific.net/KEM.1050.137.pdf

The proposed GNN model was trained using the dataset introduced in Methodology section by adopting an early-stopping strategy with a patience of 50 epochs and a checkpointing mechanism to automatically save the best-performing model based on the validation loss. A grid search was conducted to tune the main hyperparameters. After the optimal configuration was identified, the weight factor α-which balances the coordinate loss and the edge-distance loss-was systematically varied. The influence of α on prediction accuracy was evaluated using the test set, while all other hyperparameters were kept fixed. This strategy ensures that the observed performance trends are not biased by parameter tuning and that the contribution of the auxiliary task is assessed in a consistent and objective manner.

The coordinate prediction errors for different values of α are summarized in Fig. 4. When α=0, the model is trained solely on the edge-distance loss and the coordinate mean absolute error (MAE) reaches 105.6 mm , indicating that edge supervision alone is insufficient to guide the deformation prediction. As α increases from 0.1 to 0.6 , the coordinate MAE rapidly decreases and reaches its minimum value of 1.4648 mm at α=0.3. This demonstrates that combining coordinate loss with the auxiliary edge-distance loss provides more stable and physically consistent supervision, allowing the model to learn both global shape deformation and local geometric continuity. When α approaches 1.0, the coordinate MAE increases again ( 2.77 mm ), showing that excessive reliance on the coordinate loss weakens the beneficial geometric constraints provided by edge supervision. These observations confirm that the auxiliary task enhances the model's inductive bias and improves robustness, particularly when local deformation patterns play a significant role in sheet-flow evolution.

[picture]

Fig. 4. Influence of weight factors on draw-in shape prediction.

Fig. 4. Influence of weight factors on draw-in shape prediction.

Representative prediction results for α=0.0,0.3, and 1.0 are shown in Fig. 5. The predicted nodal coordinates and edge distances are shown in each case. When α=0.0, non-physical sheet contours are obtained. Although the edge distances remain reasonable, the nodal positions are poorly reconstructed. This indicates that the sheet geometry cannot be recovered from edge constraints alone.

When α=0.3, the predicted flow shape agrees well with the FE reference. Smooth deformation and uniform spacing between neighboring nodes are observed. These results suggest that the auxiliary edge-distance task provides an effective geometric constraint during message passing. As a result, the spatial continuity of the sheet is better preserved.

When α=1.0, the global shape remains acceptable. However, local distortions and irregular node spacing appear. This behavior reflects the loss of geometric regularization when the edge-distance term is removed from training.

Overall, the results show that the inclusion of the edge-distance task improves both the accuracy and the physical consistency of the GNN model. The best performance is achieved at α=0.3. This indicates that the use of multi-task learning is important for capturing global deformation behavior and local geometric relations at the same time. The proposed framework therefore provides a useful basis for future graph-based simulation of sheet metal forming.

[picture]

Fig. 5 Coordinates and relative distance comparison with the weight factor of 0.0,0.3, and 1.0 .

Fig. 5. Coordinates and relative distance comparison with the weight factor of 0.0 , 0.3 , and 1.0 .

Conclusions

The original version of this paper is available on https://www.scientific.net/KEM.1050.137.pdf

A graph neural network framework is proposed for predicting sheet flow in stamping. The model includes node geometry, edge links, and global forming forces such as BHF and DBFs. GAT-based message passing is used together with a parameter-node attention module. A multi-task scheme is also applied. Both deformed node positions and edge distances are predicted at the same time.

The results show that the edge-distance task improves the accuracy and geometric quality of the coordinate prediction. A study of the weight factor is performed. The best balance between the two tasks is found at α=0.3. At this value, the lowest coordinate error and the most consistent deformation shape are obtained.

Overall, the proposed GNN provides a fast and flexible option for deformation prediction. It serves as an effective alternative to FE simulation. It also offers a basis for future physics-based graph learning in sheet metal forming.

Acknowledgement

The original version of this paper is available on https://www.scientific.net/KEM.1050.137.pdf

The authors are thankful for USCar Program (Material testing and development of a self-trained physics-based ML model) supported by DoE in USA. Especially, the authors are thankful for Drs. Stanley Wang and Kaiping Li for their technical supports on USCar Program and critical reviews of this manuscript. This research was also supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (RS-202525406725) and supported by the InnoCORE program of the Ministry of Science and ICT (N10250154).

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The original version of this paper is available on https://www.scientific.net/KEM.1050.137.pdf

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Key Engineering Materials, ISSN: 1662-9795, Vol. 1050, pp 145-155

doi: 10.4028/p-M2Pzus

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-19

Revised:
2026-02-16

Accepted:
2026-02-26

Online:
2026-04-16

A Bayesian Data Assimilation Framework for Characterizing Recrystallization in Steels

The original version of this paper is available on https://www.scientific.net/KEM.1050.145.pdf

Bentolhoda Jamshidi 1,a*, Jos Havinga 1, b, and Ton van den Boogaard 1,c

1 Faculty of Engineering Technology, University of Twente, The Netherlands

a  b.jamshidi@utwente.nl, b  jos.havinga@utwente.nl, c  a.h.vandenboogaard@utwente.nl

Keywords: Recrystallization, Bayesian Data Assimilation, Particle Filter, Sequential Monte Carlo, Steels

Abstract

In this study, we develop a Bayesian data assimilation framework that combines a meanfield model of static recrystallization (MiReX) with a Sequential Importance Resampling (SIR) particle filter to estimate key material parameters from controlled synthetic experiments. MiReX, originally developed as a microstructurally based extension of Johnson-Mehl-Avrami-Kolmogorov kinetics, is used as a forward model in which the uncertain quantities include the grain-boundary mobility parameters (prefactor and activation energy), a stored-energy coefficient, an Avrami-type exponent, and an interface length scale. Synthetic recrystallized-fraction measurements are generated at two isothermal holding temperatures using a reference parameter set and are perturbed with Gaussian noise to mimic experimental uncertainty. Starting from broad uniform prior ranges, the particle filter propagates an ensemble of MiReX trajectories in time, updates particle weights using a Gaussian likelihood, and applies systematic resampling combined with Liu-West kernel regularization to reduce particle degeneracy while preserving posterior variance. The posterior obtained after assimilating the first temperature dataset is used as the prior for the second dataset, enabling sequential multi-temperature calibration. The synthetic experiments show that the framework recovers the reference parameters within credible intervals and provides tight uncertainty bounds on the predicted recrystallization kinetics. These results demonstrate that combining a physically based mean-field recrystallization model with sequential Monte Carlo methods provides a robust route for probabilistic parameter estimation and uncertainty quantification in microstructure evolution models.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1050.145.pdf

Static recrystallization plays a central role in controlling grain structure and mechanical properties of metals during thermo-mechanical processing. Reliable prediction of recrystallization kinetics and microstructural evolution is therefore crucial for optimizing industrial operations such as rolling, forging, and annealing. To this end, a wide range of models has been proposed, ranging from phenomenological mean-field approaches to spatially resolved phase-field simulations. The capability of these models to capture actual recrystallization kinetics depends on the modeling assumptions (e.g. averaging in mean-field approaches versus full field predictions in phase-field simulations) and on the accuracy of the internal physical model parameters. Although these models are commonly used in a deterministic setting, the uncertainty in the predictions may be significant due to the assumptions and uncertain model parameters. In this work, a mean-field recrystallization model is therefore used in a stochastic (i.e. Bayesian) framework, in which uncertain model parameters are estimated in probabilistic terms using experimental measurements. The rationale of the approach is that measurable quantities that are affected by model parameters can be used to reduce model parameter uncertainty and improve model predictions, even if no unique solution to the inverse parameter problem exists.

The multi-phase-field (MPF) method is a powerful tool for modeling recrystallization because it explicitly resolves grain-boundary migration and complex microstructural morphologies. However, its predictive capability is often limited by uncertainties in key parameters such as grain-boundary mobility, grain-boundary energy, and stored-energy distributions, which are difficult to measure experimentally and may vary spatially. In addition, MPF simulations are computationally expensive,
particularly when combined with uncertainty quantification or data-assimilation techniques. Bayesian approaches have recently been introduced to improve MPF-based modeling. Seguchi et al. employed an ensemble Kalman filter (EnKF) to estimate phase-field simulation parameters, including interface mobility and gradient energy coefficients, during eutectic alloy solidification [1], while Yamanaka et al. extended this framework to three-dimensional MPF simulations to identify anisotropic grainboundary energy and mobility functions using EnKF-based data assimilation [2]. Further developments include Bayesian estimation of anisotropic grain-boundary energies and mobilities for multiple boundary types by Miyoshi et al. [3], and molecular-dynamics-informed phase-field modeling to extract inclination-dependent grain-boundary energies by Fujiwara et al. [4].

Bayesian data assimilation has also been successfully applied to other microstructure evolution processes. Ishii et al. combined in situ electron tomography with Bayesian assimilation to identify temperature-dependent diffusion coefficients in phase-field sintering simulations [5]. In solidification studies, Ohno et al. used Bayesian inference and EnKF-based methods to estimate solid-liquid interfacial properties from atomistic and phase-field simulations [6]. Beyond phase-field models, Bayesian frameworks have been applied to calibrate constitutive and mechanistic models. Honarmandi and Arroyave used Markov chain Monte Carlo methods to calibrate strain-stress models for TRIP steels, demonstrating Bayesian inference's ability to quantify uncertainty and reveal parameter correlations [7]. These studies contribute to the increasing use of Bayesian uncertainty quantification and data assimilation in nonlinear dynamical systems. Works by van Leeuwen [8], Fearnhead and Künsch [9] , and the monograph by Grudzien and Bocquet [10] provide the theoretical basis for particle filters, ensemble Kalman methods, and generative-model-based inference. Applications continue to expand, including Bayesian estimation for recovery of elasto-plastic constitutive parameters using particle filters [11] and online Bayesian monitoring of thin-film growth during pulsed-laser deposition [12].

Alongside these developments, mean-field recrystallization models such as MiReX offer a computationally efficient alternative to MPF simulations. Based on extended Johnson-Mehl-Avrami-Kolmogorov kinetics, MiReX describes the temporal evolution of recrystallized volume fraction using physically meaningful parameters such as stored energy, grain-boundary mobility, temperature, and composition [13]. Although it does not explicitly resolve microstructure, its efficiency and interpretability make it well suited for parameter estimation and Bayesian data assimilation.

Bayesian inference provides a natural framework for parameter identification within MiReX by yielding full posterior distributions rather than single-point estimates. Sequential Monte Carlo methods, and in particular the Sequential Importance Resampling (SIR) particle filter, are well suited for the nonlinear and non-Gaussian nature of recrystallization kinetics. Particle filters can be used to evolve ensembles of states and parameters in time, assimilating recrystallization observations in a manner analogous to successful Bayesian estimation in phase-field models [1-6].

In this work, we present a Bayesian data-assimilation framework that couples the MiReX meanfield model with an SIR particle filter for parameter estimation in static recrystallization. A syntheticexperiment approach is adopted to assess the performance of the inversion method: synthetic recrystallized fraction data F(t) are generated using known model parameters and perturbed with Gaussian noise to mimic experimental uncertainty. The particle filter estimates key parameters governing recrystallization, including the mobility prefactor, grain-boundary activation energy, stored energy, Avrami exponent, and interface radius. Synthetic experiment data is generated at two isothermal temperatures, with posterior distributions propagated across temperatures to assess parameter consistency. A Liu-West shrinkage kernel is employed to reduce particle degeneracy. The paper is organized as follows. Section MiReX-based forward model and synthetic-experiment design describes the MiReX model and the synthetic-experiment design. Section Bayesian data assimilation with Sequential Importance Resampling presents the Bayesian data-assimilation methodology. Section Results presents the numerical results. Section Discussion and Conclusion discusses the implications and limitations of the study.

MiReX-Based Forward Model and Synthetic-Experiment Design

The original version of this paper is available on https://www.scientific.net/KEM.1050.145.pdf

We adopt a MiReX-inspired mean-field model as proposed by Mathevon et al. [13], that describes the time evolution of the recrystallized volume fraction F(t) under a varying temperature profile. In the original MiReX formulation, the recrystallized fraction is expressed as

FRex(t)=1−exp(−FExt(t))(1)

where FExt (t) is the extended fraction that accounts for nucleation and growth events and is typically written as

FExt(t)=NRex(∫0tM(T(τ),X)GRex(τ)dτ)nA(2)

with NRex  being the effective nucleation-site density, M(T,X) the effective grain-boundary mobility (function of temperature T and composition X ), GRex  the driving pressure for recrystallization (i.e. the internal strain energy), and nA the Avrami exponent describing growth geometry. NRex  is related to an interaction length scale Rint  and the Avrami exponent nA as below:

NRex=(1Rint)nA(3)

The effective mobility M is modeled as a segregation-corrected mobility derived from a Cahn-type description of solute drag. The "pure" mobility Mpure  is defined as

Mpure(T)=CMRTexp(−QgbRT)(4)

where Qgb is the grain-boundary activation energy, R is the gas constant, and CM is a mobility prefactor. The constant CM is a composite mobility prefactor defined as

CM=βVmδb2Dgb0(5)

grouping the grain-boundary diffusivity prefactor Dgb0, grain-boundary thickness δ, atomic spacing via the Burgers vector b, and the molar volume of ferrite Vm. Solute drag is incorporated through an effective segregation factor that accounts for the combined influence of alloying elements. Each coefficient αi represents a solute-dependent mobility slowdown coefficient, computed from nominal segregation parameters and the temperature T. The effective grain-boundary mobility is therefore written as

M=Mpure 1+Mpure ∑i=1nαiXi(6)

In the present synthetic experiments, the nominal composition and solute-drag parameters are fixed, and uncertainty is concentrated in CM and Qgb, which control the overall magnitude and temperature sensitivity of M.

To construct synthetic experiments, a reference parameter set is chosen:

θtrue =(CMtrue ,Qgbtrue ,Gtrue ,nAtrue ,Rint true )(7)

informed by previous MiReX calibrations for high-strength steels. In the simulations presented here, the true values are [13]

CMtrue =12.1,Gtrue =6.98×105 J/m3,nAtrue =1.3

Synthetic data is generated in this study, corresponding to two distinct isothermal annealing experiments, with synthetic measurements taken at the following observation times:

t11=200 s,t21=500 s,t31=1000 s for T1=923 Kt12=500 s,t22=1000 s,t32=2000 s for T2=873 K(9)

For each temperature, the nominal segregation parameters and diffusion coefficients are adjusted, leading to different effective mobilities.

At each temperature, the forward model is integrated from t=0 with the true parameters θtrue  to obtain the noiseless recrystallized fraction Ftrue (tk,T). Synthetic observations are then generated as

yk=Ftrue (tk,T)+εk(10)

where εk is Gaussian noise with zero mean and standard deviation σmeas . These synthetic datasets form the basis of the synthetic experiments used to evaluate the particle-filter-based parameter estimation framework.

Bayesian Data Assimilation with Sequential Importance Resampling

The original version of this paper is available on https://www.scientific.net/KEM.1050.145.pdf

We formulate the MiReX model into a state-space form suitable for particle filtering. The state vector at time tk is

𝐱k=[Frex(tk),CM,Qgb,G,nA,Rint](11)

where Frex (tk) is the time-evolving recrystallized fraction and the remaining entries are constant model parameters that are treated as time-invariant random variables. The system dynamics can be written as

xk=f(xk−1,Δtk)+vk(12)

where f represents one MiReX time increment as defined by Eqs. (1) and (2), with time increments Δtk=tk−tk−1 defined by the observations times given in Eq. (9). In the synthetic experiments, process noise is neglected ( vk=0 ), so uncertainty arises only from the prior and measurement noise. Observations are perturbed by Gaussian noise εk~𝒩(0,σmeas 2), yielding the measurement likelihood

p(yk∣xk)=12πσmeas exp[−(yk−Frex(tk))22σmeas 2](13)

where yk is the measured recrystallized at time tk fraction and Frex(tk) is the estimated recrystallized fraction. The posterior distribution p(xk∣y1:k) is approximated using an ensemble of Ns particles {xk(i),wk(i)}i=1Ns, such that

p(xk∣y1:k)≈∑i=1Nswk(i)δ(xk−xk(i)),(14)

where δ(·) is the Dirac delta distribution. In the prediction step, each particle is propagated deterministically:

xk(i)=f(xk−1(i),tk)(15)

Upon receiving observation yk, particle weights are updated using the measurement likelihood:

w~k(i)=wk−1(i)p(yk∣xk(i)),(16)

and then normalized,

wk(i)=w~k(i)∑j=1Nsw~k(j)(17)

A point estimate of the state can be obtained after each observation as the weighted mean of the particle set:

x^k=∑i=1Nswk(i)xk(i)(18)

After each measurement step, resampling of the particles is performed, generating a new set of particles with equal weights without changing the probability distribution of the state vector. To prevent particle impoverishment associated with static parameters, Liu-West regularization is employed. The parameter vector

θ=(CM,Qgb,G,nA,Rint)(19)

is shrunk towards the ensemble mean with added Gaussian perturbations to maintain ensemble spread:

θnew (i)=aθ(i)+(1−a)θ¯+1−a2z(i),(20)

where z(i)~𝒩(0,S) and S denotes the sample covariance matrix of the parameter ensemble. In practice, z(i) is generated as z(i)=Lcξ(i), with ξ(i)~𝒩(0,I) and LcLc⊤=S. The coefficient a determines the amount of Liu-West regularization, with values of a≤1, and a=1 corresponding to no regularization. The coefficient a is initialized at a value of a1=0.98, and after each Liu-West update the coefficient is changed with ai=1−0.8·(1−ai−1), such that it converges towards a=1, meaning that the amount of Liu-West regularization reduces over particle filter updates. After Liu-West perturbation, the parameters are constrained within prior bounds, preserving approximate first and second moments of the distribution. At the initial temperature ( 923 K ), uniform priors are assigned:

CM~U(2.42,60.52),Qgb~U(1.0×105,2.5×105)J/mol,G~U(1.0×105,3.0×106)J/m3,nA~U(0.8,2.2),Rint ~U(4.0×10−6,1.2×10−5)m.(21)

The initial recrystallized fraction is set to FRex (0)=0, and particles are initialized by independent sampling from the priors. After assimilation at one temperature, the final posterior ensemble is used as an empirical prior for the next temperature, with FRex  set to zero. This strategy treats each temperature as a separate experiment sharing the same underlying parameters but different thermodynamic conditions.

Results

The original version of this paper is available on https://www.scientific.net/KEM.1050.145.pdf

To assess the proposed Bayesian particle-filtering framework, synthetic experiments were conducted using the MiReX model of static recrystallization. Synthetic recrystallization curves were generated at two annealing temperatures, T1=923 K,T2=873 K, using the reference MiReX parameter set and alloy composition for dual-phase steel ( XMn=0.015,XCr=0.005,XSi=0.003 ). This ensures that the true parameters are physically consistent with a realistic material system.

At each temperature, the reference parameters {CM,Qgb,G,nA,Rint } were used to generate recrystallized fraction curves FRex (t). Synthetic observations were obtained by sampling these curves

[picture]

Fig. 1: Reconstructed recrystallized fraction FRex (t).

Fig. 1. Reconstructed recrystallized fraction F Rex ( t ) .

at realistic time points and adding Gaussian measurement noise with standard deviation σmeas =0.01. Since the true parameters are known, the resulting synthetic dataset provides a controlled benchmark for evaluating parameter identifiability, convergence, and multi-temperature consistency.
Reconstruction of recrystallization kinetics and filter stability. Figure 1 shows the reconstructed recrystallized fraction FRex (t) at the two temperatures. The particle ensemble accurately captures both the rapid increase of FRex (t) and its saturation towards FRex ≈1. After only a few observations, the 95% credible intervals become narrow, and the posterior mean and median closely match the noisy synthetic data, indicating rapid convergence of the particle filter despite broad initial parameter priors.

The filter is first applied at the higher temperature T1, starting from wide and uninformative parameter ranges, which leads to a large uncertainty before the first measurements are available. After each measurement update, the particle ensemble correctly matches the observed recrystallized fraction. Between measurements, the prediction uncertainty increases again due to remaining uncertainty in the model parameters. The final estimate obtained at T1 is then used as the initial condition for the analysis at the lower temperature T2, rather than restarting the filter. At the beginning of T2, the model response is still uncertain because the temperature has changed, but the filter quickly adjusts as new data are added. As more measurements are included, the uncertainty bands become narrower and the model prediction follows the synthetic reference curve more closely. This shows that using recrystallization data from multiple temperatures helps reduce parameter uncertainty and leads to more reliable predictions.
Temperature-dependent evolution of parameter posteriors. Figure 2 shows the evolution of posterior probability density functions (PDFs) for the five unknown MiReX parameters after ensemble update with the two temperatures, with vertical dashed lines indicating the reference values. As data from successive temperatures are assimilated in a sequential manner, the parameter estimates continue to narrow with added observations.

For the mobility prefactor CM, the posterior initially spans nearly the full prior range at 923 K but narrows a bit as additional temperature data are included.

Single-temperature assimilation for the activation energy Qgb leaves a broad posterior due to the well-known trade-off between prefactor and activation energy, but multi-temperature data sharply constrain Qgb. This reflects the fact that the Arrhenius mobility relation in Eq. (4) can be matched

[picture]

Fig. 2: Posterior probability density functions of MiReX parameters after sequential assimilation of multi-temperature measurements .

Fig. 2. Posterior probability density functions of MiReX parameters after sequential assimilation of multi-temperature measurements .

across multiple temperatures by a narrower set of ( CM,Qgb ) pairs, thereby breaking the degeneracy present in single-temperature fits.

The stored energy G exhibits large reduction in uncertainty, with substantial narrowing of the distribution after the observations at T1 are processed and additional narrowing after processing the observations at T2. The Avrami exponent nA converges rapidly to a narrow interval around its reference value, indicating that the transformation-curve shape is strongly constrained by the data. In contrast, the interface length Rint  remains less well identified than most other parameters, retaining a broader credible interval due to its weaker influence on the observable kinetics.

Overall, the results demonstrate that multi-temperature assimilation yields robust parameter convergence, reduced uncertainty, and improved identifiability compared to single-temperature analysis. Joint parameter structure. To analyze parameter relationships, the joint posterior distributions of parameter pairs were examined. Figure 3 shows pairwise posterior maps for the five MiReX parameters after assimilating data at T1 and at T2. The color intensity represents probability density, and the reference parameter set is marked by a cross. From T1 to T2, the posteriors generally tighten, showing improved parameter identification when the T2 dataset is included. Notably, Qgb becomes strongly concentrated near its reference value, while CM remains relatively broad, indicating weaker identifiability of CM. In contrast, the ( G,Rint  ) plane exhibits a persistent positive correlation, which becomes very narrow at T2, implying that these parameters remain strongly coupled and are not independently constrained. Overall, the T2 assimilation yields tighter posteriors but also reveals remaining parameter couplings, especially involving Rint .

[picture]

Fig. 3: Pairwise density plots of all parameter pairs ( CM,Qgb,G,nA,Rint ) for the prior and posteriors after T1 and T2.

Fig. 3. Pairwise density plots of all parameter pairs ( C M , Q g b , G , n A , R i n t ) for the prior and posteriors after T 1 and T 2 .

Generalization of the particle filter to non-isothermal conditions. So far, two constant temperatures, denoted as T1 and T2, were used to quantify uncertainty in the model parameters through sequential data assimilation. To test the generalization of the particle filter (PF) model beyond isothermal conditions, an arbitrary, non-constant temperature profile was considered. Figure 4(a) shows the applied temperature history, with dashed lines indicating T1 and T2. Figure 4(b) and Figure 4(c) show the predicted evolution of the recrystallized fraction after assimilation at T1 and after sequential assimilation at both T1 and T2, respectively. When only data from T1 is used, the prediction shows larger uncertainty, while using data from both T1 and T2 reduces the uncertainty and improves the prediction.

[picture]

Fig. 4: An arbitrary thermal profile and the evolution of the recrystallized fraction estimated after sequential data assimilation steps .

Fig. 4. An arbitrary thermal profile and the evolution of the recrystallized fraction estimated after sequential data assimilation steps .

Discussion and Conclusion

The original version of this paper is available on https://www.scientific.net/KEM.1050.145.pdf

This study presented a Bayesian data-assimilation framework that combines the MiReX mean-field model of static recrystallization with a Sequential Importance Resampling (SIR) particle filter using Liu-West regularization. The method was tested using synthetic data at multiple constant temperatures, with only a limited number of recrystallization measurements available at each temperature.

The results show that the framework can constrain the main MiReX parameters CM,Qgb,G,nA, and Rint  starting from broad prior ranges. Instead of producing a single exact solution, the Bayesian approach provides parameter distributions that explicitly quantify the remaining uncertainty. The results indicate that different combinations of CM,Qgb,G,nA, and Rint  can lead to very similar recrystallization curves for the available observations, confirming that the inverse identification problem is not unique.
Scope, identifiability, and limitations of Bayesian parameter estimation. The proposed framework performs Bayesian parameter estimation, meaning that the inferred parameter distributions are conditional on the assumed observation model, the measurement-noise model, and the MiReX mean-field formulation. As a consequence, parameter identifiability depends both on the experimental design and on the model structure.

A key result of this work is the benefit of using data from more than one temperature, especially when only a small number of measurements is available. Data from a single temperature mainly constrain the overall transformation speed but cannot clearly separate the effects of the mobility prefactor CM and the activation energy Qgb. By using data from T=923 K and T=873 K in sequence, the temperature dependence of the kinetics helps reduce uncertainty in both parameters. This shows that changing the temperature provides additional information, although some parameter coupling remains.

The analysis of parameter correlations gives further insight into which parameters can be identified. A strong correlation between CM and Qgb reflects their combined role in the temperaturedependent mobility. Although using multiple temperatures improves the estimation of these parameters, a clear correlation remains between the driving force G and the interface length Rint . This indicates that these two parameters have similar effects on the recrystallization behavior and cannot be fully separated using only recrystallized-fraction data. Additional observables, such as recrystallized grain size, nucleus density, or texture measurements, would likely improve identifiability.

The reported credible intervals are conditional on the assumed measurement-noise model. Increasing measurement noise or reducing the number of observations would broaden the posterior distribu-
tions and reduce the ability to separate correlated parameters. For real experiments, the measurement uncertainty should therefore be estimated carefully, and the choice of likelihood model should reflect the characteristics of the experimental data.

Another important limitation is related to model representativity. The inferred parameters are conditional on the validity of the MiReX mean-field formulation. If the real material behavior includes effects that are not represented in the model, such as spatial heterogeneity in stored energy, variations in nucleation behavior, or grain-boundary character effects, Bayesian calibration may compensate for model mismatch by shifting parameters away from their true physical values. In practical applications, this issue can be addressed by comparing posterior predictions with independent measurements and by explicitly evaluating model discrepancies.

Several safeguards were used in the present work to reduce the risk of misleading parameter identification. First, multi-temperature assimilation was employed to reduce parameter degeneracy and improve identifiability. Second, posterior distributions and parameter correlations were explicitly analyzed rather than relying on single best-fit values. Third, posterior predictive intervals were used to assess how well the inferred parameters reproduce the observed kinetics.

It is important to emphasize that the proposed Bayesian framework complements rather than replaces classical physically motivated identification approaches. Traditional metallurgical methods can extract kinetic parameters such as activation energy and mobility prefactors from carefully designed experiments under simplifying assumptions. The contribution of the present approach is that it allows simultaneous estimation of several physically meaningful parameters, quantifies uncertainty and parameter correlations, and naturally incorporates sequential updating when new measurements become available. The reference parameter set used in the present synthetic study was selected to be consistent with ranges reported in prior MiReX calibrations and experimental recrystallization studies for steels [13], which supports the physical plausibility of the inferred parameter values and suggests that the proposed Bayesian framework can provide physically meaningful estimates when applied to experimental data.

The main limitation of this study is the use of synthetic data. Real experiments will involve additional effects such as measurement noise, material heterogeneity, and possible differences between the model and the actual material behavior. Nevertheless, testing the method under controlled conditions is an important first step. Future work will focus on applying the framework to experimental recrystallization data and incorporating additional observables, such as texture measurements, to improve parameter identifiability and to assess the physical consistency of the inferred parameters.

Acknowledgement

The original version of this paper is available on https://www.scientific.net/KEM.1050.145.pdf

This research was carried out under project number N21022e in the framework of the Partnership Program of the Materials innovation institute M2i (www.m2i.nl) and the Netherlands Organization for Scientific Research (www.nwo.nl).

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The original version of this paper is available on https://www.scientific.net/KEM.1050.145.pdf

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The original version of this eBook is available on https://www.scientific.net/book/advanced-numerical-and-ai-strategies-for-material-forming/978-3-0364-1986-2

Key Engineering Materials, ISSN: 1662-9795, Vol. 1050, pp 157-169

doi: 10.4028/p-5QE7ks

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-19

Revised:
2026-02-17

Accepted:
2026-02-18

Online:
2026-04-16

Predicting Lateral Flow in Hot Sheet Metal Rolling Using Symbolic Regression

The original version of this paper is available on https://www.scientific.net/KEM.1050.157.pdf

Daniel Luis Mederos 1,a, Amirali Hashemzadeh 1, b, Antonella Cometa 1,c*, Celal Soyarslan 1,2, d, Frederic E. Bock 3,e, Benjamin Klusemann 3,4,f, Ton van den Boogaard 1, g

1 University of Twente, Faculty of Engineering Technology, Chair of Nonlinear Solid Mechanics, Enschede, The Netherlands

2 Fraunhofer Innovation Platform for Advanced Manufacturing, University of Twente, Enschede, The Netherlands

3 Institute of Materials and Process Design, Helmholtz-Zentrum Hereon, Geesthacht, Germany

4 Institute for Production Technology and Systems, Leuphana University of Lüneburg, Lüneburg, Germany

a d.luismederos@student.utwente.nl, b a.hashemzadeh@utwente.nl, c a.cometa@utwente.nl,

d  c.soyarslan@utwente.nl, e  frederic.bock@hereon.de, f  benjamin.klusemann@leuphana.de, g  a.h.vandenboogaard@utwente.nl

Keywords: Hot Rolling, Lateral Spread, Finite Element Modeling, Symbolic Regression, Data-Driven Modeling.

Abstract

This work investigates the use of symbolic regression (SR) to address the trade-off between predictive accuracy and computational efficiency in modeling physical phenomena by constructing compact, closed-form expressions directly from data. In this study, SR is applied to develop fast and accurate models for predicting lateral spread in the hot rolling of steel slabs. The SR models are trained on high-fidelity finite element simulation data and evaluated against established analytical models. Model selection is guided by a parsimony-based optimization strategy that balances predictive accuracy and expression complexity. The results show that the SR-derived formulations achieve lower prediction errors with reduced complexity compared to traditional analytical models. Moreover, SR maintains strong predictive performance even when trained on limited datasets, demonstrating its robustness. Overall, the findings of this work highlight the suitability of symbolic regression for computationally efficient and accurate modeling of complex physical phenomena.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1050.157.pdf

Hot rolling is a key process in steel manufacturing, in which cast slabs are transformed into plates or strips while breaking down the as-cast microstructure and improving microstructural homogeneity [1]. As the thickness is reduced, the metal undergoes elongation in the rolling direction and spreads laterally, the latter being referred to as lateral flow or spread. Unlike processes where lateral deformation is largely constrained (e.g., cold rolling), lateral spread is a dominant phenomenon in hot rolling and must be accurately controlled to achieve the desired final width. Roughing hot rolling mills play a central role in shaping steel slabs, with rougher rolls reducing thickness and edger rolls controlling width. A schematic representation of a roughing stand is shown in Fig. 1. The final slab geometry and lateral spread at the stand exit result from the combined action of the rougher and edger rolls.
Maintaining tight tolerances over the thickness and width of the rolled slabs is essential to ensure product quality and guarantee consistent performance in downstream operations. Furthermore, precise profile control minimizes product rejection or the need for additional preprocessing. Therefore, accurate prediction of lateral spread is crucial for achieving dimensional control and optimizing the hot rolling process.

[picture]

Fig. 1: Schematic of a roughing hot rolling stand, showing the vertical edger rolls and horizontal rougher rolls acting on the strip.

Fig. 1. Schematic of a roughing hot rolling stand, showing the vertical edger rolls and horizontal rougher rolls acting on the strip.

The lateral flow in hot rolling results from complex interactions between material properties, rolling parameters, and temperature, which makes it challenging to model with sufficient accuracy and precision. On the one hand, finite element (FE) simulations can capture the full three-dimensional deformation of the slab and provide detailed insight into material and process behavior [2]; however, despite their accuracy, such models are computationally demanding and thus unsuitable for direct integration into real-time control or optimization frameworks. On the other hand, analytical and semianalytical models enable a fast prediction of lateral spread and are typically derived from plasticity theory, geometric relationships, and experimental observations. Although computationally efficient, many classical formulations oversimplify the complex physics of rolling; therefore, fail to predict lateral spread with sufficient accuracy, particularly under varying material and process conditions.
To reduce the trade-off between accuracy and efficiency in predicting complex physical phenomena, data-driven strategies have emerged, often in the context of hybrid modeling approaches [3]. These frameworks describe system behavior as the combination of a baseline mechanistic model, capturing established but simplified physical assumptions, and a residual contribution that remains unexplained by the model. Chinesta et al. [4] formalized this idea through the concept of modeler ignorance. Modeler ignorance denotes an epistemic model-form uncertainty, originally defined as the discrepancy between a physics-based model and the response of the real system; it may also be understood as the systematic discrepancy between a simplified analytical formulation and a more trusted high-fidelity reference description of the process. When experimental measurements are unavailable, this reference may be provided by a detailed FE model. Building on this concept, Bock et al. [5] proposed a predictive framework in which a fast but inaccurate analytical model is corrected towards high-fidelity FE solutions using machine learning to predict residual stresses induced by laser shock peening. A similar hybrid strategy has been applied to the prediction of lateral flow in hot strip rolling by Hashemzadeh et al. [6], who introduced an Analytical Predictor-Machine Learning Corrector framework in which analytical models provide initial width predictions that are subsequently refined using a machine learning model trained on high-fidelity FE data.
In addition to conventional machine-learning algorithms, symbolic regression (SR) has gained increasing attention as an alternative data-driven modeling approach. SR aims to identify explicit mathematical relationships between input and output variables through an optimization process that balances predictive accuracy and model complexity [7]. Unlike traditional regression methods that rely on prescribed functional forms, SR automatically explores combinations of variables and mathematical operators (such as addition, multiplication, and exponentiation) to discover concise analytical expressions that may support physical interpretation.
Within the broader landscape of data-driven strategies, SR can be employed in two complementary ways: as part of a hybrid modeling framework to learn model discrepancies [8], or as a modeldiscovery tool to directly derive analytical expressions that represent the underlying physical behavior encoded in the data [9, 10]. In this work, the second approach is adopted and symbolic regression is used to construct a compact analytical model for lateral spread. In the context of hot rolling, direct
experimental measurements of lateral flow are extremely difficult due to the harsh conditions in the rolling stand, including high temperatures and limited accessibility. As a result, high-fidelity FE simulations are employed as a surrogate model for the real physical system.
The remainder of the paper is organized as follows. First, the finite element framework used to simulate the hot rolling process is described, together with the generation of a high-fidelity data set that covers both rougher and edger rolling configurations and the adopted sampling strategy. Next, the symbolic regression methodology implemented using PySR [11] is presented, and the procedure for deriving compact analytical expressions for lateral spread from the simulation data is outlined. Finally, the resulting symbolic models are evaluated on an independent validation dataset and compared with established analytical formulations in terms of predictive accuracy and model complexity.

Methodology

The original version of this paper is available on https://www.scientific.net/KEM.1050.157.pdf

FE simulations. Because lateral spread is governed by three-dimensional plastic flow, its accurate prediction requires a fully three-dimensional modeling approach. For this reason, the dataset describing lateral flow as a function of the rolling parameters is generated through three-dimensional FE simulations performed in Abaqus. In the numerical model, rolls are represented as rigid cylindrical surfaces, whereas the slab is modeled as a deformable body. Consequently, elastic roll deformations, including roll bending, are not considered. Roll bending modifies the roll gap profile across the strip width, thereby influencing the deformation mechanics and resulting lateral spread. Within the present approach, this effect could be incorporated by extending the FE model, for example through coupling with the influence function method to account for roll deflection [12]. However, this extension falls outside the scope of this study, as the symbolic regression models are assessed against analytical formulations that likewise assume rigid rolls. To capture both rolling configurations of interest in roughing mills, namely the edger and the rougher, the same FE framework is employed to perform a series of independent simulations, each corresponding to a distinct slab cross-sectional geometry. In particular, the width-to-thickness aspect ratio is systematically varied to represent edger-type and rougher-type configurations, rather than modeling their successive occurrence within a single rolling sequence, as shown in Fig. 2. Accordingly, the two rolling stages are treated independently, and deformation history is not transferred from the edger to the rougher (e.g., strain accumulation is not used as input for subsequent passes). This simplification is intentional, as the objective is to generate predictive models for lateral spread under each configuration and to enable direct comparison with analytical formulations, which likewise neglect deformation history between rolling steps. The slab length is chosen to be sufficiently large to ensure that steady-state rolling conditions are achieved, thereby eliminating transient effects near the entry and exit regions and enabling a consistent evaluation of lateral flow.

[picture]

Fig. 2: Edger and rougher rolling configurations considered in the FE model.

Fig. 2. Edger and rougher rolling configurations considered in the FE model.

The slab has a length of 600 mm and an initial width of 6 mm . It is discretized using hexahedral elements, with the element size selected on the basis of a mesh convergence study. The final mesh resolution is set to a minimum element size of 3 mm in the longitudinal direction, 1.5 mm in the width direction, and 1.28 mm in thickness. This discretization is found to provide an appropriate balance between numerical accuracy and computational cost and is therefore kept constant across all simulations.
The material behavior is described using the Johnson-Cook constitutive model [13], which relates the von Mises flow stress ( σ ) to the equivalent plastic strain ( εp ), the equivalent plastic strain rate ( ε˙p ) and the temperature (T), according to the following equation:

σ=[A+Bεpn][1+Cln(ε˙pε˙0)][1−(T−TrTm−Tr)m].(1)

Table 1: Johnson-Cook material parameters [13].

Parameter | Value | Unit | Description
A | 33.901 | MPa | Initial yield stress
B | 100.18 | MPa | Strain hardening coefficient
n | 0.4951 | − | Strain hardening exponent
C | 0.2471 | − | Strain-rate strengthening coefficient
m | 0.6444 | − | Thermal softening exponent
ε̇₀ | 0.0037 | s⁻¹ | Reference plastic strain rate
Tr | 900 | °C | Reference temperature
Tm | 1500 | °C | Melting temperature

In this study, an isothermal analysis is performed at a constant temperature of 1100∘C and constant rolling speed of 2.2772rad/s. These operating conditions are selected in accordance with representative hot rolling conditions reported in the literature [14]. Roll-slab interaction is modeled using a surface-to-surface contact formulation in Abaqus with a kinematic constraint and finite sliding. Normal contact is enforced using a hard contact formulation, while tangential behavior follows an isotropic Coulomb friction law with a constant friction coefficient of 0.2 . Explicit simulations are employed [15] due to their robustness in handling complex contact conditions typical for rolling processes. To reduce computation time while preserving numerical stability, mass scaling is applied with a factor of 900, ensuring that the kinetic energy remains below 5% of the internal energy and that the response remains quasi-static.
The simulations are executed until steady-state conditions are achieved using the built-in steady-state detection capability in Abaqus. The detection criterion is based on the evolution of the equivalent plastic strain (PEEQ in Abaqus notation).
Input-output space for data generation. The rolling geometry is primarily characterized by a small set of geometric quantities that govern material flow and lateral spread, namely the slab width ( w ), slab thickness ( h ), and roll radius ( R ). These parameters control the degree of geometric constraint and curvature imposed by the rolls and therefore play a central role in the deformation kinematics. The subscripts in and out denote entry and exit conditions, respectively, as illustrated in Fig. 2.
To enable robust and scalable data-driven modeling, both input and output quantities are expressed in non-dimensional form. Non-dimensionalization normalizes the parameter space, improves numerical conditioning, and enhances generalization performance in data-driven models [16], while also ensuring dimensional consistency within the symbolic regression framework. Based on the characteristic geometric scales of the rolling process, the following three dimensionless input parameters are
defined:

x1:=win hin ,x2:=hin R,x3:=hout hin .

The parameter intervals are chosen as:

x1∈[0.5,2],x2∈[0.1,0.25],x3∈[0.5,1].

These ranges are selected to ensure that both rougher and edger configurations are adequately represented within the data set. Specifically, edger configurations correspond to win <hin (x1<1), while rougher configurations are characterized by win >hin (x1>1). The target output of the model is the exit strip width, which is likewise expressed in non-dimensional form as:

y:=wout R.

Sampling method. Once the parameter ranges are defined, the number of simulations that can be performed is limited by computational cost, making it essential to select combinations of input parameters that efficiently cover the design space. This process, referred to as sampling, can be carried out using various strategies, including structured grids, random sampling, Latin hypercube sampling, and quasi-random sequences such as Sobol points.
The choice of sampling strategy has a direct impact on the quality and generalizability of the symbolic regression results. A key criterion for evaluating sampling methods is the discrepancy, a quantitative measure of how uniformly a set of points fills the parameter space [17]. Lower discrepancy values indicate more uniform coverage and, consequently, more representative sampling. The centered discrepancy is a specific form of discrepancy that emphasizes uniformity around the center of the sampling domain and reduces sensitivity to boundary effects. The centered discrepancy values for the different sampling strategies considered in this study are reported in Table 2.
As shown in Table 2, Sobol sampling yields the lowest discrepancy and therefore provides the most uniform coverage of the parameter space. Since Sobol sequences are most effective when the number of samples is a power of two [18], 256 simulations were performed to generate the training dataset, representing a practical balance between computational cost and coverage of the input space. In addition, 128 independent simulations were generated for validation, using the same parameter ranges and Sobol sampling strategy but with a different random seed. The influence of the number of training samples on model accuracy is examined in a subsequent sensitivity analysis.

Table 2: Centered discrepancy values for the different sampling strategies using 256 samples in a three-dimensional input space.

Sampling Method | Centered Discrepancy
Grid | 0.014900
Random | 0.002068
Latin Hypercube | 0.000446
Sobol | 0.000045

Post-processing and deformed shapes. After completion of the simulations, post-processing scripts are employed to extract the final strip width for training the predictive model. The deformed cross section is evaluated at the location where steady-state conditions are reached, that is also where lateral flow has fully developed. The exit width, denoted as wout , is measured along the centerline of the deformed cross section and serves as a scalar measure of the accumulated lateral spread. Although this constitutes a simplification, since the actual cross section is generally non-uniform due to threedimensional material flow, this assumption is adopted to ensure consistency with analytical models, which also assume a rectangular cross section.

In addition to width extraction, the steady-state deformed shapes of the slabs are analyzed to assess the influence of the process configuration on lateral flow behavior. Owing to differences in boundary conditions and dominant deformation mechanisms, the rougher and edger configurations exhibit distinct deformation patterns. Figures 3 and 4 show representative deformed slab cross sections for the two configurations, together with contour plots of the maximum equivalent plastic strain (PEEQ, in Abaqus notation) for randomly selected samples.

[picture]

Fig. 3: Deformed slab cross-sections for representative rougher configurations, showing contours of equivalent plastic strain.

Fig. 3. Deformed slab cross-sections for representative rougher configurations, showing contours of equivalent plastic strain.

[picture]

Fig. 4: Deformed slab cross-sections for representative edger configurations, showing contours of equivalent plastic strain.

Fig. 4. Deformed slab cross-sections for representative edger configurations, showing contours of equivalent plastic strain.

Lateral spread originates from the lateral material flow induced by thickness reduction under the action of the rolls. For a given undeformed slab geometry (i.e., fixed width-to-thickness ratio), the magnitude of spread increases with increasing thickness reduction. Conversely, for a fixed amount of thickness reduction, the relative importance of spread increases as the width-to-thickness ratio decreases. For large width-to-thickness ratios, deformation in the central region approaches plane-strain conditions and lateral flow is strongly constrained, resulting in limited spread. As the width-to-thickness ratio decreases, the deformation becomes increasingly three-dimensional and lateral spread becomes more pronounced.

Symbolic regression. Symbolic regression is a machine learning technique that aims to discover analytical expressions that describe physical systems directly from data, without assuming a predefined model structure [19]. In contrast to conventional regression approaches, symbolic regression searches simultaneously for the functional form and the associated parameters, enabling the identification of governing equations from observational or simulation data. In this study, symbolic regression is implemented using PySR, a high-performance framework for scientific model discovery [11]. PySR represents candidate models as hierarchical expression trees, where mathematical operators form internal nodes and variables or constants appear as terminal leaves. The search over possible expressions is performed using evolutionary algorithms, including selection, crossover, and mutation, to identify models that best map the input parameters to the target output. The search space considered in this work includes the binary operators addition, subtraction, multiplication, and division, as well as the unary operators logarithm, sine, cosine, and exponential, allowing a broad class of nonlinear analytical expressions to be explored. To quantify model complexity, each symbolic expression is represented by its corresponding complexity tree, which explicitly reflects the number and arrangement of elementary operations. As an illustration, Fig. 5 presents the complexity tree corresponding to the following expression:

αtanh(βx+γ)(2)

[picture]

Fig. 5: Complexity tree representation of the symbolic expression in Eq. (2). Multiplicative coefficients are shown in red, the additive bias in blue, and the input variable in black.

Fig. 5. Complexity tree representation of the symbolic expression in Eq. (2). Multiplicative coefficients are shown in red, the additive bias in blue, and the input variable in black.

For this illustrative example, the resulting complexity is 8 , corresponding to the number of elementary operations and terms involved in the expression. Notably, this functional form is equivalent to the computation performed by a single neuron with a hyperbolic tangent activation function in a neural network. Even at this level, the corresponding complexity tree already has a complexity of 8 . For a shallow neural network with a single hidden layer and three neurons, the explicit analytical representation would consist of the superposition of three such expressions, resulting in a complexity of 28. For deeper or wider architectures, the complexity grows substantially. In contrast, symbolic regression yields a single closed-form expression whose complexity is explicitly controlled, resulting in compact formulations that retain transparency and may support physical interpretation.
An important feature of PySR is its ability to balance model accuracy and complexity by optimizing along the Pareto front, according to the principle of the Occam's razor [19]. This enables the selection of expressions that achieve an optimal trade-off between predictive performance and analytical simplicity. In this study, a total of 256 simulations are used to train the model using PySR. The resulting symbolic models are subsequently validated using 128 independent simulations covering the same parameter ranges. Based on this validation set, the expressions are evaluated in terms of both prediction error and model complexity, and the optimal model is selected accordingly.

Analytical models. Three analytical models are used to benchmark the symbolic regression model [14]: the Ekelund, Tselikov, and Helmi-Alexander models.
The Ekelund model computes the exit width wout  from:

wout 2−win 2=[8δ−4ln(wout win )[hin +hout ]]Rδ[1.6μRδ−1.2δhin +hout ](3)

where δ=hin −hout .
The Tselikov model is given by:

Δw=wout −win =2cϵh2[Rδ−δ2μ][(1−ϵh)2ln(11−ϵh)−ϵh+1.5ϵh2](4)

where ϵh:=(hin −hout )/hin  denotes the relative draught, the parameter c, known as the spread factor, depends on the ratio between the sample width and the projected contact arc (that is, hout /Rδ ) and typically takes values in the range [0.5,1].
Finally, the Helmi-Alexander model is expressed as:

ln(wout win )=0.95ln(hin hout )(hin win )1.1exp(−0.707(win Rδ)(hin win )0.971).(5)

Results and Discussions

The original version of this paper is available on https://www.scientific.net/KEM.1050.157.pdf

SR-derived vs. analytical models. This study evaluates the performance of symbolic regression in comparison with the analytical models established by Ekelund, Tselikov, and Helmi-Alexander. The comparison is conducted using two complementary strategies: (i) assessing the predictive accuracy of symbolic expressions constrained to a complexity comparable to that of the analytical models, and (ii) examining the complexity required by symbolic regression to achieve prediction errors comparable to those of the analytical formulations.
To quantify the structural complexity of the analytical models, each formulation is represented by a complexity tree, following the structure illustrated in Fig. 5. The resulting complexity values for the analytical models, expressed as the total number of nodes in the corresponding trees, are reported in Table 3.

Table 3: Complexity of analytical models.

Model | Complexity (nodes)
Ekelund | 37
Tselikov | 30
Helmi-Alexander | 31

The first analysis focuses on the former perspective, in which symbolic regression is constrained to produce a model of similar complexity to the analytical formulations and the resulting predictive performance is assessed. Under this constraint, the PySR symbolic regression tool yields the expression reported in Eq. 6, which has a complexity of 29 and is hereafter denoted as SR 29.

woutR=[(x10.9894−0.0129)x2][(x31.3275−x2)0.0317]exp((x2x1)x2cos(x3)(1.0324−x3)).(6)

The prediction error of this equation is compared to that of the analytical models using the Mean Absolute Error (MAE) and R2, both computed based on the predicted slab width. Table 4 summarizes the
results and shows that the PySR-derived model achieves an MAE nearly one order of magnitude lower than that of the best-performing analytical formulations (Tselikov's and Helmi-Alexander's models).

Table 4: Performance comparison of models based on MAE and R2, ranked from worst to best MAE.

Model | MAE [mm] | R²
Ekelund | 2.0502 | 0.9808
Tselikov | 0.9123 | 0.9956
Helmi-Alexander | 1.7667 | 0.9741
SR 29 | 0.1485 | 0.9993

The analysis is then extended by relaxing the complexity constraint and identifying a symbolic model with predictive accuracy comparable to that of the analytical formulations. Under this condition, PySR yields the expression reported in Eq. 7, which achieves predictive performance similar to that of the best-performing analytical formulation, namely Tselikov's model (MAE =1.0124 mm,R2=0.9954 ), while exhibiting a substantially lower complexity of 7.

wout R=x1x2x3−0.3309(7)

These results demonstrate that symbolic regression consistently achieves a superior trade-off between complexity and prediction error compared to existing analytical models.
Parsimony. A parsimony metric is employed to compare PySR-generated expressions of varying complexity with the analytical models. Parsimony provides a single scalar measure that balances prediction error and model complexity, defined as:

 Parsimony = MAE +λ Complexity .(8)

Here, λ denotes the complexity penalty factor. Lower values of λ emphasize prediction error, whereas higher values penalize model complexity more strongly. In this study, λ=0.01 is selected. An overview of the results of the parsimony evaluation is provided in Table 5.

Table 5: Parsimony values for different model complexities with λ=0.01.

Model | Complexity | MAE [mm] | Parsimony
Ekelund | 37 | 2.0502 | 2.4202
Tselikov | 30 | 0.9123 | 1.2123
Helmi-Alexander | 31 | 1.7667 | 2.0767
SR 17 | 17 | 0.2374 | 0.4074
SR 19 | 19 | 0.2213 | 0.4113
SR 20 | 20 | 0.2184 | 0.4184
SR 22 | 22 | 0.2101 | 0.4301
SR 23 | 23 | 0.1865 | 0.4165
SR 24 | 24 | 0.1630 | 0.4030
SR 25 | 25 | 0.1586 | 0.4086
SR 26 | 26 | 0.1554 | 0.4154
SR 27 | 27 | 0.1534 | 0.4234
SR 28 | 28 | 0.1534 | 0.4334
SR 29 | 29 | 0.1485 | 0.4385

Among the SR-derived expressions, the model with a complexity of 24 (SR 24 in Table 5) has achieved the lowest parsimony score. The corresponding formulation is reported in Eq. 9.

woutR=exp((A−x3)cos(x3)(x2x2)x1)[sin(x3−x2)B(x1x2)].(9)

The fitted parameters in Eq. 9 are A=1.0207 and B=0.0478. The expression in Eq. 9 is identified as the best overall model when both prediction error and complexity are considered jointly.
SR-derived models for varying training dataset sizes. In the previous section, the expressions in Eq. 6, Eq. 7 and Eq. 9 were obtained using PySR based on a data set comprising 256 simulations. In this section, the influence of the size of the data set on the performance of the model is investigated by training PySR on progressively smaller subsets consisting of 128, 64 and 32 simulations and comparing the resulting models. Table 6 shows a comparison of PySR-generated models with a complexity of 29 (SR 29) trained with different numbers of simulations. The lowest MAE is obtained when using the full dataset of 256 simulations. As the number of simulations is reduced, the MAE increases, but remains lower than that of the analytical models for all dataset sizes considered (see Table 4). In addition, prediction errors remain relatively stable when the size of the data set is reduced from 128 to 64 and 32 simulations, indicating that symbolic regression can produce compact and accurate models even with limited training data.

Table 6: MAE of PySR-generated models (SR 29) for varying dataset sizes.

Number of simulations | MAE [mm]
256 | 0.1485
128 | 0.1700
64 | 0.1872
32 | 0.2334

Figure 6 presents the relationship between complexity and prediction error for the PySR-generated models trained with different dataset sizes. The graph shows that for model complexities above 20, the lowest error is obtained by the model trained on the full dataset of 256 simulations. Models trained on 128, 64, and 32 simulations exhibit higher errors, which remain comparable among these reduced dataset sizes. Across all cases, the prediction error stabilizes at a complexity of approximately 20, beyond which further increases in complexity result in negligible performance gains.

[picture]

Fig. 6: Comparison of PySR-generated models trained using different numbers of simulations.

Fig. 6. Comparison of PySR-generated models trained using different numbers of simulations.

Impact of rolling configuration on model prediction error. Model performance is further examined by distinguishing between rougher and edger rolling configurations for both analytical and PySRgenerated models. Although the analytical formulations were originally developed for rougher rolling, their MAE is also reported for edger-only and combined datasets for completeness.
Table 7 summarizes the resulting MAE values. As expected, the analytical models exhibit improved predictive performance when evaluated exclusively on rougher configurations, while their errors increase substantially for edger configurations. In contrast, the PySR-generated model with a complexity of 29 (SR 29) achieves consistently low MAE values across all configurations. Notably, the SR 29 model trained on the combined dataset maintains strong predictive accuracy, highlighting its robustness with respect to rolling configuration.

Table 7: MAE in mm for analytical and PySR-generated models across rolling configurations.

Model | Rougher | Edger | Both
Ekelund | 1.6884 | 2.6215 | 1.9982
Tselikov | 0.6158 | 1.3683 | 0.8657
Helmi-Alexander | 0.7529 | 3.6659 | 1.7201
SR 29 | 0.1423 | 0.0520 | 0.1485

Finally, Figure 7 shows the relationship between model complexity and prediction error for PySRgenerated models trained on rougher-only, edger-only, and combined datasets. At a complexity of approximately 20 , the combined model achieves low prediction errors that are close to the configurationspecific results and significantly lower than those of the analytical formulations.

[picture]

Fig. 7: Trade-off between complexity and prediction error for mathematical formula-generated models trained on rougher-only, edger-only, and combined datasets ( 256 simulations).

Fig. 7. Trade-off between complexity and prediction error for -generated models trained on rougher-only, edger-only, and combined datasets ( 256 simulations).

Conclusions

The original version of this paper is available on https://www.scientific.net/KEM.1050.157.pdf

This study presents a systematic application of symbolic regression to derive compact analytical models for predicting lateral spread in hot rolling. A structured workflow is employed, beginning with high-fidelity FE simulations and culminating in the identification of concise and accurate symbolic expressions.

A total of 256 FE simulations are performed using Sobol sampling to ensure uniform coverage of the input parameter space. The exit width of the rolled strip, normalized by the roll radius, is used as the target output for training the PySR framework. Multiple symbolic expressions are generated and subsequently validated using independent simulation datasets, and their predictive performance is assessed in comparison with established analytical models.
At comparable model complexity, the SR-derived expression achieves nearly one order of magnitude lower prediction error than the best-performing analytical formulation. Conversely, when constrained to match the accuracy of analytical models, symbolic regression yields expressions with substantially lower structural complexity. Furthermore, a parsimony-based analysis shows that the most balanced SR-derived model simultaneously achieves reduced complexity and superior predictive performance relative to all analytical models considered.
Unlike analytical formulations, which are developed specifically for rougher configurations, the symbolic regression models exhibit strong predictive performance across both rougher and edger rolling conditions. In particular, a single symbolic model trained on combined datasets maintains low prediction errors, indicating robust behavior over a wide range of geometrical configurations.
Overall, this work demonstrates that symbolic regression provides an effective alternative to traditional analytical modeling approaches by delivering compact analytical expressions with high predictive accuracy and low computational cost. The resulting formulations, characterized by explicitly controlled complexity, offer a promising basis for fast surrogate modeling of complex physical processes such as hot rolling.

Acknowledgements

The original version of this paper is available on https://www.scientific.net/KEM.1050.157.pdf

This research was carried out under project number T22008 in the framework of the Research Program of the Materials innovation institute (M2i) (www.m2i.nl) supported by the Dutch government. The authors also gratefully acknowledge Tata Steel for valuable discussions.

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The original version of this paper is available on https://www.scientific.net/KEM.1050.157.pdf

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Key Engineering Materials, ISSN: 1662-9795, Vol. 1050, pp 171-182

doi: 10.4028/p-ls5PTZ

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-19

Revised:
2026-02-16

Accepted:
2026-02-19

Online:
2026-04-16

Physics-Informed Recurrent Neural Networks with Kinematic Constraints for Large Deformation Metal Forming

The original version of this paper is available on https://www.scientific.net/KEM.1050.171.pdf

Francesco Munzone 1,a*, Javad Hazrati 1, b, Wouter Hakvoort 2,c, Redmer Van Tijum 3, d and Ton van den Boogaard 1,e

¹Nonlinear solid mechanics, Faculty of Engineering Technology, University of Twente, Enschede, Netherlands

2 Control and Mechatronics, Faculty of Engineering Technology, University of Twente, Enschede, Netherlands

3 Advanced Production Engineering, Engineering and Technology Institute Groningen, Faculty of Science and Engineering, University of Groningen, Netherlands

ª*f.munzone@utwente.nl, bj.hazratimarangalou@utwente.nl, cw.b.j.hakvoort@utwente.nl, d  redmer.van.tijum@philips.com, e  a.h.vandenboogaard@utwente.nl

Keywords: physics-informed, recurrent neural network, deep-drawing, design exploration.

Abstract

Metal-forming is a manufacturing process that involves non-linear elastoplastic deformations to shape a blank into a complex geometry. These processes are governed by numerous parameters, with significant influence on the final product. To analyse the effects of such parameters, large-scale finite element (FE) simulations are often conducted. However, these models are computationally expensive and often unsuitable for real-time analysis. To overcome these limitations, surrogate models have emerged as powerful alternatives. In this study, we propose a physics-informed recurrent neural network framework (PI-RNN) to evaluate displacements and strain tensor components. In particular, the latter are not a network output but are obtained through the application of kinematic relations. Given the initial configuration as input and the final configuration as output, it is possible to evaluate the deformation gradient. The local condition of impenetrability of matter is then injected into the loss to improve the estimation of displacements and strain tensors. The PI-RNN model is trained on data generated from FE simulations of a deep-drawing process. The accuracy of the model is evaluated on a test dataset (design points that the model does not see during training) using different error measures. The results show that the proposed PI-RNN model can reproduce the FE simulation results fairly well.

Introduction

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Most manufacturing processes aim to transform raw material into a complex-shaped final product via deformation steps. In sheet metal forming processes, the material is subjected to elasto-plastic deformations, in which many process parameters influence the quality of the final product. In manufacturing processes, design exploration, sensitivity analysis, and robust optimization are tasks that require thousands of simulations. Finite element (FE) softwares are typically employed to simulate the outcome of a forming process under different process conditions. This approach is generally considered expensive due to the number of simulations needed. A solution is offered by surrogate models, which act as approximators by learning the input-output relationship using data collected from expensive simulations. Once properly trained, they can replace the expensive model to provide fast and accurate predictions for previously unseen inputs. A widely employed class of surrogates is neural networks, which excel in learning complex and non-linear patterns, capable of handling (and preferred) in the case of large data. As for any surrogate models, neural networks learn the underlying system function based solely on data. The extraction of insight from the learned function is difficult due to the large number of parameters. Physics-informed neural networks (PINNs) offer a partial solution which relies on the automatic differentiation algorithm to compute partial derivatives of output with respect to inputs. This additional information is then injected into the loss of the network to force the solution to comply with the system equations. PINNs are widely
applied and studied in many scientific fields because of their ability to improve the generalization ability of the model based on knowledge of the system. Predictions are not purely data-driven, but also follow a logic dictated by the governing equations of the system, which usually takes the form of partial differential equations.

Elasto-plastic deformation problems are governed by complex mechanical and material equations. PINNs can exploit them to let the model find a solution that satisfies both data and governing laws. In the work of Niu et al. [1], a PINN framework is proposed to model elasto-plasticity in the case of large deformations. The model uses the initial configuration to predict displacements and the first Piola-Kirchhoff stress over the increment of a multi-step loading-unloading history. The loss includes equilibrium equations, boundary conditions, and constitutive relations, using automatic differentiation as a tool for the computation of partial derivatives. Another work by Maia et al. [2], in a multi-scale approach, substitutes the RVE micromodel of a unidirectional composite material (both path and time dependent behaviour) in the macro model using a physically recurrent neural network to describe the homogenized behaviour of the RVE in finite strain. In a later work [3], the model has been extended to consider anisotropic materials under the assumption of finite-strain.

Inspired by the current literature, we propose a physics-informed recurrent neural network (PIRNN) to learn the deformation evolution of a metal sheet subject to deep-drawing. The model is trained to learn the deformation history under different process conditions using the initial configuration as input. This allows the network to compute the deformation gradient 𝐅 through the evaluation of partial derivatives. The strain tensor components are then obtained by applying kinematic relations for large deformation. Moreover, 𝐅 is used to enforce the local condition of impenetrability of matter on the loss function as a physical constraint [4]. The surrogate model is trained to learn the effect of different combinations of process parameters. To keep this initial study manageable, only the variations of the coefficient of friction and the thickness of the sheet are considered. An evolutionary stochastic enhanced Latin hypercube is used to generate a space-filling design of experiments (DOE).

The content of the article is organized as follows. The first section introduces the DOE algorithm used for the construction of the design space and the kinematic relations necessary to construct the loss function of the proposed physics-informed recurrent neural network. In the second section, the accuracy of the model and its generalization ability are assessed on a test dataset composed of unseen combinations of features. Finally, in the third section, the conclusive remarks and potential future work are discussed.

Physics-Informed Recurrent Neural Network

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In this section, the surrogate modelling framework is discussed. Samples are selected from the design space using a search algorithm to find an optimal DOE, that is, a DOE in which the design points are spread out as evenly as possible across the design space. A physics-informed recurrent neural network is trained on the collected data. Partial derivatives of output variables with respect to inputs are exploited to evaluate the strain tensor via kinematic relations for large deformation.

Enhanced Stochastic Evolutionary Latin Hypercube.

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In surrogate modelling, a proper design of experiments (DOE) is essential to achieve high accuracy over the design space. A typical procedure for the identification of an optimal DOE starts with a random design 𝐗0. An update algorithm is in charge of manipulating it and returning a new design. Then, the two designs are ranked based on a criterion, and it is decided whether to discard the new design or substitute the old design with the new. With Latin hypercube (LH), new designs can be easily generated through the exchange of columns or rows of the structure without altering the property of the LH .

The chosen criterion quantifies the space-filling quality of the current LH design. However, ranking every possible LH is impractical, especially for high-dimensional cases. In the literature, it is possible to find search algorithms whose job is to optimize an initial LH design, increasing its spacefilling quality [5, 6]. In particular, the ESE algorithm proposed by Jin et al. (2005) [7], a variation of
the stochastic evolutionary algorithm proposed by Saab and Rao [8], uses a temperature parameter to balance global exploration and local exploitation.

The ESE algorithm consists of an inner loop and an outer loop. The inner loop is in charge of generating and accepting/rejecting new designs, based on the current threshold "temperature" Th value. The current design 𝐗 is randomly altered by swapping J columns, generating the design 𝐗try . The values of the optimal criterion allow the algorithm to accept or reject the 𝐗try , based on the Th. A function that generates a uniform random number between 0 and 1 is multiplied by Th. This allows the algorithm to choose a better or a slightly worse designs, promoting exploration of the design space. The outer loop controls the optimization process by adjusting Th. If a good design is found, Th is adjusted to locally search for the new optimal design. If no improvement is detected, Th is adjusted such that the algorithm searches for a combination of design points, escaping local minima. The parameters α1,α2, and α3 in charge of controlling Th are kept as given in the original work [7].

The optimal criterion chosen is the maximin criterion proposed by [9]. The optimal design is the one that minimizes the following function:

ϕp(X)=[∑j=1m Jj dj−p]−p(1)

where d is a list (d1, d2,…, dm) of distances in which dj represents the distance between two sample points in the design, and J is an index list (J1, J2,…, Jm) in which Jj is the number of pairs separated by the distance dj. The exponent is a positive integer. Here, a value of p=50 is chosen.

Deep drawing forming process.

The original version of this paper is available on https://www.scientific.net/KEM.1050.171.pdf

The case study is a single deep drawing stage, shown in Fig. 1. This forming process has been designed by Veldhuis et al. [10] to be sensitive to temperature, such that friction variations are caused by temperature-induced effects. Indeed, their sensitivity analysis showed that the coefficient of friction (COF) is one of the most dominant factors for the final critical-to-quality parameters (CTQs) such as the cup height, inner diameter, and outer diameter. Another parameter that showed to have a mild influence on the CTQs is the sheet thickness (th), which may vary within the metal sheet roll. Based on this, the design space is built considering the two aforementioned features.

[picture]

Fig. 1. The metal forming process used as a case study is the first stage of a deep drawing process.

Fig. 1. The metal forming process used as a case study is the first stage of a deep drawing process.

Based on the work of Jin et al. (2001) [11], a total of 40 design points (that is, FE simulations) are chosen to train the model. The test dataset is built using the same ESE-LH algorithm by sampling design points outside the training dataset. The respective range for each feature is based on the intervals reported in [10], which are [0.05, 0.2] for COF and [0.29, 0.31] for th. The FE software used is Marc Mentat 2022.1. The sheet is modelled as an axisymmetric object, with 300 elements over radial direction and 6 elements over thickness direction, for a total of 1800 quadrilateral linear elements.

Kinematic relations for large deformations.

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In sheet metal forming and forming processes in general, the workpiece is subject to large elastoplastic deformation. Infinitesimal strain theory does not hold up, and it is necessary to switch to a
more advanced theoretical formulation. Proper evaluation of strain measurement requires considering large deformation and large rotations.

The following derivations are based on Simo and Hughes work [4]. Let ℬ be the reference configuration of a body with particles 𝐗∈ℬ. With φ we describe the displacements of this body to the current configuration 𝒮 :

φ:ℬ⊂ℝ3→𝒮⊂ℝ3(2)

The function φ maps every particle 𝐗∈B to x∈𝒮. We can define the deformation gradient as the derivative of the deformation:

𝐅(𝐗)=∂φ(𝐗)∂𝐗(3)

The deformation gradient allows us to define the unphysical condition for which the volume of the particle cannot be smaller than zero:

J(𝐗)=det𝐅>0(4)

Through the deformation gradient, the left Cauchy-Green tensors are defined:

𝐁=FFT(5)

The deformation gradient is an invertible second-order tensor; thus, it can be decomposed through the polar decomposition theorem:

𝐅=RU=VR(6)

where 𝐑 is a proper orthogonal tensor, and 𝐔(𝐗),𝐕(𝐱) are symmetric, positive-defined tensors called the right and left Cauchy-Green tensors. In particular, Eq. 5 can be rewritten as follows:

𝐁=𝐕2(7)

The tensor 𝐁 is symmetric and positive definite, hence by the spectral theorem:

𝐁=∑i3λi2𝐧(i)⊗𝐧(i)(8)

Consequently, the left stretch tensor can also be written in terms of eigenvalues and eigenvectors:

𝐕=∑i3λi𝐧(i)⊗𝐧(i)(9)

A widely used strain measure is the logarithmic strain, or Hencky strain. From Eq. 9, the Hencky strain can be written as:

ln𝐕=∑i3lnλi𝐧(i)⊗𝐧(i)(10)

Physics-Informed Recurrent Neural Networks.

The original version of this paper is available on https://www.scientific.net/KEM.1050.171.pdf

Physics-informed neural networks (PINNs) are a class of neural networks in which the governing equations (or physical relations) of the system are placed into the model loss function as additional penalty terms. Generally, the physical laws describing a system are in the form of partial differential equations or involve partial derivatives. Moreover, if known, boundary conditions and initial conditions can also be as additional constraint in the loss function due to their involvement in solving PDEs. To evaluate partial derivatives, PINNs take advantage of the automatic differentiation (AD)
algorithm. Nevertheless, it is not prohibited to use numerical methods for the evaluation of such derivatives. The general form of the loss function is the following:

ℒ=ℒdata+ℒPDE+ℒBC+ℒIC(11)

where each term is referred to as labelled data ( ℒdata  ), PDEs solution ( ℒPDE  ), boundary conditions ( ℒBC ), and initial conditions ( ℒIC ).

Many variants of PINNs exist based on the type of architecture used. Here, the problem under study is a sheet metal forming problem, thus subject to plasticity and large deformation. The history of the process plays a crucial role in the evolution of the geometry. Therefore, recurrent neural networks are a reasonable choice as the base architecture for the network. The model takes as inputs the initial configuration, the displacement of the punch, the time vector, the coefficient of friction, and the sheet thickness. The initial configuration spatial coordinates, the displacement, and the strain components are sampled at the boundary nodes and at random integration points. The coefficient of friction and the thickness are used, instead, to build the DOE. The primary output of the network are the displacements 𝐱 throughout the deformation process, whereas the components of ln𝐕 are secondary quantities computed from it via the kinematic relations. The relation between the initial configuration (inputs) and the current configuration over time (output) allows us to evaluate 𝐅 through the AD algorithm, and the kinematic relations to evaluate the components of ln𝐕.

The current PI-RNN model, does not utilize the classic loss terms introduced in Eq. 11; that is, the PI-RNN does not leverage partial derivatives to solve a PDE. Both displacement and strain loss terms are evaluated through the mean square error using the data collected from FE simulations. The physical constraint is represented by the local condition of impenetrability of matter introduced in Eq. 4. Therefore, the loss function can be written as follows:

ℒ=ℒdisp +ℒstrain +ℒimp (12)

where each term is described as follows:

ℒdisp =‖xi−xi*‖2ℒstrain =‖(lnV)ij−(lnV)ij*‖2ℒimp =ln(1+e−kdet F)

(13)(14)(15)

The factor k in Eq. 15 is necessary to reduce the slope of the softplus function and make the contribution of this loss term negligible whenever the local impenetrability condition uphold (Eq. 4). In particular, for metal deformation problems det𝐅 is close to one. The values for k is manually tuned beforehand to ensure a that the transition between negative and positive values of det𝐅 is sharp, but still smooth. Here, k is set to 30 .

It is worth noting that no scaling weights are employed in the current implementation. Instead, differences in scale are addressed by standardizing both inputs and outputs. This ensures that the loss terms remain comparable in magnitude, preventing any single term from dominating the optimization process.

Learning rate tuning.

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Due to the contribution of partial derivatives in the loss function during training, PINNs becomes sensitive to the choice of learning rate. An efficient technique to assess an optimal learning rate value is to train the model for a small number of epochs, with the learning rate increasing over the iterations. As the learning rate increase, the model learns faster, resulting in lower values of the loss function, until the learning rate becomes too large, causing the model to fail.

The learning rate is increased from 10−10 to 0.1 using the following evolution formula:

β=β0(βfβ0)γ(16)

with β0 the initial learning rate, βf the final learning rate, and with γ the ratio between the current number of iterations and the maximum number of iterations. The number of epochs, thus the number of iterations, is arbitrary. A good trade-off is a maximum number of iterations that ensure a smooth evolution of β and a rapid evaluation of the loss curve in prediction. Here, it is chosen to train the model for approximately 1000 iterations, corresponding to 10 epochs. The algorithm is run on an architecture consisting of 2 hidden layers and 128 nodes. In Fig. 2, the validation loss evolution versus the learning rate is shown. Smith [12] suggests two methods to determine the optimal learning rate.

[picture]

Fig. 2. Learning rate versus validation loss used to identify the optimal learning rate range.

Fig. 2. Learning rate versus validation loss used to identify the optimal learning rate range.

The first method relies on the minimum of the curve, corresponding to the point after which the model starts to diverge and explode. In fact, in the article is suggested to set β=βmin /3. With this formula we find an optimal β=2.35·10−4. The second method, which the same author points out as the most reliable, is to select the point at which the validation loss is the steepest. However, it can be observed that the light blue curve in Fig. 2 is slightly noisy, which makes the identification of the region of interest difficult. Hence, the validation loss curve is smoothed using a Savitzky-Golay filter. The resulting smooth curve allows us to identify the point at which the loss is steepest, corresponding to a β=6.35·10−4. For most trainings performed in this article, the latter value is selected as the default learning rate enabling faster learning. However, its vicinity to the instability region may lead to model failure. Thus, the learning rate is kept flexible and open to being varied in the region defined by the two optimal values if difficulties in training are encountered.

Results

The original version of this paper is available on https://www.scientific.net/KEM.1050.171.pdf

Impact of model architecture on accuracy.

The original version of this paper is available on https://www.scientific.net/KEM.1050.171.pdf

The architecture of a neural network dictates the building blocks of the model, necessary to construct the input-output nonlinear relationship. In particular, the number of trainable model parameters is commonly used to define the complexity of the model. It is common knowledge that complex models are more versatile due to the high number of parameters but prone to overfitting. On the other hand, less complex models are less troublesome to train, but may not fully learn the underlying system relationship, resulting in underfitting.

[picture]

Fig. 3. Validation loss versus epochs for different architectures.

Fig. 3. Validation loss versus epochs for different architectures.

Different architectures are evaluated to check for complexity limits of the model. The architecture of the PI-RNN involves a set of stacked gated recurrent units (GRUs) as hidden layers and a linear layer as the output layer. The activation functions for the gates and for the recurrent unit are, respectively, the sigmoid and the hyperbolic tangent function [13]. The combination of layers and nodes is decided based on a full factorial design, in which the number of layers is l={2,3} and the number of nodes (per layer) is n={64,128,256}. The objective is not to find the optimal architecture, since this requires a much deeper analysis, but to assess the sensitivity of the model to these hyperparameters and check if model complexity leads to overfitting or underfitting. The models are implemented in the Tensorflow (v2.19) framework, trained using GPUs (NVIDIA L4). The optimizer used is Nadam with a learning rate chosen as defined in the previous section.

In Fig. 3, we can observe the validation losses over the epochs for the different architectures. The best validation loss is obtained by the model using 2 layers and 256 nodes, followed by the model with 2 layers and 128 nodes. All the remaining models seem to converge towards the same solution. From the validation losses, it can be noticed that the model responds better when the number of nodes is increased rather than the number of layers. In fact, it is typical for deeper networks to be more difficult to train, leading to accuracy degradation [14]. For the purpose of assessing model complexity, it is not necessary to tune the learning rate for each architecture, as long as the training converges towards an optimal solution. To avoid overfitting, in training, the learning rate is halved whenever the validation loss hits a plateau for more than 400 epochs (observe the drops in the losses in Fig. 3). A single GPU is able to host between 2 and 3 models before running out-of-memory. This influences computational time. Generally, a training time per epoch is observed between 45 and 50 seconds.

The learning rate is set to 6.35·10−4 for all models, except for the models using 2 layers and 128 nodes, 3 layers and 64 nodes, and 3 layers and 256 nodes, where a learning rate of 4·10−4 is used due to multiple failure of the model caused by the explosion of the loss term ℒimp . The unstable behaviour of ℒimp  is, however, still observable in Fig. 4 even after the adjustment of the learning rate. The cause seems not to be related to the model architecture, but rather on the random initialization of the model parameters. A solution may be given by initially warming up the model using lower learning rate values (chosen in the optimal region) till convergence is found. Then the learning rate is let increase over training to prevent the model to get stuck in (or unable to escape) local minima.

[picture]

Fig. 4. Validation ℒimp  versus epochs for different architectures.

Fig. 4. Validation ℒ imp versus epochs for different architectures.

Predictions on a test dataset.

The original version of this paper is available on https://www.scientific.net/KEM.1050.171.pdf

The test dataset is a composition of data collected from FE simulation performed using feature combination not seen by the model during training. The ESE algorithm is again employed to generate a test DOE. It is ensured that the resulting design points are distinct from those in the training DOE. The trained model evaluates the predictions at nodal spatial coordinates. With the only exception of the nodes at the boundaries, the nodal coordinates are unseen data points for the model. Moreover, the boundary values are still coming from a combination of friction and thickness never been observed by the model.

To assess the accuracy of the model over the design space, the fraction of variance unexplained (FVU) is used, which can be defined in terms of the coefficient of determination:

FVU=SSresSStot=∑i(yi−y^i)2∑i(yi−y¯)2=1−R2(17)

The nominator SSres  is the sum of squares of residuals, in which yi is the true value and y^i is the predicted values. The denominator SStot  is the total sum of squares, in which y¯ is the mean over all observations. In terms of R2, a surrogate model that returns a value of approximately 1 , means that SSres  is approximately zero, and the surrogate is an optimal representation of the system. On the other hand, an R2 close to zero means that the surrogate has the same representation ability as a model that always predicts y¯.

It is interesting to observe the FVU strain components over the DOE in Fig. 5. Here, the mean value y¯ is evaluated as the mean value per design point, that is, the mean evaluated using the collection of observations in space and time of one simulation. The FVU values are small overall, indicating the good approximation provided by the PI-RNN. However, the strain components lnV12 and lnV22 show a higher FVU for the combination of feature COF=0.1962 and th =0.3085 (simulation 17). Investigations on simulation 17 showed a concentration of error for both lnV12 and lnV22 in the spatial region in which the sheet comes in contact with the punch. For simplicity, only the absolute error AElnV12 is analysed. Every conclusion can also be applied to AElnV22. The error appears in the early stages of the deformation process (Fig. 6 (right)), but no irregularities are spotted in the FE results (Fig. 6 (left)).

[picture]

Fig. 5. FVU over the design space for the strain lnV11 (left), lnV12 (center), and lnV22 (right). The black squares represent the test design points.

Fig. 5. FVU over the design space for the strain ln V 11 (left), ln V 12 (center), and ln V 22 (right). The black squares represent the test design points.

[picture]

Fig. 6. FE prediction (left) and absolute error (right) for lnV12 at increment 167 of simulation 17, part of the test dataset.

Fig. 6. FE prediction (left) and absolute error (right) for ln V 12 at increment 167 of simulation 17, part of the test dataset.

Thus, the cause of such behaviour must fall on the PI-RNN and, in particular, on the training dataset. With this in mind, the FVU of each quantity is evaluated again over the design space using the training dataset. In Fig. 7 we can observe that for certain combinations of COF and th, all the fields show the same peaks of error. In particular, we will refer to simulation 16 as the combination of COF=0.1568 and th =0.2937 mm, which corresponds to the simulation returning the highest FVU values for lnV11.

[picture]

Fig. 7. FVU over the design space for the strain lnV11 (left), lnV12 (center), and lnV22 (right). The black squares represent the training design points, while the red the validation design points. The validation set is created by randomly splitting the original dataset into training and validation dataset with an 80:20 ratio.

Fig. 7. FVU over the design space for the strain ln V 11 (left), ln V 12 (center), and ln V 22 (right). The black squares represent the training design points, while the red the validation design points. The validation set is created by randomly splitting the original dataset into training and validation dataset with an 80:20 ratio.

[picture]

Fig. 8. FE prediction (left) and absolute error (right) for lnV12 at increment 167 of simulation 16, part of the training dataset.

Fig. 8. FE prediction (left) and absolute error (right) for ln V 12 at increment 167 of simulation 16, part of the training dataset.

Again, in the early stages of the deformation process, the error is located in the same region as the one observed in Fig. 6, but much higher in comparison. In particular, we can notice that in the FE simulation results, there is a jump in correspondence with the error concentration (Fig. 8 (left)), which is caused by a mesh distortion. Thus, the PI-RNN is learning the input-output relation based on unphysical conditions, leading to wrong predictions. In fact, the same mesh distortion is not observed in simulation 17, but the PI-RNN still predicts a jump in the strain components that does not exist. This can be avoided by examining the training dataset carefully, possible when dealing with a small dataset, but impractical for large datasets due to the mole of data that should be analysed. Nevertheless, more attention should be paid to the preprocessing of training data to discard possible poor FE simulations.

Fig. 9 and Fig. 10 show predictions versus true plots. The narrow blue band in Fig. 10 represents the 95% prediction interval; thus, the points falling outside the band are the remaining 5%. As observed in the FVU, the strain components show higher inaccuracy compared to the displacements. Displacements are a direct output of the PI-RNN, therefore easy to fit, while the strain comes from the various transformations of the deformation gradient, making fitting more complex, but still sufficiently accurate.

[picture]

Fig. 9. Prediction versus true of the displacements.

Fig. 9. Prediction versus true of the displacements.

[picture]

Fig. 10. Prediction versus true of the strain components. The blue band represents the 95% prediction interval.

Fig. 10. Prediction versus true of the strain components. The blue band represents the 95 % prediction interval.

Conclusion

The original version of this paper is available on https://www.scientific.net/KEM.1050.171.pdf

In this work, a recurrent neural network is used to learn the deformation evolution of a deep drawing process under different process conditions. The logarithmic strain tensor is evaluated using the deformation gradient obtained as the partial derivatives between the initial and current configuration, respectively input and output of the model. Moreover, F is used to impose locally the impenetrability of matter on the loss function as an additional constraint.

The PI-RNN is trained on data collected from FE simulations of the deep drawing process. Each simulation corresponds to a design point sampled in the design space with the help of the ESE-LH algorithm. The model is trained on displacement and strain components values sampled at the boundary nodes and at random integration points.

A preliminary analysis is performed to choose the optimal learning rate value to avoid model failure. Based on the literature, an optimal range for learning rate is found 2.35·10−4≤β≤6.35·10−4. Moreover, different architectures are tested using different combinations of layers and nodes to observe the complexity limitation of the model. All architectures converge towards a similar suboptimal minimum, with the best performance achieved by the model using 2 layers and 256 nodes. This investigation also showed the inclination of the model to fail due to instabilities of the loss term ℒimp , unrelated to the model architecture. The cause can be found in the random initialization of the model parameters.

The generalization ability of the model is tested using 20 design points unseen by the model during training. The FVU over the design space showed that the model is perfectly capable of predicting both displacement and strain components with high precision. The output standardization works well in putting the scalar quantities on a comparable scale. However, the PI-RNN generalization ability could be further improved by introducing weighting factors for each loss term to balance their contribution during the optimization process. Adaptive weighting methods based on gradient information may prove necessary to escape eventual local minima during training. The latter will be a subject of future work.

The current application of the PI-RNN considers a design space consisting of two features. In metal forming processes, many parameters play a crucial role in the shape of the final geometry. A foreseen extension of the model is the inclusion of additional features to perform sensitivity analysis and design exploration in the case of a larger DOE.

Acknowledgment

The original version of this paper is available on https://www.scientific.net/KEM.1050.171.pdf

This publication is part of the project Digital Twin (project 2.2) with project number P18-03 of the research programme Perspectief, which is (mainly) financed by the Dutch Research Council (NWO).

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The original version of this paper is available on https://www.scientific.net/KEM.1050.171.pdf

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This ePub file was created using machine-generated methods and automated software to convert the source text into an ePub format and may contain formatting inconsistencies from the original version. Formatting may vary depending on the device used.

The original version of this eBook is available on https://www.scientific.net/book/advanced-numerical-and-ai-strategies-for-material-forming/978-3-0364-1986-2

Key Engineering Materials, ISSN: 1662-9795, Vol. 1050, pp 183-192

doi: 10.4028/p-dWx8G5

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-19

Revised:
2026-02-19

Accepted:
2026-02-19

Online:
2026-04-16

An AI-Based Approach to Developing a Microstructural Model for Multi-Stage Hot Deformation Processes

The original version of this paper is available on https://www.scientific.net/KEM.1050.183.pdf

Denis Tretyakov 1,a, Nikolay Biba 2, b *, Vitaliy Belugin 1,c, Artur Gartvig 1, d, Andrei Shitikov 1,e and Sergey Stebunov 1,f

1 QForm Group FZ LLC, Fujairah - Creative Tower P.O. Box 4422 Fujairah, UAE

2 MICAS Simulation Ltd, 107 Oxford Road, Oxford, OX4 2ER, UK

atretyakov.denis.a@gmail.com, b  nick@qform3d.com, cbelugin@qform3d.com, 

d  gartvig@qform3d.com, e  a.a.sh.ggml@gmail.com, f  serg@qform3d.com

Keywords: Metals, rolling, microstructure, recrystallisation, FEM, deep learning, AI models.

Abstract

Predicting the microstructural state during manufacturing is critical, as it directly governs the material's final mechanical properties. Accurate prediction of microstructure evolution in multistage industrial hot deformation processes, such as rolling, is limited by the lack of experimental data at intermediate stages, where direct measurement is impractical. To address this, an integrated methodology combining finite element (FE) simulation in QForm UK® software, physical simulation using the Thermo-Mechanical Treatment Simulator (TMTS), and artificial intelligence (AI) is proposed and investigated.

The methodology is demonstrated for the 11 -pass hot rolling of a 41 Cr 4 steel bar. Thermomechanical loading histories from an FE model of the industrial process were used to design and simulate a targeted TMTS experiment, generating a synthetic dataset via an analytical JMAK model that combines multiple recrystallisation mechanisms. This data was used to train a recurrent neural network (RNN) with an augmented physics-informed Long Short-Term Memory (LSTM) cell to predict the totally recrystallised fraction (RX) solely from loading history data. The AI model achieved high accuracy when validated within the TMTS simulation domain, successfully capturing different recrystallisation regimes. Implementation within commercial FE software enabled direct prediction in the rolling process simulation, yielding promising predictive capability, particularly in regions with thermal histories similar to the training data, highlighting the critical importance of training data diversity.

This work establishes a proof of concept for a novel calibration methodology, where targeted physical simulation bridges the gap between industrial process complexity and data-driven AI model development, offering a practical solution for modelling scenarios where traditional experimental calibration is infeasible.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1050.183.pdf

Predicting the microstructural state of a material, or potential micro-defects, is especially challenging for complex multi-step industrial forging or rolling processes [1,2]. One of the main difficulties is the lack of experimental data for the process's intermediate stages.

For example, in the rolling process, the piece of metal moves continuously from one stand to another, but we have the material's state only at the beginning and end of the whole process. While modelling a single operation, we can assess microstructure, metal flow, and defect formation with quite high accuracy. Sometimes it is even possible to stop forging halfway to observe microstructural development, or to drill channels to visualise material flow. All this provides enough data to calibrate and validate mechanical and microstructural models of a material, or to train corresponding AI models.

In the case of multi-step processes like rolling, AI models could be highly efficient at capturing the dependence of the result on the large number of technological parameters involved. However, the shortage of experimental data needed to train these models is a real bottleneck.

One solution to this problem could be the development of a methodology and testing equipment capable of reproducing, in laboratory experiments, the most important thermomechanical histories (trajectories) observed in the real multi-step process. This paper is devoted to the proof of concept for this idea.

As an example of a complex technological process, the hot-rolling of a steel round bar on a continuous mill was chosen, and the Thermo-Mechanical Treatment Simulator (TMTS) was selected as the available equipment for simulation experiments.

A few conceptual questions were investigated:

How is it possible to design an experimental procedure on the TMTS machine capable of imitating the thermomechanical histories of the rolling process?

Is the amount, quality, and nature of data obtainable from TMTS experiments suitable and sufficient for training an AI microstructural model?

Is an AI microstructural model trained on data obtained from TMTS tests capable of providing reasonable predictions for the real technological process, to what extent and with what limitations?

The investigation of these questions was performed using synthetic data obtained from finite element (FE) simulations of both processes using the commercial metal-forming software QForm UK® and microstructural data simulated using an analytical JMAK-type model. The FE simulation was conducted using the dedicated longitudinal rolling module, as described in [3], which enables full-cycle simulation of a profile rolling process. As for the AI model, a Recurrent Neural Network (RNN) was used with an augmented long short-term memory (LSTM) cell [4], which has previously shown promising results on simpler single-step processes [5] but has been challenging to evaluate on longer, multi-step industrial processes.

Methodology - Hot Rolling

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Process description. Hot rolling is a metal forming process where a heated billet is passed through a series of rolling stands to progressively reduce its cross-section and elongate it [6]. In our case, a 100 mm square billet of 41Cr4 steel is rolled into a 28 mm diameter round bar through 11 sequential stands of a continuous mill type 250 (Fig. 1). Each stand consists of grooved rolls that deform the billet, causing a decrease in cross-sectional area known as the draft or reduction and corresponding increases in length (elongation) and width (spreading).

[picture]

Fig. 1. Rolling mill and simulation model: (a) the example of industrial rolling stands during hot deformation of the billet on a continuous mill; (b) corresponding FE representation of the first two stands and billet temperature in simulation.

Fig. 1. Rolling mill and simulation model: (a) the example of industrial rolling stands during hot deformation of the billet on a continuous mill; (b) corresponding FE representation of the first two stands and billet temperature in simulation.

During each pass, the metal flows to fill the roll groove, and its volume remains approximately constant, aside from minor losses to scale. Thus, reducing the cross-sectional area forces the length
to increase. The elongation coefficient for a pass is defined as the ratio of exit length to entry length. In this 11-pass schedule, the elongation coefficients range roughly from 1.2 to 1.4, meaning the bar's length increases by about 20−40% in each pass. Table 1 summarizes the elongation coefficients, spreading, and absolute reduction for all 11 passes of this rolling process from initial square to final round (Fig. 1b, 2).

Table 1. Rolling reduction schedule for 100 mm to 28 mm bar (11 passes)

Pass No. | Elongation coefficient | Spreading [mm] | Absolute reduction [mm] | Rolling speed [m/s]
1 | 1.28 | 11.15 | 30 | 0.36
2 | 1.22 | 11.71 | 31.15 | 0.44
3 | 1.21 | 5.57 | 18 | 0.53
4 | 1.26 | 5.31 | 15 | 0.67
5 | 1.39 | 13.34 | 31.59 | 0.93
6 | 1.33 | 12.42 | 30 | 1.23
7 | 1.38 | 14.19 | 28.76 | 1.7
8 | 1.2 | 10.24 | 21 | 2.05
9 | 1.34 | 11.38 | 21.43 | 2.74
10 | 1.21 | 7.69 | 15.5 | 3.32
11 | 1.32 | 7.99 | 15.57 | 4.37

As the cross-sectional area decreases, the linear speed of the material must increase to maintain constant volume flow through each stand. Throughout the process, the rolls are water-cooled. In the simulation model this roll cooling effect is included as a thermal boundary condition on the rollworkpiece interface to ensure realistic temperature conditions during deformation.

[picture]

Fig. 2. Simulation of passes 3 to 11 of the rolling process in QForm UK® (longitudinal cross-section). Vertical rolls are used in the model to represent billet rotation between passes. Each stand is marked according to the corresponding pass number listed in Table 1. Passes 1 and 2 are simulated as a separate operation and are shown on Fig. 1b.

Fig. 2. Simulation of passes 3 to 11 of the rolling process in QForm UK® (longitudinal cross-section). Vertical rolls are used in the model to represent billet rotation between passes. Each stand is marked according to the corresponding pass number listed in Table 1. Passes 1 and 2 are simulated as a separate operation and are shown on Fig. 1b.

Challenges in measuring in-process data on a real mill. In an actual rolling mill, it is extremely difficult to obtain measurements of the material's state between stands. Mill operations cannot be halted mid-process to extract a sample, since the workpiece is at high temperature and moving at speed. As a result, plant engineers typically know only the input conditions (billet dimensions, temperature, etc.) and the final output properties (e.g., final dimensions, surface quality, mechanical properties after rolling). What happens to the microstructure or temperature in between passes is largely inferred rather than directly observed [7]. For example, one might like to know the austenite grain size after the third stand to calibrate a microstructure evolution model, but there is no practical way to stop the process and physically examine the grain structure, as the material must continue rolling to the end. These inherent experimental difficulties mean that purely empirical approaches cannot fully explain the evolution of microstructure and temperature during multi-pass rolling. We
have only snapshots before and after the rolling sequence, but not the full picture of what happens at each stage.

Because of this lack of intermediate data, numerical simulation becomes a crucial tool. By simulating the process, we can look at the state of the material between stands, for example, predicting temperature gradients, phase transformations, or grain size after each pass, the information that cannot be measured on the operating mill. Simulation allows us to adjust and optimise the rolling schedule.

[picture]

Fig. 3. Boundary conditions applied in the QForm UK® simulation of hot rolling: (A) extension boxes used for time-scaling to account for inter-stand distances; (B) water cooling; (C) rolling direction.

Fig. 3. Boundary conditions applied in the QForm UK® simulation of hot rolling: (A) extension boxes used for time-scaling to account for inter-stand distances; (B) water cooling; (C) rolling direction.

Accounting for real process time and cooling in the simulation. One important consideration in simulating multi-pass rolling is time scaling, which ensures that the thermal history in the model matches reality. In a real mill, the stands are relatively far apart, and it takes some time for the material to travel between them, while the total bar length increases significantly. Directly simulating an entire 50−100 m long bar moving through a long rolling mill is computationally expensive. Instead of a straightforward but computationally inefficient approach, a more specialised method based on socalled "extension boxes" was used in the simulation (Fig. 3) [8]. An extension box is essentially a segment in the model that introduces a delay or extra length of material to simulate the distance between mill stands. When the workpiece enters an extension box region, no further plastic deformation occurs, but the simulation continues to solve heat transfer for a specified duration, allowing the material to cool similarly as it travels the real large distance to the next stand. In other words, we scale the simulation time to match the real process time for each interval. This ensures that phenomena such as inter-pass cooling and any metallurgical transformations (e.g., static recrystallisation or phase changes during transportation between stands) are captured.

Methodology - Thermo-Mechanical Treatment Simulator

The original version of this paper is available on https://www.scientific.net/KEM.1050.183.pdf

Description and simulation. The Thermo-Mechanical Treatment Simulator (TMTS) by Servotest Testing Systems Ltd is a sophisticated laboratory system capable of replicating industrial metal forming processes, combining high-force hydraulic deformation with precise heating (furnaces, induction) and rapid quenching to study how heat and mechanical stress affect metal properties, useful for research and development involving advanced materials like steels and alloys. It uses servocontrolled manipulators to move specimens between heating, deforming, and cooling stages, simulating complex industrial processes and has thus been widely used to simulate hot-rolling [9].

When combined with FE modelling of the rolling process, TMTS can be used to replicate the thermomechanical histories and conditions in the investigated material. To achieve these deformation histories (i.e., temperature, plastic strain, strain rate, mean stress, triaxiality, etc.), their graphs can be exported from rolling simulations, and then a TMTS experiment can be carefully designed to match these histories (Fig. 4).

The TMTS machine is well-suited for reproducing many aspects of complex multi-step industrial processes. In the context of the 11-step rolling chain considered in this study (Table 1, Fig. 2),
deformation levels beyond the sixth pass would result in extremely high strains exceeding 2.0, which, under the TMTS's uniaxial loading regime, would produce specimens thinner than 1 millimetre. To ensure realistic and reliable experimental conditions, the investigation therefore focused on the first six rolling steps. Importantly, the thermomechanical conditions in these initial stages were readily and accurately replicated in the TMTS FE simulation, enabling high-quality data generation for AI model development.

[picture]

Fig. 4. TMTS experimental setup and simulation model: (a) photo of TMTS machine tools with kind permission of Servotest Testing Systems Ltd; (b) corresponding simulation model; (c) example of tracing point placement over the workpiece's cross-section at the end of one of the deformation operations.

Fig. 4. TMTS experimental setup and simulation model: (a) photo of TMTS machine tools with kind permission of Servotest Testing Systems Ltd; (b) corresponding simulation model; (c) example of tracing point placement over the workpiece's cross-section at the end of one of the deformation operations.

Methodology - AI Microstructure Evolution Modelling

The original version of this paper is available on https://www.scientific.net/KEM.1050.183.pdf

Synthetic data used in trial. As the final goal of this investigation is to construct an AI model capable of predicting complex microstructural evolution during multi-step industrial processes, and the data is synthetic, a target model for AI prediction must first be chosen. For this, a standard version of the Johnson - Mehl - Avrami - Kolmogorov (JMAK) [10] model included in QForm UK® software was chosen [11]. As this data is synthetic, custom coefficients for the model could be chosen. Thus, coefficients were tailored so that all types of recrystallisation (Dynamic (DRX), Meta-Dynamic (MRX) and Static (SRX)) would play a significant role in the process (Table 2, Fig. 5). However, during AI model training, only the total recrystallisation (RX) fraction was used as the target output. This reflects experimentally accessible data: total RX can be measured across a specimen's crosssection using optical or electron microscopy, whereas individual mechanisms and their interactions are far more difficult to quantify.

Table 2. Model parameters for each type of recrystallisation mechanism from QForm UK® microstructure evolution module [11].

Critical strain |  | DRX |  | SRX |  | MRX |
Ac | 0.00396015 | βd | 0.693 | βs | 0.693 | βm | 0.693
Mc | 0 | Ad | 8e-11 | As | 4 | Am | 3.16
Lc | 0.1238 | Md | 0.2 | Ms | 0 | Mm | 0
Qc | 49520 | Nd | 0 | Ns | -0.75 | Nm | -0.75
Cc | 0 | Ld | 0.1 | Ls | 0.1 | Lm | 0.1
|  | Qd | 270000 | Qs | 45000 | Qm | 45000
|  | Cd | 0 | Cs | 0 | Cm | 0
|  | Kd | 1 | Ks | 1 | Km | 1

Methodologically, predicting multiple JMAK mechanisms is also considerably more challenging for a single AI model, making total RX a practical and robust target. At the same time, this strategy tests whether the model can implicitly distinguish and reproduce qualitatively different
recrystallisation behaviours using only thermomechanical loading histories and the resulting total RX evolution. No explicit information about the underlying mechanisms is provided, so any successful differentiation emerges purely from the learned correlations.

[picture]

Fig. 5. JMAK model values traced from a point in the centre of the specimen over full TMTS simulation: (a) plotted over simulation steps; (b) plotted over time. Pi - denotes the i-th deformation operation, replicating the i-th rolling pass (Table 1), with heat-treatment operations in between.

Fig. 5. JMAK model values traced from a point in the centre of the specimen over full TMTS simulation: (a) plotted over simulation steps; (b) plotted over time. Pi - denotes the i -th deformation operation, replicating the i -th rolling pass (Table 1), with heat-treatment operations in between.

AI model inputs. As the chosen AI architecture is an RNN, it takes as input complete thermomechanical histories in order to predict RX evolution along those trajectories. To be more specific the following values are passed to the model at each step directly from the FE simulation: time step [s], temperature [ ∘C ], accumulated plastic strain, strain-rate [1/s], effective stress [MPa]; and also the following histories are calculated on top of these values and also passed to the model: plastic strain increment, effective stress increment, plastic work, plastic work increment and the Zener-Hollomon parameter. Thus, the model takes 10 values as input at each time step. These values are also linearly normalised before use in the AI model. The specific set of input histories was chosen based on the author's previous microstructural modelling experience. More specific information on this choice and how these values are calculated can be found in [5].

Data preparation. Only the TMTS simulation is used to train the AI model. For this at the end of each operation two arrays of points are mapped over the middle cross-sections of the specimens working area: a 7 by 7 array in the XZ plane and a 14 by 7 array of points in the YZ plane (Fig. 4c). This is done for each of the 6 heat-treatment and 6 deformation operations for a total of 1680 training points. Each of these points' paths is tracked from the beginning of the simulation to where they were created. Then, full 10 -value model histories listed above are exported, as well as final single total RX values, which serve as the target prediction values for the AI model during training.

This is repeated two more times for simulations with doubled and quadrupled simulation step lengths for a total of 5040 training pairs of thermomechanical loading histories and corresponding total RX values. Histories from simulations with doubled and quadrupled simulation step lengths should be included in the dataset, as the AI model doesn't have the same considerations built into its architecture as a standard incremental analytical model, given that its behaviour is primarily datadriven. Expanding the dataset in this way will help stabilise the model's behaviour for histories with different step lengths, which is a necessary quality for it to be usable in conjunction with FE simulation. This data is also generally useful for expanding the AI training dataset and for verifying the input simulation. As expected, RX values are identical at the same points in simulations with different time steps.

AI model and training description. As mentioned earlier, the AI model architecture consists of an RNN model, which takes as input 10 time series of arbitrary length. The recurrent cell chosen for the model is an LSTM cell [4] with a custom physics-based augmentation. The cell maintains hidden state vectors for long-term (C) and short-term (H) memory of size 10, and is augmented such that the first value in C is strictly non-decreasing and is bounded between 0 and 1 , ensuring it respects RX physical constraints, as this value is used as the model's scalar output.

The model was trained for 10000 epochs with the Adam optimiser, minimising the mean square error between RX values and the model's output predictions at the chosen points. 90% of the dataset was used for training, with the remaining 10% reserved for validation.

This architecture was previously trained on experimental data and successfully used to predict recrystallisation during single-step hot forging of Inconel 718 [5]; however, multi-step processes remain particularly challenging due to the lack of an experimental methodology to address them, which this paper aims to address.

Results

The original version of this paper is available on https://www.scientific.net/KEM.1050.183.pdf

After the model was trained, two methods were employed to test its predictive capabilities, both in TMTS and rolling simulations:

Thermomechanical loading histories were exported from FE simulations at key points and were used as model input for direct comparisons.

The trained model was implemented as a Lua user subroutine for QForm UK®, allowing it to be tested directly in FE simulation.
AI predictions in TMTS simulation. Both training and validation accuracies during the calibration process exceeded 99%. When tested in the TMTS FE simulation, JMAK RX fields match AI-predicted RX fields nearly perfectly (Fig. 6) across the entire specimen volume and all simulation steps.

Traced data plots of JMAK and AI predicted RX values in several key points show that the AI model is able to predict RX evolution properly in completely different loading regimes (Fig. 6c). These specific points are not from the set of all training points (Fig. 4c), however they do come from the same overall domain and have very similar loading histories to some of the points used in training.

[picture]

Fig. 6. Comparison of initial JMAK model and AI predictions in TMTS simulation: (a) JMAK RX field and point locations; (b) AI predicted RX field; (c) JMAK and AI predicted RX values plotted over full simulation duration for points highlighted in (a, b). Pi - denotes the i-th deformation operation, replicating the i-th rolling pass (Table 1), with heat-treatment operations in between deformations.

Fig. 6. Comparison of initial JMAK model and AI predictions in TMTS simulation: (a) JMAK RX field and point locations; (b) AI predicted RX field; (c) JMAK and AI predicted RX values plotted over full simulation duration for points highlighted in (a, b). Pi - denotes the i -th deformation operation, replicating the i -th rolling pass (Table 1), with heat-treatment operations in between deformations.

AI predictions in the initial rolling process simulation. A qualitative analysis of the results shows that the AI model has indeed identified the different types of RX models and can distinguish between them solely based on thermomechanical loading conditions, even within a highly complex FE simulation of a multi-step hot-rolling industrial process.

Quantitatively, the results are interesting. As shown in (Fig. 7), in some points (1, 5, 7), prediction results are very accurate, with errors under 5%RX. However, other points (2,3,4,6) exhibit errors ranging from 10 to 20% in the final RX prediction. In point where prediction errors are significant, most of the deviation accumulates at the first few rolling passes, with parallel JMAK and AI predicted RX curves and no more error accumulation at later stages.

[picture]

Fig. 7. Comparison of the initial JMAK model and AI predictions in the original rolling simulation for multiple key points over the billet's cross-section.

Fig. 7. Comparison of the initial JMAK model and AI predictions in the original rolling simulation for multiple key points over the billet's cross-section.

Discussion

The original version of this paper is available on https://www.scientific.net/KEM.1050.183.pdf

This result is somewhat expected, as the TMTS simulation used for training was constructed solely from a single traced loading history from the rolling simulation, whereas different histories can be observed at different points within the billet. Indeed, the points where prediction is most accurate (Fig. 7, points 1,5,7 ) are closer to the outer edge, whereas the other points are closer to the billet's centre. This distinction is more obvious when analysing loading histories at these points (Fig. 8). The JMAK model, used for synthetic target data, is governed by three main parameters: the temperature, plastic strain, and strain-rate. The plastic strain and strain-rate histories are very close regardless of point location over the billet's cross-section (Fig. 8c,d). Temperature, however, varies significantly: points closer to the edge experience much more cooling (Fig. 8a,b). This difference is most significant during the first few rolling passes, consistent with the areas of error accumulation observed in the results for points closer to the billet centre.

The TMTS simulation was initially constructed based on a single loading history; as such, the training points mapped over its cross-section (Fig. 4c) exhibit different but similar loading histories, representative of the variety of loading regimes found in the rolling process, in terms of plastic strain and strain rate, but not so in Temperature. Moreover, temperature deviation during the TMTS simulation was generally much lower than in the rolling simulation, with a nearly uniform temperature field throughout the workpiece volume. From this, it can be inferred that the AI model had very little information from which to determine the significance of temperature for RX formation in the JMAK model, which, in reality, plays a large role. This significantly narrows the range of applicability of the trained model and highlights the importance of conducting multiple tests and or simulations with varied loading regimes to diversify the training data set. Ideally, when using the TMTS machine or similar equipment to replicate a multi-stage industrial process, histories from different parts of the billet should be considered for replication, along with some random variation in those histories, reflecting the possible instability in reproducing real-life industrial conditions.

[picture]

Fig. 8. (a) Temperature field over billet cross-section at time t* between rolling pass 3 and 4 (Table 1); curves over full 11-pass rolling simulation, characteristic for inside (3) and outside (7) points: (b) Temperature; (c) Plastic strain; (d) Strain rate.

Fig. 8. (a) Temperature field over billet cross-section at time t * between rolling pass 3 and 4 (Table 1); curves over full 11-pass rolling simulation, characteristic for inside (3) and outside (7) points: (b) Temperature; (c) Plastic strain; (d) Strain rate.

It is also of note that the JMAK model, used in this work as a substitute for experimental microstructural data, isn't able to faithfully represent all the mentioned mechanisms and cannot be truly predictive in a general sense for microstructure characteristics during complex hot metal forming processes. This has been shown by various independent benchmarking studies [1,2,12]. Real experimental data could vary significantly and could possibly be more challenging for the AI model to reproduce. However, the AI model is a much more robust and generalised tool and isn't actually tied to the JMAK model in any way. The AI model's ability to reproduce JMAK model results in complex loading regimes is only a reflection of its robust data-driven foundation. The model's applicability to experimental microstructural data has already been shown in single stage forging processes [5]. It's applicability to practical multi-stage processes as well as to a wider range of predicted values, such as grain size, will be thoroughly tested in further works.

Conclusion

The original version of this paper is available on https://www.scientific.net/KEM.1050.183.pdf

This paper presents a novel AI-based approach to developing microstructural models for multistage hot deformation processes. The core challenge, acquiring intermediate experimental data for model calibration in industrial settings, is addressed by integrating FE simulation in QForm UK®, physical laboratory testing using the TMTS machine, and a physics-informed artificial intelligence model.

The methodology was demonstrated through a case study of 11-pass hot rolling of 41Cr4 steel. A FE simulation of the process provided complete thermo-mechanical loading histories. Such histories informed a targeted Thermo-Mechanical Treatment Simulation (TMTS) test, also modelled in FE software, which replicated a critical stage of the rolling sequence to generate intermediate microstructural data. A JMAK model, incorporating DRX, SRX, and MRX mechanisms with specifically tailored coefficients, was used to synthesise a comprehensive microstructure evolution dataset. A recurrent neural network with a custom physics-augmented LSTM cell was then trained on this dataset to directly predict the total recrystallised fraction from the thermomechanical input histories.

The results demonstrate that the TMTS machine is a viable tool for reproducing thermomechanical loading conditions of multi-step hot rolling under laboratory conditions, although the use of uniaxial
compression limits the maximum achievable strain and restricts accurate replication of later rolling passes with high deformation. Additional experiments may also be required to capture the strong spatial non-uniformity of industrial billets, particularly due to temperature gradients and surface cooling. Despite these constraints, the combination of TMTS physical simulation and FE modelling proved highly effective for generating comprehensive thermomechanical and microstructural datasets, enabling a robust data-driven machine-learning approach to microstructure evolution modelling. The trained AI model was successfully implemented as a Lua subroutine in QForm UK® and validated against both TMTS and rolling simulations, confirming its accuracy and robustness. The main contribution of this work is a practical integrated framework that unifies FE modelling, physical simulation, and AI for modelling complex multi-step industrial processes where conventional experimental approaches are impractical, while emphasising the critical role of strategic experiment design in bridging laboratory and industrial conditions.

References

The original version of this paper is available on https://www.scientific.net/KEM.1050.183.pdf

Reshetov, A. and Bylya, O., 2023, April. Benchmarking mean-field models available in commercial FE software in application to two-blow forging of IN718 alloy. Materials Research Forum LLC.

Bylya, O., Reshetov, A., Stefani, N., Rosochowska, M. and Blackwell, P., 2017. Applicability of JMAK-type model for predicting microstructural evolution in nickel-based superalloys. Procedia engineering, 207, pp.1105-1110.

Stebunov, S., Biba, N., Vlasov, A., et al, 2023. Integration of automated roll pass design and simulation for the development of shape rolling technology. Materials Research Proceedings, 32, pp. 655-662.

Gers, F., 2001. Long short-term memory in recurrent neural networks, Doctoral dissertation, Verlag nicht ermittelbar.

Tretyakov, D., Bylya, O., Shitikov, A., Gartvig, A., Stebunov, S. and Biba N., 2024. The prospects of implementation of artificial intelligence for modelling of microstructural parameters in metal forming processes. Materials Research Proceedings, 41, pp. 2164-2173.

Gupta, N.K., 2021. Steel Rolling: Principle, Process & Application, London, UK.

Kim, J., Moon, Y.H., Kim, N.J. Multi-Scale Modeling of Microstructure Evolution during Multi-Pass Hot-Rolling and Cooling Process. Materials 2021, 14(11), p. 2947.

Müller, C.-E. Application of Damage Models on Long Product Rolling Simulations in QForm UK. Presented at MEFORM 2025, Freiberg, Germany.

Timothy, S.P., Yiu, H.L., Fine, J.M. and Ricks, R.A., 1991. Simulation of single pass of hot rolling deformation of aluminium alloy by plane strain compression. Materials science and technology, 7(3), pp. 255-263.

Johnson, W., Mehl, R., 1939. Reaction kinetics in process of nucleation and growth. Transactions of Transactions of the American Institute of Mining and Metallurgical Engineers, 135, pp. 416-458.

Micas Simulation LTD, "QForm UK version 11, The software documentation: Description of Microstructure evolution model", 2023.

Stefani, N., Bylya, O., Reshetov, A. and Blackwell, P., 2017. On the applicability of JMAKtype models in predicting IN718 microstructural evolution. Computer Methods in Materials Science, 17(1), pp. 59-68.

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The original version of this eBook is available on https://www.scientific.net/book/advanced-numerical-and-ai-strategies-for-material-forming/978-3-0364-1986-2

Key Engineering Materials, ISSN: 1662-9795, Vol. 1050, pp 193-204

doi: 10.4028/p-4T8OVj

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-19

Revised:
2026-02-18

Accepted:
2026-02-18

Online:
2026-04-16

Accelerating Microstructural Evolution Simulations in DIGIMU® through a Front-Tracking Lagrangian Solver: Implementation and Validation in AISI 304L Stainless Steel

The original version of this paper is available on https://www.scientific.net/KEM.1050.193.pdf

Franco Jaime 1,a*, Thomas Sourisseau 2, b, Pascal De Micheli 1,c and Marc Bernacki 3, d

1 Transvalor SA, Biot, France

2 Ugitech, Ugine Research Center, Ugine, France

3 Mines Paris - PSL, CNRS, CEMEF, UMR 7635, 06904 Sophia Antipolis, France

afranco.jaime@transvalor.com, b  Thomas.Sourisseau@swisssteelgroup.com, c  pascal.demicheli@transvalor.com, d  marc.bernacki@minesparis.psl.eu

Abstract

A new efficient numerical solver inspired by front-tracking concepts is implemented within the DIGIMU® framework to accelerate full-field simulations of microstructural evolution. The solver is applied to AISI 304L stainless steel and compared with the conventional level-set formulation under laboratory hot-torsion tests and industrial multi-pass hot rolling conditions. After a limited recalibration of grain boundary mobility and solute drag parameters, both solvers provide comparable predictions of recrystallization kinetics, grain size evolution and final microstructures. The new solver achieves a reduction in computational cost close to two orders of magnitude, while preserving the predictive capabilities of DIGIMU®, thereby enabling more efficient industrial-scale simulations. Simulated predictions will be compared to Ugitech experimental work on lab torsion tests and industrial extrusion processes.

Keywords: simulation, hot-rolling, DIGIMU ®, grain growth, recrystallization, solute drag.

Introduction

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Metallic material properties are strongly governed by their microstructures, which are themselves inherited from the thermomechanical treatments applied during processing. As a consequence, understanding, predicting, and ultimately optimizing microstructural evolution remains a central challenge and a key driver for competitiveness in modern metallurgical industries. Within this context, huge advances have been made in the last decades in order to link processing parameters, microstructural evolution, and final material properties.
However, predicting microstructural evolution during complex thermomechanical treatments, particularly multi-pass hot forming processes, remains extremely challenging. Grain growth (GG), solid-state phase transformations (SSPT), discontinuous dynamic recrystallization (DDRX) and continuous (CDRX) mechanisms, as well as post-dynamic recrystallization (PDRX), can occur simultaneously and interact in a highly complex manner. Small variations in temperature, strain, strain rate, or inter-pass times can lead to significant differences in the resulting microstructure.
Classical phenomenological models, such as those based on Johnson-Mehl-Avrami-Kolmogorov (JMAK) kinetic[1]-[3], offer very low computational cost but are generally limited to narrow validity domains defined by the calibration conditions. To overcome these limitations, more advanced modeling strategies have been developed. Mean-field models extend the applicability range by introducing statistical descriptions of grains, precipitates, and dislocation densities, but they rapidly reach their limits when dealing with complex, non-monotonic, multi-pass industrial processes [4][6]. This has motivated the development of mesoscopic, full-field approaches, in which the polycrystalline microstructure is explicitly represented within representative volume elements and local thermomechanical fields provided by macroscopic simulations are used as boundary conditions. Among the full-field approaches, several numerical frameworks have been proposed and widely studied, including Monte Carlo Potts models [7],cellular automata [8]-[10], phase-field and
multi-phase-field methods [11]-[13], front-tracking or vertex models [14], [15] and level-set (LS) formulations [16]-[19]. Each of these methods presents specific advantages and limitations in terms of numerical robustness, physical fidelity, and computational efficiency. The DIGIMU® software has been developed within this mesoscopic, full-field paradigm, relying on a finite-element (FE) formulation coupled with a level-set description of grain boundaries [20]. DIGIMU® has demonstrated strong predictive capabilities for grain growth, recrystallization, precipitate evolution, and, more recently, solute drag effects, with successful applications to industrially relevant alloys such as AISI 304L stainless steel [21], [22].
Despite these achievements, the conventional DIGIMU® framework remains computationally demanding, especially for large-scale and multi-pass industrial simulations. A significant fraction of the total computation time is associated with mesh adaptation procedures required to accurately capture evolving grain boundary networks. To address this limitation, the present work focuses on the implementation and validation of a new, highly efficient numerical solver inspired by front-tracking concepts and previously proposed by Florez et al [23].
The primary ambition of this new solver is to achieve a substantial reduction in computational time while maintaining an accuracy level comparable to that of the conventional level-set DIGIMU® framework. In the present work, both approaches are systematically compared through simulations representative of laboratory and industrial thermomechanical processing of AISI 304L stainless steel, providing a clear assessment of the performance-accuracy trade-off offered by the new numerical model.

Numerical Models in DIGIMU®

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Level-Set Model. The historical numerical core of DIGIMU® is based on a FE formulation combined with the LS method to describe evolving grain boundary networks. In this front-capturing approach, grain boundaries are implicitly defined as iso-surfaces of one or several level-set functions, whose evolution is governed by transport equations coupled with local driving forces such as grain boundary curvature, stored energy gradients, Zener pinning, and solute drag effects.
The FE-LS framework offers several important advantages. First, it naturally handles complex topological events such as grain disappearance, grain coalescence and junction evolution without requiring explicit tracking of interfaces. Second, it is particularly well suited for large deformations of the computational domain, which are typical of thermomechanical processing routes encountered in metal forming. These characteristics have made the LS approach a robust and versatile choice for industrially oriented full-field simulations.
Over the years, substantial developments have been carried out to improve the robustness and performance of the DIGIMU® LS solver, including advanced remeshing strategies and optimized algorithms for coupling microstructural evolution with thermomechanical fields [24]-[26]. Nevertheless, the computational cost remains high. In particular, a large proportion of the total simulation time is associated with remeshing operations and distance function reinitialization required to accurately resolve grain boundaries as the microstructure evolves. As a result, further performance improvements within the conventional FE-LS framework have become increasingly difficult to achieve.

Front-Tracking Lagrangian Model. To overcome the performance limitations of the level-set approach, a new numerical solver has been introduced into DIGIMU®, inspired by front-tracking and vertex-based methods. In contrast to front-capturing techniques, front-tracking approaches explicitly represent interfaces as sets of connected segments in two dimensions or surfaces in three dimensions. Geometrical quantities such as curvature and normal vectors are computed directly at the interfaces, and interface motion is described in a Lagrangian manner.
The new DIGIMU® solver adopts this philosophy while remaining compatible with a FE description of grain interiors. Geometrical and physical quantities are evaluated only at grain boundaries, and grain boundary migration is governed by an explicit Lagrangian scheme driven by local kinetic driving forces. This strategy allows to reduce the spatial discretization compared to the level-set
method and avoid some expensive steps such as the solving of the convection-diffusion equation and the distance reinitialization used in the level-set method. The cost of this approach is the burden of constant, complex remeshing to preserve the topological description during every event: grain growth or dissolution, grain collapse, grain nucleation, multiple junctions. For now, only 2D simulations are available. It is important to underline that both approaches are based on the same physical models and equations.
The new proposed solver in DIGIMU is strongly based on the Lagrangian model introduced by Florez et al. [14], called the ToRealMotion (TRM) code for "topological remeshing in Lagrangian framework for large interface motion". This solver was specifically designed to handle large interface motions, complex topological changes, and strong domain deformations. By combining explicit interface tracking with unstructured FE meshes for grain interiors, the model allows the incorporation of intragranular state variables, such as stored energy, which are essential for recrystallization modeling. A more detailed description of the TRM solver, including remeshing, repartition, grain computation at grain boundaries and multiple junction, could be found in [14, 23, 26].

Parameter Calibration and Comparison Methodology

The original version of this paper is available on https://www.scientific.net/KEM.1050.193.pdf

In order to perform a meaningful comparison between the conventional DIGIMU® solver and the newly implemented numerical framework, a strict and controlled methodology was adopted. AISI 304L austenitic stainless steel was selected as the material. Microstructure evolution was studied under thermomechanical loading conditions representative of laboratory and industrial rolling processes. All the experimental tests were carried out by UGITECH. In depth analysis and presentation of the experimental work is presented in Sourisseau et al. [22].
The simulations were conducted using identical initial microstructures, thermomechanical paths (temperature, strain, strain rate, and inter-pass times), and physical descriptions of the governing mechanisms, including grain growth, dynamic and post-dynamic recrystallization, Zener pinning, and solute drag. The only difference between the two simulation frameworks lies in the numerical formulation used to describe grain boundary migration and interface evolution.
Due to the intrinsic differences between the LS formulation and the new front-tracking inspired Lagrangian approach (TRM), small discrepancies in the effective kinetics of interface migration may arise, even when identical physical parameters are employed. For this reason, a limited and physically motivated recalibration of selected kinetic parameters was performed for the new solver.
More specifically, only three parameters were adjusted: the grain boundary mobility and two parameters ( α and β described in [21]) associated to the Cahn-Lücke-Stüwe solute drag model [27], [28]. All other material parameters, including grain boundary energy, nucleation parameters and recovery parameters were kept strictly identical between the two solvers. All other physical parameters were directly taken from previous DIGIMU® reported in [22].
The calibration procedure was carried out using reference conditions representative of laboratoryscale hot-torsion and hot-rolling tests, as described in the abstract, with the objective of matching the overall recrystallization kinetics and the order of magnitude of the resulting grain size. Once identified, the adjusted parameters were fixed and subsequently applied without modification to the industrial multi-pass hot rolling simulations.
It is important to emphasize that the computational performance of the two solvers is entirely independent of this parameter calibration. The observed differences in computation time arise solely from the numerical formulation of interface motion and mesh management strategies, and not from the choice of kinetic parameters.

Results

The original version of this paper is available on https://www.scientific.net/KEM.1050.193.pdf

Following the comparison methodology and calibration strategy described in the previous section, the predictive capabilities of the conventional LS solver and TRM solver are assessed through a series of representative thermomechanical loading cases. The comparison is carried out at different levels, including laboratory condition and industrial processing conditions. For all cases, the same material
parameters are used. A further deep description of the experimental conditions is presented in Sourisseau et al. [22].
Hot-torsion laboratory tests. The first level of comparison focuses on the hot-torsion tests. In these tests three different strain rates and temperatures were studied covering a range of use between 1020∘C−1220∘C and 0.1 s−1−5 s−1. The evolution of the recrystallized fraction as a function of strain was measured by interrupted tests followed by EBSD maps. In Fig. 1, the experimental evolution of the microstructure along with the simulations performed with the LS and TRM solver are shown. Both solvers exhibit very similar kinetics, with nearly overlapping curves over the full range of deformation. Furthermore, a good fitting with the experimental data all over the range of temperature and strain rate is achieved.
Fig. 2 presents the evolution of the average grain size predicted by both solvers under the same thermomechanical conditions. The results show a very close agreement throughout the processing path, both in terms of transient evolution and final grain size values. Minor discrepancies, when observed, remain within the expected numerical variability and do not affect the overall trends or the final microstructural state.
In addition to integral quantities, a direct comparison of the simulated microstructures provides valuable insight into the spatial fidelity of the two numerical formulations. Fig. 3 shows the final grain structures predicted by the LS based solver and the TRM solver for all the tests performed at a strain rate of 1 s−1. Both approaches lead to comparable grain morphologies, grain size distributions, and recrystallized patterns. The similarities observed in terms of grain topology and heterogeneity further confirm that the new solver preserves the essential physical mechanisms governing microstructural evolution, despite the different numerical treatment of interfaces.

[picture]

Fig. 1. Evolution of the recrystallized fraction measured experimentally, predicted by the conventional level-set solver and the TRM solver under representative hot-torsion and hot-rolling conditions for AISI 304L stainless steel.

Fig. 1. Evolution of the recrystallized fraction measured experimentally, predicted by the conventional level-set solver and the TRM solver under representative hot-torsion and hot-rolling conditions for AISI 304L stainless steel.

[picture]

Fig. 2. Comparison of the mean grain size evolution predicted by the level-set-based solver and the new solver for identical thermomechanical loading paths.

Fig. 2. Comparison of the mean grain size evolution predicted by the level-set-based solver and the new solver for identical thermomechanical loading paths.

[picture]

Fig. 3. Comparison between experimental microstructures and final grain structures predicted by the LS and (b) TRM solver for all the studied temperatures and a strain rate of 1 s−1.

Fig. 3. Comparison between experimental microstructures and final grain structures predicted by the LS and (b) TRM solver for all the studied temperatures and a strain rate of 1 s − 1 .

Industrial multi-pass case. In this section, the predictive capabilities of both solvers are evaluated through a real industrial application. The study focuses on the microstructural evolution during the multi-pass hot-rolling process at UGITECH.

The thermomechanical history of the product was obtained via FORGE® FE simulations. For this analysis, the study focuses on a representative point at the product surface. The process parameters involve a temperature range of 1000−1250∘C and strain rates varying from 0.1 s−1 to 15 s1. The resulting thermomechanical path for the multi-pass sequence is illustrated in Fig. 4.

[picture]

Fig.4. Thermomechanical path of the multi-pass hot rolling mill.

Fig.4. Thermomechanical path of the multi-pass hot rolling mill.

The evolution of the recrystallized fraction shows a good agreement between the two solvers (Fig. 5a). Regarding the mean grain size, both models yield acceptable results, with predictions falling within the range of experimental values (Fig. 5b).

[picture]

Fig.5. Comparison of the recrystallized fraction (a) and the mean grain size (b) evolution predicted by the LS solver and the TRM solver for the industrial multi pass case.

Fig.5. Comparison of the recrystallized fraction (a) and the mean grain size (b) evolution predicted by the LS solver and the TRM solver for the industrial multi pass case.

However, discrepancies arise when analyzing the grain size distribution and morphology. As shown in Fig. 6, neither LS nor the TRM solver is currently able to capture the largest grains observed in the experimental microstructure. This limitation suggests an area for further investigation to improve the solvers' sensitivity to grain size extremes. Furthermore, a comparison of the final grain structures reveals that the simulated microstructures are significantly more equiaxed than the experimental ones. The complex grain morphologies observed in the experimental images are not yet fully represented by the simulations, which tend to produce more regularized geometries.

[picture]

Fig. 6. Comparison between experimental microstructure (a) and final grain structures predicted by the LS and (b) TRM solver for all the studied temperatures and a strain rate of 1 s−1.

Fig. 6. Comparison between experimental microstructure (a) and final grain structures predicted by the LS and (b) TRM solver for all the studied temperatures and a strain rate of 1 s − 1 .

Computational performance and efficiency. While the previous sections demonstrate the equivalence of the microstructural predictions obtained with both numerical approaches, the primary motivation for the development of the new solver is the improvement of computational efficiency. This section therefore focuses on a detailed comparison of the computational cost associated with each numerical framework.
As shown in Table 1, all simulations performed with the new TRM solver were carried out using a single CPU core, resulting in computation times ranging from a few seconds to several minutes, depending on the case considered. In contrast, all simulations performed with the conventional DIGIMU® level-set solver required 16 or 24 CPU cores and led to computation times on the order of several minutes to several hours.
Table 1 summarizes the computation times and hardware configurations used for the different simulation cases. For all investigated conditions, the new solver exhibits a significant reduction in computation time, confirming the substantial performance gain achieved by the new numerical formulation.

Discussion

The original version of this paper is available on https://www.scientific.net/KEM.1050.193.pdf

The results presented in this study demonstrate that the newly implemented front-tracking-inspired solver within DIGIMU® is capable of reproducing the main microstructural features observed during the hot forming of AISI 304L stainless steel, with a level of accuracy comparable to that of the conventional level-set formulation. After a limited and physically motivated recalibration of selected kinetic parameters, both numerical approaches yield consistent predictions of recrystallization kinetics, grain size evolution, and overall microstructural morphology under laboratory and industrial processing conditions.

Table 1. Comparison of computational cost between the conventional level-set solver and the new solver for laboratory, semi-industrial, and industrial simulation cases.

Simulation | Cores | Computation time
LS | TRM | LS | TRM
Hot-torsion test | 1020°C – 1s⁻¹ | 16 | 1 | 2h 2min | 16s
1120°C – 0.1s⁻¹ | 16 | 1 | 1h 4min | 5s
1120°C – 1s⁻¹ | 16 | 1 | 30 min | 6s
1120°C – 5s⁻¹ | 16 | 1 | 7 min | 5s
1220°C – 1s⁻¹ | 16 | 1 | 7 min | 4s
Industrial | Surface | 24 | 1 | 3h 30min | 19min 35s

The need for a minor recalibration of the grain boundary mobility and solute drag-related parameters arises from the intrinsic differences in the numerical treatment of interface motion between the two solvers. While the underlying physical models remain identical, the explicit Lagrangian representation of grain boundaries modifies the effective kinetics of interface migration compared to the implicit front-capturing strategy of the level-set approach. The adjusted parameters directly control interface mobility and solute drag effects and are therefore the most sensitive to the numerical formulation and numerical diffusion in the results of the level-set formulation. However, this is not surprising, as the grain boundary mobility is always a model parameter in full-field models.
The comparison under industrial rolling conditions highlights both the strengths and current limitations of the modeling approaches. Both solvers provide comparable predictions of recrystallized fraction and mean grain size in a very wide range of temperature and strain rates on real industrial processes. However, discrepancies remain when comparing very locally simulated and experimental grain morphologies and size distributions. In this case, neither solver fully captures the largest grains, nor the complex non-convex morphologies observed experimentally. These differences point to modeling challenges that are not specific to the way of solving the equation. Probably, some more advanced work on the parameters identification could allow us to optimize our results and make them even closer to reality. However, some missing elements in the modelling equation themselves still prevent us from obtaining all the complexity of the local grain topology formalism.
From a computational perspective, the new solver delivers a substantial performance improvement, with computation times reduced by approximately one order of magnitude despite being executed on significantly fewer computational cores, maintaining the same accuracy level than the older DIGIMU solver. This gain is primarily attributed to the elimination of frequent global remeshing operations and to the coarser meshes used in the TRM solver. By explicitly tracking grain boundaries in a Lagrangian framework, the numerical effort is focused on physically relevant regions of the microstructure, resulting in a more efficient use of computational resources.
Despite the coarsening of the mesh adopted in the TRM formulation, no significant loss of accuracy is observed, as demonstrated by the comparative results. This is primarily due to the intrinsic fronttracking nature of the TRM approach, which allows all interface-related calculations to be performed directly and exactly at the grain boundary position. As a consequence, highly refined meshes within the grain interiors are no longer required. Furthermore, the TRM formulation avoids the need for repeated reinitialization of the level-set function to preserve its distance function properties, as well as the extremely fine discretization typically required around the level-set 0 iso-value to accurately capture interface motion. This leads to a substantial reduction in computational cost without compromising predictive precision.

Conclusions

The original version of this paper is available on https://www.scientific.net/KEM.1050.193.pdf

A new, highly efficient numerical solver inspired by front-tracking concepts has been implemented and validated within the DIGIMU® framework for the simulation of microstructural evolution in AISI 304L stainless steel. The solver was systematically compared with the conventional level-set formulation and to UGITECH experimental observations on laboratory hot-torsion tests and industrial multi-pass hot rolling conditions.
After a limited recalibration of the grain boundary mobility and solute drag-related parameters, the new solver was shown to reproduce the main microstructural features predicted by the reference formulation, including recrystallization kinetics, grain size evolution, and final microstructural morphology. This confirms that the change in numerical strategy does not compromise the physical consistency of the simulations. Although the level-set formulation is used as the reference numerical framework in this study, the predictions obtained with this solver are not expected to be perfectly accurate. For this reason, the comparison was not limited to the level-set formulation alone, but was also systematically carried out against experimental results, including recrystallized fraction, mean grain size evolution, and EBSD-based microstructural observations.
The main outcome of this work is the significant improvement in computational efficiency achieved with the new solver, with computation times reduced by a factor of 50 to 100 . This performance gain is solely attributed to the numerical formulation of interface motion and mesh management and is independent of the parameter calibration. Enhanced efficiency substantially extends the applicability of full-field microstructural simulations in industrial environments, making reliable simulation of complex, multi-pass thermomechanical processes more accessible within realistic computation times. The present study intentionally addresses a wide range of thermomechanical conditions, involving the simultaneous activation of several interacting mechanisms such as GG, DDRX, solute drag and post-dynamic recrystallization, over a broad domain of temperatures and strain rates representative of industrial hot-forming processes. The objective was not to achieve an extremely fine calibration for a single processing condition, but rather to demonstrate robustness and representativeness across an industrially relevant operating window. More extensive validation and further developments are currently in progress to refine the predictions for specific cases and to extend the applicability of the approach.

Acknowledgement

The original version of this paper is available on https://www.scientific.net/KEM.1050.193.pdf

The authors thank UGITECH, TRANSVALOR SA, and CEMEF - MINES Paris for their support and collaborative efforts. This work was also supported by the ANR and the companies ArcelorMittal, Aperam, Aubert & Duval, CEA, Constellium, Framatome, and Safran through the DIGIMU consortium and the RealIMotion ANR Industrial Chair.

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The original version of this paper is available on https://www.scientific.net/KEM.1050.193.pdf

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The original version of this eBook is available on https://www.scientific.net/book/advanced-numerical-and-ai-strategies-for-material-forming/978-3-0364-1986-2

Key Engineering Materials, ISSN: 1662-9795, Vol. 1050, pp 205-214

doi: 10.4028/p-ddO89X

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-19

Revised:
2026-03-04

Accepted:
2026-03-04

Online:
2026-04-16

Study on the Apparent Friction Coefficient Between a Deformable and a Rough Rigid Body under Large Sliding

The original version of this paper is available on https://www.scientific.net/KEM.1050.205.pdf

Renan B. Wojciechowski 1, b, Marta C. Oliveira 1,a* and Diogo M. Neto 1,c

¹CEMMPRE, ARISE, Department of Mechanical Engineering, University of Coimbra, Pinhal de Marrocos,3030-788 Coimbra, Portugal

a  marta.oliveira@dem.uc.pt, b  rbw@dem.uc.pt, c  diogo.neto@dem.uc.pt

Keywords: Numerical Analysis, Contact Element, Periodic boundary conditions, Sinusoidal roughness

Abstract

The mechanical analysis of sliding frictional contact under small scales is important to improve the understanding about the influence of the contact conditions on the real contact area and, consequently, on the apparent coefficient of friction. This study uses the finite element method to model the contact between an elastoplastic body and a rigid surface with a unidirectional sinusoidal topography, including large sliding. A sensitivity analysis is presented, studying the influence of the initial average pressure, local coefficient of friction and asperity wavelength on the contact conditions. The ratio between total tangential and normal force (apparent friction coefficient) reaches a steady state after a sliding distance of five roughness wavelengths, except for lower values of average initial contact pressure. Increasing the initial average contact pressure leads to an increase of the steady state apparent friction coefficient, particularly for a surface with sharper asperities. This increasing tends to stagnate also with the increase of the local friction coefficient. Withing the cases studied, increasing the initial average contact pressure from 25% to 100% of the material yield stress, leads to an increase of up to 0.07 in the apparent coefficient of friction and of the real-to-apparent contact area ratio up to 30%.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1050.205.pdf

The contact mechanics has attracted increasing attention because friction and wear are responsible for ~23% of the global energy consumption [1]. Improvements in the understanding of contact phenomena can contribute to reducing this cost and mitigating environmental impacts. Thus, design modifications in engineering systems are required, which can be supported by robust modelling and analysis of the contact mechanics to support decision-making.

The mechanical analysis of sliding frictional contact under small scales is important to improve understanding about the influence of the contact conditions on the real contact area [2] and, consequently, on the apparent coefficient of friction. The finite element method (FEM) is commonly adopted to perform this type of study, since it enables parametric analysis considering parameters such as the surface roughness or the contact pressure, to evaluate quantities such as the real contact area or the residual stresses (e.g. Gao [3] and Alfredsson [4]). This enables the development of friction laws to describe the dependency of the apparent coefficient of friction on those parameters, which can be applied in simpler, equivalent macroscopic models.

The concept of representative contact element (RCE) was introduced within the FEM to allow modelling contact phenomena at different scales, including the details of rough surfaces. For example, Wriggers [5] employed a RCE to develop a frictional law for an elastomer in contact with rigid surface. Carvalho [6] proposed a multiscale rough contact algorithm that considers RCEs at different scales, to estimate the real contact area in contact without sliding. The RCE consists of a unit cell (e.g. parallelepiped) with periodic boundary condition (PBC), i.e., the opposite parallel faces present the same displacement for counterpart nodes.

When modelling two bodies in contact, if one has a much higher stiffness than the other, it can be considered rigid. This assumption is common is sheet metal forming or when studying elastomers, such as in pulley and conveyor belt system (see [5] or in the contact between the tire and the road (see e.g. [7]). Although it is arguable if this assumption is valid for metal-metal
forming contacts, since both bodies may plastically deform at the asperity scale, some works have also adopted this approach [8]. When modelling a real surface at sufficiently small scales, the asperities should be represented rather than assuming their macro-scale regular geometry (e.g. perfectly plane). One approach is to define the surface topography through measurements (e.g. stylus profilers). However, measurements artefacts make it difficult to obtain a surface geometry that is compatible with numerical models [9, 10]. Consequently, some authors prefer to use a surface that is stochastically generated with a selection of roughness parameters [11, 12, 13]. This approach also improves the study repeatability and facilitates parametric analysis. Another way of representing the roughness is to approximate the topography by a superposition of sinusoids ([3, 5]). This allows to accurately model the elastomers' visco-elastic effects without an impractically fine finite element mesh, following the approach proposed by Wriggers [5]. In addition, a Weierstrass function can be employed to calculate the real contact area of a fractal surface, which may provide a more realistic description of engineering topographies [3,14].

The aforementioned asperities are responsible for the geometric component of the friction force during sliding (e.g. ploughing effect), while the remaining is due to the local contributions [15]. The apparent coefficient of friction (μ*) was defined by Lafaye [16] as the ratio between this total friction force and the applied normal force. As reviewed and discussed by Blau [15], the ratio μ* is a convenient and useful engineering metric, supporting the design of machines and buildings. Therefore, it has been used to characterize the slide resistance of 3D printed surfaces with bio inspired textures [17], general polymer friction [5,16] and the cutting performance of a machining tool [18].

The objective of this work is to investigate the influence of the initial average pressure pi―, local Coulomb coefficient of friction (μ) and asperity wavelength (λ) on the apparent friction coefficient, when a deformable body slide against a rigid rough surface. Therefore, a finite element model that employs a RCE to assume an effectively infinite interface is used to perform a sensitivity analysis at the microscale. The ratio between the real contact area ( Ar ) and the apparent contact area ( Aa ) is estimated based on the number of interface nodes in contact over its total. This is an indirect measure of the ratio, which assumes that the distance between the nodes remains approximately uniform during the simulation. The apparent friction coefficient μ* is evaluated during sliding as the ratio between the total tangential and normal force components. Finally, the average contact pressure during the slide ( p→ ) and shear stress and distortion are reported.

Finite Element Model

The original version of this paper is available on https://www.scientific.net/KEM.1050.205.pdf

The contact between a rough rigid surface and a flat deformable body was modelled using an RCE, under plane strain conditions. Local interfacial friction was described using Coulomb's law with different constant values for the friction coefficient. The rough rigid surface was represented by a unidirectional sinusoidal surface. Considering the roughness average of a typical machined surface Ra=3.2μ m [19], the corresponding wave amplitude is g=5.03μ m. Two different values of the ratio between the amplitude and the wavelength ( λ ) were studied, namely g/λ=0.01 and g/λ=0.02, which lie within the range suggested by Gao [3]. Accordingly, the wavelengths considered are λ=502.65μ m and λ=251.33μ m. The roughness wavelengths defined at the microscale fall within the range admissible for which macroscale constitutive models remain applicable. The rigid surface is composed by at least six waves in order to allow large sliding distances between the RCE and the rigid surface. The geometry of the rigid sinusoidal surface was discretized with Nagata patches [20].

The RCE is presented in Fig. 1. It has a parallelepiped geometry with a width equal to the wavelength λ and a height of 3λ, which ensures a uniform stress state at the upper surface, as suggested in [3]. The length along the out-of-plane direction is 8μ m. The RCE was discretized with hexahedral finite elements, using selective reduced integration, considering four zones with element sizes of λ/40,λ/20,λ/10 and λ/5. The non-conforming mesh is shown in Fig. 1 (c). Each zone has approximately the same height and they are connected through the hanging nodes feature [21]. The mechanical behaviour of the RCE is assumed isotropic and elastic-plastic, described by the von

Mises yield criterion. Accordingly, the Young's modulus is 210 GPa , the Poisson's ratio is 0.3 and the yield strength is 1.5 GPa . The plastic behaviour is described by the Swift law, such that the flow stress Y=1.505(ε0+εp)0.001 (the small hardening coefficient is chosen to approximate an almost perfectly plastic behaviour).

The numerical simulation of the contact between the RCE and the rough rigid surface comprises two stages. First, the rigid surface is pressed against the RCE by imposing a vertical displacement until an initial average contact pressure pi― is reached, as shown in Fig. 1 (a). Four different values of pi― were adopted, which were defined according to the yield strength of the material, namely 25%, 50%,75% and 100% of the yield strength. Then, large sliding is generated by imposing a sliding distance Δs=5λ to the rigid surface, as shown in Fig. 1 (b). During this second stage, the nodes on the upper face of the RCE are clamped, since it is a boundary condition easy to impose. The periodic boundary conditions associated with the RCE allow the representation of an infinite contact interface and ensure continuity of displacement and stress fields across the lateral boundaries [6]. All numerical simulations were carried out using the in-house finite element code DD3IMP [22].

[picture]

Fig. 1. Schematic of the contact between a rough rigid surface and deformable RCE: (a) rigid surface pressed against the RCE; (b) sliding of the rigid surface against the RCE; (c) nonconforming finite element mesh of the RCE.

Fig. 1. Schematic of the contact between a rough rigid surface and deformable RCE: (a) rigid surface pressed against the RCE; (b) sliding of the rigid surface against the RCE; (c) nonconforming finite element mesh of the RCE.

Results and Discussion

The original version of this paper is available on https://www.scientific.net/KEM.1050.205.pdf

Real Contact Area. The discretization of the contact surface (see Fig. 1 (c)), together with the use of a node-to-segment algorithm for handling contact, results in oscillations when the number of nodes in contact is used to evaluate the real contact area. Accordingly, the ratio Ar/Aa presented in this work must be interpreted as a discrete approximation, which must be improved in future works based on the contact variables [23]. The ratio Ar/Aa was evaluated during the pressing phase of a rigid surface against the RCE, leading to a step-like behaviour as shown in Fig. 2. This figure shows the evolution of this ratio as the normal load increases, comparing two different values of g/λ, for μ=0.10 at the contact interface. The real contact area increases approximately linearly with the applied normal load ( R2=0.98 ), which is consistent with elastic-plastic asperity deformation. The slope is higher for the lower ratio g/λ because it is more comparable to a flat surface, which would present Ar/Aa=1. In classical rough-surface contact models, only a small fraction of the apparent area actually carries load, even under high contact pressures [24]. It has been shown that for an elastoplastic rough surface there is a linear load-area relation at small contact fraction ( Ar/Aa~0.45 ), and the proportionality is determined by the mechanical properties and surface morphology. As the contact fraction increases further, dramatically higher load will be required owing to asperity interactions [25]. One can expect that the contact area fraction calculated in this study to be higher than the one obtained with 3D models due to the plane strain conditions, i.e. the real contact area is composed by stripes instead of islands.

[picture]

Fig. 2. Evolution of the ratio between the real contact area ( Ar ) and the apparent contact area ( Aa ) during the pressing of a rigid surface against the RCE considering μ=0.10 at the interface for: (a) g/λ=0.01; (b) g/λ=0.02.

Fig. 2. Evolution of the ratio between the real contact area ( A r ) and the apparent contact area ( A a ) during the pressing of a rigid surface against the RCE considering μ = 0.10 at the interface for: (a) g / λ = 0.01 ; (b) g / λ = 0.02 .

[picture]

Fig. 3. Evolution of the average contact pressure during the sliding of a rigid surface against the RCE for different levels of initial average contact pressure: (a) g/λ=0.01; (b) g/λ=0.02.

Fig. 3. Evolution of the average contact pressure during the sliding of a rigid surface against the RCE for different levels of initial average contact pressure: (a) g / λ = 0.01 ; (b) g / λ = 0.02 .

During the sliding phase of the rigid surface against the RCE, its position normal to the surface is kept constant (see Fig. 1 (b)). Fig. 3 shows the evolution of the average contact pressure during sliding, comparing different values of pi― and two levels of friction coefficient. The horizontal axis represents the normalized sliding distance ( Δs/λ ), indicating that the total applied sliding corresponds to five wavelengths of the rigid surface roughness.

As sliding stage begins, the average contact pressure decreases due to the development of a shear stress component, particularly for high values of the initial average contact pressure. Around Δs/λ=0.15, the average contact pressure starts to increase because of the misalignment between the asperity peaks and the deformable surface valleys created during the compression phase. When these points begin to realign, the pressure decreases again. This cyclic behaviour repeats, with progressively lower amplitude. In most cases, the average contact pressure can be considered stable after sliding over approximately two wavelengths of the rigid surface roughness. The effect of the friction coefficient μ on the predicted average contact pressure is low, as shown in Fig. 3. For p¯i=1.5GPa, the increase of the friction coefficient leads to a decrease of the predicted average contact pressure, particularly for g/λ=0.02. On the other hand, for pi―=0.38GPa and g/λ=0.02, the increase of the friction coefficient leads to a slight increase of the predicted average contact
pressure. Although sliding is imposed under fixed normal displacement, rather than fixed normal load, which is more commonly adopted in this type of studies, the results show that this condition also leads to stable average contact pressure.

[picture]

Fig. 4. Evolution of the ratio between the real contact area ( Ar ) and the apparent contact area ( Aa ) during the sliding of a rigid surface against the RCE, considering different values of initial average contact pressure and coefficient of friction: (a) g/λ=0.01; (b) g/λ=0.02.

Fig. 4. Evolution of the ratio between the real contact area ( A r ) and the apparent contact area ( A a ) during the sliding of a rigid surface against the RCE, considering different values of initial average contact pressure and coefficient of friction: (a) g / λ = 0.01 ; (b) g / λ = 0.02 .

Due to the use of the number of contact nodes to calculate the real contact area, during sliding sudden jumps can occur. Therefore, a moving average was applied to smooth the data during the sliding stage. Fig. 4 presents the evolution of the ratio between the real and apparent contact area during the sliding stage. Under frictionless conditions ( μ=0.0 ), the steady state is achieved for Δs/λ<5, except for pi―=0.38GPa. However, when friction is considered ( μ=0.1 ), a steady state is achieved only for g/λ=0.01, as shown in Fig. 4 (a). In contrast, for g/λ=0.02, the real contact area exhibits a slight (continuous) decrease throughout the sliding process, as illustrated in Fig. 4 (b). The increase of the friction coefficient leads to a global increase of the real contact area, particularly for high levels of initial average contact pressure. Considering pi―=1.5GPa and g/λ=0.01, the ratio Ar/Aa increases about 2.5% as μ increases from 0.0 to 0.1 .

Apparent Friction Coefficient. The apparent friction coefficient μ* was evaluated as the ratio between the tangential and normal force components during the sliding phase. Fig. 5 shows the evolution of μ* during the sliding phase, comparing four values of initial average contact pressure and three different values of μ. Globally, increasing μ leads to an increase of μ*. Moreover, the apparent friction coefficient increases when the ratio g/λ increases from 0.01 to 0.02 , as shown by comparing Fig. 5(a) with Fig. 5(b). This occurs due the higher slope between the valleys and the peaks for g/λ0.02, resulting in sharper asperities that penetrates the RCE and increases sliding resistance. Hence, increasing the initial average contact pressure also increases the apparent friction coefficient. One can expect that the apparent friction coefficient μ* calculated in this study to be higher than the one obtained with 3D models due to the plane strain conditions.

The Coulomb friction law states that the ratio between the tangential and normal force is always constant. However, this relation was not observed in the numerical results due to the roughness present at the interface of RCE. As shown in Fig. 5, for most cases μ* attains a steady state value. Therefore, the values of μ* attained for Δs/λ=5 were extracted, for all cases, and assumed as the ones corresponding to the apparent friction coefficient at the steady state regime ( μ* ss), since the variation beyond Δs/λ>4 is at most 0.006 . Fig. 6 presents μ* ss considering different values of input friction coefficient μ and four levels of initial average contact pressure. Increasing the initial average contact pressure leads to an increase of μ* ss, particularly for g/λ=0.02. The interlocking of asperities is generated when the initially flat deformable body acquires roughness when pressed against a rough rigid surface. This effect is amplified by the pressure pi―. When the average contact
pressure is low ( pi―<0.38GPa ), the μ* ss is approximately the input μ, particularly for g/λ=0.01. On the other hand, increasing both the μ and the pi―, the obtained μ* ss present a stagnation in their increase.

[picture]

Fig. 5. Evolution of the apparent friction coefficient during the sliding of a rigid surface against the RCE for different levels of initial average contact pressure: (a) g/λ=0.01; (b) g/λ=0.02.

Fig. 5. Evolution of the apparent friction coefficient during the sliding of a rigid surface against the RCE for different levels of initial average contact pressure: (a) g / λ = 0.01 ; (b) g / λ = 0.02 .

[picture]

Fig. 6 Apparent friction coefficient evaluated at the steady state as a function of the input friction coefficient, considering different levels of initial average contact pressure: (a) g/λ=0.01; (b) g/λ=0.02.

Fig. 6. Apparent friction coefficient evaluated at the steady state as a function of the input friction coefficient, considering different levels of initial average contact pressure: (a) g / λ = 0.01 ; (b) g / λ = 0.02 .

Strain and Stress Distribution. The plastic strain distribution at Δs/λ<5 was evaluated along the mid-width of the RCE, as show in the detail of Fig. 7 (b). Fig. 7 compares the plastic strain distributions for different levels of initial average contact pressure and two different values of μ. The maximum value occurs near the contact surface and decreases as the distance to the surface increases. Increasing the ratio g/λ leads to an increase of the maximum plastic strain value and reduces the height of the region with plastic deformation. Increasing the initial contact pressure also leads to an increase of the plastic strain due to the increase of the level of stress at the interface generated by the increase of compression force. In the same way, the increase the friction coefficient leads to an increase of the plastic strain, particularly for the ratio g/λ=0.01. Additionally, the increase of μ also moves the maximum plastic strain closer to the surface, as shown in Fig. 7. This agrees with the Hertz solution, where the maximum stress moves from the subsurface to the surface when μ increases.

It must be noted that even relatively moderate normal load leads to extremely high local deformations: accumulated plastic strain reaches 300%. Normal metal materials are not supposed to bear such high deformations. Consequently, a damage or/and a fracture model should be incorporated in the analysis to get more physical results [26]. Nevertheless, the plastic strain levels are strongly dependent on the hardening law. Although not shown here, a small increase of the hardening coefficient drastically reduces the maximum values attained.

[picture]

Fig. 7. Plastic deformation distribution at the mid-width of the RCE after sliding against a rigid surface for different levels of initial average contact pressure: (a) g/λ=0.01; (b) g/λ=0.02.

Fig. 7. Plastic deformation distribution at the mid-width of the RCE after sliding against a rigid surface for different levels of initial average contact pressure: (a) g / λ = 0.01 ; (b) g / λ = 0.02 .

The sliding of the rigid surface from the right side to the left side induces a shear deformation on the deformable body. This effect was quantified by the total tangential displacement, d, of the left lower node. Fig. 8 presents the evolution of this distance during the sliding stage, highlighting the global increase of d. The distortion of the deformable body is higher for g/λ=0.02, as shown in Fig. 8 (b). Increasing either the initial contact pressure or the friction coefficient yields an increase of the distortion. For cases that achieve a steady state apparent friction coefficient (see Fig. 5), the distance d is continuously increasing during the sliding stage. The fluctuations are consequence of the asperity entering in contact with the monitored node, which produces an increase of the local tangential loading.

[picture]

Fig. 8. Evolution of the tangential displacement of node located at the left lower corner of the RCE while sliding against a rigid surface for different levels of initial average contact pressure: (a) g/λ=0.01; (b) g/λ=0.02.

Fig. 8. Evolution of the tangential displacement of node located at the left lower corner of the RCE while sliding against a rigid surface for different levels of initial average contact pressure: (a) g / λ = 0.01 ; (b) g / λ = 0.02 .

Fig. 9 presents the distribution of the shear stress ( τxy ) evaluated at the RCE after sliding against the rigid rough surface, considering different conditions for g/λ=0.01. The shear stress distribution is approximately symmetric in relation to the mid-width plane, when the initial average contact pressure is pi―=0.75GPa and the friction coefficient is zero (see Fig. 9 (a)). This distribution is like the one obtained in the Hertz contact theory, because the contact occurs locally in a small area. Increasing the initial average contact pressure leads to the distortion of the deformable body due to the sliding and, consequently, of the shear stress distribution, as shown in Fig. 9 (b). Using the same value of pressure, the increase of the friction coefficient amplifies the geometrical distortion, as shown in Fig. 9 (c).

[picture]

Fig. 9. Shear stress distribution on the RCE after sliding against the rigid surface with g/λ=0.01 : (a) μ=0.0 and pi―=0.75GPa; (b) μ=0.0 and pi―=1.5GPa; (c) μ=0.1 and pi―=1.5GPa.

Fig. 9. Shear stress distribution on the RCE after sliding against the rigid surface with g / λ = 0.01 : (a) μ = 0.0 and p i ― = 0.75 G P a ; (b) μ = 0.0 and p i ― = 1.5 G P a ; (c) μ = 0.1 and p i ― = 1.5 G P a .

Conclusions

The original version of this paper is available on https://www.scientific.net/KEM.1050.205.pdf

This paper presented a numerical investigation of the frictional contact conditions arising from surface roughness in the interaction between a deformable body and a rough rigid surface under large sliding. The finite element method was employed to simulate the contact problem, making use of a representative contact element (RCE) and plane strain conditions to reduce the computational cost. The influence of the local friction coefficient, the initial average contact pressure and roughness wavelength was analysed, with particular emphasis on the steady-state sliding regime.

Since sliding is imposed under a fixed normal displacement, the average contact pressure remains approximately constant throughout the sliding stage. The results show that the main parameter governing the real contact area is the initial average contact pressure, whereas both the friction coefficient and the roughness ratio g/λ have only a minor influence on this quantity. In contrast, the apparent friction coefficient is significantly affected by all three parameters, increasing with higher local friction coefficient, higher initial average contact pressure, and larger values of g/λ. For a surface roughness ratio of g/λ=0.02 and a local friction coefficient of μ=0.05, the predicted apparent friction coefficient is approximately μ*=0.09, when the initial average contact pressure is about half of the material yield stress.

Although the applied initial average contact pressure is at most equal to the yield stress of the material, the roughness of the rigid contact surface induces high levels of plastic deformation in the vicinity of the contact interface. Indeed, for a surface roughness ratio of g/λ=0.01, the maximum equivalent plastic strain exceeds 50% when the initial average contact pressure is larger than 75% of the yield stress. The high shear stress levels at the contact interface induce significant distortion of the RCE in the vicinity of the contact zone. This effect is amplified for higher local friction coefficients, higher initial average contact pressures, and larger values of the roughness ratio g/λ. In this exploratory work, an almost perfectly plastic behaviour was assumed, to simplify the analysis. This enabled the validation of the contact area fraction under loading, but disabled further comparisons for the sliding stage.

Acknowledgements

The original version of this paper is available on https://www.scientific.net/KEM.1050.205.pdf

This research is sponsored by national funds through FCT - Fundação para a Ciência e a Tecnologia, under projects UID/00285/2025 and LA/P/0112/2020. It was also supported by the project Sim2Adapt (ref. 2022.08459.PTDC). All the supports are gratefully acknowledged.

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Key Engineering Materials, ISSN: 1662-9795, Vol. 1050, pp 215-225

doi: 10.4028/p-j0sDDJ

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-19

Revised:
2026-02-17

Accepted:
2026-02-18

Online:
2026-04-16

Using Kolmogorov-Arnold Networks for Discrepancy Modeling of Lateral Flow in Hot Rolling of Steel Slabs

The original version of this paper is available on https://www.scientific.net/KEM.1050.215.pdf

HASHEMZADEH Amirali 1,a*, BOCK Frederic E. 2, b, COMETA Antonella 1,c, SOYARSLAN Celal 1,3, d, KLUSEMANN Benjamin 2,4,e, VAN DEN BOOGAARD Ton 1,f

1 Chair of Nonlinear Solid Mechanics, University of Twente, The Netherlands

2 Institute of Materials and Process Design, Helmholtz-Zentrum Hereon, Geesthacht, Germany

3 Fraunhofer Innovation Platform, University of Twente, The Netherlands

4 Institute for Production Technology and Systems, Leuphana University Lüneburg, Lüneburg, Germany

a a.hashemzadeh@utwente.nl, b frederic.bock@hereon.de, c a.cometa@utwente.nl,

d c.soyarslan@utwente.nl, e benjamin.klusemann@leuphana.de, f a.h.vandenboogaard@utwente.nl

Keywords: Hot Rolling, Finite Element Models, Lateral Flow, Discrepancy Modeling, KANs

Abstract

Kolmogorov-Arnold networks (KANs) have emerged as a promising counterpart to multilayer perceptrons (MLPs) which offer a more interpretable functionality for different machine learning (ML) applications. Their main difference lies in the definition of KAN layers, using learnable activation functions, which has made these networks optimal for physics-based applications. In this work, we focus on analyzing the performance of KANs in capturing the physics of the hot rolling process, which is an integral part of steel manufacturing industry. Initially, we introduce non-dimensional parameters to encapsulate geometrical factors in the process. We perform space-filling sampling in the space spanned by these parameters. The sampled points yield the necessary parameters for the finite element (FE) simulations, forming the ground truth (GT) data for the network. A closed-form analytical model for spread is considered from previous studies in the literature, and its predictive performance is assessed against the FE results. In defining the input space for the network, different alternatives are compared and it was seen that input space containing the non-dimensional features and the predictions of the analytical model reduced overfitting and better generalization. The effect of KAN hyperparameters are evaluated, and the network with tuned parameters demonstrate optimal performance on the test set. Lastly, after applying symbolification for this network, a closed-form expression is obtained that captures the discrepancy between the analytical model and the GT results, and its performance is tested against test set data.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1050.215.pdf

MLPs or multilayer feedforward networks have become the essential constituents of ML applications using deep learning. This was primarily due to the universal approximation theorem [1], which states that a feedforward network with a linear output and at least one hidden layer with any type of squishing activation functions can approximate any continuous and Borel-measurable nonlinear function with a mapping between finite-dimensional spaces. The error of this approximation can be reduced arbitrarily if a sufficiently large number of neurons are used in the hidden layer. Often to optimize their performance, the architecture of MLPs includes multiple hidden layers with fixed activation functions, which enables a nested formulation of the problem. However, such a formulation has also brought forward various limitations associated with these networks. For instance, catastrophic forgetting in accommodating variations in the data distribution [2], prioritizing learning data with higher frequency as opposed to ones with lower frequency, a phenomenon referred to as spectral bias [3,4], which results in convergence problems for the network if the problem physics includes a high frequency
solution, such as wave propagation [5]. Moreover, especially in deep neural networks (DNNs), MLPs are prone to suffer from exploding gradients problem which can make the use of stochastic gradient descent (SGD) unfeasible [6]. Moreover, the blackbox nature of MLPs makes them less interpretable [7].
In search of circumventing or improving existing problems with MLPs, KANs have been introduced as a promising alternative [8]. Development of KANs has been based on the Kolmogorov-Arnold's Representation Theorem (KART) [9], which as opposed to MLPs, translated into having learnable activation functions on the edges rather than nodes. One of the main advantages of KANs, that made them attractive alternatives to MLPs, is the fact that KANs result in interpretable models with much less parameters required for the same task [10]. Consequently, KANs have been applied to benchmark problems, commonly tested with MLPs, to compare their performance. For instance, in the study of [11], it was found that in physics-informed KAN using Jacobi orthogonal polynomials enhanced the limitations of traditional physics-informed neural networks (PINNs), together with less ill-conditioning for flow-field predictions in a fluid dynamical application. Moreover, in [12], a physics-informed KAN was proposed to predict the temperature and velocity fields in a turbulent flow using 3D experimental ground-truth (GT) data. To achieve this, the loss function included terms penalizing the residuals of the governing equations for velocity, as well as boundary conditions.
The initial version of KANs allows for great flexibility, which has been explored in different studies. For instance, the main development of KANs introduced B-spline functions together with a smoothing term as bases to expand the edge activation functions. However, other orthogonal basis functions have been implemented and compared, and each has shown to contain advantages for different applications. Chebyshev polynomials [13], ReLU [14], Jacobi polynomials [15], Gaussian radial basis function (RBF) [16], and Fourier bases [17] include some of the alternatives that have been introduced as possible replacements of B-spline functions. These alternatives have proved to reduce the computational time and the required parameters associated with B-splines while maintaining the overall network accuracy and precision for different applications. Also, in [18], a method was introduced for better tackling the model complexity in KANs, based on the differential evolution (DE) algorithm for selecting the optimal KAN structure. This method showed enhancements in convergence, prediction performance, and interpretability, compared with traditional KANs.
KANs have attracted attention for their potential benefits across different research fields, from material modeling to industrial applications. For instance, Zhang et al. in [19] have shown that integrating KANs into PINNs result in more accurate predictions of nonlinear deflections of ionic polymermetal composites (IPMC) and an improved convergency compared with MLP-based PINNs. Also in [20], KANs were used to construct physically admissible and polyconvex free-energy functions that were later used for modeling compressible hyperelastic materials. With the aim of reducing the training time for KANs, Howard et al. [21] introduced an architecture for KANs, based on domaindecomposition of the analysis region, allowing for multiple KANs to be trained in tandem in smaller domains, yielding more accurate solutions. As another application, in [22], the graph convolutional networks (GCNs) were replaced by spline-based KAN layers, resulting in graph Kolmogorov-Arnold networks (GKANs), forming a new way of inter-layer information processing, which proved to have a superior accuracy compared to GCNs when comparable number of parameters were considered. MLPs have been used for modeling different aspects of hot rolling process, including lateral flow or spread, which refers to the transverse deformation that occurs during thickness reduction. A hybrid CNN-long short-term memory (LSTM) model was proposed to predict lateral flow in hot strip finishing mills by integrating spatial and temporal features from rolling data [23]. Zhong et al. [24] enhanced the Shibahara spread model by incorporating equipment wear and interference factors, optimizing its parameters with Bayesian-optimized differential evolution and adaptive gradient descent, achieving a 9.77% improvement in width prediction.

In this work, we explore the potential of KANs to render interpretable models for spread in hot rolling of steel slabs. For reaching this purpose, initially, the GT data is generated via FE simulations. Next, an analytical model resulting in a closed-form solution for the spread is compared to the simulations results. Lastly, KANs are utilized for modeling the discrepancy between the analytical models and GT data, yielding interpretable models for width prediction that can serve as online monitoring tools.

Methodology

The original version of this paper is available on https://www.scientific.net/KEM.1050.215.pdf

KART and KANs: The main development of KANs has been established based on KART. According to this theorem [9], every continuous multivariate function f:[0,1]n→ℝ for n∈ℕ on a close and bounded interval can be expressed as a composition of finite number of continuous functions ϕi,j and addition operation:

f(x1,…,xn)=∑i=12n+1Φi(∑j=1nϕi,j(xj)),(x1,…,xn)∈[0,1]n(1)

In this equation, ϕi,j:[0,1]n→ℝ and Φi:ℝ→ℝ correspond to the continuous inner and outer map functions, respectively. Also, ϕi,j are global, hence they are independent of f, and Φi depend on the mapping f that is to be approximated. Theoretically, it is possible to express any nonlinear function f with only 2 layers, if those layers are wide enough; however, this might make the ϕi,j functions non-smooth and hence pathological [25]. One of the main innovations that was proposed in [8] was how to make a KAN layer and deeper KAN networks. Using the global nature of the inner functions ϕi,j, B-spline functions were used as bases to expand them. The number of the chosen splines ( N ) and their order (k) are considered as tunable hyperparameters:

ϕ(x)=∑n=0N−1cnBnk(x)(2)

where Bnk(x) represents the nth B-spline of order k and cn 's are parameters that are optimized in the back propagation. At one KAN layer, the ranges of considered grid points are updated according to the outputs of the previous layer. Moreover, to gain more control over the magnitudes of the resulting activation functions, a residual term was introduced in the expansion of the inner functions:

ϕ~(x)=∑n=0N−1cnBnk(x)+wbx1+e−x(3)

where wb is another trainable parameter. Between the nodes of KAN layers l and l+1, the inputs are transformed via the inner functions to form the outputs, as it can be seen in Fig. 1.

[picture]

Fig. 1: Structure of a KAN layer.

Fig. 1. Structure of a KAN layer.

If the nodes on layer l and l+1 are represented by xl and xl+1, respectively, where:

xl:=[xl,1,xl,2,xl,3,…]T,xl+1:=[xl+1,1,xl+1,2,xl+1,3,xl+1,4,…]T.

then, the transformation equation between these two layers can be written as:

xl+1=(ϕl,1,1(·)ϕl,1,2(·)⋯ϕl,1,nl(·)ϕl,2,1(·)ϕl,2,2(·)⋯ϕl,2,nl(·)⋮⋮⋮ϕl,nl+1,1(·)ϕl,nl+1,2(·)⋯ϕl,nl+1,nl(·))⏟:=Φlxl⇒xl+1,j=∑i=1nlϕl,j,i(xl,i).(4)

where for layer l,nl represents the number of inputs and Φl defines the collective activation functions. As a result, the overall mapping for the KAN network containing a total number of L layers performing on the input vector 𝐱inp  can be expressed as:

KAN(xinp )=(ΦL∘ΦL−1∘⋯∘Φ2∘Φ1)(xinp ).(5)

During the training of the KAN networks, for having sparse and simpler models, the concepts of sparsification and pruning were introduced to the network. In the sparsification of KAN networks, the L1 norm was initially defined for the individual and collected inner activation functions over the edges. Within a KAN layer l with ninp  and nout  number of inputs and outputs, respectively:

‖ϕ‖1:=1ninp ∑m=1ninp |ϕ(xm)|,‖Φl‖1:=∑i=1ninp ∑j=1nout ‖ϕl,i,j‖1.(6)

Also, an entropy regularization term for gaining a sufficient effect on sparsity was introduced:

S(Φl):=−∑i=1ninp ∑j=1nout ‖ϕl,i,j‖1‖Φl‖1ln(‖ϕl,i,j‖1‖Φl‖1)(7)

As a result, for a KAN with a total of L layers, the total loss objective ℒ is the sum of the network prediction loss ℒpred  and the regularization loss.

ℒ=ℒpred +λ[μ1∑l=1L‖Φl‖1+μ2∑l=1LS(Φl)].(8)

For a regression task with a total number of N samples, if we denote the GT target with y and if (•)(i) represents the ith  sample, ℒpred  reads:

ℒpred :=1N∑i=1N(KAN(xinp (i))−y(i))2.(9)

The overall model sparsity is dependent on the parameter λ in Eq. 8. In addition to regularization, pruning is performed on the node level as a strategy to detect the nodes that do not contribute to the overall network performance. For doing so, for each node i at layer l, the incoming and outgoing inner activation functions are evaluated, and the maximum value is taken for all the input and output connections (represented by Il,i and Ol,i in Fig. 2). The node is deactivated if the difference between these scores are less than a threshold value, resulting in smaller overall networks. The value of the threshold is considered a hyperparameter.

[picture]

Fig. 2: Pruning of KANs at node level.

Fig. 2. Pruning of KANs at node level.

FE simulations:

The original version of this paper is available on https://www.scientific.net/KEM.1050.215.pdf

[picture]

Fig. 3: The geometric parameters of the rolling process.

Fig. 3. The geometric parameters of the rolling process.

For generating the GT data, initially, the process parameters, including the initial width ( W0 ), initial and final thicknesses ( h0 and h1, respectively), and rolling radius ( R ) were transformed into nondimensional parameters using Buckingham- Π theorem [26]. The resulting three dimensionless groups x1:=W0/h0,x2:=h0/R, and x3:=h1/h0 also define the input space for the KANs, with bounds x1∈[1,5],x2∈[0.1,0.25], and x3∈[0.5,1). In the second step, to capture the full problem nonlinearity resulting from the FE simulations, a space-filling sampling was performed in the ( x1,x2,x3 ) space within the considered intervals. The space-filling quality of a particular sampling is quantified via its central discrepancy (CD) value [27] as the lower values correspond to better space-filling quality. We considered Latin Hypercube Sampling (LHS), Sobol, Halton, and random sampling methods, and it was observed that for lower sample sizes, Sobol sampling resulted in a lower CD value, which made it the optimal choice for the subsequent analysis (see Fig.4). Also, we considered a total of 128 sample points for the subsequent analysis.

[picture]

Fig. 4: CD vs. n for different sampling methods.

Fig. 4. CD vs. n for different sampling methods.

For obtaining the value of the geometrical parameters, required in the FE simulations, the value of the rolling radius was set to a value of 200 mm . The FE simulations were performed using the explicit solver of Abaqus with the assumption of an isothermal process at a temperature of 1100∘C, typical for industrial hot rolling applications. For the considered geometries, a global mesh size of 3 mm and
mass scaling was applied, while keeping slab kinetic energy within 5% of its internal energy. The angular velocity ( ω ) of the rollers were set to a constant value of 2.7227rad/s. Also, the rollers were assumed to be analytical rigid surfaces. For modeling the material behavior in the slab, the values of Poisson's ratio (ν) and coefficient of friction (μ) were set to 0.28 and 0.2 , respectively. Although the elastic modulus of steel decreases significantly at elevated temperatures, a constant value of E=200 GPa was adopted, as elastic effects are negligible compared to plastic deformation under hot rolling conditions. Also, the Johnson-Cook (JC) model was selected for modeling the hardening behavior, accounting for plastic strain (ε¯pl), plastic strain-rate (ε¯˙pl), and temperature (non-dimensionalized as T^:=[T−Tref ]/[Tmelt −Tref ]):

σ¯(ε¯pl,ε¯˙pl,T^)=[A+Bε¯pln][1+Cln(ε¯˙pl/ε˙0)][1−T^m].(10)

The required parameter set ( A,B,C,m,n,Tmelt ,Tref ,ε˙0 ) was adopted from [28] (corresponding to steel grade 2), as shown in Table 1.

Table 1: Selected JC parameters for steel grade 2 in [28].

A [MPa] | B [MPa] | n [−] | C [−] | m [−] | ε̇₀ [s⁻¹] | Tref [°C] | Tmelt [°C]
33.901 | 100.18 | 0.4951 | 0.2471 | 0.6444 | 0.0037 | 900 | 1500

Moreover, since it was necessary to evaluate the spread value for each simulation only under steadystate (SS) condition, the SS detection utility of Abaqus, available for the explicit solver, was utilized [29]. For this purpose, among the different possible norms, equivalent plastic strain (SSPEEQ) was chosen with the default tolerance of 0.001 . Also to form the SS detection control volume, exit plane and cutting plane were located at the roller center, and at x=60 mm from the roller center, respectively.

[picture]

Fig. 5: FE simulation of the rolled slab with depicted results at the last time-step for a ) transverse displacement distribution (U3),b) deformed cross-section at SS , and c ) distribution of the equivalent plastic strain (PEEQ). In a ) and b ), absolute values of U3 are represented with units of mm .

Fig. 5. FE simulation of the rolled slab with depicted results at the last time-step for a ) transverse displacement distribution ( U 3 ) , b ) deformed cross-section at SS , and c ) distribution of the equivalent plastic strain (PEEQ). In a ) and b ), absolute values of U 3 are represented with units of mm .

Results and Discussion

The original version of this paper is available on https://www.scientific.net/KEM.1050.215.pdf

Traditionally, there have been different closed-form solutions for predicting spread in hot rolling applications either grounded on assumptions to simplify the multi-physics problem or on empirical correlations. Accordingly, they introduce partial physics to the considered problem with reasonable generalization capability. One of such models is the proposed model by Shibahara [30]:

ln(W1, Shibahara W0)=ln(h0h1)exp(−1.64(W0h0)0.376(W0Rδ)0.016[W0h0](h0R)0.015[W0h0])(11)

Considering the FE solutions obtained for the sampled configurations, Shibahara's model indicates a drastic deviation from the GT with R2≈−0.159, MSE ≈18.3058 mm2, as shown in Fig. 6. The pronounced observable discrepancy between Shibahara's model and the GT is the objective to be defined as the output for KANs.

[picture]

Fig. 6: Prediction vs. true plot for Shibahara model compared to the GT data.

Fig. 6. Prediction vs. true plot for Shibahara model compared to the GT data.

B-spline activation functions of order 3 were used as the bases to expand the network activation functions. For the input-output (I/O) configuration of KANs, three definitions for the input space were tested. In the first setup, analytical spread predictions were used, without any further information. In the second setup, only the input space features {x1,x2,x3} were considered. In the last configuration, the combined set of the input feature set and the predictions of the analytical model were introduced to the network. The output for all of these cases were defined as GT spread values. Furthermore, as one of the tunable hyperparameters of the network, we allowed the network width to vary in a range {16, 32, 64, 128}. The network was trained for a total of 80 steps. Moreover, a train-validation-test ratio of 80-10-10 was selected as splitting criterion for the considered 128 samples. In Fig. 7, the obtained learning curves for training and validation losses are depicted for different network widths and input definitions.

[picture]

Fig. 7: Learning curves for the networks when the network input ( I ) is considered to be a)I={ΔWShibahara },b)I={x1,x2,x3} with λ=0.1,c)I={x1,x2,x3,ΔWShibahara } with λ=0.01, and d)I={x1,x2,x3,ΔWShibahara } with λ=0.1.

Fig. 7. Learning curves for the networks when the network input ( I ) is considered to be a ) I = { Δ W Shibahara } , b ) I = { x 1 , x 2 , x 3 } with λ = 0.1 , c ) I = { x 1 , x 2 , x 3 , Δ W Shibahara } with λ = 0.01 , and d ) I = { x 1 , x 2 , x 3 , Δ W Shibahara } with λ = 0.1 .

As can be seen in Fig. 7a ), the input space definition I={ΔWShibahara } where WShibahara  is appropriately nondimensionalized, yields higher values of training and validation loss errors compared to the cases in b)−d ). Moreover, although the expanded definitions in b ) and c ) results in lower root mean square error (RMSE) values, overfitting remains a problem in these cases. Choosing the expanded input space definition, including the input space features and analytical model predictions, together with a smaller regularization parameter of 0.01 , results in models with less overfitting, while retaining the model accuracy (Fig. 7d)). The number of grid extensions is one of the trainable hyperparameters referring to fine-graining the spline grids in KANs. For the network with I={x1,x2,x3,ΔWShibahara } and λ=0.1, fine-graining was performed for the considered width spectrum with a grid range of g∈{2,3,4,5,8,10,15,20}. As it can be seen in Fig. 8, the stair-case loss reduction can be seen up to a grid-size of 3 .

[picture]

Fig. 8: Learning curves for network grid and width updates.

Fig. 8. Learning curves for network grid and width updates.

As an ultimate metric to evaluate the selected KAN network, the selected KAN network with 4 inputs and overall regularization parameter of 0.1 and a grid size of 2 , the network was applied to the test set, for which the results are shown in Fig. 9. In Fig. 9a ), the learning curves between the training and validation loss are not suggestive of overfitting in the network and in part b ) the network shows an acceptable performance on the test set with R2 and RMSE metrics equal to 0.99 and 0.38 mm , respectively. Also, the pruned network is shown in Fig. 9d ), from which a closed-form expression can be obtained in the symbolification step. The resulting nondimensional expression for ΔWFE reads:

ΔWFE(x1,x2,x3,ΔWShibahara )=−0.164x12+0.108x1−17.378x2+0.026x3−0.132ΔWShibahara 2+3.255ΔWShibahara +1.530.(12)

The complexity of the symbolified expression is dependent on the second regularization parameter, corresponding to μ2 in Eq. 8. We set this parameter to 2 and in Fig. 9c), the performance of the closed-form of the discrepancy model is evaluated against the test set, which is closely aligned with the network, prior to symbolification.

Conclusions

The original version of this paper is available on https://www.scientific.net/KEM.1050.215.pdf

In this work, we investigated the applicability of KANs for obtaining interpretable models that capture spread in hot rolling processes of steel slabs. Initially, the process-specific geometrical parameters were used to define the sampling input space in terms of non-dimensional parameters. To better represent the problem non-linearity, different space-filling sampling methods, including random, Halton, LHS, and Sobol, were compared in terms of their CD value in lower data regimes. A total of 128 sampled points were considered, and they were used to define the configurations subjected to FE simulation. For increasing computational efficiency, the spread values at SS were captured for each simulation and the results were post-processed to form the GT for the network, with a train-validationtest ratio of 80-10-10. The obtained spread results were compared to Shibahara's closed-form solution,

[picture]

Fig. 9: Performance of a trained KAN network for capturing the width with a ) learning curve, b ) prediction vs. true plot before pruning and symbolification, c ) prediction vs. true plot for the test set, after pruning and symbolification (Eq. 12), and d ) network pruning, considering different input contributions.

Fig. 9. Performance of a trained KAN network for capturing the width with a ) learning curve, b ) prediction vs. true plot before pruning and symbolification, c ) prediction vs. true plot for the test set, after pruning and symbolification (Eq. 12), and d ) network pruning, considering different input contributions.

which was the subject of network correction. As the output of the network, GT spread values were selected and B-spline functions of order 3 were chosen as the trainable activation functions. Also, with the focus on grid-extension and network width as the tunable hyperparameters, it was observed that expanded input space definition with the inclusion of the non-dimensional process parameters and the analytical model predictions resulted in a reduced over-fitting on the validation set. Moreover, the same effect was observed via reducing the overall regularization coefficient, which was set equal to 0.1 . With the tuned grid-size and network width of 2 and 4 , the network also showed an optimal performance on the test set with R2 and RMSE measures of 0.99 and 0.3750 mm , respectively. Lastly, the output of the network was symbolified and resulted in a closed-form expression for the existing model discrepancy, with a similar prediction performance on the test set to the KAN.

Acknowledgment

The original version of this paper is available on https://www.scientific.net/KEM.1050.215.pdf

This research was carried out under project number T22008 in the framework of the Research Program of the Materials innovation institute (M2i) (www.m2i.nl) supported by the Dutch government. The authors also gratefully acknowledge Tata Steel for valuable discussions.

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The original version of this paper is available on https://www.scientific.net/KEM.1050.215.pdf

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The original version of this eBook is available on https://www.scientific.net/book/advanced-numerical-and-ai-strategies-for-material-forming/978-3-0364-1986-2

Key Engineering Materials, ISSN: 1662-9795, Vol. 1050, pp 227-234

doi: 10.4028/p-E8wywR

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-21

Revised:
2026-03-02

Accepted:
2026-03-02

Online:
2026-04-16

Modelling Deformation Mechanisms Decomposition, Separation and Compaction in Mechanical Joining Processes of Fiber Reinforced Thermoplastics on Meso Scale

The original version of this paper is available on https://www.scientific.net/KEM.1050.227.pdf

Benjamin Gröger a,1*, Johannes Gerritzen b,1, Andreas Hornig c,1,2,3 and Maik Gude d,1 

¹Institute for Lightweight Engineering and Polymer Technology (ILK), TUD Dresden University of Technology, Holbeinstrasse 3, 01307 Dresden (Germany)

2 Department of Engineering Science, Solid Mechanics and Materials Engineering, University of Oxford, Oxford, OX1 3PJ, United Kingdom

3 Center for Scalable Data Analytics and Artificial Intelligence Dresden/Leipzig (ScaDS.AI), TUD Dresden University of Technology, Strehlener Straße 12-14, Dresden, 01069, Germany

abenjamin.groeger@tu-dresden.de, bjohannes.gerritzen@tu-dresden.de, c  andreas.hornig@tu-dresden.de, d  maik.gude@tu-dresden.de (*corresponding author)

Keywords: Composite-Metal Joining, Simulation, Fluid-Structure-Interaction, Finite Element Method (FEM)

Abstract

The mechanical joining of continuous fiber-reinforced thermoplastics (cFRTP) and metal sheets represents a promising approach for manufacturing hybrid lightweight structures. To reduce the time and cost associated with extensive experimental investigations, numerical modeling strategies are increasingly applied. In this numerical study, a further step in the modelling strategy for the direct pin-pressing (DPP) process of cFRTP and metal sheets is presented. The study focuses on modeling and simulating the occurring deformation mechanisms of decomposition, compaction, and separation of individual rovings on the mesoscale to analyze the resulting material structure. For this purpose, two simplified models were derived. The textile architecture is represented based on micrographs of cross-sections and discretized using the finite element method. The deformation of individual rovings during joining leads to a deformation of their initial elliptical cross section. To capture this level of resolution, both a cohesive zone and a pure contact approach are applied within the rovings. The highly viscous thermoplastic melt is modeled as a fluid employing the Arbitrary Lagrange-Eulerian (ALE) method. Matrix and roving meshes are coupled to account for fluidstructure interaction (FSI) during process. The study shows that coupling of matrix and rovings is necessary to obtain more accurate predictions of the deformation behaviour. Furthermore, the cohesive zone approach is better suited to simulate the emerging deformation mechanisms.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1050.227.pdf

Continuous fiber reinforced plastics (cFRTP) are widely used in industrial applications due to their outstanding specific strength and stiffness. In contrast to thermosetting matrix systems, cFRTP enable shorter cycle times and easier forming during processing, primarily due to their temperaturedependent forming behavior, especially above the melting temperature. Additionally, the handling of the semi-finished products so called organo-sheets, is easier and faster at room temperature. As a result, the use of hybrid structures combining metals and cFRTP has increased, which leads to challenging tasks for joining dissimilar materials [1] and subsequently to develop methodologies to reliably join metals and cFRTP [1,2]. Mechanical joining technologies generate a form closure by plastic deformation with the occurrence of fiber failure and delamination (e.g. [3]). Alternatively, when the thermoplastic matrix is above its melting temperature, fiber rearrangement processes can occur (cf. [4,5]). The rearrangement process is driven by process route (e.g. heating strategy) [6], tool design and concept [7,8], material system [9] as well as process parameters (e.g. tool velocity, process temperature). To date, investigations have been primarily based on experimental approaches due to the lack of suitable numerical methods for simulating such local forming processes. However,
experimental setups require fundamental decisions regarding the tool concept and process configuration, which limit parameter variation and constrain the development process.

During the joining process of metal sheets and organo-sheets above melting temperature fiber deformation and rearrangement [10] as well as matrix flow process [11] due to the motion and deformation of the joining partner and tools occur. Numerical simulations of the joining process are capable of predicting the resulting material structure of the cFRTP, including changes in fiber orientation and the formation of matrix-rich zones. Accurate prediction of the resulting material structure requires a scale-specific modeling approach, in which fibers and matrix are modeled separately [12,13], commonly referred to as direct numerical simulation. However, modeling individual fibers, the matrix, and the associated fluid-structure interaction (FSI) on the microscale requires very high computational efforts. Therefore, modeling approaches on the sub-mesoscale based on fiber bundles [14-16] or meso scale [17] for cFRTP are appropriate. These approaches enable the prediction of fiber rearrangement and deformation mechanisms as well as matrix flow processes. When using fabrics, deformation mechanisms of the rovings, such as decomposition, compaction, and separation, can be observed (Fig. 1).

The numerical method for these flow process simulations are often based on Arbitrary-LagrangeEulerian (ALE) method [18], which allows to simulate a flow of fluid or ductile materials [12,17,19]. Within the fluid zone, solid structures (e.g. fibers, fiber bundles, tools) can be embedded and coupled. The coupling can be achieved either via velocity-field-based force transfer [12,20] or penalty-based or constraint-based coupling schemes [21].

In order to develop a modelling strategy for mechanical joining technologies, the novel DPP developed in [4] is chosen as a representative example. Based on the experimental investigation of this DPP by Popp et al. [6,7,9], a first numerical simulation approach on the sub-mesoscale using fiber bundles was proposed by Gröger et al. [19].

In the present work, the modeling strategy is extended to the mesoscale by explicitly accounting for the textile architecture and the roving geometry in order to predict the deformation mechanisms of decomposition, compaction, and separation (Fig. 1). Modeling the individual yarns with deformable cross-sections results in severe numerical instabilities due to the large deformation degrees of freedom. Therefore, a dedicated numerical study is conducted to systematically investigate and model these dominant mechanisms.

[picture]

Fig. 1. Computed tomography scan of the resultant material structure after DPP with the occurring mechanism of compaction, decomposition and separation of the rovings

Fig. 1. Computed tomography scan of the resultant material structure after DPP with the occurring mechanism of compaction, decomposition and separation of the rovings

Materials and Methods

The original version of this paper is available on https://www.scientific.net/KEM.1050.227.pdf

The presented paper investigates a non-crimp fabric (NCF) made of glass fibers previously studied in [19], combined with a polypropylene (PP) matrix material (Borealis BJ100-HP), using DPP joining process improved by Popp et al. [7]. The principle of the joining process is illustrated in Fig. 2.

[picture]

Fig. 2. Illustration of the DPP process in three steps according to Popp et al. [7], adapted from [19]

Fig. 2. Illustration of the DPP process in three steps according to Popp et al. [7], adapted from [19]

Joining process. During the joining process, the organo-sheet is placed in a positioning device and elastically supported by a PEEK insert. A mask with a circular opening is positioned above the sheet, and an infrared (IR) spot selectively heats the organo-sheet to approximately 180∘C at the bottom and 300∘C at the top. This creates a zone of molten matrix in the area underneath the mask opening, while the surrounding material remains solid to suppress undesired fiber rearrangement. After heating, a steel sheet with a pin is pressed into the cFRTP at a defined velocity with v=100 mm/s. Subsequently, a constant compression force is maintained during a relaxation phase, completing the joining process as the organo-sheet cools below its melting temperature.

Material modelling. The matrix is modeled as a compressible shear-thinning fluid (*MAT_ALE_VISCOUS) with material parameters of the PP at 200∘C from [19]. The shear-ratedependent viscosity is implemented via a tabulated function following a regularized CROSS model [22]. The corresponding parameters η∞ as infinite-shear viscosity, η0 as zero-shear viscosity, and the both CROSS parameters λ and n are listed in Table 1 and were experimental determined in [19]. In addition, the dependency of density on hydrostatic pressure, commonly referred to as an equation of state (EOS), is described by the Murnaghan model, given as

Δp=k0[(ρρ0)γ−1].(1)

Here, ρ0 denotes the initial density at atmospheric pressure and 200∘C, while k0 and γ are material coefficients of the EOS model. The EOS parameters are fitted by experimental trails with the high pressure capillarity rheometer GÖTTFERT RHEOGRAPH 75 [19]. Due to the use of the ALEmethod for the PP melt flow process an EOS is mandatory.

Table 1. Coefficients of the Cross model describing the shear-rate-dependent viscosity and parameters of the Murnaghan EOS for the PP matrix at 200∘C, taken from [19]

Cross model | EOS
mathematical formula | mathematical formula | mathematical formula | n | mathematical formula | mathematical formula | mathematical formula
2.68 | 328.22 | 0.040 | 0.59 | superscript number | 103.1 | 6.519

As concluded by Gröger et al., the textile architecture must be represented in detail for the mesoscale modeling approach [19]. To allow for an efficient assessment of the rovings' behavior, they are modeled as solid, with orthotropic plasticity as surrogate for the material flow. For this, a Hill type yield surface is chosen (*MAT_122_3R_3D). Given the challenge of determining material parameters above the melting temperature experimentally, parameters from a preliminary numerical study presented in [23] are adopted and extended. They identified a maximum flow stress of σmax= 0.308 MPa , both for shear as well as transverse loading. Hence, omax is taken as surrogate yield strength under all uniaxial loadings apart from that in the fiber direction. Since high temperature compression tests revealed negligible flow in this direction [24], an arbitrary high strength of 1 GPa is assumed, effectively limiting flow in this direction, as shown in [25]. Since the analyses in [23] did not exhibit an increase of deformation resistance with increasing flow, no hardening was
implemented. The resulting surrogate material parameters are summarized in Table 2. With the direction-dependent elastic constants and Hill yield potential parameters F,G,H,L,M, and N.

Table 2. Orthotropic elastic and plastic material properties of the homogenized GF/PP roving

| Elastic constants
Property | superscript number | superscript number | mathematical formula | superscript number | superscript number | superscript number | superscript number | superscript number | superscript number
Value | 44,380 | 10.8 | 10.8 | 3.91 | 3.91 | 3.84 | 0.49 | 0.49 | 0.49

Plastic HILL yield potential parameters

Property | mathematical formula | mathematical formula | mathematical formula | mathematical formula | mathematical formula | mathematical formula | mathematical formula
Value | 0.999 | mathematical formula | mathematical formula | 0.5 | 0.5 | 0.5 | 0.308

The geometric dimensions of the NCF are determined by computed tomography (CT) scans and are illustrated in Fig. 3. Gröger et al. [19] measured the characteristic dimensions of the NCF and assumed a total layer thickness of 0.25 mm in order to model eight layers, resulting in an overall organo-sheet thickness of 2 mm . The height of an individual roving was measured to be 0.32 mm .

[picture]

Fig. 3. CT scan of the pre-consolidated organo sheet made of non-crimped glass fiber fabric (white) and PP (gray) with the characteristic dimensions (average) measured in [19]

Fig. 3. CT scan of the pre-consolidated organo sheet made of non-crimped glass fiber fabric (white) and PP (gray) with the characteristic dimensions (average) measured in [19]

Numerical Setup

The original version of this paper is available on https://www.scientific.net/KEM.1050.227.pdf

The numerical setup follows the results of Gröger et al. [19], using LS-DYNA R16.1.0 mpp with double precision. The process model includes a conical pin with a root diameter dpin  of 1.25 mm , a tip diameter DPin  of 1.50 mm , and a height hPin  of 1.80 mm (Fig. 4 top). To reduce the computational effort, the setup is simplified to a single-roving (1R) and a two-roving (2R) model. Each roving with a total length of 0.8 mm is discretized using four finite elements along the fiber direction and two elements in the thickness direction (Fig. 4, bottom). To enable the modeling of roving decomposition and separation, two different approaches are investigated. In the first approach, cohesive zone (CZ) elements are implemented between the mid-section of the roving and two additional planes along the thickness direction (Fig. 4 top right, purple elements). The bilinear material behavior of the CZ elements is defined using the parameters reported in [23]. The parameters are based on representative volume cells on micro scale with PP melt and glass fibres under shear and normal stress. In the second approach, the roving elements are separated geometrically and interact solely via a frictionless contact definition, without the use of CZ elements. Both the CZ-based and the separation-based (Sep) models employ a coupling between the fluid matrix and the solid structures using a penalty based coupling method (*CONSTRAINED_LAGRANGE_IN_SOLID) [21]. The rovings are fully embedded in the matrix material, which is allowed to flow through the elements of the ALE domain. The vertical boundaries of the ALE domain are constrained to prevent material flow in the normal direction. In addition, the lateral motion of the rovings is restricted by rigid walls on both sides transverse to the fiber direction, while the roving ends are only permitted to move within the cross-sectional plane. To avoid excessive element distortion at the roving ends, the rotationally symmetric pin geometry is
simplified to a cuboid extending along the full roving length. The height and width of the cuboid correspond to the pin height hPin  and the tip diameter DPin . This cuboid is driven downward in a displacement-controlled manner with the same velocity v as the original pin. The rovings are positioned between the tool above and a rigid block below, at which the reaction forces Fz are recorded for comparison. While the 1R model primarily focuses on roving compaction beneath the pin, the 2R model emphasizes the interaction between neighbouring rovings, particularly with respect to decomposition and separation mechanisms.

[picture]

Fig. 4. Simplified models of the DPP with increasing complexity from 1R to 2R model

Fig. 4. Simplified models of the DPP with increasing complexity from 1R to 2R model

Results

The original version of this paper is available on https://www.scientific.net/KEM.1050.227.pdf

The reaction force Fz is shown in Fig. 5. For the 1R model, both investigated approaches exhibit only a slight increase in reaction force with increasing compaction defined by the tool displacement Δz and the initial tool-PEEK-insert distance z0 via compaction =z0−Δzz0.

[picture]

Fig. 5. Reaction force Fz at the rigid block for each model and the two modelling approaches of CZbased and separation-based model

Fig. 5. Reaction force F z at the rigid block for each model and the two modelling approaches of CZbased and separation-based model

However, the CZ-based approach allows a higher degree of compaction before numerical instability occurs. A similar trend is observed for the 2 R model. For the 2 R model, both approaches initially show an increase in reaction force due to the first contact between the tool and the rovings. The subsequent decrease in force at a compaction ε of approximately 0.1 can be attributed to the lateral displacement of the upper roving out of the cavity. At a compaction level of around 0.32 , the separation-based model becomes numerically unstable and collapses, accompanied by an abrupt increase in reaction force. In contrast, the CZ-based model remains stable up to a compaction of approximately 0.75 and exhibits an increasing but unsteady force response, which is mainly caused by the continued contact with the tool and the progressive compaction of the lower roving.

For further evaluation of the CZ-based and separation-based models, the resulting deformation of the rovings and the matrix is illustrated in Fig. 6. In the 1R-CZ model, the roving retains a continuous
structure despite the higher degree of compaction. In contrast, the separation-based model exhibits a lower level of compaction, and the roving remains separated even though the matrix restricts the motion via fluid-structure coupling. The matrix flow appears physically plausible, as the molten material is displaced outward from the cavity.

[picture]

Fig. 6. Deformation of the rovings and matrix flow during compaction for each model

Fig. 6. Deformation of the rovings and matrix flow during compaction for each model

For the 2R models, increased motion and deformation of the roving elements are observed. In the CZ-based approach, the roving located directly beneath the tool is strongly compressed while remaining continuous. The adjacent roving moves laterally away from the tool, which indicates that the coupling forces between the matrix and the roving may be underestimated. In the separation-based model, both rovings exhibit pronounced separation and reduced compaction. The outward motion of roving elements from the cavity is considered responsible for the abrupt increase in reaction force and the subsequent numerical collapse. This behavior becomes even more pronounced when no coupling between the roving elements and the matrix is considered, confirming the importance of fluid-structure interaction for numerical stability and realistic material deformation. Furthermore, the dynamic effects caused by the high tool velocity during joining induce rigid body motion once interactions between rovings occur. This has to be considered in full-scale simulations, as it may result in excessive and nonphysical element distortion of the roving elements. Therefore, this qualitative numerical study provides essential insight for future full-process simulations and further model development.

Summary

The original version of this paper is available on https://www.scientific.net/KEM.1050.227.pdf

The presented results indicate that the CZ-based approach is more suitable for simulating the resulting material structure of the DPP process, particularly with respect to roving compaction and decomposition when considering fluid-structure interaction. Moreover, the CZ-based model exhibits improved numerical stability compared to a purely separation-based approach. The proposed mesoscale modeling strategy represents a computationally efficient compromise between microscale and macroscopic approaches and enables the investigation of local deformation mechanisms in cFRTP joining processes. Nevertheless, further refinement of the roving material model is required to reduce element distortion at free roving edges. Planned squeeze experiments will provide material parameters for thickness compaction and serve as a basis for future model validation. In the long term, the approach offers significant potential for simulation-based process development of hybrid metal-cFRTP joints.

In future work, the material model of the roving needs to be further refined in order to reduce element distortion at the free edges of the rovings. For this purpose, squeeze experiments will be conducted to identify material parameters governing compaction in the thickness direction.

Acknowledgements

The original version of this paper is available on https://www.scientific.net/KEM.1050.227.pdf

The authors gratefully acknowledge

The computing time made available to them on the high-performance computer at the NHR Center of TU Dresden. This center is jointly supported by the Federal Ministry of Research, Technology and Space of Germany and the state governments participating in the NHR (www.nhr-verein.de/unsere-partner).

The subproject C04 of the Collaborative Research Centre TRR285/2 for CT data and measurements.

Fundings

The original version of this paper is available on https://www.scientific.net/KEM.1050.227.pdf

This research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - TRR 285/2 - 418701707 subproject A03.

References

The original version of this paper is available on https://www.scientific.net/KEM.1050.227.pdf

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This ePub file was created using machine-generated methods and automated software to convert the source text into an ePub format and may contain formatting inconsistencies from the original version. Formatting may vary depending on the device used.

The original version of this eBook is available on https://www.scientific.net/book/advanced-numerical-and-ai-strategies-for-material-forming/978-3-0364-1986-2

Key Engineering Materials, ISSN: 1662-9795, Vol. 1050, pp 235-247

doi: 10.4028/p-SA56ki

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-21

Revised:
2026-02-16

Accepted:
2026-02-17

Online:
2026-04-16

Combined Experimental and Simulation-Based Approach for Optimizing Roll Forming of Pipes for Hydrogen Infrastructure

The original version of this paper is available on https://www.scientific.net/KEM.1050.235.pdf

Murli Manohar Pandey 1,a*, Anja Rautenstrauch 1, b*, Andreas Kunke 1,c and Till Clausmeyer 1, d

1 Professorship Forming Technology, Institute for Machine Tools and Production Processes, Chemnitz University of Technology, Reichenhainer Straße 70, 09126 Chemnitz, Germany

a  murli-manohar.pandey@mb.tu-chemnitz.de, b  anja.rautenstrauch@mb.tu-chemnitz.de, c  andreas.kunke@mb.tu-chemnitz.de, dtill.clausmeyer@mb.tu-chemnitz.de

Keywords: roll forming, modeling, FEM simulation, pipe.

Abstract

Roll forming pipes for hydrogen infrastructure poses particular challenges for process design, especially with regard to geometric accuracy and the avoidance of forming defects that could compromise the integrity of the pipelines. Geometric accuracy is crucial to ensure uniform pressure distribution within the pipe. Conventional trial-and-error approaches to developing roll flower designs are time-consuming and cost-intensive, especially when working with high-strength steel grades. This work presents an integrated methodology for roll forming of monolithic sheet by incorporating real-world machine stiffness and experimental anisotropy. A finite element model was developed for S235 and S355 steels, validated through three-point bending and Digital Image Correlation (DIC). While database-derived models (JMatPro) underestimated yield stress by up to 30%, the experimental model precisely predicted strain distribution trends with high precision, with local deviations remaining below 0.02 absolute strain. A central novelty is the integration of in-situ 3D laser scans of the roll forming mill under load, allowing the simulation to account for elastic machine deflections of up to 0.9 mm . The resulting simulation framework enables the accurate prediction of pipe geometry and process-induced residual stresses, which are critical for the longterm integrity of pipelines against hydrogen-induced cracking.

Introduction.

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Roll forming is a key manufacturing process for pipeline tubes in energy infrastructure. In hydrogen applications, the process must satisfy especially high demands for leak tightness, material integrity, and geometric accuracy, since hydrogen can alter mechanical properties and increase susceptibility to hydrogen-induced cracking. Bolobov et al. show that compressed hydrogen affects the mechanical properties of pipeline steels [1], while Jia et al. and the BAM project report emphasize hydrogen-induced cracking and the sensitivity of welded microstructures to hydrogen exposure [2, 3]. Consequently, process-induced stresses and strain distributions must be controlled to ensure longterm pipeline safety.

Numerical simulation has therefore become increasingly important for roll forming process design. Egger et al. demonstrate the effective use of FE simulation for tube manufacturing including roll forming and welding [4], and Song et al. show that complex material behavior in high-strength steels can be captured by modern FE models [5]. Dong et al. further highlights the need for experimental validation in bimetallic pipe forming simulations [6]. However, many studies still rely on simplified or database-based material models, which are often not validated for the bending-dominated strain paths of roll forming. Yu et al. point out that supplementing material data and verifying simulations experimentally improves prediction accuracy [7].

Another challenge is the influence of machine and tool stiffness. Lamprecht et al. show that mill compliance significantly affects forming results and requires parameter identification [8], while the article from voestalpine highlights the importance of realistic process parameters for reliable FE simulations [9]. Validation in literature is often limited to final geometry or single process variables, as shown by Sheu and Jian in ERW tube simulations [10]. Studies on complex profiles and highstrength materials from Mahajan et al., and Suckow et al. underline the potential of FE simulation,
but a comprehensive coupling of experimental material characterization and process validation is still rarely established [11, 12].

Nevertheless, there is a challenge in appropriately considering the machine tool elasticity in simulations as well as process design. Moreover, the 'loaded' roll gap representing the active deformation state with the sheet inserted, deviates significantly from the theoretical tool geometry. While many studies assume an 'infinitely rigid' toolset, this simplification often yields geometric inaccuracies in the final product. This work addresses this limitation through an integrated methodology combining Finite Element Analysis (FEA) with real-world geometric deviations captured via portable 3D scanning. By quantifying the differential between unloaded and loaded machine states, a stiffness-compensated simulation model is proposed. This approach is essential for optimizing hydrogen-ready pipe forming, where even sub-millimeter geometric deviations can induce localized stress concentrations and premature failure.

Two complementary strategies were employed to derive the flow curves of material model. Initially, synthetic flow curves and mechanical properties were generated using JMatPro. Subsequently, these were compared against experimental data obtained via uniaxial tensile testing. The validity of the material models was assessed using three-point bending tests, where simulation results, specifically bending forces, profiles, and strain distributions, were compared against experimental measurements to evaluate predictive accuracy in bending-dominated forming processes.

The resulting comprehensive simulation framework enables the accurate prediction of global pipe geometry, cross-sectional strain distributions, and springback behavior. Furthermore, it identifies critical zones for potential edge waves and stress concentrations. The comparison between JMatProbased and experimentally derived models quantifies discrepancies in plastic flow behavior and their subsequent impact on geometric accuracy. Ultimately, this close coupling of experiment and simulation explicates deformation mechanisms and significantly reduces the reliance on physical prototyping.

Material modeling

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Accurate representation of JMatPro-based material behavior is crucial for simulating the roll forming process. In the approach, the material model for pipe steels such as S235 and S355 was developed taking into account the alloy composition and mechanical properties obtained from standardized tests. The materials to be investigated are the steel alloys presented in Table 1.

Table 1. Alloy composition of test material.

Material | Alloy composition [%] | Standard
S235 | Al: 0.0021; Cr: 0.17; Cu: 0.35; Mn: 0.6; Mo: 0.02; Nb: 0.001; Ni: 0.11; Si: 0.21; Ti: 0.0011; V: 0.004; C: 0.07; N: 0.0057; P: 0.026; S: 0.035 | DIN EN 10025-2
S355 | Al: 0.0025; Cr: 0.06; Cu: 0.2; Mn: 1.09; Mo: 0.017; Nb: 0.002; Ni: 0.09; Si: 0.2; Ti: 0.002; V: 0.002; C: 0.18; N: 0.0082; P: 0.015; S: 0.012 | DIN EN 10025-2

Table 2. Mechanical properties from tensile test.

Material | E [GPa] | Rp0,2 [MPa] | Rm [MPa] | A [%] | Ag [%]
S235 | 244.12 | 291.5 | 417.64 | 34.48 | 17.73
S355 | 240.64 | 393.5 | 485.27 | 28.18 | 13.98

Tensile tests were carried out in accordance with DIN 50125 to determine the yield strength and strength properties. The mechanical parameters elastic modulus E, yield strength Rp0,2, tensile
strength Rm, elongation at break A, and plastic uniform elongation Ag for 2 mm thick samples were derived from the test results and are presented in Table 2. The Ludwik relationship was used in this work to mathematically extrapolate the flow curves obtained from physical tests beyond necking region, for higher strains. It is a proven empirical model for mapping the hardening of metallic materials and describes the plastic flow behavior via the relationship as shown in Eq. 1.

kf=kf0+C*φn.(1)

Here kf is the true yield stress, kf0 is the initial yield stress, φ is the true plastic strain, C is the strength coefficient, and n is the work hardening exponent. The parameters C and n are determined to be 414 MPa and 0.406 for S235 material and 404 MPa and 0.430 for S355 material respectively, by regression analysis from the test data. The resulting flow curves are shown in Fig. 1.

The experimentally obtained flow curves exhibit generally larger stresses than the JMatProgenerated data. For S235, the experimental flow curve starts at approximately 300 MPa , whereas the JMatPro-based curve begins at around 210 MPa . A similar deviation is observed for S355, where the experimental flow curve starts at about 395 MPa , compared to 360 MPa from JMatPro.

[picture]

Fig. 1. Material flow curves derived and employed in Simufact forming.

Fig. 1. Material flow curves derived and employed in Simufact forming.

These differences indicate an underestimation of the yield stress when using database-based material models and may lead to deviations in simulation results, particularly with respect to forming forces, stress and strain distribution, and springback. Therefore, the use of experimentally determined flow curves is beneficial for achieving more realistic process simulations.

To characterize the directional dependence of the material, tensile tests were performed on samples at 0∘ (rolling direction), 45∘ and 90∘. A key parameter is the Lankford coefficient r , which describes the ratio of transverse strain to thickness change and is derived from the longitudinal and transverse strains measured during the tensile test.

r=ln(b1/b0)ln(s1/s0).(2)

The Lankford coefficient r of the different orientations is used to calculate the mean anisotropy rm and the planar anisotropy rplan , which describe the average forming behavior and directional deviations.

rm=r0∘+2r45∘+r90∘4rplan =r0∘−2r45∘+r90∘22

(3)(4)

These characteristic values, as shown in Table 3, are crucial for the numerical simulation of sheet metal forming processes, as they supplement the material model with regard to the directional dependence of plastic flow and deformation properties.

Table 3. Material Anisotropy.

Material | Rolling direction [°] | Lankford coefficient r | Mean anisotropy rm | Plane anisotropy rplan
S235 | 0 | 0.370 | 0.391 | -0.039
45 | 0.410
90 | 0.374
S355 | 0 | 0.495 | 0.535 | -0.004
45 | 0.539
90 | 0.568

Validation of the material model

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To validate the material model, three-point bending tests were performed in accordance with DIN EN ISO 7438 on a universal testing machine from Hegewald & Peschke ( 150 kN maximum force). The aim of the tests was to experimentally determine the maximum bending force, the resulting strains, and the resulting bending angle, and then to compare these values with the simulation results from Simufact Forming. This comparison made it possible to verify whether the material model correctly represents the real mechanical behavior of the steels under bending-dominated loading. Since, rollforming is mainly dominated by bending operation, this serves as first validation.

The non-contact optical 3D measurement system ARAMIS Adjustable 24 M was used for the precise recording of the shape changes during the bending tests. The system is based on digital image correlation (DIC) and point and pattern tracking of the sample surface (Fig. 2). To prepare the sample surface, a stochastic, high-contrast speckle pattern was applied, which the ZEISS CORRELATE software requires in order to precisely track displacements and strains. The pattern makes it possible to locate specific points on the surface in successive images and calculate their 3D coordinate changes.

During the tests, the system tracks surface changes in real time, allowing local strains, the bending angle, and geometric deviations caused by the process to be determined with precision. The data was then evaluated in ZEISS CORRELATE, yielding precise and reproducible results for the validation of the material models.

[picture]

Fig. 1. Test setup Three-point bending tests according to DIN EN ISO 7438.

Fig. 1. Test setup Three-point bending tests according to DIN EN ISO 7438.

The experimental and numerical results show consistent trends in bending angle and force development for both S235 and S355 steels shown in Table. 4, where tensile test-based flow curves were used in the bending simulation. While the quantitative deviations in maximum force, specifically
the overestimation for S355, point to the limits of hardening parameters under pure bending, the consistent trends in bending angle and strain distribution confirm the model's overall reliability. Hill48 model was used for the yield criterion with Lankford coefficients (Table 3) as input parameters.

Table 4. Bending test - comparison of simulation vs. experiment.

Parameters | Experiment | Simulation
Material | Rolling direction [°] | Feed motion of the bending punch [mm] | Bending angle [°] | Max. bending force [N] | Bending angle [°] | Max. bending force [N]
S235 | 0 | 4 | 20.5 | 1151 | 24.27 | 931
0 | 8 | 45.5 | 1235 | 51.60 | 1135
0 | 12 | 69 | 1155 | 73.83 | 1101
0 | 16 | 87 | 1104 | 93.72 | 1101
45 | 16 | 88.5 | 1225 | 93.72 | 1101
90 | 16 | 88 | 1196 | 93.65 | 1101
S355 | 0 | 4 | 20 | 1167 | 22.33 | 1417
0 | 8 | 45 | 1235 | 45.27 | 1505
0 | 12 | 68.5 | 1195 | 67.69 | 1258
0 | 16 | 88 | 1177 | 87.34 | 1258
45 | 16 | 88 | 1196 | 87.35 | 1256
90 | 16 | 88 | 1189 | 87.34 | 1258

The precise prediction of strain (error <2% ) verifies that the material model, incorporating batchspecific experimental flow curves and Hill48 anisotropy, effectively reflects the mechanical behavior required for the roll forming simulation. Fig. 3 shows the bending force-stroke curves obtained from the three-point bending tests. The experimental force curves for S235 and S355 show a significant overlap, with both materials reaching peak forces of approximately 1200 N . In contrast, the simulations reveal material-dependent deviations: the bending force for S355 is consistently overestimated, while the simulation for S235 underestimates the force during the initial stroke phase.

[picture]

Fig. 2. Force-stroke curve in comparison simulation and experiment.

Fig. 2. Force-stroke curve in comparison simulation and experiment.

Fig. 4 depicts the strain distribution from experimental and numerical three-point bending tests for both investigated materials. For steel grade S235, a maximum strain of 0.2 was measured experimentally, while the simulation predicted a comparable value of 0.18 , Similarly, for S355, the experimentally determined strain of 0.21 is closely matched by the simulated value of 0.23 . The small deviations observed indicate that the material model is able to reliably reproduce the strain distribution under bending-dominated loading conditions. Overall, the results demonstrate that the simulation provides an accurate representation of the experimental behavior.

[picture]

Fig. 3. Equiv. plastic strain distribution in bending test after springback- experiment vs. simulation (sample 6 m/min to rolling direction) - a) S235 exp., b) S235 sim., c) S355 exp., d) S355 sim.

Fig. 3. Equiv. plastic strain distribution in bending test after springback- experiment vs. simulation (sample 6 m / m i n to rolling direction) - a) S235 exp., b) S235 sim., c) S355 exp., d) S355 sim.

Comparison of material models

The original version of this paper is available on https://www.scientific.net/KEM.1050.235.pdf

A detailed model was created in Simufact Forming 2025.2 for the numerical investigation of the roll forming process shown in Fig. 5.

[picture]

Fig. 4. Tool model.

Fig. 4. Tool model.

The experimental validation was performed on a roll forming mill comprised of 15 stands arranged in a progressive flower design, utilizing 8 active (AR) and 7 passive (PR) roll stands. To bridge the gap between the theoretical design and production reality, the mill geometry was captured using a portable 'T-Scan hawk2' 3D scanner. This enabled in-situ measurement without machine disassembly, capturing roll profiles, positions, and alignments in the actual production environment. The measurement was performed in both unloaded and loaded states to capture process-induced elastic deflections due to forming forces. The geometric comparison of loaded state measurement to unloaded state is presented in Fig. 6. Deflection was determined by comparing loaded positions to unloaded baseline.

[picture]

Fig. 5. Geometric deviation of the rolling mill in loaded state (with 2 mm sheet).

Fig. 5. Geometric deviation of the rolling mill in loaded state (with 2 mm sheet).

Along with the deflection evaluation, the axial position of individual roll as well as their relative distances to passive rolls were also evaluated. Maximum deviation of 2.5 mm was observed at passive stand 5, heavily containing positional error rather than elastic deflection. All the other stands were relatively stiffer, with approx. 0.9 mm maximum elastic deflection at roll stand 2 as shown in Fig. 6.

While many studies assume an 'infinitely rigid' toolset, leading to geometric errors, this model explicitly accounts for machine compliance. To maintain computational efficiency, the rollers are modeled as rigid bodies in Simufact Forming 2025.2, but their positions are adjusted to the measured 'loaded' roll gap identified during scanning. The numerical domain was discretized using solid hexahedral elements, with 5 physical layers across the 2 mm sheet thickness to capture throughthickness stress gradients. The blank geometry for the simulation was defined with dimensions of 800 mm in length, 183.78 mm in width. Furthermore, an inverse kinematic approach was adopted, where the sheet remains stationary while the forming stands translate; this method is mathematically equivalent to the physical process ( drax ) and isolates geometric deformation from inertial effects. Although friction plays a minor role in roll forming simulation [13] and can be neglected, a Coulomb friction coefficient of 0.01 was applied to stabilize the model and isolate forming behavior. Table 5 summarizes the specific parameters employed in the roll forming simulation.

Table 5. Process definition for sheet roll forming (Mechanical model).

Simulation type | Implicit 3D
Ambient temperature °C | 20
Workpiece starting temperature °C | 20
Workpiece: Thermal transfer coefficient W/m²K | 50
Tool starting temperature °C | 20
Tool: Thermal transfer coefficient W/m²K | 50
Process temperature | cold
Element type | Solid hexahedral
Feed rate mm/s | 100

The aim of the simulation was to optimize the roll gaps and to compare material models, particularly the flow curves stored in JMatPro, with the experimentally determined material data from tensile test. The model enables the prediction of pipe geometry, strain distribution, and springback along the pipe length and across the width. By comparing the JMatPro-based simulation results with
the experimentally based material models, differences in plastic flow behavior and in the accuracy of the prediction for the pipe geometry can be quantified.

[picture]

Fig. 6. Comparison of plastic strain simulated with material card JMatPro and experimental material data - a) S235 JMatPro. b) S235 exp. data. c) S355 JMatPro. d) S355 exp. Data.

Fig. 6. Comparison of plastic strain simulated with material card JMatPro and experimental material data - a) S235 JMatPro. b) S235 exp. data. c) S355 JMatPro. d) S355 exp. Data.

Plastic strains from the simulations using the JMatPro material data were compared with those based on experimentally determined material data (see in Fig. 7). For both S355 and S235, the results show good agreement. Edge strains for S355 range from 0.21 to 0.30, while S235 exhibits edge strains between 0.13 and 0.18 in both simulation approaches. These findings confirm that the JMatPro-based material map reliably reproduces the experimentally observed plastic deformation and is suitable for predictive simulations.

[picture]

Fig. 7. Comparison of simulated pipe geometry with experimentally manufactured pipes - a) S235 JMatPro, b) S235 exp. data, c) S355 JMatPro, d) S355 exp. Data.

Fig. 7. Comparison of simulated pipe geometry with experimentally manufactured pipes - a) S235 JMatPro, b) S235 exp. data, c) S355 JMatPro, d) S355 exp. Data.

For consistency and to eliminate end effects, geometric comparison was performed on the mid sections of the manufactured pipes, utilizing the segment length of 220 mm to the simulated pipe geometry using both experimental and JMatPro-based material data (see in Fig. 8). For S235, surface deviations range from -0.2 to -0.4 mm , with edge deviations up to 1 mm . S355 shows similar deviations for both material maps.

[picture]

Fig. 8. Average surface deviation of simulated pipe segments (using different material models) from experimentally manufactured pipes (nominal).

Fig. 8. Average surface deviation of simulated pipe segments (using different material models) from experimentally manufactured pipes (nominal).

Overall, both approaches reproduce the experimental pipe geometry well, with the largest discrepancies consistently occurring at the pipe edges. While deviations across the surface remain limited, S235JMatPro variant shows the largest deviation, reaching maximum deviation of 1 mm . The bow effect would explain the longitudinal deviations, while the asymmetric positioning of rolls in experiment might have caused the circumferential deviation. The simulation thus provides a basis for validating the material models in the context of pipe forming and shows the extent to which the theoretical flow curves from JMatPro reflect the actual behavior under the specific roll forming conditions as seen in Fig. 8 and Fig. 9.

Iterative Adjustment of Roll Gap for Realistic Simulation

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The initial input data for the distance between the active top and bottom roller axes ( 1−4 ) are based on the design concept used for roll tooling layout and serve as the reference for all subsequent iterations. The simulation was refined through a series of iterations to bridge the gap between design concept and production reality.

The design of the geometric parameters such as the distance of rollers in axial direction and selecting the roll gap depends on the target geometry of the profile and the material. Even smaller changes of the sheet thickness and the material grade or even the material batch require adaptations.

For this study. we refer to the initial state as "basic - design concept". Refinements of these settings for the roll gap are referred to as " 1 - Real Data unloaded "i.e. the measured positions in the unloaded state, " 2 - Real Data loaded "i.e. similar to 1, but with the sheet inserted, " 3 - modified positions of stand 5−8 " and " 4 - modified positions of stand 30% " refer to settings for subsequent simulations with adjusted geometric parameters. Therefore, 4 iterations of simulations were conducted. In the setup of the simulations, one faces the challenge that using elastically deformable rollers is numerically too costly. However, applying rigid rollers with too small roll gaps, not accounting for the deformation in the experiments leads to contact problems and finally simulation terminations. As
discussed, "finding the nominal" and the "appropriate loaded" gap Is the main challenge discussed in this paper.

The roll gaps of AR1 to AR4 are dominated by the sheet thickness, while the roll gaps of AR5 to AR8 are dominated by end profile geometry. In Iteration 1 and 2, initial setting of real unloaded/loaded geometries resulted in simulation termination. This failure highlighted that the "asbuilt" machine state lacked the necessary roll gap to accommodate the 2 mm sheet thickness in the roll passes 1 to 4 . Only after a targeted adjustment of roll passes 1 to 4 in Iteration 3, explicitly accounting for the 2 mm sheet thickness in the roll gap design, was it possible to achieve a stable and executable simulation of the roll forming process.

Table 6. Distance between active top and bottom roller axis 90∘

Iteration / active roller | AR1 | AR2 | AR3 | AR4 | AR5 | AR6 | AR7 | AR8
Basic - Design concept | 242.0 | 242.0 | 242.0 | 242.0 | 269.0 | 266.0 | 263.6 | 261.0
1 - Real Data unloaded (±0.05 mm) | 241.2 | 241.4 | 241.4 | 241.6 | 268.0 | 265.1 | 262.3 | 259.3
2 - Real Data loaded (±0.05 mm) | 242.0 | 241.6 | 241.7 | 241.8 | 268.2 | 265.5 | 262.7 | 259.8
3 - Modified positions of stand 1 - 4 | 242.1 | 242.0 | 242.0 | 242.0 | 268.2 | 265.5 | 262.7 | 259.8
4 - Modified positions of stand 5 - 8 | 242.1 | 242.0 | 242.0 | 242.0 | 268.9 | 265.9 | 263.6 | 260.5

Upon completion of Iteration 3, the resulting profile was analyzed and found to exhibit torsional deformation as shown in Fig. 10. This observation provided the direct motivation for the asymmetric flow adjustments implemented in Iteration 4, which subsequently eliminated the torsion and improved the overall stability of the roll forming process, hence, a more balanced deformation sequence.

[picture]

Fig. 9. Comparison of simulated pipe geometry from Iteration 3 (pre-optimization) to Iteration 4 (post-optimization)

Fig. 9. Comparison of simulated pipe geometry from Iteration 3 (pre-optimization) to Iteration 4 (post-optimization)

Following the iteration 4, it was observed that the component did not exhibit torsion. A closer examination of the forming behavior in roll pass 8 revealed that while the profile in Iteration 3 maintained symmetrical contact with the rollers, it resulted in higher forming forces and residual torsion, as shown in Fig. 11.

[picture]

Fig. 10. Plastic strain of the formed pipes in the rolling gap 8 - a) Iteration 3 b) Iteration 4

Fig. 10. Plastic strain of the formed pipes in the rolling gap 8 - a) Iteration 3 b) Iteration 4

In contrast to traditional design goals of perfect symmetry, Iteration 4 resulted in an asymmetrical deformation pattern in the pipe. Despite asymmetrical flow, this configuration required significantly less forming force while maintaining full contact of the pipe with the rollers. This indicates that the adjustments made in Iteration 4 altered the stress distribution and material flow, reducing the force demand in roll pass 8 while slightly changing the deformation pattern, as shown in Fig. 11 and Fig. 12.

[picture]

Fig. 11. Vertical forces in Roll forming stand AR5 to AR8 in Iteration 3 and 4

Fig. 11. Vertical forces in Roll forming stand AR5 to AR8 in Iteration 3 and 4

A more balanced stress state in the pipe is desirable because it prevents the formation of localized high-energy zones that can promote hydrogen accumulation and subsequent embrittlement. Consequently, the modified stress distribution achieved in Iteration 4 not only improves forming efficiency but may also reduce the risk of hydrogen-induced damage in the final component.

These findings highlight the importance of accurately capturing stress and material flow in the simulation model, since only a realistic representation of the forming process allows meaningful evaluation of design modifications. In general. the initial, more detailed examinations of the roller model indicate that the accuracy of the simulation closely reflects the component results when the models are adjusted as realistically as possible. Consequently, further investigations regarding the spacing and positions of active and passive rollers should be conducted to improve the predictive capability of the simulation.

Conclusion and Outlook

The original version of this paper is available on https://www.scientific.net/KEM.1050.235.pdf

Roll forming of pipes is a key manufacturing process for hydrogen infrastructure and requires stringent safety, tightness, and dimensional accuracy. Conventional trial-and-error development is time-consuming and costly, particularly for high-strength steels. This study successfully established an integrated methodology that bridges the gap between theoretical tool design and the physical production environment. By incorporating batch-specific experimental anisotropy and loaded state machine deflection data, the predictive capability of the simulation was significantly enhanced, moving beyond the inaccuracies inherent in "infinitely rigid" tool assumptions.

Database-derived material models (JMatPro) were found to be insufficient for precise process design, as they underestimated the yield stress of S235 and S355 steels by up to mathematical formula. In contrast, the experimentally derived Ludwik-based model accurately reflected the mechanical behavior in bending-dominated paths, predicting strain distribution with local deviations remaining below 0.02 absolute strain.

Portable 3D laser scanning identified process-induced elastic deflections of up to 0.9 mm in the rolling mill under load. The integration of these "loaded" gap measurements into the simulation framework proved essential for model stability; neglecting these deflections led to unrealistic contact problems and simulation terminations. A critical discovery in the final iteration 4 was that deviating from idealized symmetrical deformation patterns can be beneficial. Implementing a stiffnesscompensated, asymmetrical flow successfully eliminated longitudinal torsion and significantly reduced the vertical forming forces required in the final roll passes.

The refined simulation provides a robust basis for controlling process-induced residual stresses. By achieving a more balanced stress distribution, the risk of localized high-energy zones, which are susceptible to hydrogen accumulation and subsequent hydrogen-induced cracking, is effectively minimized.

Building on this, future work will focus on systematically varying and optimizing the spacing and relative positioning of active and passive rolls. The goal is to better control material flow, minimize asymmetric stress states, and reduce geometric deviations at an early stage. This in-depth analysis can further enhance the predictive capability of the simulation and provide a robust basis for the optimal design of future roll forming processes.

Acknowledgment

The original version of this paper is available on https://www.scientific.net/KEM.1050.235.pdf

The authors gratefully acknowledge the financial support provided by the Saxon State Ministry for Science. Culture and Tourism (Sächsisches Staatsministerium für Wissenschaft und Kunst. SMWK) through the CETPartnership program and EU funding under the research and innovation framework programs. This work was carried out within the framework of the project HOOPLA (Grant No. 100694930).

References

The original version of this paper is available on https://www.scientific.net/KEM.1050.235.pdf

V. I. Bolobov, Il'nur U. Latipov, G. G. Popov, G. V. Buslaev, Y. V. Martynenko, Estimation of the Influence of Compressed Hydrogen on the Mechanical Properties of Pipeline Steels, Energies 14(19) (2021) 6085. https://doi.org/10.3390/en14196085

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J. Song. J. Lan. L. Zhu. Z. Jiang. Z. Zhang. C. Ma. Finite Element Simulation and Microstructural Analysis of Roll Forming for DP590 High Strength Dual Phase Steel Wheel Rims. Materials 17(15) (2024) 3795.

Z. Dong. Z, Xu. W, Wang. Z, Bi. J. Zhang, Numerical Simulation and Experimental Confirmation of a Bimetallic Pipe Forming Process, Materials 13(16) (2020) 3561.

Y. Yu, W. Haibo, L. Qiang, G. Yanzhi, Finite element simulation of flexible roll forming with supplemented material data and the experimental verification, Chin. J. Mech. Eng. 29 (2016) 1 12.

M. Lamprecht, E. Kocbay, M. Leonhartsberger, Y. Vetyukov, F. Bleicher, Nonlinear mechanical model of the shaft of a roll forming mill and parameter identification, Int. J. Adv. Manuf. Technol. 112 (2021) 3363 3375.

voestalpine Roll Forming Corporation, Finite Element Analysis in Roll Forming Applications. ThomasNet Knowledge Center, 2022.

J.J. Sheu, E.-X. Jian, The ERW Tube Cold Roll Forming Simulation with Different Cage Roll Arrangement and Fin-Pass Design, Proceedings of the 14th International Conference on the Technology of Plasticity - Current Trends in the Technology of Plasticity. ICTP 2023. Lecture Notes in Mechanical Engineering. Springer. Cham. 2024.

P. Mahajan, A. Abrass, P. Groche, FE simulation of roll forming of a complex profile with the aid of steady state properties, Steel Res. Int. 89 (2018) 1700350. . 201700350

T. Suckow, J. Schroeder, P. Groche, Roll forming of a high strength AA7075 aluminum tube, Prod. Eng. Res. Devel. 15 (2021) 573-586.

Q.V. Bui, J.P. Ponthot, Numerical simulation of cold roll-forming processes, J. Mater. Process. Technol. 202 (2008) 275-282.

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The original version of this eBook is available on https://www.scientific.net/book/advanced-numerical-and-ai-strategies-for-material-forming/978-3-0364-1986-2

Key Engineering Materials, ISSN: 1662-9795, Vol. 1050, pp 249-257

doi: 10.4028/p-F27Mqr

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2026-01-19

Revised:
2026-03-01

Accepted:
2026-03-03

Online:
2026-04-16

Integrated Artificial Neural Network-Based Approach for Predicting Surface Roughness Parameters in Laser Surface Texturing of Ti6Al4V

The original version of this paper is available on https://www.scientific.net/KEM.1050.249.pdf

Agata Sposato 1,a*, Giovanni Carpino 2,b, Maria Rosaria Saffioti 1,c, Giovanna Rotella 2, d, Domenico Umbrello 1,e

¹Department of Mechanical, Energy and Management Engineering, University of Calabria, Via Ponte Pietro Bucci, Cubo 45C, 87036, Rende, CS, Italy

2 Department of Engineering, University LUM Giuseppe Degennaro, Casamassima, BA 70010, Italy

a agata.sposato@unical.it, b carpino@studenti.lum.it, c mariarosaria.saffioti@unical.it, d rotella@lum.it, e  domenico.umbrello@unical.it

Keywords: Laser surface treatment, Surface roughness, Artificial neural network.

Abstract

Laser surface texturing (LST) is an effective technique for tailoring the surface properties of Ti6Al4V alloy, widely employed in biomedical applications where surface topography plays a key role in osseointegration and functional performance. Nevertheless, the strong nonlinear relationship between laser process parameters and resulting surface roughness still limits predictive control of laser-textured surfaces. This work presents an experimental study aimed at investigating the influence of laser surface texturing parameters on the surface morphology of Ti6Al4V. Key process variables, including laser power, scanning speed, pulse frequency, pulse duration and overlap percentage are systematically varied using a fiber laser system. The textured surfaces are characterized through three-dimensional surface roughness parameters, namely Sa, Sz, Sku, Svk, and Ssk, providing a detailed quantitative description of surface topography relevant for biomedical applications. The resulting experimental dataset represents a fundamental basis for the subsequent development of artificial intelligence models, based on neural networks, for predicting surface roughness parameters as a function of laser processing conditions. The proposed approach supports data-driven optimization of laser surface texturing processes within intelligent and sustainable manufacturing frameworks.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1050.249.pdf

Among the most used materials in the field of medicine, titanium alloys, more specifically, Ti6Al4V stand out due to their excellent balance of high specific strength, corrosion resistance, and good biocompatibility [1]. The aforementioned characteristics of titanium make Ti6Al4V a benchmark material used in orthopedic implants, dental fixtures, and trauma devices. Nevertheless, the long-term performance of titanium-based implants cannot be described only through bulk characteristics; rather, their surface characteristics, which include topography, roughness, wettability, and microstructural integrity, play a significant role. With this idea in mind, surface engineering techniques have started to receive more attention as a potential tool to improve osseointegration, tribological performance, and biological response of titanium-based implants without altering the bulk mechanical characteristics of the component. Among the available surface modification techniques, LST has emerged as a highly flexible and controllable approach. This is because laser processing allows for the precise control of surface morphology, energy delivery, and repeatability, all while maintaining compatibility with industrial-scale production and sustainable manufacturing concepts [3]. New developments in laser technology, such as ultrafast and fiber lasers, have extended the applicability of LST to include the precise manipulation of surface features on the micro- and nano-scale without the need for a heat-affected zone [4]. The versatility of the laser processing techniques has led to the recognition of LST as a key technology in a variety of industrial fields, such as the biomedical, aerospace, and manufacturing industries. In the field of biomedicine, laser surface texturing of titanium alloys has primarily been used to modify the surface roughness in an effort to enhance the functional properties of the surface in order to improve cell adhesion, bone ingrowth, and mechanical interlocking in the surrounding tissues. Experimental investigations have shown that the surface
patterns created using laser surface texturing can significantly influence the biological responses. In particular, Rotella et al. highlighted the relevance of laser-based surface functionalization for titanium screws used in orthopaedic implants, showing that carefully designed surface textures can improve implant-bone interaction while preserving structural integrity [5]. These findings confirm that surface roughness parameters are not merely descriptive quantities, but key design variables for biomedical performance. Additionally, the laser surface treatment of Ti6Al4V has also been investigated from the point of view of the process-structure relationship. Numerical as well as experimental analyses have confirmed that the power, scanning velocity, pulse width, and hatch spacing of the laser beam strongly influence the geometry of the molten pool, as well as the cooling rate and the microstructural transformations that occur during the process. The finite element method has also been found useful for the simulation of the thermal fields as well as the molten pool during the laser surface treatment, providing useful information about the surface morphology and the microstructure that occur during the process, as reported elsewhere [6]. However, despite the significant advancements that have taken place, the highly nonlinear relationship between the surface roughness characteristics and the various parameters of the laser beam is still considered the main challenge that is faced during the surface roughness characterization process. The surface roughness characterization has shifted more towards the 3D surface characteristics, such as Sa, Sz, Sku, Svk, and Ssk, as compared to the 2D surface characteristics, such as Ra and Rz, for functional surfaces that come into contact with tissues. However, the correlation of these measures of surface roughness with the parameters of the laser process through analytical and physics-based models is extremely difficult due to the inherent complexities involved in the interaction of the laser with the material and the stochastic nature of the process involving the change of the material from the solid to the liquid phase and the subsequent ejection and solidification of the molten material. To overcome these difficulties, the application of Artificial Intelligence (AI) and Machine Learning (ML), which have gained significant attention in the area of advanced manufacturing processes, has come into prominence through several comprehensive reviews on the application of AI in the area of smart production and its ability to extract relevant patterns from complex experimental results and contribute to the decision-making process in the optimization of the process [7,8]. AI has been reported to have strong application in the prediction of the results of the process in the area of manufacturing, including the prediction of surface roughness in the process. Batu et al. demonstrated the effectiveness of machine learning models in predicting surface roughness of additively manufactured components, showing significant improvements over traditional regression approaches when dealing with nonlinear and multivariate relationships [9]. These results are particularly relevant for laser-based processes, where multiple interacting parameters govern surface formation. Similar trends have also been reported in tool condition monitoring and predictive maintenance, where AI-based frameworks have been used to integrate the data from the sensors to improve the reliability of the processes and the quality of the products [10,11]. Recent studies have also focused on the role of AI in the domain of laser processing. Murzin provided a comprehensive overview of AI-driven innovations in laser processing of metallic materials, underlining how neural networks and hybrid data-driven models can capture complex dependencies between laser parameters and resulting material properties [12]. Such approaches are increasingly recognized as essential tools for advancing laser surface engineering beyond empirical methodologies. While there is an increasing amount of published research on laser surface texturing and AI-based manufacturing techniques, the integration of surface roughness characterization and AI-based modeling is limited, especially for the case of biomedical-grade titanium alloy Ti6Al4V. Most published research works have focused on the qualitative nature of the surface texture and/or limited parameters of surface roughness without fully addressing the range of 3D surface parameters. Moreover, most AI-based roughness prediction models have been developed for machining or additive manufacturing processes, rather than for laser surface texturing of titanium alloys intended for biomedical use. Recent research on laser surface pre-treatment and functionalization has emphasized the significance of surface state management in the development of durable bonds and performance, thus supporting the need to implement precise and predictable surface modification techniques [13]. Moreover, the sustainability factor has also encouraged the application of laser and

AI techniques, as they can contribute to the development of efficient, waste-free, and energy-aware manufacturing processes [14, 15]. Within this framework, the present study positions itself as a foundational experimental contribution aimed at systematically correlating laser surface texturing parameters with a comprehensive set of surface roughness metrics (Sa, Sz, Sku, Svk, Ssk) for Ti6Al4V alloy. By generating a high-quality experimental dataset, this work provides the necessary basis for the subsequent development of AI models, specifically artificial neural network (ANN), capable of predicting surface roughness outcomes as a function of laser process parameters. Such predictive capability is expected to support the design of optimized, functionally tailored titanium surfaces for biomedical applications, bridging the gap between laser processing science and intelligent manufacturing systems.

Materials and Methods

The original version of this paper is available on https://www.scientific.net/KEM.1050.249.pdf

The experimental campaign was carried out on Ti6Al4V alloy specimens in the form of flat sheets with a thickness of 1 mm . The samples were cut to suitable dimensions for laser processing and surface characterization. Before the laser treatment, the surfaces had to be cleaned to eliminate possible contaminants and ensure the repeatability of the interaction conditions during the laser treatment. Laser surface texturing was performed using an air-cooled ytterbium fiber laser system (Lasit Compact Mark G8), with a wavelength of 1065 nm . The laser treatment was provided with an F-theta lens with a focal length of 160 mm and a nominal focal spot diameter of 55μ m. The experimental setup used for the laser treatment is schematically shown in Figure 1.

[picture]

Fig. 1. Experimental set up of laser texturing process.

Fig. 1. Experimental set up of laser texturing process.

The laser surface texturing process was carried out by varying the major parameters of the process in order to observe the effect of the parameters on the surface morphology obtained by the laser surface texturing process. The parameters varied in the experiment were the laser power ( P ), scanning speed (v), frequency (f), pulse width ( τ ), and overlap percentage (l). The range of the parameters varied in the experiment is shown in Table 1. The selected ranges were chosen to ensure stable processing conditions while enabling the generation of a wide variety of surface topographies.

Table 1. Investigated laser process parameters.

Laser parameters | Range | Unit
Power (P) | 20 – 100 | W
Scanning speed (v) | 50 – 1600 | mm/s
Pulse frequency (f) | 1.82 – 72.73 | kHz
Pulse amplitude (τ) | 4 – 250 | ns
Overlap % (l) | 0 – 90 | %

After laser processing, surface topography was characterized using a non-contact threedimensional white light confocal profilometer (Rtc/UP-24). The measurement process was carried out following the ISO 25178-2:2012 standard for consistency in the obtained results. To ensure the
reliability of the obtained results, the roughness of the laser-textured surfaces was measured at five different locations for each sample. The reported roughness values correspond to the arithmetic mean of the repeated measurements. The surface morphology was quantitatively described using the following surface roughness parameters according with ISO 25178; Sa (arithmetical mean height) represents the average absolute deviation of the surface from the mean plane and provides a general indication of surface roughness amplitude; Sz (maximum height) corresponds to the sum of the maximum peak height and maximum pit depth within the sampling area, describing extreme surface features; Ssk (skewness) indicates the asymmetry of the height distribution and allows distinguishing peak-dominated from valley-dominated surfaces; Sku (kurtosis) describes the sharpness of the height distribution and provides information on the presence of sharp peaks or deep valleys; Svk (reduced valley depth) is particularly relevant for functional surfaces, as it characterizes the depth of valleys contributing to fluid retention and wear behavior. The combined evaluation of amplitude and functional parameters enables a more comprehensive understanding of the laser-induced surface morphology.

Results

The original version of this paper is available on https://www.scientific.net/KEM.1050.249.pdf

Surface analysis. Before developing the artificial neural network model, a preliminary statistical analysis of the experimentally measured surface roughness parameters was carried out in order to assess data dispersion, variability, and distribution. This stage is crucial to grasp the intrinsic complexity of the laser surface texturing process and to justify the use of data-driven modeling techniques. In particular, the most important descriptive statistics concerning the five surface roughness parameters (Sa, Sz, Ssk, Sku, and Svk) of the 151 laser-textured samples made of the titanium alloy Ti6Al4V are given in Table 2. For each surface roughness parameter, the minimum, maximum, average values, and standard deviations are reported.

Table 2. Statistical summary of surface roughness parameters.

Roughness parameters | Min | Max | Mean | Std. Dev. | Unit
Sa | 0.60 | 6.51 | 2.15 | 1.44 | μm
Sz | 11.04 | 40.96 | 22.39 | 7.22 | μm
Ssk | -1.37 | 1.38 | 0.25 | 0.32 | -
Sku | 0.09 | 9.40 | 3.20 | 1.20 | -
Svk | 0.51 | 9.95 | 1.82 | 1.29 | μm

The wide range and high standard deviation observed for the roughness parameters indicate a strong sensitivity of surface morphology to laser processing conditions, supporting the need for advanced nonlinear modeling techniques. To further investigate the variability and distribution characteristics of the measured surface roughness parameters, graphical analyses were performed. Figure 2 shows the distribution of Sa, Sz, Ssk, Sku, and Svk across the entire experimental dataset.

[picture]

Fig. 2. Distribution of surface roughness parameters.

Fig. 2. Distribution of surface roughness parameters.

Boxplots were selected to highlight median values, interquartile ranges, and the presence of potential outliers, providing a concise visualization of the statistical dispersion associated with each roughness descriptor. To provide an intuitive understanding of the influence of individual laser parameters on surface roughness, selected mono-parametric scatter plots were generated.

[picture]

Fig. 3. Trend of the Sa and Sz measurements at varying scanning speed and laser power.

Fig. 3. Trend of the Sa and Sz measurements at varying scanning speed and laser power.

Figures 3-4 illustrate representative relationships between key laser processing parameters and selected roughness parameters. Only selected combinations (power and scanning speed) are shown to avoid redundancy and to focus on the most physically meaningful trends. For example, as shown in Figure 3 the variation of the surface roughness parameters Sa and Sz as a function of laser power for different scanning speeds. Overall, both Sa and Sz exhibit a clear sensitivity to laser power, with higher power levels generally leading to smoother surface, while lower speed lead to rougher surface. This can particularly be observed at lower and intermediate scanning speeds, where the increase in energy input results in greater melting and material redistribution, thus accentuating the surface features. However, increasing the scanning speed results in the reduction of the surface roughness values, which can be attributed to the reduced energy density provided to the surface and the corresponding reduced interaction time between the laser beam and the material.

Figure 4 shows the trends of the surface roughness parameters Ssk, Svk, and Sku as a function of laser power for different scanning speeds. The statistical parameters Ssk and Sku further highlight changes in surface morphology induced by the combined variation of power and scanning speed. Ssk values fluctuate around zero, indicating transitions between peak-dominated and valley-dominated surface profiles depending on the processing conditions. In particular, lower scanning speeds and higher power levels tend to promote more asymmetric height distributions, reflecting the formation of irregular peaks or deeper valleys. Similarly, Sku values suggest variations in the sharpness of the height distribution, with certain parameter combinations producing surfaces characterized by more pronounced localized features.

The reduced valley depth Svk follows trends broadly consistent with those observed for Sa and Sz, increasing at higher power levels and lower scanning speeds. This behavior suggests that higher energy input enhances the formation of deeper surface valleys, which may have implications for functional properties such as lubrication retention or osseointegration in biomedical applications.

Overall, the observed trends confirm that surface morphology is governed by a complex interaction between laser power and scanning speed, with nonlinear and coupled effects that justify the subsequent development of a multivariate ANN-based predictive model.

[picture]

Fig. 4. Trend of the Ssk, Sku and Svk measurements at varying scanning speed and laser power.

Fig. 4. Trend of the Ssk, Sku and Svk measurements at varying scanning speed and laser power.

Artificial neural network. All experimental data were collected in a structured dataset consisting of 151 cases, which was used to develop and validate the proposed ANN model. The objective of the network was to establish a quantitative relationship between laser processing parameters and the resulting surface topography descriptors. The ANN was configured as a feed-forward multi-output architecture including five input parameters (laser power, scanning speed, pulse amplitude, overlap and frequency) and five output variables corresponding to the measured roughness parameters (Sa, Sz, Ssk, Sku and Svk). This multi-output configuration enables the simultaneous prediction of correlated surface descriptors, preserving their intrinsic physical interdependence. The network consisted of one hidden layer with a number of neurons ranging between 5 and 20 in order to identify the optimal configuration balancing accuracy and generalization capability. A sigmoid activation function was adopted to capture the nonlinear interactions between process parameters and surface features. The dataset was randomly divided into training ( 90% ) and testing ( 10% ) subsets. During training, weights and biases were iteratively optimized through a backpropagation-based procedure aimed at minimizing the prediction error between experimental and simulated outputs. The comparison between experimental measurements and ANN predictions (Figure 5) highlights a satisfactory overall agreement. In particular, Sz and Svk exhibit strong trend reproduction across different running conditions, with limited dispersion between predicted and measured values. Importantly, no systematic overestimation or underestimation is observed, confirming the absence of prediction bias and the robustness of the trained model. For Sa , slightly larger local deviations can be observed in specific tests; however, the absolute discrepancy remains moderate relative to the experimental variability, indicating that the main surface evolution mechanisms are correctly captured. In the case of Ssk, percentage-based metrics are inherently less representative due to the parameter's proximity to zero and sign inversion. Nevertheless, the predicted skewness values remain within the same order of magnitude as the experimental data, demonstrating the model's capability to reproduce distribution asymmetry without systematic distortion. The largest deviations are observed for Sku in correspondence with isolated experimental peaks. Since kurtosis is highly sensitive to sporadic extreme asperities, these discrepancies are likely associated with limited representation of rare surface events within the dataset. This suggests that further dataset enrichment may improve prediction accuracy for higher-order statistical descriptors.

[picture]

Fig. 5. Comparison between experimental and ANN-predicted surface roughness parameters (a) Sa, (b) Sz, (c) Ssk, (d) Sku, (e) Svk.

Fig. 5. Comparison between experimental and ANN-predicted surface roughness parameters (a) Sa, (b) Sz, (c) Ssk, (d) Sku, (e) Svk.

It is important to emphasize that the dataset consists of a limited number of experimental cases; nevertheless, the adopted network architecture and training strategy ensured good generalization capability and avoided overfitting. The ability of the ANN to simultaneously predict multiple correlated roughness descriptors highlights the robustness of the proposed modeling approach. Beyond predictive capability, the developed ANN framework represents a powerful tool for process optimization. Once the model is fully trained, the addition of more experimental data will enrich the model, which can then be employed for the determination of the optimal set of parameters for the
laser process, so as to achieve the desired surface characteristics. Additionally, the inclusion of more functional performance indicators, such as coating adhesion, tribological properties, or biocompatibility, will facilitate the implementation of an efficient data-informed, sustainable surface engineering approach, for the intelligent design of the laser process for the titanium alloy material systems.

Conclusion

The original version of this paper is available on https://www.scientific.net/KEM.1050.249.pdf

The current research has confirmed the possibility of using a multi-output Artificial Neural Network for predicting the laser-induced surface topography parameters of titanium alloys. The findings of the current research can be summarized as follows:

Effective multi-parameter prediction: The proposed ANN model has been successfully used for establishing the nonlinear relationship between five laser process parameters and five surface roughness descriptors (Sa, Sz, Ssk, Sku, and Svk).

Good generalization capability: despite the relatively limited dataset ( 151 experimental cases), the model showed satisfactory agreement between predicted and experimental values, particularly for Sz and Svk, with no evidence of systematic prediction bias.

Robust trend reproduction with identified limitations: the ANN accurately captured the main surface evolution trends across different processing conditions. Greater discrepancies were found for higher-order statistical properties, such as Sku, in association with isolated extreme surface events, emphasizing the sensitivity of kurtosis to extreme asperities.

Potential for data-driven process optimization: the developed modeling framework holds a promising position for intelligent laser parameter control and surface tailoring.
Future studies will aim to extend the experimental data set and incorporate additional functional performance indicators for enhanced accuracy in predicting surface properties for sustainable surface engineering strategies.

Acknowledgements

The original version of this paper is available on https://www.scientific.net/KEM.1050.249.pdf

Authors sincerely thank @STAR and MaTeRiA laboratories, funded by "Progetto STAR 2 PIR010000800", for providing equipment employed for experimental analysis. This research was supported by Ministry University Research within the PRIN 2022 (project ADVANCE2022W9SHCJ) and the PNRR (project FAIR Spoke 9 AI- PE00000013) programs funded by the NextGenerationEU.

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Keywords Index

#

. Stamping Residual Learning-Based Synthetic Data for Hybrid Metamodeling in the Stamping of an Automotive Door Panel

A

Aerospace Aluminum Alloys Parameter Optimization and Data-Driven Soft-Sensor Framework for Torque Prediction in Bobbin-Tool Friction Stir Welding of AA2024

Ai Models An AI-Based Approach to Developing a Microstructural Model for Multi-Stage Hot Deformation Processes

Ale Method Robust and Feature-Aware Arbritary Lagrangian-Eulerian Method for Material Forming Applications

Algebraic Multi-grid Coarsening Algebraic Hierarchical Graph Neural Networks for Forming Simulation of Thermoplastic Composite Materials

Artificial Neural Network Integrated Artificial Neural Network-Based Approach for Predicting Surface Roughness Parameters in Laser Surface Texturing of Ti6Al4V

B

Bayesian Data Assimilation A Bayesian Data Assimilation Framework for Characterizing Recrystallization in Steels

Bayesian Optimization Multi-Objective Bayesian Optimization of Dual-Phase Steel Microstructures for Minimal Damage Initiation

Bobbin-tool Friction Stir Welding Parameter Optimization and Data-Driven Soft-Sensor Framework for Torque Prediction in Bobbin-Tool Friction Stir Welding of AA2024

C

Cellular Automata Investigation of Code Optimization Strategies for Enhancing the Performance of Static Recrystallization Cellular Automata Models

Composite-metal Joining Modelling Deformation Mechanisms Decomposition, Separation and Compaction in Mechanical Joining Processes of Fiber Reinforced Thermoplastics on Meso Scale

Computation Efficiency Investigation of Code Optimization Strategies for Enhancing the Performance of Static Recrystallization Cellular Automata Models

Contact Element Study on the Apparent Friction Coefficient Between a Deformable and a Rough Rigid Body under Large Sliding

D

Data-driven Modeling Predicting Lateral Flow in Hot Sheet Metal Rolling Using Symbolic Regression

Data-driven Simulation Algebraic Hierarchical Graph Neural Networks for Forming Simulation of Thermoplastic Composite Materials

Deep Drawing Al-Based Predictions of Forming Effects for Enhanced Crash Simulation

Deep Learning An AI-Based Approach to Developing a Microstructural Model for Multi-Stage Hot Deformation Processes

Deep-drawing Physics-Informed Recurrent Neural Networks with Kinematic Constraints for Large Deformation Metal Forming

Deposition Efficiency Numerical Investigation of Deposition Efficiency Influencing Factors in the Friction Surfacing Process

Design Exploration Physics-Informed Recurrent Neural Networks with Kinematic Constraints for Large Deformation Metal Forming

Discrepancy Model Residual Learning-Based Synthetic Data for Hybrid Metamodeling in the Stamping of an Automotive Door Panel

Discrepancy Modeling Using Kolmogorov-Arnold Networks for Discrepancy Modeling of Lateral Flow in Hot Rolling of Steel Slabs

Draw-in Prediction Graph Neural Network for Draw-in Prediction in Sheet Metal Stamping

Dual-phase Steel Multi-Objective Bayesian Optimization of Dual-Phase Steel Microstructures for Minimal Damage Initiation

E

Explainable Ai Explainable Machine Learning and 3D Visualization for Rotary Tube Bending: Expert Evaluation of a Web-Based Tool

F

Fem An AI-Based Approach to Developing a Microstructural Model for Multi-Stage Hot Deformation Processes

Fem Simulation Combined Experimental and Simulation-Based Approach for Optimizing Roll Forming of Pipes for Hydrogen Infrastructure

Finite Element Method (fem) Modelling Deformation Mechanisms Decomposition, Separation and Compaction in Mechanical Joining Processes of Fiber Reinforced Thermoplastics on Meso Scale

Finite Element Modeling Predicting Lateral Flow in Hot Sheet Metal Rolling Using Symbolic Regression

Finite Element Models Using Kolmogorov-Arnold Networks for Discrepancy Modeling of Lateral Flow in Hot Rolling of Steel Slabs

Fluid-structure-interaction Modelling Deformation Mechanisms Decomposition, Separation and Compaction in Mechanical Joining Processes of Fiber Reinforced Thermoplastics on Meso Scale

Forming Algebraic Hierarchical Graph Neural Networks for Forming Simulation of Thermoplastic Composite Materials

Forming Effects Al-Based Predictions of Forming Effects for Enhanced Crash Simulation

Friction Surfacing Numerical Investigation of Deposition Efficiency Influencing Factors in the Friction Surfacing Process

Fsw Robust and Feature-Aware Arbritary Lagrangian-Eulerian Method for Material Forming Applications

G

Graph Network Simulators Algebraic Hierarchical Graph Neural Networks for Forming Simulation of Thermoplastic Composite Materials

Graph Neural Networks Graph Neural Network for Draw-in Prediction in Sheet Metal Stamping

H

Hot Rolling Long Short-Term Memory to Predict the Quality Parameters in the Hot Rolling Process; Predicting Lateral Flow in Hot Sheet Metal Rolling Using Symbolic Regression; Using Kolmogorov-Arnold Networks for Discrepancy Modeling of Lateral Flow in Hot Rolling of Steel Slabs

Hot Strip Rolling Prediction of Strip Width Deviation in Hot Strip Roughing Mills Based on Machine Learning Regression

Hybrid Metamodel Residual Learning-Based Synthetic Data for Hybrid Metamodeling in the Stamping of an Automotive Door Panel

K

Kans Using Kolmogorov-Arnold Networks for Discrepancy Modeling of Lateral Flow in Hot Rolling of Steel Slabs

L

Laser Surface Treatment Integrated Artificial Neural Network-Based Approach for Predicting Surface Roughness Parameters in Laser Surface Texturing of Ti6Al4V

Lateral Flow Using Kolmogorov-Arnold Networks for Discrepancy Modeling of Lateral Flow in Hot Rolling of Steel Slabs

Lateral Spread Predicting Lateral Flow in Hot Sheet Metal Rolling Using Symbolic Regression

Long Short-term Memory Long Short-Term Memory to Predict the Quality Parameters in the Hot Rolling Process

M

Machine Learning Prediction of Strip Width Deviation in Hot Strip Roughing Mills Based on Machine Learning Regression; Explainable Machine Learning and 3D Visualization for Rotary Tube Bending: Expert Evaluation of a Web-Based Tool; Parameter Optimization and Data-Driven Soft-Sensor Framework for Torque Prediction in Bobbin-Tool Friction Stir Welding of AA2024; Graph Neural Network for Draw-in Prediction in Sheet Metal Stamping

Material Forming Robust and Feature-Aware Arbritary Lagrangian-Eulerian Method for Material Forming Applications

Mesh Adaptation Robust and Feature-Aware Arbritary Lagrangian-Eulerian Method for Material Forming Applications

Metals An AI-Based Approach to Developing a Microstructural Model for Multi-Stage Hot Deformation Processes

Microstructure An AI-Based Approach to Developing a Microstructural Model for Multi-Stage Hot Deformation Processes

Microstructure Evolution Investigation of Code Optimization Strategies for Enhancing the Performance of Static Recrystallization Cellular Automata Models

Microstructure Optimization Multi-Objective Bayesian Optimization of Dual-Phase Steel Microstructures for Minimal Damage Initiation

Modeling Combined Experimental and Simulation-Based Approach for Optimizing Roll Forming of Pipes for Hydrogen Infrastructure

N

Numerical Analysis Study on the Apparent Friction Coefficient Between a Deformable and a Rough Rigid Body under Large Sliding

Numerical Study Numerical Investigation of Deposition Efficiency Influencing Factors in the Friction Surfacing Process

P

Particle Filter A Bayesian Data Assimilation Framework for Characterizing Recrystallization in Steels

Periodic Boundary Conditions Study on the Apparent Friction Coefficient Between a Deformable and a Rough Rigid Body under Large Sliding

Physics-informed Physics-Informed Recurrent Neural Networks with Kinematic Constraints for Large Deformation Metal Forming

Physics-informed Ai Al-Based Predictions of Forming Effects for Enhanced Crash Simulation

Pipe Combined Experimental and Simulation-Based Approach for Optimizing Roll Forming of Pipes for Hydrogen Infrastructure

Prediction Prediction of Strip Width Deviation in Hot Strip Roughing Mills Based on Machine Learning Regression

Predictive Quality Prediction of Strip Width Deviation in Hot Strip Roughing Mills Based on Machine Learning Regression

Process Monitoring Parameter Optimization and Data-Driven Soft-Sensor Framework for Torque Prediction in Bobbin-Tool Friction Stir Welding of AA2024

Process Parameter Optimization Parameter Optimization and Data-Driven Soft-Sensor Framework for Torque Prediction in Bobbin-Tool Friction Stir Welding of AA2024

Q

Quality-relevant Material Parameters Long Short-Term Memory to Predict the Quality Parameters in the Hot Rolling Process

R

Recrystallisation An AI-Based Approach to Developing a Microstructural Model for Multi-Stage Hot Deformation Processes

Recrystallization A Bayesian Data Assimilation Framework for Characterizing Recrystallization in Steels

Recurrent Neural Network Physics-Informed Recurrent Neural Networks with Kinematic Constraints for Large Deformation Metal Forming

Roll Forming Combined Experimental and Simulation-Based Approach for Optimizing Roll Forming of Pipes for Hydrogen Infrastructure

Rolling An AI-Based Approach to Developing a Microstructural Model for Multi-Stage Hot Deformation Processes

Rotary Tube Bending Explainable Machine Learning and 3D Visualization for Rotary Tube Bending: Expert Evaluation of a Web-Based Tool

S

Sequential Monte Carlo A Bayesian Data Assimilation Framework for Characterizing Recrystallization in Steels

Sheet Metal Forming Graph Neural Network for Draw-in Prediction in Sheet Metal Stamping

Simulation Modelling Deformation Mechanisms Decomposition, Separation and Compaction in Mechanical Joining Processes of Fiber Reinforced Thermoplastics on Meso Scale

Sinusoidal Roughness Study on the Apparent Friction Coefficient Between a Deformable and a Rough Rigid Body under Large Sliding

Smoothed Particle Hydrodynamics Numerical Investigation of Deposition Efficiency Influencing Factors in the Friction Surfacing Process

Soft-sensor Framework Parameter Optimization and Data-Driven Soft-Sensor Framework for Torque Prediction in Bobbin-Tool Friction Stir Welding of AA2024

Static Recrystallization Investigation of Code Optimization Strategies for Enhancing the Performance of Static Recrystallization Cellular Automata Models

Steels A Bayesian Data Assimilation Framework for Characterizing Recrystallization in Steels

Surface Roughness Integrated Artificial Neural Network-Based Approach for Predicting Surface Roughness Parameters in Laser Surface Texturing of Ti6Al4V

Symbolic Regression Predicting Lateral Flow in Hot Sheet Metal Rolling Using Symbolic Regression

Synthetic Data Residual Learning-Based Synthetic Data for Hybrid Metamodeling in the Stamping of an Automotive Door Panel

T

Torque Prediction Parameter Optimization and Data-Driven Soft-Sensor Framework for Torque Prediction in Bobbin-Tool Friction Stir Welding of AA2024

U

User Evaluation Explainable Machine Learning and 3D Visualization for Rotary Tube Bending: Expert Evaluation of a Web-Based Tool

V

Virtual Vehicle Development Al-Based Predictions of Forming Effects for Enhanced Crash Simulation

W

Width Deviation Prediction of Strip Width Deviation in Hot Strip Roughing Mills Based on Machine Learning Regression

Authors Index

A

Abdullah, I. On-Line Prediction of Dry Zones During Composite Process Through Digital Shadow Approach

Afrasiabi, M. Numerical Investigation of Deposition Efficiency Influencing Factors in the Friction Surfacing Process

Ahn, T. Graph Neural Network for Draw-in Prediction in Sheet Metal Stamping

Alves, Z. Robust and Feature-Aware Arbritary Lagrangian-Eulerian Method for Material Forming Applications

B

Belugin, V. An AI-Based Approach to Developing a Microstructural Model for Multi-Stage Hot Deformation Processes

Beraudo, C. Robust and Feature-Aware Arbritary Lagrangian-Eulerian Method for Material Forming Applications

Bergmann, L. Parameter Optimization and Data-Driven Soft-Sensor Framework for Torque Prediction in Bobbin-Tool Friction Stir Welding of AA2024

Bernacki, M. Accelerating Microstructural Evolution Simulations in DIGIMU® through a Front-Tracking Lagrangian Solver: Implementation and Validation in AISI 304L Stainless Steel

Biba, N. An AI-Based Approach to Developing a Microstructural Model for Multi-Stage Hot Deformation Processes

Biermann, I. Multi-Objective Bayesian Optimization of Dual-Phase Steel Microstructures for Minimal Damage Initiation

Bock, F. Parameter Optimization and Data-Driven Soft-Sensor Framework for Torque Prediction in Bobbin-Tool Friction Stir Welding of AA2024; Predicting Lateral Flow in Hot Sheet Metal Rolling Using Symbolic Regression; Using Kolmogorov-Arnold Networks for Discrepancy Modeling of Lateral Flow in Hot Rolling of Steel Slabs

Boogard, T. A Bayesian Data Assimilation Framework for Characterizing Recrystallization in Steels; Predicting Lateral Flow in Hot Sheet Metal Rolling Using Symbolic Regression; Physics-Informed Recurrent Neural Networks with Kinematic Constraints for Large Deformation Metal Forming; Using Kolmogorov-Arnold Networks for Discrepancy Modeling of Lateral Flow in Hot Rolling of Steel Slabs

Bot, P. On-Line Prediction of Dry Zones During Composite Process Through Digital Shadow Approach

Bouattour, G. Parameter Optimization and Data-Driven Soft-Sensor Framework for Torque Prediction in Bobbin-Tool Friction Stir Welding of AA2024

Brüggemann, H. Long Short-Term Memory to Predict the Quality Parameters in the Hot Rolling Process

C

Carpino, G. Integrated Artificial Neural Network-Based Approach for Predicting Surface Roughness Parameters in Laser Surface Texturing of Ti6Al4V

Chafle, R. Numerical Investigation of Deposition Efficiency Influencing Factors in the Friction Surfacing Process

Clausmeyer, T. Combined Experimental and Simulation-Based Approach for Optimizing Roll Forming of Pipes for Hydrogen Infrastructure

Cometa, A. Predicting Lateral Flow in Hot Sheet Metal Rolling Using Symbolic Regression; Using Kolmogorov-Arnold Networks for Discrepancy Modeling of Lateral Flow in Hot Rolling of Steel Slabs

D

Dahlinger, P. Algebraic Hierarchical Graph Neural Networks for Forming Simulation of Thermoplastic Composite Materials

Denker, J. Prediction of Strip Width Deviation in Hot Strip Roughing Mills Based on Machine Learning Regression

Dev, S. Graph Neural Network for Draw-in Prediction in Sheet Metal Stamping

E

Elbossily, A. Numerical Investigation of Deposition Efficiency Influencing Factors in the Friction Surfacing Process

Engel, B. Explainable Machine Learning and 3D Visualization for Rotary Tube Bending: Expert Evaluation of a Web-Based Tool

Ewen, D. Prediction of Strip Width Deviation in Hot Strip Roughing Mills Based on Machine Learning Regression

F

Fehlemann, N. Multi-Objective Bayesian Optimization of Dual-Phase Steel Microstructures for Minimal Damage Initiation

Fouché, O. On-Line Prediction of Dry Zones During Composite Process Through Digital Shadow Approach

Franke, R. Prediction of Strip Width Deviation in Hot Strip Roughing Mills Based on Machine Learning Regression

Fraser, K. Numerical Investigation of Deposition Efficiency Influencing Factors in the Friction Surfacing Process

Freymuth, N. Algebraic Hierarchical Graph Neural Networks for Forming Simulation of Thermoplastic Composite Materials

G

Galdos, L. Residual Learning-Based Synthetic Data for Hybrid Metamodeling in the Stamping of an Automotive Door Panel

Garcia, J. Robust and Feature-Aware Arbritary Lagrangian-Eulerian Method for Material Forming Applications

García, E. Residual Learning-Based Synthetic Data for Hybrid Metamodeling in the Stamping of an Automotive Door Panel

Gartvig, A. An AI-Based Approach to Developing a Microstructural Model for Multi-Stage Hot Deformation Processes

Gerritzen, J. Modelling Deformation Mechanisms Decomposition, Separation and Compaction in Mechanical Joining Processes of Fiber Reinforced Thermoplastics on Meso Scale

Gröger, B. Modelling Deformation Mechanisms Decomposition, Separation and Compaction in Mechanical Joining Processes of Fiber Reinforced Thermoplastics on Meso Scale

Gude, M. Modelling Deformation Mechanisms Decomposition, Separation and Compaction in Mechanical Joining Processes of Fiber Reinforced Thermoplastics on Meso Scale

H

Hakvoort, W. Physics-Informed Recurrent Neural Networks with Kinematic Constraints for Large Deformation Metal Forming

Hashemzadeh, A. Predicting Lateral Flow in Hot Sheet Metal Rolling Using Symbolic Regression; Using Kolmogorov-Arnold Networks for Discrepancy Modeling of Lateral Flow in Hot Rolling of Steel Slabs

Havinga, J. A Bayesian Data Assimilation Framework for Characterizing Recrystallization in Steels

Hazrati, J. Physics-Informed Recurrent Neural Networks with Kinematic Constraints for Large Deformation Metal Forming

Hoang, T. Algebraic Hierarchical Graph Neural Networks for Forming Simulation of Thermoplastic Composite Materials

Hornig, A. Modelling Deformation Mechanisms Decomposition, Separation and Compaction in Mechanical Joining Processes of Fiber Reinforced Thermoplastics on Meso Scale

Hribsek, M. Multi-Objective Bayesian Optimization of Dual-Phase Steel Microstructures for Minimal Damage Initiation

Huang, L. Graph Neural Network for Draw-in Prediction in Sheet Metal Stamping

I

Ilinich, A. Graph Neural Network for Draw-in Prediction in Sheet Metal Stamping

J

Jaime, F. Accelerating Microstructural Evolution Simulations in DIGIMU® through a Front-Tracking Lagrangian Solver: Implementation and Validation in AISI 304L Stainless Steel

Jamshidi, B. A Bayesian Data Assimilation Framework for Characterizing Recrystallization in Steels

K

Kallabis, S. Prediction of Strip Width Deviation in Hot Strip Roughing Mills Based on Machine Learning Regression

Kallien, Z. Numerical Investigation of Deposition Efficiency Influencing Factors in the Friction Surfacing Process

Klusemann, B. Numerical Investigation of Deposition Efficiency Influencing Factors in the Friction Surfacing Process; Parameter Optimization and Data-Driven Soft-Sensor Framework for Torque Prediction in Bobbin-Tool Friction Stir Welding of AA2024; Predicting Lateral Flow in Hot Sheet Metal Rolling Using Symbolic Regression; Using Kolmogorov-Arnold Networks for Discrepancy Modeling of Lateral Flow in Hot Rolling of Steel Slabs

Knoche, J. Explainable Machine Learning and 3D Visualization for Rotary Tube Bending: Expert Evaluation of a Web-Based Tool

Kuhlenkötter, B. Prediction of Strip Width Deviation in Hot Strip Roughing Mills Based on Machine Learning Regression

Kunke, A. Combined Experimental and Simulation-Based Approach for Optimizing Roll Forming of Pipes for Hydrogen Infrastructure

Kärger, L. Algebraic Hierarchical Graph Neural Networks for Forming Simulation of Thermoplastic Composite Materials

L

Lackmann, L. Prediction of Strip Width Deviation in Hot Strip Roughing Mills Based on Machine Learning Regression

Laerhoven, K. Explainable Machine Learning and 3D Visualization for Rotary Tube Bending: Expert Evaluation of a Web-Based Tool

Lauzeral, N. On-Line Prediction of Dry Zones During Composite Process Through Digital Shadow Approach

Lecointe, D. On-Line Prediction of Dry Zones During Composite Process Through Digital Shadow Approach

Lepenies, I. Al-Based Predictions of Forming Effects for Enhanced Crash Simulation

Lian, J. Long Short-Term Memory to Predict the Quality Parameters in the Hot Rolling Process

M

Madej, Ł. Investigation of Code Optimization Strategies for Enhancing the Performance of Static Recrystallization Cellular Automata Models

Majumder, T. Long Short-Term Memory to Predict the Quality Parameters in the Hot Rolling Process

Marchand, C. On-Line Prediction of Dry Zones During Composite Process Through Digital Shadow Approach

Mederos, D. Predicting Lateral Flow in Hot Sheet Metal Rolling Using Symbolic Regression

Micheli, P. Accelerating Microstructural Evolution Simulations in DIGIMU® through a Front-Tracking Lagrangian Solver: Implementation and Validation in AISI 304L Stainless Steel

Mitsch, J. Algebraic Hierarchical Graph Neural Networks for Forming Simulation of Thermoplastic Composite Materials

Munzone, F. Physics-Informed Recurrent Neural Networks with Kinematic Constraints for Large Deformation Metal Forming

Muñiz, L. Residual Learning-Based Synthetic Data for Hybrid Metamodeling in the Stamping of an Automotive Door Panel

Münstermann, S. Multi-Objective Bayesian Optimization of Dual-Phase Steel Microstructures for Minimal Damage Initiation

N

Neto, D. Study on the Apparent Friction Coefficient Between a Deformable and a Rough Rigid Body under Large Sliding

Neumann, G. Algebraic Hierarchical Graph Neural Networks for Forming Simulation of Thermoplastic Composite Materials

Niget, F. On-Line Prediction of Dry Zones During Composite Process Through Digital Shadow Approach

Nouira, F. Parameter Optimization and Data-Driven Soft-Sensor Framework for Torque Prediction in Bobbin-Tool Friction Stir Welding of AA2024

O

Oh, S. Graph Neural Network for Draw-in Prediction in Sheet Metal Stamping

Okariz, J. Residual Learning-Based Synthetic Data for Hybrid Metamodeling in the Stamping of an Automotive Door Panel

Oliveira, M. Study on the Apparent Friction Coefficient Between a Deformable and a Rough Rigid Body under Large Sliding

P

Pacchione, M. Parameter Optimization and Data-Driven Soft-Sensor Framework for Torque Prediction in Bobbin-Tool Friction Stir Welding of AA2024

Pandey, M. Combined Experimental and Simulation-Based Approach for Optimizing Roll Forming of Pipes for Hydrogen Infrastructure

R

Rautenstrauch, A. Combined Experimental and Simulation-Based Approach for Optimizing Roll Forming of Pipes for Hydrogen Infrastructure

Ripert, U. Robust and Feature-Aware Arbritary Lagrangian-Eulerian Method for Material Forming Applications

Rotella, G. Integrated Artificial Neural Network-Based Approach for Predicting Surface Roughness Parameters in Laser Surface Texturing of Ti6Al4V

S

Saffioti, M. Integrated Artificial Neural Network-Based Approach for Predicting Surface Roughness Parameters in Laser Surface Texturing of Ti6Al4V

Scharifi, E. Long Short-Term Memory to Predict the Quality Parameters in the Hot Rolling Process

Seitz, J. Prediction of Strip Width Deviation in Hot Strip Roughing Mills Based on Machine Learning Regression

Shitikov, A. An AI-Based Approach to Developing a Microstructural Model for Multi-Stage Hot Deformation Processes

Siddig, N. On-Line Prediction of Dry Zones During Composite Process Through Digital Shadow Approach

Sitarz, S. Investigation of Code Optimization Strategies for Enhancing the Performance of Static Recrystallization Cellular Automata Models

Sitko, M. Investigation of Code Optimization Strategies for Enhancing the Performance of Static Recrystallization Cellular Automata Models

Sourisseau, T. Accelerating Microstructural Evolution Simulations in DIGIMU® through a Front-Tracking Lagrangian Solver: Implementation and Validation in AISI 304L Stainless Steel

Soyarslan, C. Predicting Lateral Flow in Hot Sheet Metal Rolling Using Symbolic Regression; Using Kolmogorov-Arnold Networks for Discrepancy Modeling of Lateral Flow in Hot Rolling of Steel Slabs

Sposato, A. Integrated Artificial Neural Network-Based Approach for Predicting Surface Roughness Parameters in Laser Surface Texturing of Ti6Al4V

Stebunov, S. An AI-Based Approach to Developing a Microstructural Model for Multi-Stage Hot Deformation Processes

T

Thakare, N. Long Short-Term Memory to Predict the Quality Parameters in the Hot Rolling Process

Tijum, R. Physics-Informed Recurrent Neural Networks with Kinematic Constraints for Large Deformation Metal Forming

Tretyakov, D. An AI-Based Approach to Developing a Microstructural Model for Multi-Stage Hot Deformation Processes

Trinidad, J. Residual Learning-Based Synthetic Data for Hybrid Metamodeling in the Stamping of an Automotive Door Panel

U

Umbrello, D. Integrated Artificial Neural Network-Based Approach for Predicting Surface Roughness Parameters in Laser Surface Texturing of Ti6Al4V

W

Wang, H. Graph Neural Network for Draw-in Prediction in Sheet Metal Stamping

Wojciechowski, R. Study on the Apparent Friction Coefficient Between a Deformable and a Rough Rigid Body under Large Sliding

Würth, T. Algebraic Hierarchical Graph Neural Networks for Forming Simulation of Thermoplastic Composite Materials

Y

Yazdani, A. Explainable Machine Learning and 3D Visualization for Rotary Tube Bending: Expert Evaluation of a Web-Based Tool

Yoon, J. Graph Neural Network for Draw-in Prediction in Sheet Metal Stamping