Cover

ISBN print: 978-3-0364-0989-4
ISBN eBook: 978-3-0364-1989-3
DOI book: 10.4028/b-Wx1eZ0
© 2026 Trans Tech Publications Ltd, All Rights Reserved

Lionel Fourment on:
Optimization, AI and Inverse
Analysis in Forming

Edited by
António Gil Andrade-Campos

Lionel Fourment on: Optimization, AI and Inverse Analysis in Forming

Special topic volume with invited peer-reviewed papers only

Edited by

Copyright © 2026 Trans Tech Publications Ltd, Switzerland

All rights reserved. No part of the contents of this publication may be reproduced, processed or transmitted in any form or by any means without the written permission of the publisher.

Trans Tech Publications Ltd
Seestrasse 24c
CH-8806 Baech
Switzerland
https://www.scientific.net

Volume 1049 of
Key Engineering Materials
ISSN print 1013-9826
ISSN web 1662-9795

Full text available online at https://www.scientific.net/book/lionel-fourment-on-optimization-ai-and-inverse-analysis-in-forming/978-3-0364-1989-3

Distributed worldwide by
Trans Tech Publications Ltd
Seestrasse 24c
CH-8806 Baech
Switzerland
Phone: +41 (44) 9221022
e-mail: sales@scientific.net

Key Engineering Materials, ISSN: 1662-9795, Vol. 1049, pp -1-0

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

Accepted:
2026-04-08

Online:
2026-04-15

Preface

The original version of this paper is available on https://www.scientific.net/KEM.1049.-1.pdf

Results for the simplest material tests and the most complex materials forming processes often depend on complex and unknown physical models and on inherent variability of design variables. The sensitivity of results to material and process variables can be used to determine unknown variables from known results (inverse analysis and parameter estimation) as well as to determine variable settings to achieve optimal results. Numerical methods for process optimization and for solving inverse problems are being intensely and continuously developed, as the reliability and the computational efficiency of simulation software is reaching unprecedented frontiers, particularly when AI technology is applied. Innovation in this field of research, applied to all materials forming engineering disciplines, is of utmost importance for ESAFORM conference.

Papers from both numerical and experimental standpoints that fall within the scope of this issue are listed as follows.
Contributions on the following subjects are presented:

Methods for Metamodeling, Digital Twins, Control and Optimization of Forming Processes:

The original version of this paper is available on https://www.scientific.net/KEM.1049.-1.pdf

shape and topological optimization;

optimization of manufacturing processes and machines;

process control and digital twins;

new metamodeling techniques;

Artificial intelligence, machine learning, deep learning and other data-driven approaches.

Inverse Analysis, Calibration and Model Discovery:

The original version of this paper is available on https://www.scientific.net/KEM.1049.-1.pdf

identification of constitutive, friction, heat transfer or damage parameters;

identification of boundary conditions or unknown process conditions;

design of experimental procedures, specimens and measurement techniques for inverse analysis;

numerical methods and algorithms for inverse analysis;

Model discovery, including data-driven models, selection and identification.

Stochastic Approaches:

The original version of this paper is available on https://www.scientific.net/KEM.1049.-1.pdf

reliability assessment;

robust design;

optimisation under uncertainty.

Editor

The original version of this paper is available on https://www.scientific.net/KEM.1049.-1.pdf

A. Gil Andrade-Campos

The original version of this paper is available on https://www.scientific.net/KEM.1049.-1.pdf

University of Aveiro

Co-editors

The original version of this paper is available on https://www.scientific.net/KEM.1049.-1.pdf

Matteo Strano
Politecnico di Milano
Halilovič Miroslav
University of Ljubljana
Sam Coppieters
KU Leuven
Luca Quagliato
University of Trento

Table of Contents

Preface

Advanced Control of Stretch-Reducing Mills Using Artificial Intelligence

Laura Antonini, Mariangela Quarto, Claudio Giardini, Giuseppe Pellegrini, Marco Zambelli

Proxy-Physics-Informed and Transfer Learning Networks for Radial-Axial Ring Rolling (RARR) under Varying Data Availability

Luca Quagliato, Johannes Seitz, Mattia Perin, Dominik Malejka, Tobias Moser, Caterina Zanella, Bernd Kuhlenkötter

Material Parameter Identification Using Bayesian Data Assimilation and Biaxial Tensile Test

Akinori Yamanaka, Shun Shimaoka

Innovation of the "Cut-Clamp-Play" Concept for Robust and Efficient Sheet Metal Material Characterization

Andraž Maček, Bojan Starman, Miroslav Halilovič, Fabrice Pierron, Pascal Lava, Sam Coppieters

Method of an Automated Tool Design of Punch-Bending Processes Using the Asset Administration Shell

Henning Peters, Andreas Mazur, Brijesh Chaudhary, Lukas Kersting, Ansgar Trächtler, Barbara Hammer

Uncertainty Detection in Sheet Metal Bending Processes with Machine Learning

Tomás Parreira, Daniel Cruz, Armando Marques, Pedro Prates, Marta Oliveira, Abel Santos, Bernardete Ribeiro, André Pereira

AI-Driven Design and Optimization of Bending Processes

Diogo Fernandes, Tomás Parreira, Daniel Cruz, Armando Marques, Pedro Prates, Marta Oliveira, Abel Santos, Diogo Neto, André Pereira

Hybrid Numerical and Data-Driven Modelling for Defect Prediction in Screw Press Hot Bulk Forging of the EN AW-6060 Part

Artem Alimov, Yuyao Jiang, Markus Gardill, Sebastian Härtel

Machine Learning Algorithms Applied to the Identification of CPB06 Yield Criterion Parameters

Armando Marques, Tomás Parreira, Bernardete Ribeiro, Pedro Prates, André Pereira

Geometry-Based Concept for Automatic Cut Planning and Flattening of Sheet Metal Components

Jannis Korn, Clemens Acksteiner, Sebastian Langula, Christina Guilleaume, Alexander Brosius

Towards Single-Test Anisotropy Calibration: Sensitivity, Identifiability and Validation of YLD2000-2d

Bojan Starman, Andraž Maček, Miroslav Halilovič, Pascal Lava, Fabrice Pierron, Sam Coppieters

Towards DIC-Subset-Independent Machine Learning Models for Constitutive Parameter Identification in Sheet Metal Forming

Dário Mitreiro, João Henriques, Pedro André Prates, António Andrade-Campos

Application of Reference-Free Strain Measurement to Assess the Deformation Level of Pre-Deformed Components

Marco Rossi, Giulia Tanoni, Matteo Anedda, Maria Caterina Fioretti, Emanuele Principi

Improving FEMU Calibration from a Single Heterogeneous Test through a Data-Driven Approach: An Exploratory Study

Mafalda Gonçalves, Rui Amaral, Sandrine Thuillier, António Andrade-Campos

Robust Performance Optimization Using Bayesian Optimization and Extreme Value Theory

Augustin Persoons, Valentin Duarte Rocha, Laurence Giraud-Moreau, Pascal Lafon

Keywords Index

Authors Index

The original version of this eBook is available on https://www.scientific.net/book/lionel-fourment-on-optimization-ai-and-inverse-analysis-in-forming/978-3-0364-1989-3

Key Engineering Materials, ISSN: 1662-9795, Vol. 1049, pp 1-11

doi: 10.4028/p-5o49gU

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

Submitted:
2025-12-11

Revised:
2026-02-11

Accepted:
2026-03-02

Online:
2026-04-15

Advanced Control of Stretch-Reducing Mills Using Artificial Intelligence

The original version of this paper is available on https://www.scientific.net/KEM.1049.1.pdf

Laura Antonini 1,a*, Mariangela Quarto 1, b, Claudio Giardini 1,c, Giuseppe Pellegrini 1, d and Marco Zambelli 1,e

1Department of Management, Information and Production Engineering, University of Bergamo, Via Pasubio 7/b, 24044 Dalmine, BG, Italy

alaura.antonini@unibg.it, b  mariangela.quarto@unibg.it, c  claudio.giardini@unibg.it, d giuseppe.pellegrini@unibg.it, e marco.zambelli@unibg.it

Keywords: Stretch-reducing mill, Artificial Neural Network, Intelligent process control

Abstract

The stretch-reducing mill is a forming process to manufacture tubes by progressive metal deformation. This process is characterized by high complexity and a high number of variables that are strongly interconnected. To overcome the limitations and substantial simplifications of the traditional modelling and offer higher flexibility and suitability for real-time control, an Artificial Neural Networks approach is employed. By defining three parallel networks, they were predicted the milling status, the tube thickness and the angular speeds of the stands composing the process. With the models results, an optimization algorithm is employed to determine the best configuration of angular speeds of the stands to obtain a defined final tube thickness. The Artificial Neural Networks show extremely low RMSE across training, validation, and test sets, confirming their ability to model complex nonlinear dependencies. The optimisation stage reaches the target thickness with only 0.0079% error while preventing unstable operating conditions. The overall methodology provides a tool for implementing the intelligent and data-driven control of the stretch-reducing mill.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1049.1.pdf

The stretch-reducing mill (SRM) is a forming process widely applied for producing tubes. The deformation is caused by a sequence of stands whose angular speeds determine the stress distribution along the tube. Due to the complexity of the process, small variation in inter-stand tensions can generate significant effect on stability, potentially leading to slippage or drawing [1].

In a stretch-reducing mill, one of the only manageable variables is the angular speed of the rolling stands (ω). During the process, the material flow rate, resulting from the multiplication between the area and the speed, remains constant due to mass conservation. The speed difference between stands and the tube creates some tensions that can lead to two different limit milling conditions: if the tube moves slower than the maximum tangential stand speed, a slippery condition occurs; if the tube moves faster, the system enters a drawing situation [1]. Another important phenomenon that cannot be ignored is the creep, referring to a viscous deformation occurring under constant load at high temperature [2].
Therefore, the tube thickness decreases in the space between stands, while it increases inside the single stand. This leads to an unpredictable variation of the wall thickness: the process must be carefully monitored and controlled to avoid instability, slippage, or excessive plastic or viscous deformation.

The traditional analytical or numerical models of SRM provide a valuable outcome but often require great simplifications and consider many interdependent variables, requiring high computational power. The variation of the angular speeds ( ω ) of the stands affects the next ones, creating a cascade effect on the entire process that is difficult to represent and predict with the traditional physical models [3].

In this complex contest, data-driven methods such as Artificial Neural Networks (ANNs) fit particularly well. These models are able to describe the dynamic behaviour of the SRM process, learning hidden relationships among variables while ensuring technological and mechanical consistency. They provide more flexibility in modelling, especially with non-linear and multivariate processes [4]. The applications of Artificial Neural Networks in manufacturing context can enhance
almost real-time and accurate problems detection, enhancing continuous improvement strategies [5]. By optimization algorithms, as the backpropagation [6], they provide systems able to perform intelligent process control and support during the decision-making process.

The main purpose of this work is to present a novel methodology that integrates physics-informed data generation, three parallel ANNs modelling various aspects of SRM process, and an optimization algorithm that computes valid stress distribution to obtain the desired final tube thickness. Differently from other works, the proposed framework perform classification, regression of the thickness and angular speeds predictions, improving robustness near limit situations. The simulation tool adopted in this work is not a finite element but, starting from analytical formulations, it is built a physicsinformed simulation tool for data generation. Then, it is developed an ANN-based approach with multiple parallel networks to manage the consecutive rolling stands and their interconnections. The models are evaluated in order to provide accurate predictions and guarantee model adaptability and scalability. Subsequently, an optimization algorithm [6] is applied to support the definition of the best combination of angular speeds ( ω ) to be set to obtain the desired tube thickness with the minimum error. This approach is designed for high interpretability, physical consistency, and potential realtime deployment.

The Milling Process

The original version of this paper is available on https://www.scientific.net/KEM.1049.1.pdf

The SRM is a forming process composed of a sequence of rolling stands that progressively reduce the tube diameter while adjusting the wall thickness [7]. Since there is no mandrel, deformation in SRM is managed exclusively by kinematics of the stands, which makes the control problem particularly sensitive to distribution of the inter-stands stresses.

In addition to the stands angular speed, there are other forces and parameters to keep into account when analysing a stretch-reducing mill process. The mechanical equilibrium can be described through the balance of stresses in the normal and tangential directions. More in detail, as shown in Fig. 1, R is the roll radius at the point of contact with the material, r represents the radial position of the element within the workpiece, p is the contact pressure exerted by the roll on the material, α represents the angle of contact or inclination between the roll and the material at a given point, d α is the infinitesimal angle swept by the roll in the axial direction (along the arc of contact), dϑ is the infinitesimal angle along the circumferential direction of the material, h represents the thickness of the material at the section under consideration, σϑ is the circumferential stress, σa is the axial stress and dσa is the incremental change in axial stress across the differential element.

[picture]

Fig. 1.Stress state representation in the tube.

Fig. 1. Stress state representation in the tube.

The balance in the pressure direction can be described in (1):

p·R· dα·r· dϑ−σa· h·r· dϑ·sin dα2−(σa+dσa)·h·r· dϑ·sin dα2+2σϑ·sindϑ2· h·R· dα·cosα=0(1)

Given f as the friction coefficient between the roll and the material, dh as the incremental change in thickness of the material, dr as the incremental change in radial position of the differential element, the tangential direction on the meridional plane can be described as in (2):

f·p·R·dα·r·dϑ+σa·h·r·dϑ·cosdα2−(σa+dσa)(h+dh)(r+dr)·dϑ·cosdα2+2·σϑ·R·dα·h·sinα·dϑ2=0(2)

The plastic deformation condition can be described by the ( 3 ), where σ0 is the equivalent stress of the material:

(σa−σϑ)2+(σϑ+p)2+(−p−σa)2=2·σ02(3)

Therefore, considering σz as the stress in the axial direction, the stress-strain relationship is reported in (4):

dhh2σz−σa−σϑ=drr2σϑ−σa−σz(4)

By analysing the behaviour of the function with respect to α, it is possible to identify the neutral section ( αn ) in a graphical way as the curve cross (Fig. 2).

[picture]

Fig. 2. σα distribution in an isolated stand.

Fig. 2. σ α distribution in an isolated stand.

Therefore, considering h0 and r0 as the thickness and the radius of the material at the entry of the deformation zone, and he and re as the thickness and the radius of the material at exit section in the deformation zone, the final thickness value results as:

he=h0r0re(5)

These set of equations represent the tube thickness evolution along the milling process. It is important to also consider the creep phenomenon as its influence corresponds to a thickness reduction.
Finally, considering that the volumetric flow rate of the tube that must be constant, given by the multiplication between the area and the speed, it possible to determine the angular speed of each stand ωi.
In practice, these relationships can be employed to define a certain axial stress distribution between the stands to derive the thickness variation and identify possible boundary conditions (drawing and slipping) due to a too high inter-stand tension.
The framework described represents the foundation for generating the data employed in the neural networks presented in the following sections.

Methodology

The original version of this paper is available on https://www.scientific.net/KEM.1049.1.pdf

Framework overview

The original version of this paper is available on https://www.scientific.net/KEM.1049.1.pdf

The dataset employed in this work was generated through a simulation pipeline based on the mechanical and technological characteristics of the milling process. It included the infeed and outfeed thickness of each stand, along with the dimensional setup of each stand. Based on this configuration, they were calculated the infeed and outfeed thickness of the tube at each stand. After the generation of the dedicated dataset, the entire developed neural networks were applied to it to simulate and evaluate the rolling process behaviour.

At the end, an optimisation algorithm was applied to determine the best inter-stand tension values, by minimizing the error between the target and the predicted final tube thickness.
The overall workflow is reported in Fig. 3.

[picture]

Fig. 3. Study workflow.

Fig. 3. Study workflow.

Data generation

The original version of this paper is available on https://www.scientific.net/KEM.1049.1.pdf

The dataset employed in this work was designed following the physical characteristics of the stretch-reducing milling process. It was defined a physics-based data generator derived from established analytical formulations of SRM process. Although the data were simulated, this study allowed to analyse the milling operating situations. The numerical model did not have the goal to outperform the state-of-the-art finite element model, but it was designed to generate training data for real-time control. A structured pipeline was defined in MATLAB®R2024b.

Starting from the first phase, it was created a configuration file reporting the dimensional properties of the process including the characteristics of the system (number of stands, etc.), the tube initial diameter, and the material (temperature, creep behaviour, etc). Then, all the feasible combinations of inter-stand stresses were generated with a loop search the entire stress range, specifying the maximum and minimum value of these stresses ( σmin ,σmax  ) and the step size ( Δσ ). These parameters were selected based on the specific analysis to limit high stress between stands that could damage or even rupture the tube. They ensured physical conditions while excluding excessive tensions. Therefore, slippage and drawing conditions were excluded to ensure only realistic combinations. Each combination was used as input into the simulation tool that computes the evolution of process variables during the process. For each combination, it was calculated also the angular speed of all the stands (ωi), the infeed and outfeed thicknesses of the tube, and the state of each stand (slippage or drawing). Simulations for which the physical conditions were not followed were filtered out to keep the data quality.
In the end data were extracted and formatted to create the final dataset counting around 2600 parameters combinations and containing: inter-stand tensions, the state of the stands as categorical values ( 0 as the acceptable situation, 0.5 as slippage, 1 as drawing), the computed angular speeds ( ωi ),
the infeed tube thickness for each stand, and the final outfeed thickness at the end of the whole process.

From this dataset, the ANNs could learn the hidden relationships between variables ensuring high generalization capabilities.

Artificial Neural Networks

The original version of this paper is available on https://www.scientific.net/KEM.1049.1.pdf

The main purpose of the work is the optimization of the stress distribution between the stands of the stretch-reducing mill to achieve the desired final thickness of the rolled material. Three artificial neural networks (ANN) were developed in cascade to simulate the process behaviour, while an iterative optimisation strategy was employed to find the best inter-stand tension values.

The networks were developed through the use of Python programming language and the PyTorch libraries. Data were scaled using the Min-Max normalization ranging from zero to one. All the networks took as input the inter-stand tensions obtained by physical-based simulation. The networks had the same activation function: tangent sigmoid for the hidden layer, while a pure linear function was applied in the output layer. It was split into 70% for training, 15% for validation, and 15% for testing in each network. As optimizer it was selected the L-BFGS (Limited-memory Broyden-Fletcher-Goldfarb-Shanno) [6] due to its suitability to small-to-medium sized networks and with limited data availability. Following a hyper parametrization process, the learning rate, which explains how quickly a machine learning model updates its parameters during training, was set to 0.5 , a high value mitigated by the optimizer. The maximum iterations were defined as 500 and the history size as 10 .

As anticipated, three ANNs were designed to model the stretch-reducing mill process.
The first network was used to predict the milling condition (status1-6) where values close to zero were considered acceptable, close to 1 for drawing or close to 0.5 for slippage. The inputs were the inter-stand tensions ( T1;2, T2;3, T3;4, etc.). To balance model simplicity and performance, the network configuration was a feedforward net with 15 neurons in the hidden layer. The network scheme is represented in Fig. 4, where on the left are reported inputs, net and outputs, while on the right are shown the specifications with the representation of the activation functions.

[picture]

Fig. 4. First network: tensions to status.

Fig. 4. First network: tensions to status.

The second one employed (Fig. 5) the inter-stand tensions ( T1;2, T2;3, T3;4, etc.) to predict the tube thicknesses ( t2,t3,t4, etc.). The hidden layer dimension was defined as 15 neurons based on the same reasoning as for the first net. To reflect the real behaviour of the milling process, the network was composed by 6 sub-nets, one for each stand, which use the output of the previous net as part of the
input layer. The first stand received as input the initial tube thickness ( t0 ), the first inter-stand tension (T1;2) and predict the tube thickness (t1) after the first stand. This predicted value, along with the previous and the subsequent tensions, became the input for the second sub-net, and so on. The last sub-net received as input only the previously predicted thickness and the previous tensions, and it predicted the final tube thickness (tf). Additionally, each sub-net took into account also the creep effect between stands.

[picture]

Fig. 5. Second network: tensions to thickness.

Fig. 5. Second network: tensions to thickness.

The third network (Fig. 6) took as input the inter-tensions ( T1;2, T2;3, T3;4, etc.) to forecast the angular speeds ( ω1−6 ) of the stands. Always considering the balance between performance and task complexity, the hidden layer was defined as 10 neurons. The angular speeds are derived considering one stand speed as fixed reference, typically set to 1 , and expressing all other stand speeds relatively to it. This reflects a physical constraint of the process, where only one stand's speed is physically fixed, and the others are adjusted accordingly. Each stand is represented by a single network that predict the corresponding angular speed (ωn).

[picture]

Fig. 6. Third network: tensions to angular speeds.

Fig. 6. Third network: tensions to angular speeds.

To evaluate the performance of the ANN, several well-known indices were computed. The network loss was defined as the Mean Squared Error (MSE) that quantifies the average squared difference between the predicted and target values. Based on MSE, the Root Mean Squared Error (RMSE) was also calculated together with the Pearson Coefficient (R) to assess the linear correlation between the predicted and actual values.

Optimization algorithm

The original version of this paper is available on https://www.scientific.net/KEM.1049.1.pdf

On the neural networks results it was applied the bisection algorithm [6] to optimize the stress between stands to minimize the predicted and the target final thickness (tf) of the rolled material. The use of a single scaling factor ensured stability and physical consistency, while limiting the flexibility of the optimization.
The algorithm iteratively searched for the optimal set of inter-stand tensions and therefore of angular speeds. The bisection algorithm is employed to find the root of an equation of the form f(x)=0, when the function is continuous over an interval [a,b] and changes sign at the extremes, hence f(a)·f(b)<0 [6]. According to the Intermediate Value Theorem [8], if a continuous function changes the sign in the interval, at least one solution exists in that interval. The algorithm is based on this principle to progressively narrow the interval until the root is located with the desired precision. Instead of optimising every tensor singularly, the algorithm introduces a scale parameter φ that multiplies the maximum allowed values for each stand, collecting the values in the vector ub=[Tmax,1, Tmax,2,…]. In this way the vector of the maximum tension is defined as x=φ*ub, reducing the problem to a single variable. At the beginning φ was set equal to 0.5 , and the algorithm modify φ at each iteration to minimize the objective function. The optimization continues until the absolute error falls below a certain defined threshold of 0.001 . With a penalty mechanism, non-physical plausible solutions are discarded: if the absolute maximum of the status exceeds 0.3 , a penalty term of 1.000 was included into the objective function.

In this case study the objective function was defined as minimum of the absolute values of the difference between the estimated thickness and the target one:

 objf = min ∣ target_thickness − estimated_thickness (x)∣(6)

This simple but effective optimization algorithm ensures stable and controlled convergence to a nearoptimal solution within a limited number of iterations.

Case study presentation

The original version of this paper is available on https://www.scientific.net/KEM.1049.1.pdf

In this work it was analysed a 6 stands SRM by using three Artificial Neural Networks. The dataset of these 6 stands milling process was generated following the workflow presented above. The interstand tensions related to this case study were 5( T1;2, T2;3, T3;4, T4;5, T5;6). The exit thickness of each stand serves as the entry thickness for the following stand (t1,t2,t3,t4,t5). The variable references are reported in Table 1.

Table 1. Variables references.

Inter-stand tensions

Stand

statuses

Stand

angular

speeds

Tube thickness

entering the

stand

Final tube

thickness

By employing the inter-stand tensions as input, the three neural networks predicted the milling status (drawing, slippage or accepted), the final tube thickness and the angular speed of the stands. Then, the metrics were calculated to evaluate the model performance, and it was applied the optimization algorithm to define the best combination of angular speeds to set to obtain the desired final tube thickness.

Results

The original version of this paper is available on https://www.scientific.net/KEM.1049.1.pdf

In order to evaluate the neural networks, they were computed the MSE, RMSE and R values.
The first network was employed to predict the milling status (acceptable, slippage or drawing) starting from the inter-stand tensions. Looking at Fig. 7 and Fig. 8 it is possible to notice that almost all the values are arranged almost perfectly along the bisector line, indicating a strong correspondence between the predicted and the real values. The R coefficients are close to one, confirmed also by the distribution of the values along the line. The analysis stand by stand shows a coherent behaviour with all the stands following the same trend. Table 2 shows the high precision of first network predictions reporting uniformly low MSE and RMSE values across all data. This consistency indicates that the ANN successfully learns the physical thresholds that identify different milling states (acceptable, slippage and drawing), despite of the inherent nonlinearities of SRM kinematics.

[picture]

Fig. 7. First network: from inter-stand tensions to the mill status of a mill with 6 stands.

Fig. 7. First network: from inter-stand tensions to the mill status of a mill with 6 stands.

[picture]

Fig. 8. First network: performance across the different 6 stands.

Fig. 8. First network: performance across the different 6 stands.

Table 2. First network performance.

Net 1 | All | Training | Validation | Testing
| MSE | RMSE | MSE | RMSE | MSE | RMSE | MSE | RMSE
| 1.5·10⁻⁵ | 3.90·10⁻³ | 1.0·10⁻⁵ | 3.21·10⁻³ | 9.0·10⁻⁶ | 2.99·10⁻³ | 4.5·10⁻⁵ | 6.67·10⁻³

The second ANN used the inter-stand tensions to predict the tube thickness, with a recurrent component to simulate the intermediate tube thicknesses. In Fig. 9, the four scatterplots report a good overlap between the predicted and the actual values. Each stand follows the same tendency without deviations or outliers (Fig. 10). The complexity of thickness predictions arises from non-linear relationships among tensions and parameters across stands. The model reaches a MSE so small to be approximated to zero, while the RMSE values demonstrate high level of precision in prediction (Table 3). The near-net zero RMSE highlights the ability of the ANN to reproduce the complex mechanisms that manage thickness evolution in multi-stands SRM. The network is able to implicitly understand the relationships between parameters and constraints. In addition, the absence of outliers suggests that the network learned to predict thickness as a distributed phenomena and not at a stand-by-stand effect.
This level of accuracy confirms the suitability of the ANN for modelling the behaviour that usually require high computational effort. This level of accuracy allows for reliable integration of the model into control and optimization routines, where small prediction errors translate directly into more stable and physically achievable operating points.

[picture]

Fig. 9. Second network: from inter-stand tensions to tube thickness.

Fig. 9. Second network: from inter-stand tensions to tube thickness.

[picture]

Fig. 10. Second network: performance across the different 6 stands.

Fig. 10. Second network: performance across the different 6 stands.

Table 3. Second network performance.

Net

All
Training
Validation
Testing

MSE | RMSE | MSE | RMSE | MSE | RMSE | MSE | RMSE
0 | superscript number | 0 | superscript number | 0 | superscript number | 0 | superscript number

The third network aimed to predict the angular speeds from the inter-stand tensions. The R coefficients are all close to one (Fig. 11), demonstrating that the network is able to learn the relationship between the inter-stand tensions and the angular speeds. The stand analysis presents data that are distributed along the bisector line, without deviations (Fig. 12). The MSE and RMSE values highlighted the model accuracy confirming that the ANN is able to describing the nonlinear kinematic interactions that manage speed adjustments along the SRM (Table 4). Since the angular speed is the main controllable variable in a real SRM system, this level of accuracy if essential. Given the level of accuracy of the predictions obtained, it is possible to define feasible and well-balanced speed settings during optimization, improving process stability.

[picture]

Fig. 11. Third network: from inter-stand tensions to angular speed.

Fig. 11. Third network: from inter-stand tensions to angular speed.

[picture]

Fig. 12. Third network: performance across the different 6 stands.

Fig. 12. Third network: performance across the different 6 stands.

Table 4. Third network performance.

Net

3

All
Training
Validation
Testing

MSE | RMSE | MSE | RMSE | MSE | RMSE | MSE | RMSE

Overall, the results obtained across the ANNs demonstrate the ability of the proposed methodology to accurately reproduce the complex interactions managing the SRM behaviour. The low RMSE values and the strong linear fit with the target data (in all ANNs) are particularly significant. In practical, even small deviations in thickness, angular speed, or milling state can spread along the stand sequence and compromise process stability. Therefore, the resulting predictive stability has a direct physical relevance. The results obtained confirm that physical complexity does not always involve model complexity. The achieved accuracy therefore provides a reliable basis for downstream optimization, enabling the definition of physically consistent tension and velocity settings, which would be difficult to achieve using traditional analytical models alone.
When all the process parameters data were predicted, it was applied the bisection algorithm to optimize the process by identifying the optimal set of inter-stand tensions to set in the process to obtain the desired final thickness. As described above, the objective function was defined as the minimum of the absolute value of difference between the target and the predicted thickness values (Eq. 6), ensuring feasibility and accuracy. A penalty was introduced in order to filter out the solutions associated with slippage or drawing.
By analysing the results (Table 5), the optimal solution identified returned a small error of 0.0079%, demonstrating the method accuracy. From the figure (Fig. 13) it is better understandable that the tensions follow a symmetric distribution, avoiding any big value differences that could tear off or damage the tube. The angular speed showed a coherent adaptation, offsetting the effect of the tensions. The status values indicated acceptable milling condition. These results showed that the optimization achieved convergence while preserving realistic milling conditions.

Table 5. Process parameters optimized with the optimization applied on a 6 stands stretch-reducing mill.

| Stand1 | Stand2 | Stand3 | Stand4 | Stand5 | Stand6
Tensions |  | 0.9 | 2.8 | 7.5 | 2.8 | 0.9
Status | 0.000 | 0.000 | 0.0001 | -0.000 | -0.019 | 0.290
Angular speed | 1.000 | 0.988 | 0.888 | 1.120 | 1.070 | 1.132
Thickness | 8.605 | 8.794 | 8.873 | 8.932 | 8.948 | 8.949
Calculated thickness | 8.949
Desired thickness | 8.950
Error | superscript number

[picture]

Fig. 13. Optimization results for a 6 stands stretch-reducing mill.

Fig. 13. Optimization results for a 6 stands stretch-reducing mill.

The three ANNs integrated with the optimization algorithm demonstrated high predictive accuracy while maintaining physical feasibility, confirming their suitability for this modelling this process. The observed consistency between training, validation, and testing performance across all ANNs indicates that the developed methodology generalises well beyond the data seen during training. This behaviour is supported by the structure of the dataset, which covers a wide range of feasible tension set, including boundary configurations (close to slippage or drawing). The lack of discrepancy between validation and test errors suggests that there is no overfitting, and that the ANNs remain robust even in regions of the domain that are physically critical or less frequently sampled. This level of generalisation is essential for ensuring reliable use of the models within optimisation-based control strategies.

The obtained results are consistent across stands and predictions, highlighting the capability of the network to understand complex milling relationships. More in detail, the robustness in near limit conditions suggests that the models learned to capture physical feasibility limits. This aspect is crucial to allow operators to understand when a process is going out of control before it actually happens.

Conclusions

The original version of this paper is available on https://www.scientific.net/KEM.1049.1.pdf

The present paper proposed an ANN-based modelling and optimization for stretch-reducing mill validated on physic-informed data. Three parallel neural networks were computed and demonstrated accuracy in predicting the milling status, the tube thickness, and the angular speeds of the stands. The integration of three specific ANNs with an optimization algorithm is able to identify setpoints that are physically feasible, and to reach the desired thickness with a minimal error. The methodology shows a powerfulness in supporting the decision-making process for the intelligent control of the milling process, and it can be extended to real-time supervisory systems, multi-objectives control and predictive quality. The strong generalization capability of the ANNs, underlined by the consistency of training, validation and tests errors, confirms the reliability and efficiency of the developed methodology. By combining physics-informed data generation with accurate data-driven prediction, the proposed methodology represents a promising candidate for next-generation intelligent forming process control. This study is based on simulated data, but experimental validation and multiparameter optimization remain open directions for future studies in which the proposed framework could be integrated.

References

The original version of this paper is available on https://www.scientific.net/KEM.1049.1.pdf

V. B. Ginzburg, "Steel-Rolling Technology," 1989. doi: 10.1201/9781466593510.

C. Giardini, A. Bugini, and G. Pellegrini, "Creep Influence in a Stretch Reducing Mill Computer Simulation," in Proceedings of the Twenty-eighth International, London: Macmillan Education UK, 1990, pp. 455-460. doi: 10.1007/978-1-349-10890-9_62.

A. Ojeda-López, M. Botana-Galvín, L. González-Rovira, and F. J. Botana, "Numerical Simulation as a Tool for the Study, Development, and Optimization of Rolling Processes: A Review," Metals (Basel), vol. 14, no. 7, p. 737, Jun. 2024, doi: 10.3390/met14070737.

E. Öznergiz, C. Özsoy, I. I. Delice, and A. Kural, "Comparison of empirical and neural network hot-rolling process models," Proc Inst Mech Eng B J Eng Manuf, vol. 223, no. 3, pp. 305-312, Mar. 2009, doi: 10.1243/09544054JEM1290.

B. Ördek, Y. Borgianni, and E. Coatanea, "Machine learning-supported manufacturing: a review and directions for future research," Prod Manuf Res, vol. 12, no. 1, Dec. 2024, doi: 10.1080/21693277.2024.2326526.

M. J. Kochenderfer and T. A. Wheeler, Algorithms for Optimization, Second edition. 2019. Accessed: Dec. 11, 2025. ISBN: 9780262039420. Available:

G. Pellegrini, C. Giardini, and A. Bugini, "An Automatic Speed Programme Generation for SRM," in Proceedings of the Thirtieth International MATADOR Conference, London: Macmillan Education UK, 1993, pp. 211-216. doi: 10.1007/978-1-349-13255-3_28.

M.N. Vrahatis, "Generalization of the Bolzano theorem for simplices," Topol Appl, vol. 202, pp. 40-46, Apr. 2016, doi: 10.1016/j.topol.2015.12.066.

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/lionel-fourment-on-optimization-ai-and-inverse-analysis-in-forming/978-3-0364-1989-3

Key Engineering Materials, ISSN: 1662-9795, Vol. 1049, pp 13-28

doi: 10.4028/p-B4tPZr

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

Submitted:
2025-12-18

Revised:
2026-02-12

Accepted:
2026-03-02

Online:
2026-04-15

Proxy-Physics-Informed and Transfer Learning Networks for Radial-Axial Ring Rolling (RARR) under Varying Data Availability

The original version of this paper is available on https://www.scientific.net/KEM.1049.13.pdf

Luca Quagliato 1,a*, Johannes Seitz 2, b, Mattia Perin 3,c, Dominik Malejka 2, d, Tobias Moser 2,e, Caterina Zanella 1,f, and Bernd Kuhlenkötter 2, g

1 Department of Industrial Engineering, University of Trento, 38123, Trento, Italy

2 Chair of Production Systems, Ruhr-University Bochum, 44780, Bochum, Germany

3 Department of Management and Engineering, University of Padova, 36100, Vicenza, Italy

aluca.quagliato@unitn.it, b  seitz@lps.ruhr-uni-bochum.de, c  mattia.perin@phd.unipd.it, d  malejka@lps.ruhr-uni-bochum.de, e  moser@lps.rub.de, f  caterina.zanella@unitn.it, gkuhlenkoetter@lps.rub.de

Keywords: Radial-Axial Ring Rolling (RARR), Proxy-Physics Informed Neural Networks (P-PINNs), Transfer Learning, Gematrical Modeling, Forces Modeling.

Abstract

This contribution presents a proof-of-concept and benchmark analysis between two different approaches for multiple domain modeling, employing numerical and experimental results relevant to the Radial-Axial Ring Rolling (RARR). First, a finite element method (FEM) simulation data-based Deep Neural Network (DNN) was employed as base model for a subsequent transfer learning (TL), carried out by employing 40 experimental (EXP) data belonging to a different RARR domain. The DNN-TL model is benchmarked against the proposed solution, in terms of a ProxyPhysics Informed Neural Network (P-PINN), defined using proxy equations for the outer diameter (OD) expansion, the radial force (RF), and the axial force (AF) and trained on 30 FEM and 30 EXP data. The proxy equations for the P−PINN are based on known knowledge and analytical formulations developed for the RARR. The results show that employing the whole 218 cases strong FEM database for the training of the base DNN model results in high prediction accuracy on the FEM cases and can be adapted well through TL. When employing only 30 FEM cases for the training of the base DNN model results in slight improvements for RF and in a significant drop of performance for AF , after TL. Instead, in a limited data scenario, the P-PINN model, powered equally by data and proxy equations steering the learning process, is capable of modeling well and simultaneously both FEM and EXP cases with reasonable accuracy, averaging at 3.7% for OD and 24.0% for RF/AF.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1049.13.pdf

Radial-Axial Ring Rolling (RARR) enables the production of seamless rings for machinery, metallurgy and aerospace applications [1,2]. Historically, RARR research progressed from analytical models to increasingly detailed finite element method (FEM) simulations [3-9]. In parallel, manufacturing research has recently shifted toward data-driven modeling, where machine learning (ML) is used to support process understanding, prediction and optimization of outputs such as geometry, loads, energy consumption and defect minimization [10,11]. For RARR, ML has been applied to predict loads and torque [12] and energy consumption [13]. Moreover, Mirandola et al. [13] reported that the high process complexity and large number of independent variables present in RARR databases can cause relatively simple ML architectures to break down when experimental variability is introduced.

Although ML can provide quick and flexible surrogates, its accuracy in interpolation, and especially extrapolation, depends strongly on the quality, variance and coverage of the training data [10, 11]. Consequently, recent manufacturing-oriented ML studies increasingly emphasized careful data pre-processing and structuring, and the use of hybrid formulations that introduce physical or expert information during training [14-16]. For instance, Lee et al. [14] used Gaussian Process Regression (GPR) to support deep-drawing blank design, demonstrating how probabilistic models can guide design decisions while quantifying uncertainty. Similarly, Lee et al. [15] proposed an

Extreme Gradient Boosting (XGB) inspired optimization algorithm aimed at RARR applications, exemplifying current trends toward combining powerful regressors with explicit optimization loops. Complementarily, Modanloo et al. [16] introduced a Gradient-Enhanced Expert-Informed Neural Network (GE-EINN) that translates expert knowledge into differential constraints, improving convergence and extrapolation accuracy also for small-scale EXP and FEM hybrid datasets.

Within the ring rolling domain, the technical depth of data-driven workflows has similarly increased while remaining grounded in manufacturing targets. Lee et al. [15] coupled an XGB regressor with a Differential Evolution (DE) optimizer to create case-specific processing maps, focusing on minimizing mandrel work for flat rings. The XGB-DE model was trained on FEM data and validated against experimental cases. Lafarge et al. [17] explored data-based model predictive control for ring rolling, integrating measured process signals with predictive models to support closed-loop operation. Fahle et al. [18] addressed quality monitoring by modeling ring ovality from rolling time-series data through a semi-supervised scheme and further employed a Generative Adversarial Network (GAN) to augment scarce datasets with synthetic samples and reported the ability to generate distinct samples with comparable variance. Finally, Cappellini et al. [19] formulated a multi-objective optimization workflow that trades off production energy and time using surrogate response surfaces, illustrating how validated FEM models can be leveraged to generate structured databases for process analysis and optimization.

Besides the groundwork laid so far, a key issue for deployment is model longevity [11]. Trained models often degrade when applied to new mills, products or operating windows, making domain adaptation essential [10,11]. Transfer learning (TL) addresses this challenge by reusing a model trained in a source domain and fine-tuning it using comparatively limited data from a target domain. In RARR, Seitz et al. [20] demonstrated TL across two mills with a pronounced size ratio (20:1), reducing training effort for the target configuration; the study also discussed the use of GANgenerated synthetic data to alleviate underfitting in low-data scenarios. However, TL remains fully data-driven, and its success depends on having a robust pre-trained backbone and sufficient variance in the target domain database [10,11]. Synthetic data generation and oversampling can be practical for regularization and dataset expansion [21,22], but generated samples typically remain close to the distribution of the data used to train the generator. Hence, synthetic data cannot reliably create new information density in poorly sampled regions. To further reduce dependence on large datasets, physics-informed neural networks (PINNs) have been proposed, adding residual terms derived from governing equations to the loss function [23]. Expert-informed variants, such as GE-EINNs [16], extend this idea by encoding heuristic trends as differentiable constraints. However, the lack of differentiable equations typical of RARR, and other manufacturing processes, limits the direct applicability of standard PINNs/EINNs as benchmarks for the multi-output prediction problem considered in this research.

To address these limitations, this contribution propose a Proxy-Physics Informed Neural Network (P-PINN) that follow the PINN training philosophy [23] but replace exact governing equations with proxy relations encoding cause-effect trends between features and target variables. These proxy residuals provide a tunable bias through the loss function and offer a control for model customization, with the aim of improving generalization in low-data and cross-domain settings. The proposed PPINN was benchmarked against a supervised DNN with transfer learning (DNN-TL). The base DNN models were trained on FEM datasets of varying size and then fine-tuned using experimental data, while performances were evaluated for three key process outputs: OD,RF ) and AF using mean average percentage error (MAPE), together with training time and model complexity. Results show that when a sufficiently large FEM database is available, DNN-TL delivers the lowest deviations (≈3.7% for OD,≈12.7% for RF and ≈17.3% for AF). Conversely, when only 30 FEM cases are available, the base DNN becomes unreliable for force prediction (deviations >60% ), and TL inherits this weakness, leading to noisy AF predictions ( ≈36% MAPE). Under the same low-data scenario, P -PINNs trained with 30 FEM and 30 EXP cases use the proxy residuals as an initial backbone and achieve more consistent cross-domain predictions ( ≈3.7% OD and ≈23−25%RF/AF ), positioning PPINNs as an alternative to TL in the case of data scarcity or PINN framework incompatibility.

FEM and EXP database generation

The original version of this paper is available on https://www.scientific.net/KEM.1049.13.pdf

To train the models proposed in this research time-series belonging to both FEM simulations and RARR experiments were employed. The thermo-mechanical FEM simulations were implemented in the Simufact Forming 15 by employing a model developed by one of the authors and extensively validated against analytical solutions and experimental results [5, 6], as in Fig. 1(a).

In all simulations the following settings were employed: 325 mm for main roll radius, 125 mm for the mandrel radius, 596 mm for the axial rolls' length and 35∘ for their vertex angle, 175 mm for the radius of the centering rolls, 150∘C for the tools' temperature and 50∘C for the environmental temperature. As per the ring dimensions, the following ranges were considered: 429.3~990.2 mm and 211.0~646.6 mm for the outer and inner diameters, 112.2~423.0 mm for the height for the preforms and 556.7~1992.8 mm and 397.0~1812.9 mm for the outer and inner diameters, 99.8~388.0 mm for the height for the final rings, for a total of 218 FEM simulations.

[picture]

Fig. 1. (a) Example of FEM model implementation with results for the RARR process and (b) RARR process experiment at the Chair of Production Systems of Ruhr-University Bochum.

Fig. 1. (a) Example of FEM model implementation with results for the RARR process and (b) RARR process experiment at the Chair of Production Systems of Ruhr-University Bochum.

As materials, a 42 CrMo 4 medium-carbon low-alloy steel, an E355 carbon-steel, a S355−J2 structural grade steel, an AlMgSil 6000 series aluminum, and a nickel-based Inconel 718 superalloy were employed. The material properties of all five materials employed in the FEM simulations were modeled by employing the Hensel-Spittel (H-S) flow stress, Eq. (1), summarized by material in Table 1. In the FEA simulations considered in this research, a shear friction law was employed by setting the friction factor to m=0.85 for the contact between the ring and the mandrel and the ring and the main roll whereas to m=0.6 for the contact between the ring, axial rolls and centering rolls [5,6].

σf=C1·eC2·T·ε(n1·T+n2)·e(L1·T+L2)·ε˙(m1·T+m2)[MPa](1)

Table 1. Hensel-Spittel flow stress material constants employed in the FEM simulations.

Materials | C₁ | C₂ | n₁ | n₂ | L₁ | L₂ | m₁ | m₂
42CrMo4 | 5290.5 | -3.7E-3 | -3.3E-4 | 2.1E-1 | -8.3E-5 | 2.9E-2 | 3.0E-4 | -1.6E-1
E355 | 2549.5 | -3.4E-3 | -3.2E-4 | 1.9E-1 | -4.9E-5 | 1.1E-2 | 1.2E-4 | -4.3E-1
S355-J2 | 2478.7 | -2.9E-3 | -4.2E-4 | 3.7E-01 | -7.5E-5 | 3.2E-2 | 2.6E-4 | -1.4E-1
AlMgSi1 | 378.5 | -4.9E-3 | -1.1E-4 | -2.6E-2 | 6.0E-5 | -2.6E-2 | 3.5E-4 | -3.2E-2
Inconel 718 | 10501.1 | -3.1E-3 | -1.8E-4 | 5.4E-1 | -2.2E-5 | 2.4E-2 | -2.7E-6 | 9.8E-2

The 218 FEM cases were discretized with 3D 8-node brick hexahedra (MARC® element type 7) using a 16/12/16 mm (axial/radial/tangential) mesh with elements' number scaling with the initial ring size and no remeshing. Simulations were solved with MUMPS using a Lagrangian formulation
for large deformations. Contact employed a 1/20 element-size criterion, and relative velocities <0.001 mm/s were treated as sticking conditions. Each run took ~10 h and the cases originate from a repository at the University of Padova [13]. Full settings are available from the corresponding author.

The experimental database comprises 40 rolling series, 17 from the Chair of Production Systems (Ruhr-University Bochum) and 23 from an industrial collaboration at the University of Padova (mill details confidential). The former tests were run on a ThyssenKrupp Wagner mill (20/16-1000/160; max forces 200 kN radial and 160 kN axial; capacity ≈1 m diameter and 160 mm height). Data were acquired via the CARWIN control system: RF/AF through analog-to-digital converters, ring thickness/diameter via an ultrasonic displacement transducer and an elastically supported metering roll, and process time from the PLC sampling rate. Each case provides force/geometry time series plus static initial/final geometries, material properties and process parameters. All experiments used S235 steel with H-S model constants (Table 1): 1967.97, -3.0E-8, -6.9E-4, 0.58, -9.9E-5, 0.049, 4.2E-4, -0.27. Preforms covered OD 262~2148 mm, inner diameter 158~1357 mm, height 71~515 mm ; final rings OD 387~4192 mm, ID 289~3741 mm, height 51~450 mm.

For each FEM and EXP case, the same feature set composed of initial and final ring geometry (outer diameter, wall thickness and height), billet temperature, main-roll rotational speed during rolling, and material encoded by H-S coefficients was collected. The targets in terms of outer diameter (OD) evolution and the radial (RF) and axial (AF) force time series were collected as well. Figure 2 reports the corresponding FEM and EXP distributions as mean curves with Min-Max envelopes.

[picture]

Fig. 2. Mean curves and Min-Max ranges for FEM and EXP time-series for three target variables employed in this study, namely (a) OD, (b) RF, and (c) AF.

Fig. 2. Mean curves and Min-Max ranges for FEM and EXP time-series for three target variables employed in this study, namely (a) OD, (b) RF, and (c) AF.

Base DNN and transfer learning models

The original version of this paper is available on https://www.scientific.net/KEM.1049.13.pdf

Preprocessing. Since ring-rolling FEM time series differ in duration, sampling rate, and noise, each raw simulation was converted into a uniform fixed-size sample. Process inputs (main-roll rotational speed, mandrel feed, upper axial-roll feed, and initial ring temperature) were time-normalized and sampled at 32 evenly spaced instants, while targets (OD, RF, AF) resampled at 100 instants. Static descriptors, initial/final ring geometry (OD, thickness, height) and material encoding via the flow-stress law of Eq. (1), were appended, while all signals smoothed with a Savitzky-Golay filter.

DNN for Multi-Output Prediction (base DNN). A feed forward DNN, as the one employed in [16] is employed in this research. For numeric stability, RF and AF were mapped through an asinh transform, as in Eq. (2), where y is the physical force value in kN,y′ is the transformed value, and s≥1 is the scaling coefficient.

y′=asinh(ys)→y=s·sinh(ys)(2)

Differently from the DNN in [16], the loss function employed in this research utilizes some specific features to help steering the learning process, as hereafter detailed. First, considering the substantial difference between OD and RF/AF behaviors, while all losses are based on the mean
square error (MSE), their formulations differ. As concerns OD, the loss is defined as in Eq. (3) while the single components are detailed in Eq. (4).

Loss(OD)=Lbase +LBC+Lspan +Larea +Lmono +Lcurv Lbase =1N∑k(y^k−yk)2;LBC=(y^0−y0)2+(y^N−1−yN−1)2Lspan =(y^N−1−y^0−(yN−1−y0))2;Larea =(∑ky^k−∑kyk)2Lmono =∑kmax(0,yk−yk+1)2;Lcurv =∑k(y^k+1−2y^k+y^k−1)2

(3)(4)

For OD, the loss for the predicted curve y^ combines six terms to enforce accuracy and physical plausibility, namely: a pointwise MSE term Lbase  against the target y, boundary constraints LBC to match known initial/final OD, and Lspan  to prevent cumulative drift. Global behavior is additionally controlled through an area-matching term Larea , a monotonicity penalty Lmono  that discourages negative OD increments, not physically possible in RARR, and a curvature term Lcurv  that suppresses higher-order oscillations, consistent with the expected pseudo-linear OD expansion trend. For RF and AF, the loss is tailored to emphasize the mid-cycle active region where forces matter most. Thus, points between 25% and 75% of normalized time are weighted more, Eq. (5), sudden peaks are smoothed/stabilized, Eq. (6), and Larea  is also included. Hence, performance is then evaluated as MAPE@active over that active region avoiding excessively effort on fitting the near-zero regions.

Lw(RF/AF)=(∑kwk(y^k−yk)2∑kwk) with 0.25≤wk≤0.75Lp(RF/AF)=[maxk(y^k)−maxk(yk)]2

(5)(6)

The three encoded output sequences, OD residual, asinh-transformed RF, and asinh-transformed AF , are concatenated into a single multi-output prediction vector, employed during the learning process of the base DNN model, and subsequently also in the TL stage.

Transfer learning (TL) module. Considering the feed forward DNN model, as presented in the previous section, to adapt it to the EXP domain, a transfer learning module was developed. The TL workflow predicts first a zero-shot prediction aimed at understanding the differences between the origin (FEM) and target (EXP) domains, and then progressively updates the DNN weights and biases to adapt them to the target domain. To ensure compatibility between FEM-trained models and EXP training data, all EXP targets are transformed into the same encoded space used during FEM training. The OD curve is first residualized as in Eq. (7) while RF and AF are transformed by employing the same asinh scaling coefficients, as in Eq. (8), respectively.

ODresEXP(k)=ODEXP(k)−ODFEM(k)RFtrEXP(k)=asinh(RFEXP(k)sRF);AFtrEXP(k)=asinh(AFEXP(k)sAF)

(7)(8)

The OD baseline ODFEM(k) is defined as the linear interpolation between the known initial and final ring diameters and serves as an initial guess of the trajectory for this target variable. By considering Eq. (7) and (8), the TL training proceeds in the same latent space previously defined during the base DNN training, meaning that the domain is adapted rather than retrained. After
transforming each experimental target curve into the same encoded representation used during FEM training, the three sequences are concatenated into a single target vector, as in Eq. (9), standardized by using the FEM latent space statics in terms of mean ( μyFEM ) and variance ( σyFEM ), as in Eq. (10).

ytrEXP=[ODres,trEXP,RFtrEXP,AFtrEXP]∈ℝ3,NystdEXP=ytrEXP−μyFEMσyFEM

(9)(10)

At this point, fine-tuning is applied by constructing TL heads for each of the target variables, namely OD, RF, and AF, separately. It should be noted that even though separate sections of the network are devoted to different target variables, the model is still a multivariate regression algorithm. Given a standardized EXP input ( Xstd EXP ), the TL phase aims to adjust the heads, defined by the parameter θg, so that the predictions match the encoded experimental targets ystdEXP, as per Eq. (11).

z^gEXP=fg(Xstd EXP,θg) with g∈{OD,RF,AF}(11)

At this point, the losses are constructed with the same strategy employed for the base DNN model and reported in Eq. (3) and (4), yet the main aim changes. For OD, their purpose shifts from enforcing global consistency to applying localized corrections that compensate for small experimental deviations while for RF and AF, since EXP forces contain longer low-force phases and sharper active windows, the weighting isolates the physically meaningful segment more aggressively than in the FEM stage, preventing the model from being dominated by negligible force regions.

Proxy-Physics Informed Neural Networks (P-PINNs) model

The original version of this paper is available on https://www.scientific.net/KEM.1049.13.pdf

P-PINN governing equations. This section presents all the details relevant to the P-PINN model, developed for this research due to the lack of RARR-related physical equations translatable into learnable constraints. Moreover, as previously mentioned, EINN formulations [16] were also not suitable for the task at hand considering the high dimensionality of features and target variable that are described in ℝ3,N and not in ℝ2,N, as in [16]. In this regard, the P-PINN model is grounded on two separate input groups, employed for the prediction of OD, RF, and AF. The former, defined as in Eq. (12), comprises the so-called deformation sequence, composed of the resampled rotational speed rpm(k), mandrel feeding distance ΔT(k), upper axial roll feeding distance ΔH(k), and resampled axial height (H(k)) calculated from the feeding speed. The latter, as presented in Eq. (13), is a scalar descriptor including the initial/final geometry of the ring ( D0,T0,H0,Df,Tf,Hf ), a material strength proxy ( Sproxy  ), and the 8 constants for the Hensel-Spittel flow stress model ( HS1,…,HS8, ), previously defined in Eq. (1). Xseq  and xscal  of Eq. (12) and (13) are then stacked in a unified latent vector, as in Eq. (14), together with the labels identifying the material ( emat  ) and the domain ( edom ), either FEM or EXP, for data management and plotting purposes.

Xseq =[rpm(k),ΔT(k),ΔH(k),H(k)]T∈ℝ4,Nxscal =[D0,T0,H0,Df,Tf,Hf,Sproxy ,HS1,,…,HS8]z=[Xseq ,xscal ,emat ,edom ]

(12)(13)(14)

P-PINN OD modeling. After constructing the training dataset, the learning process is different for OD and for RF/AF. For the former, rather than predicting OD(k) directly as a single sequence, the P-PINN model predicts incremental radial growth and then constructs OD from these increments. First, through a small MLP (multi-layer perceptron), the OD raw head is constructed to roughly
estimate how much OD should grow at each step based on the information in Z, as in Eq. (15). For all means and purposes, Eq. (15) represents a standard data-driven learning process, as the one shown for the base DNN in the previous section.

ΔODiraw= head OD(z)i with i=1,…,N(15)

However, the estimation of Eq. (15) is completely unconstrained, meaning that ΔODiraw  could be negative, not following the true D0−Df behavior or inconsistent OD domain. To avoid this, physical-proxy constraints are applied to ΔODiraw  to both make it physically meaningful and learn the patterns of the EXP domain. The constraints of Eq. (16), enforced sequentially, ensures that the predictions are positive and follow a positive growth for OD, as per Eq. (17). For OD, the P-PINN predicts incremental radial growth from a preliminary shape evolution ( ΔOiraw  ) by enforcing physical-proxy constraints, as per Eq. (16) at model level in the forward pass of the network rather than through PDE residuals or soft physics penalties typical of standard PINNs.

ΔODi=softplus(ΔODiraw)>0;ΔODi*=ΔODiDf−D0∑j=1NΔODiOD(k)=D0+∑i=1kΔODi*

(16)(17)

P-PINN RF/AF modeling. For RF and AF, proxy modeling is formulated more explicitly than for OD, reflecting their different feature dependence and relying on prior analytical RARR force models developed by the authors [5, 6]. In P-PINNs, physics is not imposed through PDE residuals or loss penalties, instead, the governing structure is embedded as forward-pass proxies, so that the proxies capture the dominant geometric and material drivers while the data-driven residual component mainly provides fine tuning. Accordingly, the RF proxy in Eq. (18) is expressed as a weighted combination of the principal contributors, namely the thickness variation ΔT(k), diameter growth △OD(k), and main-roll speed pm(k), scaled by an effective flow stress σeff  computed from the H-S law. The weights ω1,ω2,ω3 and a small scaling factor ssoft RF are learned to calibrate the RF magnitude. As concerns AF, the modeling is similar is shape but different in meaning than that of RF. As shown in Eq. (19), the ReLU function is not included in AFproxy (k) to allow for negative values in case of specific machine or simulation settings.

RFproxy (k)=σeff [ω1ReLU{ΔT(k)}+ω2ReLU{ΔOD(k)}+ω3ReLU{rpm(k)}]·ssoft RFAFproxy (k)=0.75·σeff [ω4c(k)+ω5ΔOD(k)+ω6rpm(k)]·ssoft AF

(18)(19)

Moreover, AFproxy (k) is also based on the outer diameter △OD(k) and main roll rotational speed rpm(k), as RFproxy (k), but includes c(k)=max(H0−Hk,0), defined as the compression gap. In this regard, c(k) quantifies the instantaneous axial displacement between the ring and the axial roll, estimated through the upper axial roll feeding speed, and serves as the primary driver of axial force in the Proxy-PINN by activating AF only when true geometric compression, and not noise, occurs. It ought to be pointed out that ΔOD(k) in Eq. (18) and (19) is estimated through Eq. (17), thus it is an internal model estimation and is not based on the ground truth OD. Accordingly, RF and AF are predicted, at each epoch of the calculation, by a mix of data- and physics-driven equations (20).

RF(k)=RFproxy (k)+gRF·ΔRF(k);AF(k)=AFproxy (k)+gAF·ΔAF(k)(20)

In Eq. (20), RFproxy (k) and AFproxy (k), are fully proxy-driven and serve as the backbone of the learning process, while the latter parts of both equations are DNN residual-based. The data-driven contribution is learned, through the scalars gRF and gAF, during the training of the model.

In terms of losses, since most of the constraints are applied at model level in the forward section of the network, the loss for OD is only based on the batch size ( B ) and the number of discretized points ( N ), as in Eq. (21), which is based on an MSE.

Loss(OD)data =1BN∑j=1B∑k=1N(OD^s(j)(k)−ODs(j)(k))2(21)

As concerns RF and AF, they use the same data loss structure on scaled outputs, as in Eq. (22), thus only the former is hereafter summarized. First, the data-driven part of the loss is constructed considering discrepancy between predicted and measured RF and AF curves using an asinh-scaled MSE with FEM/EXP domain-dependent weights as ωFEM(j)=0.5 and ωEXP(j)=2 and where arf is the asinh scaling constants used to normalize the force magnitudes before applying MSE. The asinh scaling reduces the dynamic range of RF/AF and improves numerical conditioning for large and noisy force peaks. The second part of the loss is represented by the P-PINN consistency term, where the predictions are steered to be closer to the surrogate proxies. The utilization of this additional control over the learning process applied at loss level required the utilization of the PINN module over the standard DNN module. For RF, and for AF, the proxy part of the loss is defined as in Eq. (23). The last part of the loss is introduced to stabilize and capture physical smoothness of force evolution, thus the main behavior over noise, as per Eq. (24).

RF^s(j)(k)=asinh(RF(j)(k)arf) so that: Loss(RF)data =1B∑j=1Bωdom (j)[1N∑k=1N(RF^s(j)(k)−RFs(j)(k))2]Loss(RF)proxy =1BN∑j=1B∑k=1N(RF^s−RF^s, proxy )2Loss(RF,AF)smooth =‖Δ2RF^s‖22+‖Δ2AF^s‖22

(22)(23)(24)

In conclusion, the total loss for the P-PINN model is the sum of Loss(OD)data  of Eq. (21), the datadriven loss for RF and AF as shown for the former in Eq. (22) in terms of Loss(RF)data , the proxy PINN part of RFLoss(RF)proxy  as in Eq. (23), and the corresponding Loss(AF)proxy  part constructed in the same way, and the Loss(RF,AF)smooth part of Eq. (24) helping to promote a behavior towards the main trend.

Models' Pre-Processing and Tuning

The original version of this paper is available on https://www.scientific.net/KEM.1049.13.pdf

This last section before the results chapter summarizes all the technical information related to the base DNN, the TL, and the P-PINN models defined in the previous sections. All computations were performed in Python on a MacBook Pro (Apple M4), without external GPU acceleration.
DNN for Multi-Output Prediction (base DNN). The base DNN operates on the FEM-only matrix considering a combination of scalars, namely the initial/final geometry of the ring, the initial ring temperature, and time-sequences describing the main roll rotational speed, the thickness reduction rate and height compression rate, namely the mandrel and upper axial roll feeding speeds. The network outputs the full trajectories of OD,RF and AF on a fixed grid of N=100 points per target,
i.e. 300 output channels and inputs are standardized with a StandardScaler. OD is handled in physical units by subtracting a linear baseline (added back after inference). RF/AF are first scaled by a pergroup factor, then mapped with an asinh transform to compress noisy peaks and finally standardized. Training is performed in this transformed space, and predictions are post-processed by destandardization, inverse-asinh, and reconstruction in physical units. Hyperparameters are tuned separately for each target, consistently with the different pre-processing approaches, using a grid search of 60 trials combined with k=3 cross-validation and a 90%/10%train/ validation split. Given the sparsity of EXP data, no independent test set is held out during tuning. In contrast, the selected model is evaluated across 5 random seeds to assess generalization. The same tuning protocol is used for DNN-TL and P-PINN to ensure a fair comparison. Here follows the search ranges, while the full grid available on request. OD: hidden units 256−512, dropout 0.00−0.35, learning rate 3×10−5−5×10−3, weight decay 10−7−10−3, batch size 48-256, epochs 450−1500. RF: hidden units 384−1024, dropout 0.00−0.40, learning rate 3×10−5−10−2, weight decay 10−8−10−3, batch size 64-384, epochs 450-1500. AF: hidden units 384-512, dropout 0.05−0.25, learning rate 10−4−3×10−3, weight decay 10−7−10−4, batch size 64-192, epochs 600-1200. All runs used the AdamW optimizer coupled with an early stopping to limit overfitting and select the best model based on the validation performance.

Transfer learning (TL) module. The transfer learning (TL) stage adapts the pre-trained base DNN to the experimental (EXP) domain while preserving the structure learned from the FEM simulations. TL is performed separately for OD, RF, and AF output heads. Inputs are normalized using the original FEM-trained StandardScaler, and the TL targets are standardized using the stored FEM statistics for each group. OD is fine-tuned in residual form, exactly as in pre-training, while RF and AF remain in the asinh-scaled space. Each TL head is trained on an 80%/20% split of the filtered EXP dataset using Adam W with learning rate 3×10−4, weight decay 10-5, batch size 128, and gradient-norm clipping at 1.0. Training proceeds for a maximum of 1,000 epochs with early stopping based on the validation loss and a patience of 35~60 epochs. After TL, predictions are de-standardized, inverse-transformed, and reconstructed in physical units. For the TL model, a manual hyperparameters tuning was carried out on a subset of those employed for the base DNN model.

P-PINN model. The P-PINN is trained on 30 and 30 randomly sampled FEM and EXP cases. Inputs consist of a deformation sequence and scalar descriptors. The sequence block is encoded by a 1D CNN with three Conv1d layers ( 4→32→64→128 channels) and ReLU activations. Material and domain labels are appended to tag each case during learning. As for the base DNN, OD is kept in physical units, while RF/AF are mapped to asinh space. In the total loss, AF is prioritized by using weights of 8:1 vs OD and 4:1 vs RF. Optimization uses AdamW for stability under mixed data-driven and proxy-physics terms and consistency with DNN-TL, with gradient-norm clipping =5 to avoid instabilities. The best model selection retains the parameters at the minimum validation data-loss. Hyperparameters were tuned with Optuna under 3-fold cross-validation, rather than grid search, because P-PINN hyperparameters are more strongly coupled due to loss-weight scheduling and proxy-term interactions. Each trial samples: learning rate [5×10−4,3×10−3] (log-uniform), weight decay [10−6,10−3] (log-uniform), batch size {4,8,12}, and epochs {4000,12000} in 1000 -step increments. The intervention/decay schedule for LossRFproxy  and LossRF,AFsmooth  is also optimized via λstart proxy ∈[0.1,0.6] and λend proxy ∈[0.02,0.3], with the schedule switch epoch selected from {500,750, 1000,1250,1500,1750,2000} to control when training transitions from capturing global behavior to correcting smaller deviations. The Optuna objective is the fold-average of MAPE@active summed over OD, RF, AF, evaluated on the full FEM and EXP sets. The best setting yields lr≈6.35×10−4, wd ≈6.4×10−6, B=4,12,000 epochs, with λstart proxy ≈0.34,λend proxy ≈0.03, and a switch at epoch 2000. The model is then retrained once on the full FEM + EXP training set with this configuration, and the best checkpoint is used for benchmarking against DNN-TL.

Results

The original version of this paper is available on https://www.scientific.net/KEM.1049.13.pdf

Given the large result set, each model is presented through representative OD/RF/AF charts and a numerical comparison based on MAPE@active averaged over five runs and five random seeds. The chapter compares the base DNN trained on 218 vs 30 FEM cases, the corresponding TL models adapted with 40 EXP cases, and a P-PINN trained on a balanced 30 FEM +30 EXP subset, to contrast data-rich and data-limited regimes and benchmark P-PINN against DNN/TL under both.

Base DNN performance. Two base-DNN configurations were evaluated: training on the full FEM set ( 218 simulations) and on a reduced subset ( 30 randomly selected cases). Both tests were repeated five times with different random seeds and train/validation splits to assess repeatability. Fig. 3 shows representative validation cases, while performance is reported as MAPE@active.

[picture]

Fig. 3. Examples of true vs predicted curves by the base DNN model trained with 218 cases for (a) the outer diameter OD, (b) the radial force RF, and (c) the axial force AF considering 218 input cases. (d), (e), and (f) show the same comparison but considering 30 training cases for the base DNN model.

Fig. 3. Examples of true vs predicted curves by the base DNN model trained with 218 cases for (a) the outer diameter OD, (b) the radial force RF, and (c) the axial force AF considering 218 input cases. (d), (e), and (f) show the same comparison but considering 30 training cases for the base DNN model.

Prior to consolidation, five FEM files were discarded because some features could not be reliably extracted/encoded, mainly due to unresolved material-name mapping, leaving 213 FEM cases for the "full-data" training. Across the five seeded trials, the DNN trained on 213 FEM cases predicts all three targets well, achieving a MAPE@active of 1.98% (OD), 15.5% (RF), and 17.4% (AF). Reducing
training data to 30 FEM cases degrades performance considerably, Fig. 3(d)-(f). OD remains reasonable but is slightly overestimated ( 6.2% ), whereas RF and AF are strongly underestimated, with 62.8% and 63.9% MAPE@active, respectively. Overall, the results confirm that purely datadriven accuracy depends critically on database size/coverage, motivating hybrid formulations, id est P-PINN, for low-data or boundary scenarios.

DNN-TL performance. As concerns the TL application, the two base DNN models reported in the previous section were employed by applying the same 40 cases strong EXP database. In this regard, the first attempt is related to the utilization of the base DNN model trained with 218 FEM cases, which showed the best performance in its own latent space, as reported in Fig. 4.

[picture]

Fig. 4. True vs predicted curves by the DNN-TL model for an example EXP case (#121) for (a) the outer diameter OD , (b) the radial force RF , and (c) the axial force AF considering 218 input cases. (d), (e), and (f) show the same comparison but considering 30 cases for the training of the base DNN.

Fig. 4. True vs predicted curves by the DNN-TL model for an example EXP case (#121) for (a) the outer diameter OD , (b) the radial force RF , and (c) the axial force AF considering 218 input cases. (d), (e), and (f) show the same comparison but considering 30 cases for the training of the base DNN.

The above-mentioned model, hereafter referred to as DNN(218)-TL, showed comparable performance in comparison to the base DNN model from which it is transfer learned, being the MAPE@active, for the three target variables, equal to 3.7%OD,12.69% for RF , and 17.3% for AF , respectively. The results for an example case are reported in Fig. 4(a), (b), and (c). However, when the base DNN model is switched from DNN(218)-TL, to the one trained with only 30 FEM cases, defined hereafter as DNN(30)-TL, an interesting scenario arises. First, thanks to its monotonic
behavior and the similarity between the FEM and EXP domains for this target variable, OD seems to be insensitive to the change in the base DNN model with the MAPE@active changing to 3.2. On the other hand, while RF predictions improve to 10.6%, those for AF degrade to 36.3%. Since all the results of Fig. 4 refer to the same case, by comparing Fig. 4(c) with Fig. 4(f) is it possible to grasp the increase noise in the predicted curve, a symptom of the lack of stability in the base DNN model.

Large AF errors are not purely computational but reflect the intrinsic complexity of axial loading in RARR. AF arises from contact between curved, out-of-plane ring and axial-roll surfaces, where force transmission depends on rapidly changing contact states and secondary deformation. These mechanisms are weakly constrained when the base DNN is trained on only a few FEM cases, and the mismatch is amplified in experiments, making AF particularly sensitive to the representational capacity inherited by TL. Consistently, applying the 30-FEM base DNN directly to EXP data yields MAPE@active = 20.3% (OD) but extreme errors for forces ( >800%RF,~300%AF ), highlighting a strong FEM-EXP domain gap not evident from Fig. 2 alone.

Overall, the 218-FEM base DNN is a robust FEM-domain regressor and its DNN(218)-TL remains reliable for OD/RF/AF. With only 30 FEM cases, the base model captures mainly the global trends, making TL more flexible and even improve RF adaptation, but it does not provide a sufficiently structured backbone to transfer-learn AF, whose performance degrades relative to DNN(218)-TL. Hence, while TL can still work under limited FEM data, the weak FEM representation particularly limits AF transfer. Hence, the proposed P-PINN is introduced next to jointly model FEM and EXP within a single, proxy-constrained framework.

P-PINNs performance. Since the P-PINN is trained on a randomly selected 30 FEM +30 EXP mix, the results in this section refer to both numerical and experimental cases, which are learned and predicted simultaneously by a single model. Representative validation curves from the five repetitions (different random seeds) are shown in Fig. 5, while the mean performance across seeds is MAPE @ active =3.66%(OD),22.9%(RF), and 25.1%(AF). Notably, RF and AF errors are comparatively balanced, addressing the DNN(30)-TL behavior where RF was adapted much better than AF. However, deviations remain higher than DNN(218)-TL, indicating that proxy relations alone cannot fully compensate for the information density provided by the full FEM database used to build the TL backbone.

Comparing the FEM example in Fig. 5(a)-(c) with the EXP example in Fig. 5(d)-(f) shows that the proxy equations can represent the FEM domain, but joint FEM+EXP training prevents the build in source domain → adapt to target advantage of DNN-TL, which mainly affects the EXP accuracy. For OD, enforcing known initial/final geometry, as in analytical approaches [5,6] translated in here into proxy forms, stabilizes the trajectory but may overconstrain the mid-expansion evolution, producing larger mid-phase errors for smoother expansions, Fig. 5(d). For RF/AF, predictions are substantially less noisy than DNN(30)-TL, yet they can still exhibit sharp peaks and misestimate rises/drops, suggesting that limited data coverage cannot be fully replaced by proxies. Nonetheless, the model captures the overall trends and peak magnitudes of both forces with reasonable accuracy across both FEM and EXP, using a single formulation.

Although the P-PINN model is more accurate in the presence of small-scale databases, it is also more computationally demanding, requiring a proxy equations' definition phase, similarly to EINN [16], a more complex tuning, which was carried out by the Optuna hyperparameters optimization scheme, and a running time of approximately 30 minutes, in contrast to the 10 minutes of the base DNN trained on 218 cases and TL which is almost real time.

[picture]

Fig. 5. True vs predicted curves by the P-PINN model for an example FEM (#193) and one EXP case (#317) for (a)-(d) the outer diameter OD, (b)-(e) the radial force RF, and (c)-(f) the axial force AF considering 30 FEM and 30 EXP cases for the training.

Fig. 5. True vs predicted curves by the P-PINN model for an example FEM (#193) and one EXP case (#317) for (a)-(d) the outer diameter OD, (b)-(e) the radial force RF, and (c)-(f) the axial force AF considering 30 FEM and 30 EXP cases for the training.

Discussion

The original version of this paper is available on https://www.scientific.net/KEM.1049.13.pdf

This section expands on the previous results chapter by discussing the advantages, limitations, and improvement paths of the proposed P-PINN formulation, and by providing practical guidance on when to prefer P-PINNs versus transfer learning.

Consistent with prior TL studies in RARR [20], DNN(218)-TL achieves higher accuracy with lower modeling complexity than P-PINN. This outcome is expected since proxy equations capture dominant trends, but cannot reproduce the full variability embedded in a large well-contructed database. The broad geometric/material coverage of the 218-case FEM set is therefore central to the robustness of TL across source and target domains. Conversely, when data are limited and the featuretarget variables relationships are highly nonlinear, typical for RARR [13], purely data-driven models become difficult to train reliably, especially for the most complex targets, such as AF . In this lowdata regime, hybrid approaches can provide useful robustness by combining prior knowledge with data, at the cost of higher model complexity and computational effort. Since proxies are not exact closed-form laws, they also require a design phase to avoid overwhelming the data-driven component.

In this regard, and from a modeling perspective, P -PINNs use the proxy terms as an initial structural backbone that guides optimization toward physically admissible solutions, while later training stages rely increasingly on data-driven residuals for fine-scale corrections. Operationally, this behaves like an integrated transfer-learning scheme where the base model is given by the proxies, and domain adaptation is performed by the residual network. In this work, this was implemented through a two-stage proxy schedule λstart proxy  and λend proxy , with λstart proxy  about an order of magnitude larger, effectively introducing curriculum learning. Before the switch, a stronger proxy coupling biases training toward proxy-consistent behavior to establish the global trends. Then, after the switch (2,000 epochs in this study) the proxy influence is relaxed so the data can refine deviations rather than reconstruct the overall dynamics from scratch. Because the balance between proxy and data residuals is problem-dependent, both λ values were tuned via Optuna.

Considering what stated in the last paragraph, the degradation of AF in DNN(30)-TL is not only a matter of data volume, but also of losing latent geometric structure that the base DNN learns when trained on the full FEM set. In contrast, P-PINN partially preserves this architecture through the proxy structure, yielding more balanced RF/AF errors under limited data. In addiiton to that, the slightly better RF adaptation observed in DNN(30)-TL relative to DNN(218)-TL is consistent with a known TL effect where a mild underfitting in the source model can increase flexibility in the target domain, as also reported in other TL contexts [24, 25]. However, when underfitting becomes severe, as for AF, transfer performance deteriorates, leading to high deviations in both the source and target domains.

In this regard, and conceptually speaking, P -PINNs can be seen as embedded TL where proxies act as a fixed, non-trainable prior, while the residual network performs data-driven adaptation. Accordingly, P-PINNs are not intended to replace TL when large, high-quality FEM databases are available. Rather, they provide a viable option when data scarcity, domain mismatch, or longevity requirements limit purely data-driven approaches. Overall, P-PINNs occupy a middle ground between TL and physics-informed or expert-informed modeling, offering controlled customization and robustness while still requiring data for calibration. While the present formulation remains a proof of concept, the results indicate that P-PINNs can prioritize controlled generalization over maximum achievable accuracy in practically relevant, data-limited settings.

Conclusions

The original version of this paper is available on https://www.scientific.net/KEM.1049.13.pdf

This contribution detailed a proof-of-concept study for the development of customized neural networks that employs the PINNs structure while grounded on proxy equations, thus P-PINNs. The proposed P-PINN model was benchmarked against the well-established transfer learning approach. In low data scenario, for both the origin and target domains, P-PINNs allow for a better and wiser modeling, resulting in a stable training and better overall performance. On the other hand, for industrial applications, or more broadly when large datasets are available, TL is still confirmed as the tool of choice, being easier to set up and overall, better performing. To this end, some key issues raised in this contribution will require additional work in the future. First and foremost, the definition of the proxy equations would benefit from a semi-automatization, possibly based on a known set of analytical or empirical equations. Moreover, the threshold underlining the choice between P-PINN and TL should also benefit from a deeper and more systematic modeling, allowing end-users to select the best modeling approach based on model selection maps, helping in this initial yet critical choice.

References

The original version of this paper is available on https://www.scientific.net/KEM.1049.13.pdf

J. Seitz, Q. Wang, L. Goldbach, T. Moser, A. Brosius, B. Kuhlenkötter. A Review of DataCentric Optimization and Machine Learning in Ring Rolling, in: Drossel, WG., Ihlenfeldt, S., Dix, M. (eds) Production at the Leading Edge of Technology. WGP 2024. Lecture Notes in Production Engineering. Springer, Cham, 2025, pp. 242-252.

J.M. Allwood, A.E. Tekkaya, T.F. Stanistreet, The development of ring rolling technology, Steel Res. Int. 76 (2005) 111-120.

J.B. Hawkyard, W. Johnson, J. Kirkland, E. Appleton, Analyses for roll force and torque in ring rolling, with some supporting experiments, Int. J. Mech. Sc. 15 (1973) 873-893.

A.G. Mamalis, W. Johnson, J.B. Hawkyard, Pressure distribution, roll force and torque in cold ring rolling, J. Mech. Eng. Sc. 18 (1976) 196-204.

L. Quagliato, G.A. Berti, D. Kim, N. Kim, Slip line model for forces estimation in the radialaxial ring rolling process, Int. J. Mech. Sc. 138-139 (2018) 17-33.

G.A. Berti, L. Quagliato, M. Monti, Set-up of radial-axial ring-rolling process: Process worksheet and ring geometry expansion prediction, Int. J. Mech. Sc. 99 (2015) 58-71.

L. Guo, H. Yang, Towards a steady forming condition for radial-axial ring rolling, Int. J. Mech. Sc. 53 (2011) 286-299.

N. Kim, H. Kim, K. Jin, Optimal design to reduce the maximum load in ring rolling process, Int. J. Prec. Eng. Manuf. 13 (2012) 1821-1828.

Z.W. Wang, J.P. Fan, D.P. Hu, C.Y. Tang, C.P. Tsui, Complete modeling and parameter optimization for virtual ring rolling, Int. J. Mech. Sc. 52 (2010) 1325-1333.

J. Cao, M. Bambach, M. Merklein, M. Mozaffar, T. Xue, Artificial intelligence in metal forming, CIRP Annals - Manuf. Tech. (2024) 1-27.

L. Quagliato, J. Seitz, M. Perin, Machine learning modeling for material science and manufacturing: overview and perspectives for the future, AI Mater. (2025) 0005.

A. Parvizi, H.R.R. Raftar, Application of artificial neural network and genetic algorithm to predict and optimize load and torque in T-section profile ring rolling, Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sc. 233 (2019) 5966-5976.

I. Mirandola, G.A. Berti, R. Caracciolo, S. Lee, N. Kim, L. Quagliato, Machine learning-based models for the estimation of the energy consumption in metal forming processes, Metals 11 (2021) 833.

S. Lee, Y. Lim, L. Galdos, T. Lee, L. Quagliato, Gaussian process regression-driven deep drawing blank design method, Int. J. Mech. Sc. 265 (2024) 108898.

S. Lee, J. Park, N. Kim, T. Lee, L. Quagliato, Extreme gradient boosting-inspired process optimization algorithm for manufacturing engineering applications, Mat. & Des. 226 (2023) 111625.

V. Modanloo, S. Jang, T. Lee, L. Quagliato, Gradient enhanced-expert informed neural network (GE-EINN) for forming depth prediction from a small-scale metal stamping dataset, J. Manuf. Proc. 140 (2025) 224-240.

R. Lafarge, S. Hütter, M. Tulke, T. Halle, A. Brosius, Data-based model predictive control for ring rolling, Prod. Eng. 15 (2021) 821-831.

S. Fahle, T. Glaser, A. Kneißler, B. Kuhlenkötter, Improving quality prediction in radial-axial ring rolling using a semi-supervised approach and generative adversarial networks for synthetic data generation, Prod. Eng. 16 (2022) 175-185.

C. Cappellini, C. Giardini, S. Bocchi, A multi-objective optimization workflow of ring-rolling process parameters based on production energy and time, Procedia CIRP 122 (2024) 683-688.

J. Seitz, T. Moser, S. Fahle, C. Prinz, B. Kuhlenkötter, Transfer learning approaches in the domain of radial-axial ring rolling for machine learning applications, Proceedings of the 5th Conference on Production Systems and Logistics (CPSL) (2023).

L. Quagliato, S. Kim, O.R. Hassan, T. Lee, Heel pad's hyperelastic properties and gait parameters reciprocal modelling by a Gaussian mixture model and extreme gradient boosting framework, Biomedical Signal Processing and Control 107 (2025) 107818.

V. Modanloo, M. Elyasi, T. Lee, L. Quagliato, Modeling of tensile strength and wear resistance in friction stir processed MMCs by metaheuristic optimization and supervised learning, The International Journal of Advanced Manufacturing Technology (2025).

S. Cuomo, V. Schiano Di Cola, F. Giampaolo, G. Rozza, M. Raissi, F. Piccialli, Scientific machine learning through physics-informed neural networks: Where we are and what's next, Journal of Scientific Computing 92 (2022) 88.

A. Deng, X. Li, D. Hu, T. Wang, H. Xiong, C.-Z. Xu, Towards inadequately pre-trained models in transfer learning, in: Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), 2023, pp. 19397-19406.

J.M. Springer, S. Goyal, K. Wen, T. Kumar, X. Yue, S. Malladi, G. Neubig, A. Raghunathan, Overtrained language models are harder to fine-tune, arXiv preprint arXiv:2503.19206, 2025.

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/lionel-fourment-on-optimization-ai-and-inverse-analysis-in-forming/978-3-0364-1989-3

Key Engineering Materials, ISSN: 1662-9795, Vol. 1049, pp 29-35

doi: 10.4028/p-Wg5agb

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

Submitted:
2025-12-19

Revised:
2026-03-01

Accepted:
2026-03-02

Online:
2026-04-15

Material Parameter Identification Using Bayesian Data Assimilation and Biaxial Tensile Test

The original version of this paper is available on https://www.scientific.net/KEM.1049.29.pdf

Akinori Yamanaka a1,a* and Shun Shimaoka 1, b

¹Department of Mechanical Systems Engineering, Tokyo University of Agriculture and Technology, 2-24-16, Naka-cho, Koganei, Tokyo, Japan

a  a-yamana@cc.tuat.ac.jp, b  s252517q@st.go.tuat.ac.jp, (*corresponding author)

Keywords: Bayesian data assimilation, Material modeling, Parameter identification

Abstract

This study proposes a Bayesian data assimilation approach to estimate material model parameters based on deformation fields measured via digital image correlation in a biaxial tensile test using a cruciform specimen. The anisotropy parameters and exponent of the Yld2000-2d yield function for a A5052P-H32 aluminum alloy are identified. The results indicate that the proposed method can estimate parameters with high accuracy-comparable to those identified via conventional multiaxial testing methods-while requiring only a single biaxial test. The proposed method offers an efficient framework for material modeling by minimizing a cost function via Bayesian optimization, enabling parameter identification from a single biaxial tensile test for sheet metal forming applications.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1049.29.pdf

To efficiently design sheet metal forming processes, one must improve the accuracy of numerical simulation models that reproduce the deformation behavior of metallic materials. Achieving this accuracy requires the precise identification of parameters for yield functions and hardening laws. In this study a material model that describes the elastoplastic deformation behavior of metals is used. The conventional approach to material modeling is based on mechanical testing under various stress states, including multiple uniaxial tensile tests and multiaxial stress tests such as biaxial tensile tests using cruciform specimens. In this approach, the parameters of the material model are identified by fitting the model to stress-strain responses or equiplastic work contours obtained from these experiments [1]. However, performing multiple mechanical tests under different loading conditions is time-consuming and experimentally demanding.

In this study, a method is proposed to estimate the parameters of a material model by applying a data assimilation approach based on Bayes' theorem-hereinafter referred to as Bayesian data assimilation (DA) [2]-to heterogeneous strain fields that arise on the surface of a cruciform specimen during a biaxial tensile test. The strain fields are measured using digital image correlation (DIC). It is demonstrated that this method enables the estimation of material model parameters using only the results of a single biaxial tensile test. Notably, the Bayesian DA method employed in this study, which is based on Bayes' theorem, is advantageous over conventional inverse methods such as the finite-element model updating [3] and virtual fields methods [4]. Specifically, DA allows the uncertainty inherent in experimental data to be rigorously incorporated into the Bayesian inference. Herein, the anisotropy parameters and exponent of the Yld2000-2d yield function [5] are estimated via biaxial tensile tests conducted using a simple cruciform specimen for a 5000 -series aluminum alloy sheet with strain measurements obtained using DIC. Furthermore, it is demonstrated that employing the parameters estimated using the proposed DA method improves the predictive accuracy of the deformation behavior observed experimentally.

Experimental Methods

The original version of this paper is available on https://www.scientific.net/KEM.1049.29.pdf

Biaxial Tensile Test. To obtain experimental data for estimating the parameters of the material model through DA, a biaxial tensile test using a cruciform specimen was conducted. A hydraulically controlled biaxial tensile testing machine, as shown in Fig. 1(a), was employed. The loads applied to
the specimens were measured using two load cells. To continuously capture the deformation of the specimen surface, a charge-coupled device (CCD) camera (DFK23U274, Image Source) was positioned above the specimen and sequential images were recorded at a sampling frequency of 5 Hz . The testing machine was operated using an in-house control code developed in LabView. The material used in this study was a commercially available A5052P-H32 aluminum alloy with a nominal thickness of 1.0 mm . The geometries of the specimens are shown in Fig. 1(b). The specimen was clamped over a 30 mm region. A random speckle pattern was applied to one side of the specimen using spray paint for strain measurements using DIC. As shown in Fig. 1(b), two strain gauges were attached to opposite sides of the specimen to control the testing machine. In the biaxial tensile test, the true stress ratio in the x - and y-directions was set to σxx:σyy=1:1.

[picture]

Fig. 1 (a) Biaxial testing machine and (b) dimensions of biaxial tensile specimen. Black rectangle in (b) shows strain gauge for controlling biaxial testing machine.

Fig. 1. (a) Biaxial testing machine and (b) dimensions of biaxial tensile specimen. Black rectangle in (b) shows strain gauge for controlling biaxial testing machine.

DIC. To measure the deformation field (i.e., the displacement and strain fields) using DIC, the resolution of the CCD camera was set to 1600×1200 pixels. To compute the deformation field from the captured images, image correlation software (GOM Correlate Professional 2021 Hotfix8 (GOM GmbH )) was used. The subset and step sizes were set to 19 and 15 pixels, respectively. Figure 2 shows the region of interest (ROI) specified for measuring the deformation field. A cruciform-shaped ROI was defined at the center of the specimen, excluding 1.5 mm from the edge of the specimen. At the measurement points indicated by red dots in Fig. 2, the displacements were recorded and used as the boundary conditions in the biaxial tensile test simulation.

[picture]

Fig. 2 Region of interest (ROI) to measure deformation field using digital image correlation.

Fig. 2. Region of interest (ROI) to measure deformation field using digital image correlation.

Numerical Methods

The original version of this paper is available on https://www.scientific.net/KEM.1049.29.pdf

Biaxial Tensile Test Simulation Using Yld2000-2d Yield Function. ABAQUS/Standard was used to perform a biaxial tensile test simulation corresponding to the experimental test. The geometry of the finite-element model was created by removing the gripping regions from the specimen shape, as
shown in Fig. 1 (b). The model was discretized into 8,418 elements with an approximate element size of 0.5 , and reduced-integration shell elements were employed. The prescribed displacements were applied to each arm of the finite-element model. The Young's modulus and Poisson's ratio of the specimens were set to 63.347 GPa and 0.33 , respectively. The Swift hardening law, expressed as σ=c(a+εp)n, was employed to describe the strain-hardening behavior, where σ is the equivalent stress and εp is the equivalent plastic strain. The parameters of this hardening law were identified from a uniaxial tensile test conducted using A5052P-H32 aluminum alloy, which resulted in c=336.90MPa, a=5.01×10−3, and n=0.143. To implement the Yld2000-2d yield function in ABAQUS, the Unified Material Model Driver for Plasticity user subroutine [6] was employed. The Yld2000-2d yield function contains eight anisotropy parameters; it sets αi(i=1−8)=1.0 and M=8.0 as the initial estimation. The total simulation time was 650 s , and the strain fields were output every 65 s . The strain distributions at these time instants were compared with the experimentally measured strain fields obtained via DIC in the data assimilation process.

Bayesian DA Based on DMC-TPE. In this study, DA to minimize the cost function using the Data assimilation method Minimizing the Cost function with Tree-structured Parzen Estimator (DMCTPE) method [2] was employed. In DMC-TPE, the evolution of the system state and its relationship with the experimental observations are formulated within a state-space model. The state vector xt and observation vector yt at time t are defined as follows:

xt=(εxx1(t),…,εxxl(t),εyy1(t),…,εyyl(t),εxy1(t)…,εxyl(t),α1,…,α8,M)Tyt=(εxx1,obs(t),…,εxxl,obs(t),εyy1,obs(t),…,εyyl,obs(t),εxy1,obs(t)…,εxyl,obs(t))T

(1)(2)

Here, εijk(t) denotes the ij(i,j=x or y) component of strain at node k(k=1,2,…l) in the finiteelement model of the cruciform specimen at time t; l is the number of nodes in the ROI; and εijk,obs(t) represents the corresponding strain component observed experimentally at the spatial location associated with εijk(t).

As illustrated in Fig. 3, the misfit between the experimental results and numerical simulations is evaluated using the cost function, which includes the time integration of the misfit over time from t=0 to t=tend .

J(x0)=12(x0−x0b)TB−1(x0−x0b)+12∑t=0tend(yt−Ht(xt))TRt−1(yt−Ht(xt))(3)

Here, the superscript T denotes the transposed matrix, Ht is the observation operator that extracts the components from xt corresponding to yt, and x0b is the initial estimate for x0;B and Rt are the error covariance matrices associated with the uncertainties of x0b and yt, respectively, and are referred to as the background and observation error covariance matrices, respectively. The first term on the righthand side of Eq. (3) represents the deviation from the initial estimate, weighted by the background error covariance matrix, and therefore reflects the uncertainty associated with the prior parameter estimates. The second term corresponds to the time-integrated misfit between the experimentally measured strain field (obtained via DIC) and the strain field computed from the finite-element simulation, weighted by the observation error covariance matrix. The x0 that minimizes the cost function J(x0) is regarded as the optimal state vector, which is denoted by x0a, and the parameters contained in x0a are regarded as the optimal estimates. In this study, the minimization of 𝐉 was performed using Bayesian optimization based on the TPE, which sequentially explores the admissible parameter space. The optimization is performed without explicitly computing the gradient of the cost function, as gradient evaluation for the present finite-element-based formulation is computationally complex. Instead, a surrogate model of the cost function is constructed using Gaussian process regression. The surrogate model is used to iteratively propose new candidate values of x0 that are
expected to reduce the cost function, and this procedure is repeated until convergence. Therefore, x0a is obtained by repeating the following steps: (i) execute the biaxial tensile test simulation using the current estimate for x0, (ii) compute J(x0) based on the simulation results, and (iii) update the estimate for x0 to reduce J(x0). In this study, B is defined as follows:

B=diag.(0,…,0,(Sα1)2,(Sα2)2,…,(Sα8)2,(SM)2)(4)

The initial strain in the specimen is assumed to be zero, and its variance is set to zero. Therefore, the first through 3l-th components of B are zero. Sαi(i=1,2,…,8) and SM denote the standard deviations representing the uncertainties in the initial estimates for the anisotropy parameters αi and the exponent M of the Yld2000-2d yield function, respectively. Assuming that the standard deviations are 30% of the corresponding initial estimated values, Sαi=0.30 and SM=2.40 are used. Rt is defined as follows:

Rt=diag·((Sεxx)2,…,(Sεxx)2,(Sεyy)2,…,(Sεyy)2,(Sεxy)2,…,(Sxy)2)(5)

Here, Sεij(i,j=x or y) denotes the standard deviation of the strain components. In this study, the strain distribution on the specimen surface prior to the start of testing was measured using DIC, and the standard deviations computed from these measurements were considered observation errors. Sεxx=4.56×10−4,Sεyy=4.25×10−4, and Sεxy=2.97×10−4 are used.

The initial state of the specimen and the initial estimates for the parameters are expressed as follows:

x0b=(0,…,0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1,0,8.0)T(6)

Eq. (6) indicates that the initial strain in the specimen is zero and that the initial estimates for the parameters were set to αi(i=1−8)=1.0 and M=8.0, assuming isotropic plastic deformation. During the minimization calculation of the cost function using Bayesian optimization, one must prescribe admissible ranges for the parameters to be estimated. In this study, these ranges were set to 0.7≤αi(i=1,2,…,8)≤1.3 and 6.0≤M≤12.0.

[picture]

Fig. 3 Schematic illustration of Bayesian data assimilation (DA) based on DMC-BO.

Fig. 3. Schematic illustration of Bayesian data assimilation (DA) based on DMC-BO.

Results and Discussion

The original version of this paper is available on https://www.scientific.net/KEM.1049.29.pdf

Full-field Strain Measurement. Fig. 4(a) shows the distribution of the logarithmic strain measured using DIC. The arms of the specimens exhibited a uniaxial tensile state, whereas the central region showed a non-uniform strain distribution. The shear-strain components were concentrated near the corners of the specimens. Fig. 4(b) presents a scatter plot of the strain components in the principal strain space to evaluate the deformation field developed on the surface of the cruciform specimen. As shown in Fig. 4(b), many points corresponded to a uniaxial tensile state. By contrast, the strain values
associated with the equibiaxial tension were relatively low, and only negligible levels of simple pure shear strain were observed.
Parameter Identification and Verification. Among the 100 iterative calculations, the anisotropy parameter αi and the exponent M of the Yld2000-2d yield function that resulted in the minimum value of the cost function were regarded as the optimal estimates. This computation was repeated five times as independent trials. This computation was repeated five times as independent trials using different random seeds for generating the prior samples in the Bayesian optimization procedure. Because Bayesian optimization relies on prior sampling, the results may depend on the initial random realization; therefore, multiple trials were conducted to assess the robustness of the parameter identification. Table 1 presents the optimal estimates obtained from the five trials. These optimal estimates were compared with previously identified reference values based on conventional mechanical testing [7]. The reference parameters reported in [7] were identified based on stress-strain data obtained from uniaxial tensile tests and hydraulic bulge tests. Although some of the optimal estimates exceeded 1.0, the corresponding reference values did not. By contrast, the estimated exponent had a value similar to the reference value.

Fig. 5 shows the yield surfaces computed using the Yld2000-2d yield function with both the optimal estimate and reference value. For the parameters obtained via DA, the yield surfaces were computed for each of the five independent trials, and the standard deviation of these surfaces is presented. The yield surface constructed using the reference value exhibited a more outwardly expanded shape and pronounced protrusion along the equibiaxial loading direction.

Fig. 6 presents the strain distributions obtained from the biaxial tensile test simulations using both the optimal estimate and reference value. Clearly, using the optimal estimate enables the quantitative reproduction of the deformation field observed in the experiment. This shows that the proposed DA approach can successfully estimate the parameters of the Yld2000-2d yield function with the same accuracy as those identified based on multiple material testing via only a single biaxial tensile test.

[picture]

Fig. 4 (a) Distribution of logarithmic strains when true stress along x - and y-directions was σxx=σyy= 242.2 MPa . Distribution of principal strain components is shown in (b).

Fig. 4. (a) Distribution of logarithmic strains when true stress along x - and y -directions was σ x x = σ y y = 242.2 MPa . Distribution of principal strain components is shown in (b).

Table 1 Optimally estimated anisotropic parameters and exponent of Yld2000-2d yield function obtained using proposed DA and reported in [7].

| α₁ | α₂ | α₃ | α₄ | α₅ | α₆ | α₇ | α₈ | M
Data assimilation | 1.003 | 1.072 | 0.992 | 1.051 | 1.024 | 0.978 | 0.819 | 1.149 | 11.231
Reference | 0.963 | 0.976 | 0.863 | 0.996 | 0.996 | 0.887 | 0.953 | 1.158 | 11.564

[picture]

Fig. 5 Yield surface calculated using parameters estimated optimally using DA and reference values shown in Table 1.

Fig. 5. Yield surface calculated using parameters estimated optimally using DA and reference values shown in Table 1.

Conclusion

The original version of this paper is available on https://www.scientific.net/KEM.1049.29.pdf

In this study, a Bayesian DA method is proposed to estimate the parameters of a material model by assimilating the deformation field obtained via a biaxial tensile test using a cruciform specimen into a biaxial tensile test simulation. Using the proposed method, the anisotropy parameters and the exponent of the Yld2000-2d yield function were estimated for a A5052P-H32 aluminum alloy sheet. The results indicated that the parameters can be estimated with an accuracy comparable to that of the conventional approach, which requires multiple material tests, while requiring only a single biaxial tensile test. These findings contribute to the development of efficient material modeling and the design of sheet-metal forming processes.

[picture]

Fig. 6 Distribution of logarithmic strains when σxx=σyy=242.2MPa (a) measured using DIC and (b) calculated using reference values and (c) estimated values shown in Table 1.

Fig. 6. Distribution of logarithmic strains when σ x x = σ y y = 242.2 M P a (a) measured using DIC and (b) calculated using reference values and (c) estimated values shown in Table 1.

References

The original version of this paper is available on https://www.scientific.net/KEM.1049.29.pdf

D. Yanaga, T. Kuwabara, N. Uema, M. Asano, Material modeling of 6000 series aluminum alloy sheets with different density cube textures and effect on the accuracy of finite element simulation, Int. J. Solid Struct. 49 (2012) 3488-3495.

K. Sasaki, A. Yamanaka, Bayesian data assimilation for phase-field fracture simulation and full-field strain measurement using digital image correlation, Theor. Appl. Frac. Mech. 141 (2026) 105347.

Y. Zhang, A. Van Bael, A. Andrade-Campos, S. Coppieters, Parameter identifiability analysis: Mitigating the non-uniqueness issue in the inverse identification of an anisotropic yield function. Int. J. Solids Struct. 243 (2022) 111543.

A. Lattanzi, F. Barlat, F. Pierron, A. Marek, M. Rossi, Inverse identification strategies for the characterization of transformation-based anisotropic plasticity models with the nonlinear VFM. Int. J. Mech. Sci. 173 (2020) 105422.

F. Barlat, J. C. Brem, J. W. Yoon, K. Chung, R. E. Dick, D. J. Lege, F. Pourboghrat, S. H. Choi, E. Chu, Plane stress yield function for aluminum alloy sheets-Part 1: Theory. Int. J. Plasticity. 19 (2003) 1297-1319.

H. Takizawa, T. Kuwabara, K. Oide, J. Yoshida, Development of the subroutine library 'UMMDp' for anisotropic yield functions commonly applicable to commercial FEM codes. J. Phys. Conf. Ser. 734 (2016) 032028.

Y. Saito, H. Takizawa, Modeling of yield surfaces for A5052 aluminum alloy sheets with different tempers by simplified identification method and its experimental validation. Mater. Trans. 64 (2023) 605-1613.

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/lionel-fourment-on-optimization-ai-and-inverse-analysis-in-forming/978-3-0364-1989-3

Key Engineering Materials, ISSN: 1662-9795, Vol. 1049, pp 37-42

doi: 10.4028/p-TF8dpD

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

Submitted:
2025-12-23

Revised:
2026-02-17

Accepted:
2026-03-02

Online:
2026-04-15

Innovation of the "Cut-Clamp-Play" Concept for Robust and Efficient Sheet Metal Material Characterization

The original version of this paper is available on https://www.scientific.net/KEM.1049.37.pdf

Andraž Maček 1,a, Bojan Starman 2, b, Miroslav Halilovič 2,c, Fabrice Pierron 3, d, Pascal Lava 3,e and Sam Coppieters 1,f*

1 KU Leuven, Gent, Belgium

2 Faculty of Mechanical Engineering, University of Ljubljana, Ljubljana, Slovenia

3 MatchID, Gent, Belgium

a  andraz.macek@kuleuven.be, b  bojan.starman@fs.uni-lj.si, c  miroslav.halilovic@fs.uni-lj.si, d fabrice.pierron@matchid.eu, e pascal.lava@matchid.eu, f sam.coppieters@kuleuven.be (*corresponding author)

Keywords: MT2.0, Virtual Fields Method, DIC, Material Characterization, Plastic Anisotropy.

Abstract

Material Testing 2.0 (MT2.0) couples full-field deformation measurements (Digital Image Correlation, DIC) with inverse identification methods (Virtual Fields Method, VFM) to extract constitutive parameters from a small number of heterogeneous experiments. This paper presents the Cut-Clamp-Play concept: an integrated industrial MT2.0 solution that unifies specimen design, automated testing hardware, and a computationally efficient VFM identification chain to deliver fast, user-friendly sheet-metal characterization. A perforated cruciform specimen is optimized for parameter identifiability of the Yld2000-2d anisotropic yield function and used in a single biaxial test. A working prototype has been built at KU Leuven and used to collect representative DIC data; the measured displacement/strain response is double-symmetric, confirming correct mechanical operation. Projected and early prototype results indicate that the Cut-Clamp-Play approach can reduce operator actions by roughly 70% and produce identification results within one hour for typical sheet-metal cases, while further work is required to make the fully automated "Play" stage robust for industrial deployment.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1049.37.pdf

Accurate constitutive models are essential for predictive engineering simulation in automotive, aerospace, and consumer industries. Traditional material characterization (MT1.0) relies on multiple quasi-homogeneous tests and manual post-processing, which is time-consuming and resource intensive. MT2.0 replaces large test campaigns with a small number of carefully designed heterogeneous experiments - potentially two instead of seventeen [1] - combined with full-field measurement and inverse identification, enabling substantial reductions in experimental effort and improved parameter identifiability [2].

Two practical barriers limit industrial adoption of MT2.0: (i) computational cost of nonlinear inverse identification for anisotropic plasticity, and (ii) high user dependency across specimen preparation, DIC acquisition, and inverse processing. The Cut-Clamp-Play concept addresses these barriers by combining an identifiability-optimized perforated cruciform specimen, a computationally efficient stress-reconstruction strategy embedded in VFM, and an integrated testbed that minimizes operator actions. This paper summarizes the concept, specimen design and identifiability rationale, prototype implementation, representative DIC measurements, and the remaining steps required to industrialize the "Play" automation stage.

Methodology

The original version of this paper is available on https://www.scientific.net/KEM.1049.37.pdf

System concept and scope. The Cut-Clamp-Play concept targets sheet-metal biaxial characterization with minimal operator expertise. The operator performs three actions: Cut an optimized perforated cruciform from sheet stock, Clamp it into the fixture, and Play the automated test sequence. The system executes the biaxial loading, acquires stereo DIC images, computes kinematic fields, and runs a VFM-based identification chain designed for speed and robustness. A working prototype implementing the mechanical fixture and optical acquisition has been built and used for initial validation at KU Leuven.

Identification framework. Identification is based on the nonlinear Virtual Fields Method (VFM) applied to measured kinematic fields. For a chosen set of virtual fields u*, the equilibrium statement reads

[−∫Sσ·ε* dS−∫LfT―·u* dl]2=0(1)

where σ is the Cauchy stress and ε* the virtual strain associated with u*. The stresses σ are reconstructed using the measured strain field ε and the constitutive material model (Yld2000-2d [3]) parameterized by θ parameters. By using optimization methods, we identify a set of θ parameters which fulfil equilibrium (Equation 1) as close as possible. A detailed theoretical development of the sensitivity-based VFM employed in this work is provided by Marek [4].

Following Equation 1, the stresses need to be reconstructed in every VFM iteration for every measured material point, for every time step and in the case of gradient-based optimization techniques for every constitutive parameter. Meaning that in a typical identification procedure we need to reconstruct the stresses approximately 100 -milon-times. To reduce runtime, the explicit stress-reconstruction operator NICE [5] is employed to compute stresses and sensitivities explicitly at the measurement points (both spatially and temporally), thereby avoiding nested implicit stress solves within the identification loop. The NICE operator has been seamlessly integrated into the MatchID software package and is available in version 2026.1, implemented in both the Results module (for stress calculation) and the VFM module.

Specimen design and identifiability. A perforated cruciform specimen was chosen because it can reproduce, within a single heterogeneous region of interest, the variety of stress states that are normally obtained from multiple homogeneous tests. The geometry conceptually originated from engineering judgement: combine three standard uniaxial tests and one biaxial test (four homogeneous tests) into a single heterogeneous biaxial tensile experiment by arranging local features that produce distinct local stress-strain responses. In practice, the perforations and fillets create zones with different stress types and strain gradients so that, under a single global biaxial loading path, the region of interest simultaneously samples stress states analogous to the separate homogeneous tests (e.g., uniaxial along rolling direction (RD), uniaxial along transverse direction (TD), uniaxial along 45∘ direction to RD, and biaxial tension, as shown in Figure 1). This engineered heterogeneity increases the information content of one experiment and reduces the number of separate tests required to identify anisotropic plasticity parameters.

The specimen geometry was further constrained by a simple force-equilibrium requirement under the assumption of a homogeneous, isotropic material: all ligaments that carry load in the rolling and transverse directions are designed to be subjected to the same nominal stress, therefore they must have the same cross-sectional area. Ligaments oriented at 45∘ to the RD carry the same axial force component only if their cross-section is scaled by 1/2 relative to the rolling/transverse ligaments. This scaling follows directly from resolving forces on a 45∘ ligament and enforcing equal nominal stress levels across ligament families.

After imposing these cross-section constraints the remaining geometric degrees of freedom reduce to ligament lengths and internal/external radii (fillets). These parameters were selected to maximize the approach to an equibiaxial stress state in the region of interest while preserving a steep-gradientfree measurement area for DIC and delaying localization. In practice the ligament lengths control the strain gradient and the spatial extent in the region of interest, whereas the fillet radii control stress concentration and the smoothness of the strain field used for correlation.

Design selection proceeded by evaluating the resulting stress maps under the intended global biaxial load and adjusting ligament lengths and fillets until the biaxial regions exhibited near-equibiaxial stress conditions. The final geometry balances three practical requirements: (1) broad, smoothly varying strain fields with small strain gradients for robust DIC correlation; (2) delayed necking by reducing stress concentrations; and (3) sufficient heterogeneity to encode the equivalent information of the four homogeneous tests into one experiment.

[picture]

Fig. 1. Preliminary specimen design of perforated biaxial test, following the engineer's practical insight of replacing 3 standard uniaxial and one equibiaxial (e.g., bulge) test to one heterogeneous biaxial tensile test.

Fig. 1. Preliminary specimen design of perforated biaxial test, following the engineer's practical insight of replacing 3 standard uniaxial and one equibiaxial (e.g., bulge) test to one heterogeneous biaxial tensile test.

Prototype and Experimental Procedure

The original version of this paper is available on https://www.scientific.net/KEM.1049.37.pdf

Prototype implementation. A prototype cruciform biaxial testing machine and fixture were fabricated and assembled at KU Leuven (see Figure 2). The prototype includes gripping clamps, alignment guides, and stereo camera mounts positioned to capture the gauge region. Mechanical stops and repeatable fixturing ensure consistent specimen placement between tests. The prototype was used to collect representative DIC data for validation and demonstration purposes.

[picture]

Fig. 2. Prototype biaxial testing machine with integrated cameras for stereo DIC: (a) complete testbed with camera mounts and electrical cabinet and (b) clamped specimen with camera setup. The machine represents a research platform for performing biaxial experiments on sheet-metal. The control logic of the machine enables early adoption into the "Cut-Clamp-Play" concept using an additional computer seen in the bottom-left corner of the image (a).

Fig. 2. Prototype biaxial testing machine with integrated cameras for stereo DIC: (a) complete testbed with camera mounts and electrical cabinet and (b) clamped specimen with camera setup. The machine represents a research platform for performing biaxial experiments on sheet-metal. The control logic of the machine enables early adoption into the "Cut-Clamp-Play" concept using an additional computer seen in the bottom-left corner of the image (a).

Measurement and test protocol. Representative tests follow a controlled biaxial loading path selected to excite anisotropic responses in the gauge region. Stereo DIC captures surface displacements at a sampling rate sufficient to resolve the quasi-static loading sequence. Typical processing yields displacement fields u and derived strain fields ε across the region of interest. Identifiability checks are performed prior to full identification to confirm that the measured fields contain sufficient information for the targeted parameter set of the Yld2000-2d constitutive model.

Initial Results

The original version of this paper is available on https://www.scientific.net/KEM.1049.37.pdf

Measured DIC response. The representative DIC fields from the prototype exhibit clear double symmetry in both displacement and strain maps across the whole specimen region (see Figure 3). This symmetry confirms that the mechanical fixture and loading paths produce the intended boundary conditions and that the optical setup provides consistent measurements suitable for inverse identification.

[picture]

Fig. 3. Measured DIC strain response for a representative test: (a) strain εxx in RD, (b) strain εxx in TD, (c) shear strain εxy and (d) Von Mises Equivalent Strain εeq. The response is double-symmetric about both principal axes, confirming correct mechanical alignment and symmetric loading in the prototype.

Fig. 3. Measured DIC strain response for a representative test: (a) strain ε x x in RD, (b) strain ε x x in TD, (c) shear strain ε x y and (d) Von Mises Equivalent Strain ε e q . The response is double-symmetric about both principal axes, confirming correct mechanical alignment and symmetric loading in the prototype.

Identification performance. Early numerical benchmarks and synthetic tests using the NICE-extended stress reconstruction indicate approximately 30% reduction in CPU time for the VFM identification chain compared to a baseline implicit integration approach. On the prototype hardware and typical desktop, end-to-end identification (from processed DIC fields to parameter estimates with confidence intervals) is achievable within the target of one hour for standard sheet-metal cases, subject to DIC processing time and chosen temporal discretization.

Operator action reduction. Laboratory trials with technicians unfamiliar with MT2.0 workflows show a substantial reduction in required operator actions when using the Cut-Clamp-Play protocol and prototype fixture. Prognosed reductions in manual steps will be approximately 70% relative to a conventional MT2.0 chain that requires specimen-specific fixturing, manual camera alignment, and multi-software data transfers.

Discussion

The original version of this paper is available on https://www.scientific.net/KEM.1049.37.pdf

Key advantages:

Efficiency: The combination of an identifiability-optimized specimen and a fast stress-reconstruction strategy enables dense temporal discretization and rapid parameter estimation without prohibitive runtimes.

Usability: The Cut-Clamp-Play workflow minimizes operator expertise required for routine sheet-metal characterization, lowering the barrier for industrial adoption.

Validation readiness: Double-symmetric DIC responses from the KU Leuven prototype confirm mechanical and optical integrity of the testbed, supporting reliable inverse identification.

Limitations and remaining work:

"Play" robustness: While the prototype automates many steps, further engineering is required to make the fully automated "Play" stage robust under industrial variability (e.g., variable speckle quality, lighting conditions, and non-ideal user actions). Additional work is needed on automated speckle quality assessment, adaptive correlation settings, and assessing robust experiment parameters.

Model-identifiability trade-offs: Increasing constitutive complexity (more parameters) reduces identifiability; practical deployments must balance model expressiveness and the number of reliably recoverable parameters. Current targets aim for reliable identification of at least 7 of 9 Yld2000-2d parameters for typical sheet metals.

DIC sensitivity: Although the workflow reduces operator influence, DIC image quality remains a limiting factor. Robust pre-test checks and automated image-quality metrics are necessary to ensure consistent identification accuracy.

Expanding and validating the whole inverse identification on other metallic sheets.

Conclusions

The original version of this paper is available on https://www.scientific.net/KEM.1049.37.pdf

The Cut-Clamp-Play MT2.0 concept demonstrates a practical pathway to industrialize full-field inverse material characterization for sheet metals. By combining an identifiability-optimized perforated cruciform specimen, a computationally efficient VFM identification chain (NICE-extended), and a prototype cruciform testbed built at KU Leuven, the approach reduces operator actions by roughly 70% and achieves representative end-to-end identification within one hour for typical cases. Measured DIC fields from the prototype show double symmetry, confirming correct mechanical operation. Remaining development focuses on making the automated "Play" stage robust enough to handle industrial variability and on broadening validation across materials and process conditions. Once these steps are completed, the Cut-Clamp-Play system can substantially lower the barrier for MT2.0 adoption in industrial simulation workflows.

Acknowledgements

The original version of this paper is available on https://www.scientific.net/KEM.1049.37.pdf

The authors acknowledge the KU Leuven and University of Ljubljana team, and industrial partner MatchID who supported early testing. The work was financially supported by VLAIO under Project No. HBC.2024.0807.

References

The original version of this paper is available on https://www.scientific.net/KEM.1049.37.pdf

Zhang Y., Yamanaka A., Cooreman S., Kuwabara T., Coppieters S., Inverse identification of plastic anisotropy through multiple non-conventional mechanical experiments. Int J Solids Struct 2023;285:112534. https://doi.org/10.1016/j.ijsolstr.2023.112534.

Pierron F., Grédiac M., Towards Material Testing 2.0. A review of test design for identification of constitutive parameters from full-field measurements. Strain 57, 2021, . 12370

Barlat F. et al., Plane stress yield function for aluminum alloy sheets-part 1: theory. International Journal of Plasticity 19, 1297-1319 (2003).

Marek A., Davis F.M., Pierron F., Sensitivity-based virtual fields for the non-linear virtual fields method. Comput Mech 60, 409-431 (2017).

Halilovič, M., Starman, B., Coppieters, S., 2024. Computationally efficient stress reconstruction from full-field strain measurements. Comput. Mech., 74(4), .

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/lionel-fourment-on-optimization-ai-and-inverse-analysis-in-forming/978-3-0364-1989-3

Key Engineering Materials, ISSN: 1662-9795, Vol. 1049, pp 43-54

doi: 10.4028/p-YFY9Oq

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

Submitted:
2026-01-16

Revised:
2026-02-24

Accepted:
2026-03-02

Online:
2026-04-15

Method of an Automated Tool Design of Punch-Bending Processes Using the Asset Administration Shell

The original version of this paper is available on https://www.scientific.net/KEM.1049.43.pdf

Henning Peters 1,a*, Andreas Mazur 2,b, Brijesh Chaudhary 1,c, Lukas Kersting 1,d, Ansgar Trächtler 1,3,e, Barbara Hammer 4,f

1 Fraunhofer Institute for Mechatronic Systems Design (IEM), Paderborn, Germany

2 Bielefeld University, Center for Cognitive Interaction Technology (CITEC), Germany

3 Heinz Nixdorf Institute (HNI), Paderborn University, Germany

a  henning.peters@iem.fraunhofer.de, b  amazur@techfak.uni-bielefeld.de, c  brijesh.chaudhary@iem-extern.fraunhofer.de, dlukas.kersting@iem.fraunhofer.de,

e ansgar.traechtler@hni.uni-paderborn.de, f bhammer@techfak.uni-bielefeld.de

*Corresponding author

Keywords: punch-bending, digital twin, asset administration shell, process optimization.

Abstract

A common challenge of punch-bending is the control and compensation of process deviations which are induced, e.g., by different material batches. The resulting deviations are currently countered iteratively using expert knowledge in the design phase as well as during the production process. However, this is not always successful due to the complex interactions between semi-finished product properties and forming tools. To automate the optimization of tool geometries and process adjustments, an accurate prediction model of correlations between process/tool parameters and product properties is necessary. In that regard, an application programming interface (API) aligned with the Asset Administration Shell (AAS) specification is developed utilizing a hybrid data-driven approach. It enables the transformation of various data formats e.g., from sensor and simulation data into a machine-readable structure. The API is linked to a digital twin and a database, to automatically gather requested information and provide the new structured data to a machine learning (ML) model. To validate the developed method, hybrid punch-bending data is automatically transferred to an ML model which then returns tool geometry suggestions for a defined product geometry.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1049.43.pdf

Punch-bending is a forming process in which semi-finished products like metal strips or wires are formed into products like brackets, mounts, electronic contacts or spring elements. The respective punch-bending machines consist of several bending operations to achieve complex geometries [ 1,2 ].

A common challenge of punch-bending is the control and compensation of process deviations. Those are induced in the wire and therefore in the process in form of curvature and residual stresses during wire production and wire coiling [1, 3]. In industrial applications, those deviations are currently countered iteratively using expert knowledge and setup guidelines in the design phase as well as during the production process. However, this is not always successful due to the complex interactions between semi-finished product properties and forming tools [1,4,5].

To enable the optimization of tool geometries and process adjustments, an accurate prediction model of correlations between process/tool parameters and product properties is necessary. Therefore, a hybrid data-driven modeling approach including classical models (analytical, empirical, numerical) and real data from the manufacturing process is chosen. Classical models enable investigations using a wide range of tool configurations without having to manufacture them, as well as capturing effects that are difficult or impossible to measure. However, they are based on assumptions and simplifications, which means that deviations from reality may occur [3,6]. By including real process data within the data-driven approach, deviations and disturbances which are not considered by classical modeling approaches can extend the data set to receive a more holistic understanding of the
process and therefore a more robust model [7]. However, this hybrid approach requires a large amount of data in different data formats. In this regard, an automated acquisition, preprocessing and transfer of hybrid data is required, which can be realized by a digital twin (DT).

The first step towards automated data acquisition was presented by the authors in [8], by defining a possible structure of the DT as well as by connecting the real process with the DT. However, this current approach only includes real process data. To add further data sources regarding the hybrid approach, an automated integration of hybrid data into the DT has to be investigated.

Related Work

The original version of this paper is available on https://www.scientific.net/KEM.1049.43.pdf

The concept of the digital twin is often seen as an enabler for IIoT and Industry 4.0 applications. However, because there are several different definitions regarding the concept of the digital twin, only a very specific and limited utilization is possible, which makes it significantly more difficult to transfer to other applications. To counteract this, multiple standardized specifications were developed to enable an interoperability between different assets. Via semantic modeling the specifications describe the properties and the behavior of an asset as well as the relationship to other assets [9, 10]. Depending on the specification they also provide defined interfaces to interact with the digital representation [11].

The Asset Administration Shell (AAS) is one of the developed specifications which is subsequently refined by the Industrial Digital Twin Association (IDTA) [10]. In [12] the combination of multimodal, i.e. hybrid, data and its integration into automation structures is defined as research challenge for data-driven modeling. Due to the defined technology-neutral standard of the AAS, interoperability with regard to the exchange of information and data from different sources is made possible [13]. Therefore, the AAS can be used to automatically acquire hybrid data from the punchbending process and thus use it for the hybrid data-driven modeling approach.

Several cases have already been examined with regard to the utilization of the AAS in the context of Industry 4.0 applications. One focus of research is manufacturer- and supplier-independent data exchange. Here, the technology-neutral nature of AAS is utilized to enable shared production and development between multiple stakeholders [14, 15]. In this regard, the concept of the digital product passport represents an implementation of this data exchange. [16]. The aspect of manufacturer- and supplier-independent data exchange between different machines in production is another research topic, which enables the interoperability between automated industrial systems [17, 18]. Other prominent applications include skill-based manufacturing [19], process planning [20], process monitoring [21] and predictive maintenance [22].

However, to the best of the authors knowledge, the application regarding a hybrid data-driven tool design in forming technology has not yet been researched.

Data Sources

The original version of this paper is available on https://www.scientific.net/KEM.1049.43.pdf

In order to pursue the hybrid data-driven approach to optimizing tool design for the punch-bending process, a large amount of data in various formats is required. The hybrid data used in this paper can be divided into three groups, which are shown in Table 1: real process data, simulation data and historical data. The table also shows the quantities contained in each data source as well as their data formats. The table also lists the tool configurations that are considered as input data and are available in the same data format for each previous data source.

Table 1. Data sources, quantities and data formats of the hyrid data-driven modeling approach of punch-bending processes.

Data Source | Quantity | Data Format
Process data (sensor data) | Forces, bending angle, positions, curvature | Time series data, single sensor data, image data
Simulation | Forces, bending angle, residual stresses | Time series data, single sensor data, nodal data
Historical data (analytical model, reference studies) | Forces, bending angle | Tabular data
Tool configuration | Tool geometries, process settings | Single parameter values

Process data is generated within the real punch-bending process which is displayed in Figure 1. It consists of a material coil, a mechatronic straightener and a punch-bending machine. The mechatronic straightener includes seven straightening rolls. The three upper rolls are each connected to a stepper motor and can therefore be moved via the machine control. Within this punch-bending process, multiple sensors are installed to capture the effects appearing during the straightening and bending operations. The straightening machine is equipped with laser triangulation sensors to determine the positions of the upper straightening rolls. Load cells are also attached to the upper rolls to measure the resulting straightening forces. Both the position and force signals are sent to the machine control as time series data. This is implemented using Beckhoff's TwinCAT automation software. Using an optical measuring system, the wire curvature can be determined offline. First, several coordinate points on the wire geometry are determined and then converted into a corresponding curvature. This is then available as a single sensor data point. The bending angle of the punch-bending product is determined inline using an optical measuring system as well. The image recognition of the camera software identifies the relevant geometric features, outputs them as single sensor data and forwards them afterwards to the automation software.

[picture]

Fig. 1. Overview of the punch-bending process, which is used for process data generation.

Fig. 1. Overview of the punch-bending process, which is used for process data generation.

Simulation data is generated via the FE-Software Abaqus. Besides forces and the bending angle also residual stress or strain is captured. Here, the calculated values are assigned to the individual nodes of the mesh so that the determined effects are available as nodal data. For historical data, data from analytical models and data from experimental reference studies are summarized. In this way, previously gathered knowledge can also be incorporated into data-driven modelling. It is assumed that this data is available in tabular form.

The hybrid data-driven approach for optimizing the tool design for punch-bending processes thus has several different data formats, which, given the large amount of data required, necessitates automated acquisition, preprocessing and transfer of the hybrid data in a uniform machine-readable
format. This could be realized by a digital twin and a suitable automation structure. In [8], the authors already described a structure between the digital twin, its included database and the real punchbending process. Consequently, only real data was included. To enable the processing of hybrid data, an extension of the existing automation approach is necessary.

Modification of the Automation Structure for Hybrid Data

The original version of this paper is available on https://www.scientific.net/KEM.1049.43.pdf

The modification approach is shown in Figure 2. Regarding this hybrid data-driven approach, simulation data as well as historical data are to be integrated in the same machine-readable format to enable the processing by subsequent clients like preprocessing or the ML module.

[picture]

Fig. 2. Schematic representation of the hybrid automation structure, incorporating the API to enable the transformation of various data formats into a machine-readable structure.

Fig. 2. Schematic representation of the hybrid automation structure, incorporating the API to enable the transformation of various data formats into a machine-readable structure.

As described in previous work the AAS specification is integrated by using the software called AASX Package Explorer. It enables the development of the digital representation of the punchbending process and its assets. Furthermore, additional meta data of the assets can be stored in the respective submodels to obtain a holistic representation and comprehensive information storage. This information model is then uploaded to the AAS Server (I). There are several open-source AAS implementations. An overview and comparison is given in [23]. Based on the study conducted in [23], it is evident that Eclipse BaSyx matches the requirements for usage within the hybrid datadriven model best among the implementations examined. Therefore, it will be used as the implementation of the AAS for the digital twin in this paper which allows the interaction with the created digital representation. For the digital twin of the punch-bending process the communication was implemented via HTTP/REST in a Docker environment. Here, two versions were developed: A local/unsecured and a secured version. All components therefore can be either addressed by their local port or via a nginx reverse proxy to also implement secured communication via HTTPS.

An important subcomponent of the digital twin is the database (II). It has to store all acquired process data as well as initial process and tool parameters which are then transferred to the ML module. BaSyx also requires a database to store the AAS meta data. For this, the NoSQL data management system MongoDB is currently the only persistent data storage implementation in BaSyx (V2). Due to the properties of a NoSQL database, MongoDB enables the storage of multiple data formats. Also, time series data can be stored in so called time series collections, which offer an optimized storage for time series data compared to the standard MongoDB collections. Therefore, MongoDB was also selected for the storage of process data, process parameters and meta data of the punch-bending process. However, in order to combine the various data formats from different data sources into a standardized format for hybrid data-driven modelling, a database structure was
developed to assign the used tool configuration to the respective experiment or simulation. This is realized by an identification system which is shown in Figure 3. First, each tool or process property serving as input data receives its own Input ID (marked in yellow). Tool and process properties include, e.g., straightening roll infeed, the radius of the die or the angle of the bending punch, but also material properties. Sensor signals, simulation results or results from reference studies represent output data and are assigned a Output ID accordingly. The IDs with their corresponding assignments are then stored in a respective database collection. The Input Parameter ID combines a Input ID with a specific value (marked in turquoise). The combination of all tool parameters of one measurement is then summarized in a Input Configuration ID (marked in brown). If one or multiple tool parameters are changing, new Input Parameter IDs as well as a new Input Configuration ID, if not already existing, are created. With the initialization of a new measurement, a new Measurement ID is created and linked to a specific Input Configuration ID to ensure that recorded sensor data is linked to the tool and process properties (marked in pruple). Acquired data of the sensors, either real or simulated, are then assigned to the Measurement ID and their respective Output ID within the so-called Output Data Collection (marked in light green).

[picture]

Fig. 3. Schematic representation of the identification system which enables a standardized format for hybrid data-driven modeling by transforming parameter and values into Input, Input Parameter, Input Configuration, Output and Measurement IDs.

Fig. 3. Schematic representation of the identification system which enables a standardized format for hybrid data-driven modeling by transforming parameter and values into Input, Input Parameter, Input Configuration, Output and Measurement IDs.

As described in [8] the tool and process properties are listed in the digital twin, where they are extracted. Before an experiment is conducted on the real system, the values for each tool are entered by the operator, so that the IDs can be generated. This was conducted via the TwinCAT automation software of the machine. For the mere consideration of real sensor data this approach was sufficient. However, when considering the inclusion of hybrid data sources, which means the addition of simulation and historical data, this procedure is no longer reasonable. This is because the processing of simulated and historical data would otherwise depend on the machine control, even if it is not needed at that point in time. Furthermore, the generation of IDs via the machine control was subject to a certain degree of failure due to the cyclical processing of the automation software. Therefore, a centralized and modular approach is necessary which is displayed schematically with the application programming interface (API) (III) in Figure 2. The API is created in Python, ensuring the transferability of the methodology. The primary function of the API is to adopt the mechanisms which were implemented in the automation software by generating and assigning IDs for real process data as well as for simulation and historical data. The API centralizes the transformation of hybrid data formats into a standardized, machine-readable format, thereby enabling hybrid data-driven modeling
of punch-bending processes. Therefore, a connection between API and AAS server is set up. Furthermore, the API also acts as an interface to the database. For example, requests regarding specific IDs can be addressed.

Process data (IV) is recorded and transferred via the TwinCAT automation software. Within the automation software, a connection to the database is set up using the Database Server function. For each sensor, so-called AutoLog groups are created which enable an automated transfer of the process data to the database. The IDs provided by the API are also included in the data logging of the respective sensor signals so that the data can be assigned to the respective measurement. Simulation data ( 𝐕 ) of the punch-bending process is created with the FE-software Abaqus. As shown schematically in Figure 3, the simulation data for each simulation is stored in an inp and an odb-file. The inp-file is a text document containing the input data for the simulation. The odb-file, on the other hand, is a proprietary file containing the simulation results. Here, the data has to be transformed into tabular data, which is then inserted into the database structure and the AAS server via the API. Historical data (VI) is already provided in tabular form. Accordingly, this data is transferred to the database structure and the AAS server using the same systematic approach via a Python script and the API.

Subsequently, the acquired hybrid data is transferred to the ML module (VII) which contains an ML ensemble model as well as an algorithm to compute counterfactual explanations. The ML algorithms allow the generation of process- and tool design suggestions, as already presented by the authors in [24]. Suggested process parameter adjustments are then returned to the digital twin to implement direct adjustments like straightening roller infeed to the system, or to forward tool geometries to the operator. The following section will go into more detail about the functionality of the automated data-driven modeling of punch-bending processes, from initializing a measurement to receiving a setup suggestion.

Method of Automated Data Processing for Hybrid Data-Driven Models

The original version of this paper is available on https://www.scientific.net/KEM.1049.43.pdf

To achieve an automated data processing of hybrid data the approach is divided into a procedure for process data and a procedure for simulation and historical data. This is because the real process data is transferred to the database and the digital twin via the automation software, while simulation and historical data is transferred via data tables.

Procedure for process data.

The original version of this paper is available on https://www.scientific.net/KEM.1049.43.pdf

Within this procedure, seen in Figure 4, the section Recording sensor data is based on the methodology previously implemented in the automation software, which was described by the authors in [8]. Instead of the automation software, the API is utilized to generate the respective IDs. First, measurements are initiated via the digital twin by specifying the input parameters which are to be examined. Using the synchronization function between the digital twin and the system the parameters are transferred to the system which requests the Input, Input Parameter, Input Configuration, Output and Measurement ID using the respective HTTP requests defined by the API. If an ID is non-existent, a new ID is generated and inserted into the designated database collection. Otherwise, the existing ID is returned. Additionally, the initialized measurement as well as its meta data like the data source is listed as a segment in the digital twin to be accessible by the ML module.

After one or multiple measurements are completed, in section Transfer data to ML module, the ML module checks for new measurements in the digital twin. For that the ML module requests the API to initiate the search for new Measurement IDs which are forwarded to the digital twin. Using an additional database collection, it is checked whether a measurement was already sent to the ML module or not. If this is not the case, the new Measurement IDs are sent back to the API. Afterwards, the data encrypted using IDs is translated into its plain names and actual values before being transmitted as a list of dictionaries to the ML module.

Depending on whether the ML module has been initialized, two possible scenarios must be considered: scenario (a) describes the case in which the ML module has never been used before. Here, all gathered data is used to train the ensemble model. Scenario (b) represents the case in which the

ML module has already been initialized. Then, only new data that so far has not been used to train the model is used for finetuning. Using the trained ensemble model, the algorithm for computing counterfactual explanations can search for a possible process configuration suggestion, given a desired target geometry. Afterwards, the suggestion is sent back to the digital twin.

[picture]

Fig. 4. Sequence diagram of the developed method of automated data processing for hybrid datadriven models using process data.

Fig. 4. Sequence diagram of the developed method of automated data processing for hybrid datadriven models using process data.

Procedure for historical and simulation data.

The original version of this paper is available on https://www.scientific.net/KEM.1049.43.pdf

For historical and simulation data the first section differs compared to the procedure for process data, which is displayed in Figure 5. Where process data is transferred directly to the database and sorted into the database structure, historical and simulation data must first be imported and then converted into the database structure. Due to the different data formats, the data has to be provided in a uniform tabular format like a csv-file, which is the case for the historical data used. The simulation data of the punch-bending process, on the other hand, is acquired with the FEM-Software Abaqus. The simulation data for each simulation, as mentioned before, is stored in an inp and an odb-file. The inpfile contains input data like material properties, tool geometries or process settings in a text document format. The proprietary odb-file, contains the simulation output like the product angle, forming forces or residual stresses. Both files are first converted by a Python script, using the Python library abqpy [25] for the odb-file. Here, the data is transformed into tabular data, which is then inserted into the database structure and the AAS server via the API. The data table, either historical or simulation data, then provides the necessary data to the API. Each line represents a measurement taken (cf. Figure 3). Therefore, the input has to be separated from the output data so that the ML module is able to learn the resulting correlations. For this purpose, keywords were defined for input and output data, which are compared with the keys used for the historical or simulation data. After that, input and output parameters are assigned to the Input Collection and respectively the Output Collection with their specific IDs, which are generated by the API. Accordingly, the Input Parameter and Input Configuration IDs are generated. For each line, one Measurement ID is assigned. Similar to process data, the Measurement IDs and the respective meta data are posted into the digital twin, so that the ML module can access the measurements. Subsequently, the historical or simulation data is stored in the Sensor Data Collection.

The sections Transfer data to ML module and Transfer tool suggestions are almost identical compared to the procedure for process data (cf. Figure 4). The Measurement IDs are requested by the

ML module. They are then translated by the API into the actual keys and values, so that the ML module, after learning the correlations, can provide tool suggestions to be adopted by the operator.

[picture]

Fig. 5. Sequence diagram of the developed method of automated data processing for hybrid datadriven models using simulation and historical data.

Fig. 5. Sequence diagram of the developed method of automated data processing for hybrid datadriven models using simulation and historical data.

Validation of the Method

The original version of this paper is available on https://www.scientific.net/KEM.1049.43.pdf

To validate the method described, a case study was conducted using the developed ML model and data set presented by the authors in [24]. The data set falls into the category of historical data and displays a single-stage bending operation of a punch-bending process. The input parameters used are the Young's modulus, the wire thickness as well as the angle and radius of the bending tools. As outcome serves the resulting angle of the punch-bending product. Because it is a historical data set, it already comes in a tabular format (csv) (see data sources). Therefore, it does not need to be transformed as in the case of simulation data.

Validation procedure.

The original version of this paper is available on https://www.scientific.net/KEM.1049.43.pdf

Following the developed procedure (cf. Figure 5), the data is inserted via a Python script. Depending on the file name and the parameters used, the data set is sorted into either the Simulation or Historical data category based on predefined keywords. These keywords include data format-specific terms that can appear both in the file name and among the listed input or output parameters. Because the data set, e.g., lacks keywords regarding node coordinates as output parameters, the data set considered in this paper is correctly assigned to the category historical. Compared to the retrospective assignment of the data source for historical and simulation data, process data is assigned to the Real Process category at the moment a measurement is initialized. This is achieved through the continuous connection of the API to the real process and its data acquisition. This meta data can later be used by the ML module to differentiate between the data of the hybrid data sources. In a similar way the parameters are compared with input-specific or output-specific keywords in order to categorize the parameters as input or output data. Accordingly, the input parameters Young's modulus, wire thickness, tool angle and tool radius are assigned to the Input Collection and the output parameter product angle to the Output Collection. The respective IDs as well as the Input Parameter, Input Configuration, Measurement and Output IDs are then generated by the API.

To represent the imported measurement within the digital twin, the IDTA time series data submodel template [26] is used. For this work the focus lies on the own added submodel element Source which displays the data source of the data set, which is in this case historical (cf. Figure 6, step I). Each measurement is therefore posted into the digital twin as submodel element collection
containing the respective submodel elements. Afterwards, the data is inserted into the database structure.

After this procedure, the data sets of every data source are stored in the same data format within the developed database structure. The subsequent steps are therefore identical for all data sources and can thus be applied to the validation of the methodology for automated processing of hybrid data. Within the scope of this paper, the focus will therefore remain on the historical data set from [24]. Accordingly, the measurement list within the digital twin is checked for recently added Measurement IDs. Using an additional database collection, the measurement list of the digital twin is compared with already transferred Measurement IDs to prevent duplicating data sets. New Measurement IDs are then sent to the API, which requests the respective incorporated IDs to translate those to actual names and values, before they are transferred to the ML module. The data is used to train a surrogate model enabling the calculation of counterfactual explanations (cf. Figure 6, step II). Using the trained model, a target angle is inserted (Figure 6, step III) to generate process configuration suggestions based on the counterfactual explanations (cf. Figure 6, step IV). A target product angle of 120∘ leads, e.g., to a process configuration displayed in Figure 6 (step V) which is sent to BaSyx. This completes the developed procedure. Thus, a data set has been successfully integrated into the database structure and then automatically transferred to the ML module in order to eventually obtain a prediction for a suitable process configuration for the punch-bending process.

[picture]

Fig. 6. Validation process for automated data processing for hybrid data-driven modelling of punchbending processes: From automated integration via the ML module to the final parameter suggestion in the digital twin.

Fig. 6. Validation process for automated data processing for hybrid data-driven modelling of punchbending processes: From automated integration via the ML module to the final parameter suggestion in the digital twin.

Discussion

The original version of this paper is available on https://www.scientific.net/KEM.1049.43.pdf

The results successfully show the validity of the presented automation approach for punch bending. The import and transfer of data set used for validation, containing 5 parameters and 663 data points in the context of the experiment, takes around 34 seconds. As an example, a simulation data set of a single-staged bending operation provided with Abaqus, containing 26 parameters and 663 data points takes around 196 seconds. When repeating the step while retaining the previous data, only a small increase in time of 1-2 seconds is recorded for the historical data, while the simulation data already requires 260 seconds for the repetition. One limiting factor of this approach is the current reference of measurements in the digital twin. BaSyx stores such a submodel and all its contents in a single MongoDB document in BSON format, which has a maximum capacity of 16 MB . Accordingly, this approach reaches its limits with larger data sets.

Conclusion and Outlook

The original version of this paper is available on https://www.scientific.net/KEM.1049.43.pdf

The article presented a method of automated data processing for optimizing the tool design of punchbending processes by using a hybrid data-driven approach. Previous work initially introduced the digital twin based on the AAS specification as an automation concept. In order to complement the hybrid data-driven approach, the previously developed integration of process data into the automation structure was supplemented by simulation and historical data. Therefore, an API was developed which acts as an interface between hybrid data sources, the database and the AAS server. The API centralizes the transformation of hybrid data formats into a standardized, machine-readable format, thereby enabling a hybrid data-driven modeling of punch-bending processes to achieve an automated tool design. Furthermore, a direct connection between ML module to the digital twin has been established. It was demonstrated using the developed methodology, that due to the standardized format process parameters and product properties of the punch-bending process can be automatically transmitted to the ML module. The correlations learned in this way then resulted in a suggested process configuration for a specified target angle. Automatically returned to the digital twin, the suggested process configuration can then be adopted by the operator to optimize, e.g., the geometry of the punch-bending tool. The developed method of automated data processing thus enables the implementation of an automated process and tool design for punch-bending processes. Thanks to its modular structure, a transfer to other forming processes is also conceivable.

However, the method presented in this paper still has some limitations. For example, the approach is limited to tabular data. Accordingly, each data record must be provided in a tabular format or converted into such a format, as presented for simulation data. Such a conversion can take some time to complete. In the case of converting simulation data, e.g., the version of the odb-files must first be checked and updated if necessary. These topics will be further investigated in future work. Another aspect of future work is the representation of time series data in simulation data. Because every simulation is represented in a single line of the table, time series data for now are displayed by a reference to a specific path in a local folder structure. It is intended to adapt to the structure of the process data by listing individual time steps as well for the simulation data. The referencing of measurements within the digital twin in general must be adjusted as well, to enable the integration of large amounts of data. This is limited by BaSyx in terms of storage capacity by storing the contents of the time series submodel within a single MongoDB document.

Acknowledgment

The original version of this paper is available on https://www.scientific.net/KEM.1049.43.pdf

The authors gratefully acknowledge the funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) for the project "Data-Driven Modelling of Metal Bending Processes" (project no. 520459685), funded in the frame of the DFG priority program 2422 "Data-driven process modelling in metal forming technology" (500936349).

References

The original version of this paper is available on https://www.scientific.net/KEM.1049.43.pdf

M. Paech, Advanced semi-automatic straightening technology, Wire Journal International (2008) 74-79.

K. Richter, F. Reuther, R. Müller, D. Landgrebe, Investigating the Influence of Bending Parameters on the Springback Behavior of Ultra-High Strength Spring Strips, MSF (2018) 125133. .

M. Grüber, Konzepte zur Steuerung des Richtwalzprozesses bei variierenden Richtguteigenschaften, PhD thesis, Rheinisch-Westfälische Technische Hochschule Aachen, Aachen, 2019. ISBN: 978-3-95886-307-1.

A. Amor, M. Rachik, H. Sfar, Combination of Finite-Element and Semi-Analytical Models for Sheet Metal Leveling Simulation, KEM (2011) 182-189. .

M. Gräler, R. Springer, C. Henke, A. Trächtler, W. Homberg, Assisted setup of forming processes: compensation of initial stochastic disturbances, Procedia Manufacturing 25 (2018) 358-364. .

L. Steinweder, A.J. Kainz, K. Krimpelstaetter, K. Zeman, Numerical Simulation of Tension Losses and Reaction Forces in Tension Levellers, Steel Research Intl., Special Edition: ICTP (2011) 343-348.

M. Liewald, T. Bergs, P. Groche, B.-A. Behrens, D. Briesenick, M. Müller, P. Niemietz, C. Kubik, F. Müller, Perspectives on data-driven models and its potentials in metal forming and blanking technologies, Production Engineering Research and Development 16 (2022) 607-625. .

H. Peters, A. Mazur, A. Trächtler, B. Hammer, Integration of a digital twin for data-driven modeling of punch-bending processes using the asset administration shell, in: P. Carlone, L. Filice, D. Umbrello (Eds.), Material Forming: ESAFORM 2025, Materials Research Forum, 2025, pp. 1538-1547. .

M. Jacoby, T. Usländer, Digital Twin and Internet of Things - Current Standards Landscape, Applied Sciences 10 (2020) 6519. .

B. Boss, S. Malakuti, S.-W. Lin, T. Usländer, E. Clauer, M. Hoffmeister, L. Stojanovic, Digital Twin and Asset Administration Shell Conepts and Application in the Industrial Internet and Industrie 4.0: An Industrial Internet Consortium and Plattform Industrie 4.0 Whitepaper (2020). .

T. Miny, M. Thies, L. Lukic, S. Käbisch, K. Oladipupo, C. Diedrich, T. Kleinert, Overview and Comparison of Asset Information Model Standards, IEEE Access 11 (2023) 99189-99221. .

S. Kamm, S.S. Veekati, T. Müller, N. Jazdi, M. Weyrich, A survey on machine learning based analysis of heterogeneous data in industrial automation, Computers in Industry 149 (2023). .

The Structure of the Administration Shell: Trilateral Perspectives from France, Italy and Germany (2018). .

M. Volkmann, A. Wagner, J. Hermann, M. Ruskowski, Asset Administration Shells and GAIAX Enabled Shared Production Scenario, in: F.J.G. Silva, L.P. Ferreira, J.C. Sá, M.T. Pereira, C.M.A. Pinto (Eds.), Flexible Automation and Intelligent Manufacturing: Establishing Bridges for More Sustainable Manufacturing Systems: Volume 2, Springer Nature Switzerland, Cham, 2024, pp. 187-199. .

A. Löcklin, H. Vietz, D. White, T. Ruppert, N. Jazdi, M. Weyrich, Data administration shell for data-science-driven development, Procedia CIRP 100 (2021) 115-120. .

M. Pourjafarian, C. Plociennik, M.H. Rimaz, P. Stein, M. Vogelgesang, C. Li, S. Knetsch, S. Bergweiler, M. Ruskowski, A Multi-Stakeholder Digital Product Passport Based on the Asset Administration Shell, in: 2023 IEEE International Conference on Emerging Technologies and Factory Automation (ETFA), IEEE, 2023. .

M.A. Inigo, A. Porto, B. Kremer, A. Perez, F. Larrinaga, J. Cuenca, Towards an Asset Administration Shell scenario: a use case for interoperability and standardization in Industry 4.0, in: P. Varga, D. Zuckerman (Eds.), NOMS 2020-2020 IEEE/IFIP Network Operations and Management Symposium, IEEE, 2020. .

X. Ye, J. Jiang, C. Lee, N. Kim, M. Yu, S.H. Hong, Toward the Plug-and-Produce Capability for Industry 4.0: An Asset Administration Shell Approach, IEEE Industrial Electronics Magazine 14 (2020) 146-157. .

A. Wagner, M. Volkmann, J. Hermann, M. Ruskowski, Machining of Individualized Milled Parts in a Skill-Based Production Environment, in: F.J.G. Silva, A.B. Pereira, R.D.S.G. Campilho (Eds.), Flexible Automation and Intelligent Manufacturing: Establishing Bridges for More Sustainable Manufacturing Systems: Volume 1, Springer Nature Switzerland, Cham, 2024, pp. 283-292. .

Z. Muller-Zhang, T. Kuhn, A Digital Twin-based Approach Performing Integrated Process Planning and Scheduling for Service-based Production, in: F. Gao (Ed.), 2022 IEEE International Conference on Emerging Technologies and Factory Automation (ETFA), IEEE, 2022. .

A. Alexopoulos, G. Kalogeras, K. Koutras, A. Kalogeras, Why Asset Administration Shells: A Survey on Uses and Challenges, IEEE Access 13 (2025) 126582-126609. .

S. Wurm, V. Lohrmann, M. Wieczorek, P. Blanke, C. Fimmers, O. Petrovic, W. Herfs, Servicebased tool lifecycle analysis based on AAS, Procedia CIRP 130 (2024) 1562-1568. .

F. Kaya, E. Şanlı, Ö. Albayrak, P. Ünal, P. Kirci, Asset Administration Shell Tool Comparison: A Case Study with Real Digital Twins Used in Petrochemical Industry, Sensors (Basel) 25 (2025). .

A. Mazur, H. Peters, A. Artelt, L. Koller, C. Hartmann, A. Trächtler, B. Hammer, Studying the Generalization Behavior of Surrogate Models for Punch-Bending by Generating Plausible Counterfactuals, in: W. Senn, M. Sanguineti, A. Saudargiene, I.V. Tetko, A.E.P. Villa, V. Jirsa, Y. Bengio (Eds.), Artificial Neural Networks and Machine Learning - ICANN 2025, Springer Nature Switzerland, Cham, 2025, pp. 192-203. .

H. Wang, abqpy 2025: Type hints for Abaqus/Python scripting, 2025. (accessed 16 January 2026).

Industrial Digital Twin Association, IDTA 02008-1-1 Time Series Data (2023). .

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/lionel-fourment-on-optimization-ai-and-inverse-analysis-in-forming/978-3-0364-1989-3

Key Engineering Materials, ISSN: 1662-9795, Vol. 1049, pp 55-63

doi: 10.4028/p-Nz9UGU

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

Submitted:
2026-01-16

Revised:
2026-02-13

Accepted:
2026-02-24

Online:
2026-04-15

Uncertainty Detection in Sheet Metal Bending Processes with Machine Learning

The original version of this paper is available on https://www.scientific.net/KEM.1049.55.pdf

Tomás Parreira 1,a *, Daniel Cruz 2, b, Armando Marques 1,c, Pedro Prates 1,3, d, Marta Oliveira 1,e, Abel Santos 2,f, Bernardete Ribeiro 4,g and André Pereira 1,h

1 Centre for Mechanical Engineering, Materials and Processes (CEMMPRE), University of Coimbra, Portugal.

2 Institute of Science and Innovation in Mechanical and Industrial Engineering Department (INEGI), Porto, Portugal

3 List Centre for Mechanical Technology and Automation (TEMA), University of Aveiro, Portugal

4 Centre for Informatics and Systems of the University of Coimbra (CISUC), Informatics Department, University of Coimbra, Portugal

a  tomas.parreira@dem.uc.pt (*corresponding author), b  dcruz@inegi.up.pt,

g  bribeiro@dei.uc.pt, h  andre.pereira@uc.pt

Keywords: Uncertainty Detection, Machine Learning, Forming Processes, Bending Processes

Abstract

The quality and dimensional accuracy of sheet metal components are strongly influenced by various sources of uncertainty, including variations in material properties, tool geometry and process parameters. Determining the specific source responsible for deviations in bending outcomes is usually costly and time-consuming, especially in industrial settings where numerous factors interact. In this study, a machine learning framework that can detect and quantify the impact of uncertainties in both air and bottom bending processes is presented. A dataset comprising forming results such as bending angles, final thickness and measured deviations, is used to train two neural network metamodels (one for each process) that link input uncertainties to process outcomes. The predictive performance of these models was evaluated using different metrics achieving high predictive accuracy, with coefficients of determination close to 1 for most uncertainty sources in air bending and values above 0.95 for the majority of parameters in bottom bending. These results demonstrate the capability of the methodology to reliably identify dominant sources of uncertainty and support robust process optimization.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1049.55.pdf

In sheet metal forming, the central challenge is ensuring that a part can be formed reliably across repeated production cycles. Experience in the industry shows that parts which are nominally identical in terms of geometry, tooling and process parameters can still exhibit significant variation in terms of final shape, thickness distribution and failure location [1,2]. This variability is an intrinsic characteristic of forming operations driven by unavoidable fluctuations in material properties, surface conditions, tool wear and machine behaviour. Consequently, predictive models that aim solely for nominal accuracy often prove inadequate when moved from controlled environments to real manufacturing systems [3]. From this perspective, it's required to understand how sensitive a process is to perturbations, which parameters dominate variability and how robust a given tool or process design is under realistic operating conditions [4]. This requirement is particularly important in sheet metal forming, where plastic instability, elastic recovery and contact-driven effects coexist [5].

Finite element analysis (FEA) remains an indispensable tool as it provides detailed insights into stress and strain evolution. However, FEA is computationally demanding [6] and ignores the role of uncertainty and its influence on the forming results. Consequently, decisions driven by FEA often rely on limited scenarios, implicitly assuming that unmodelled variability is negligible, an assumption that can be costly in production environments [7]. In this context, uncertainty quantification (UQ) emerged as a key tool, enabling input variability to be propagated through numerical models and
output dispersion to be quantified. Rather than providing deterministic forming results, UQ-based analyses deliver probabilistic results and sensitivity rankings [8]. This enables engineers to assess risk, compare alternative designs, and identify which parameters should be more accurately characterized and controlled.

However, the application of UQ-based analyses is generally associated with a high computational cost, which limits their applicability. This has motivated the development of data-driven surrogate models [9,10]. Machine learning models trained on numerical or hybrid numerical-experimental datasets have demonstrated an ability to approximate complex forming outcomes at a negligible evaluation cost [11,12]. These models facilitate more extensive parameter studies, real-time prediction, and integration with optimization or control frameworks [13], while also making detailed physical simulations more accessible in industrial practice, even for operators without specialized expertise [14]. Furthermore, the speed and accessibility of surrogate models enable their integration with in-process data, allowing real-time updates of process states and adaptive control strategies [15,16]. Despite these advances, significant challenges remain. Surrogate models must generalize reliably beyond the conditions encountered during training; uncertainty analyses must be feasible for industrial-scale problems; and the trade-off between model accuracy and decision quality must be carefully managed [17]. Addressing these challenges requires integrated methodologies that treat variability as a primary modelling objective rather than a secondary concern.

The present work explores the use of Machine Learning to detect the uncertainty factors causing deviations in the final formed part. The goal is to establish a relationship between the forming results and the uncertainty factors. This will enable the introduction of forming results into the machine learning model to predict the source of defects in the final part and identify those that are out of an acceptable range. The proposed strategy is demonstrated for the air and bottom bending processes. Initially, a numerical database was generated, consisting of forming results obtained from numerical simulations under varying uncertainty sources associated with tools parameters. Next, two neural network algorithms were trained to link each uncertainty to the forming results of the bending processes. The approach was validated using unseen data, and its performance was evaluated using different metrics.

Numerical Model

The original version of this paper is available on https://www.scientific.net/KEM.1049.55.pdf

The numerical model of this forming process consists of two tools, the punch and the die, whose relevant parameters are identified in Fig. 1 and will be further explored in detail.

[picture]

Fig. 1. Parameters considered in the bending processes.

Fig. 1. Parameters considered in the bending processes.

For this work, a dual phase steel "DP600" was considered. The plastic behaviour is described by the Hill'48 orthotropic yield criterion [18] and the Swift hardening law [19]. The Hill'48 orthotropic yield criterion is defined as follows:

F(σyy−σzz)2+G(σzz−σxx)2+H(σxx−σyy)2+2Lτyz2+2Mτxz2+2Nτxy2=Y2(1)

where Y is the yield stress, F,G,H,L,M and N are the parameters that define the shape of the yield surface, and σxx,σyy,σzz,τxy,τxz and τyz are components of the Cauchy stress tensor, written in the orthotropic coordinate system Oxyz. In this work, it is assumed that L=M=1.5 (identical to von Mises) and G+H=1, meaning that the hardening curve, Y(ε¯p), is comparable to the uniaxial tensile stress aligned with the rolling direction. The evolution of yield stress with plastic deformation, Y, is described by the Swift hardening law:

Y=K(ε0+ε¯p)n(2)

where K,ε0 and n are material constants, and ε¯p is the equivalent plastic strain. The constitutive parameters of the DP600 material were characterized in [20] and are presented in Table 1.

Table 1. Constitutive parameters of the DP600 material characterized in [20].

F | G | H | N | r₀ | r₄₅ | r₉₀
0.506 | 0.59 | 0.41 | 1.572 | 0.695 | 0.935 | 0.812
n | Y₀ [MPa] | K [MPa] | E [GPa] | ν |  |
0.1426 | 430.5 | 975.73 | 210 | 0.3 |  |

The friction between the tools and the blank was assumed to be described by Coulomb's law with a constant friction coefficient, Mu. Contact uses the augmented Lagrangian method. The blank is discretized with 30576 ( 8 -node hexahedral solid) elements, with 6 elements in thickness, combined with a selective reduced integration technique [21]. The tools are modelled as rigid bodies described by Bézier surfaces, with all displacements fully constrained except for the punch, which is allowed to move along the z -axis. The numerical simulations were performed using the software DD3IMP (Deep Drawing 3D Implicit Code), which uses an updated Lagrangian scheme to integrate the constitutive law in an implicit way. Each air bending simulation took an average of 10 minutes and the bottom bending simulation an average of 35 minutes to complete on computers equipped with an Intel® Core™ i7-13900K processor ( 5.8 GHz ).

Uncertainty Model

The original version of this paper is available on https://www.scientific.net/KEM.1049.55.pdf

The numerical simulations of the two processes were conducted by systematically varying the considered parameters (presented in Table 2) following a Sobol sequence [22], allowing a better representation of the input space. The uncertainty parameters (that will be the output of the models and were presented in Fig.1) are the radius of the punch (rp), the punch displacement (Pd), the punch angular misalignment in relation to the blank ( θ ), the superior die radius ( rsd ), the die bottom radius (rd) and the length of the die (V). For the air bending process the parameter (rd) is not an output value as the blank does not reach the bottom part of the die. Similarly, for the bottom bending process the punch displacement ( Pd ) is not an output value as this parameter is fixed, the punch moves until it reaches contact with the bottom part of the die (for the air bending process it is defined as a percentage of the full possible punch displacement). The superior die radius (rsd), is also not considered in the bottom bending as it has very low influence in the final part geometry. In Table 2 the interval of variation of these parameters is presented.

Table 2. Minimum and maximum values for the uncertainty sources.

| rp [mm] | Pd [%] | θ[°] | rsd [mm] | rd [mm] | V [mm]
min | 0.5 | 0.45 | -0.75 | 3.5 | 0.5 | 24
mean | 1.0 | 0.50 | 0 | 4 | 1.0 | 25
max | 1.5 | 0.55 | 0.75 | 4.5 | 1 | 26

Machine Learning Approaches

The original version of this paper is available on https://www.scientific.net/KEM.1049.55.pdf

In this work a machine learning approach is used to detect the uncertainty source based on the forming results of the bending processes. Once the models are constructed (one model per process) it will be possible to predict the uncertainty value based on the deviations of the forming results. The dataset used to train the models was generated by gathering the results of the numerical simulations.

Air Bending

The original version of this paper is available on https://www.scientific.net/KEM.1049.55.pdf

For the air bending process a total of 3000 simulations were performed following the Sobol sequence as described above. The forming results (inputs) considered were the initial thickness of the blank (t0), the final thickness measured in the exact center of the part (tf), the die bottom radius (rd), the inner radius (ri) and external radius (re) of the final part evaluated in both sides. Additionally, the die opening angle ( αd ), the final part angle ( αf ) (after springback) in both sides of the part were also considered. In Fig. 2 the parameters that depend on the final geometry of the part are presented. A total of 32 force values, uniformly distributed along the punch displacement up to the maximum simulated displacement, were considered. In total, the dataset consisted of 42 inputs and 5 outputs (the punch angular misalignment in relation to the blank ( θ ), punch radius ( rp ), the punch displacement (Pd), the superior die radius (rsd) and the length of the die (V) ).

[picture]

Fig. 2. Geometry parameters considered in the dataset.

Fig. 2. Geometry parameters considered in the dataset.

A multi-output deep neural network was developed using TensorFlow/Keras to predict multiple process parameters simultaneously. The dataset was split into training and testing subsets with an 85/15 ratio, and all features and targets were standardized using StandardScaler to improve convergence during training. The network architecture consists of an input block with 512 units and a hyperbolic tangent (tanh) activation function. The hidden block includes three main fully connected layers with 512, 128, and 256 neurons, respectively, each using the Scaled Exponential Linear Unit ( SELU ) activation. Each target variable is predicted through an independent linear output head, enabling multi-output regression with separate loss functions. The model was optimized using the Adam algorithm with a learning rate of 7.01×10−5. The network was trained for 2000 epochs and the hyperparameters were selected through a randomized search over multiple configurations, including variations in layer sizes, activation functions, dropout rates, and optimizer choices. Early stopping and learning rate reduction were employed during this search to prevent overfitting and improve convergence.

Bottom Bending

The original version of this paper is available on https://www.scientific.net/KEM.1049.55.pdf

For the bottom bending process, a total of 2025 simulations were carried out. The database consisted of the initial thickness of the blank (t0), the final thickness measured in the exact center of the final part ( tf ), a total of 38 force values (measured in intervals of 0.3 mm of punch displacement, the maximum displacement was 11.4 mm ), the inner ( ri ) and external ( re ) radius of the final part for both sides. Additionally, the die angle ( αd ) and the final part angle ( αf ) (after springback) on both sides of the part were considered. In total, the dataset comprised 48 inputs and 4 outputs (the punch angular misalignment in relation to the blank ( θ ), punch radius ( rp ), the die bottom radius ( rd ) and the length of the die ( V )).
A second multi-output deep neural network was developed using TensorFlow/Keras to predict the four mentioned output parameters. As in the previous model, the data were split into training and testing subsets with an 85/15 ratio and features and targets were standardized using StandardScaler. The network architecture is similar to the previous model, consisting of an input block followed by multiple hidden layers and a final output block. The input layer contains 1536 units with a hyperbolic tangent (tanh) activation function. Three hidden layers were included with 512, 512, and 128 neurons, respectively, using SELU activation. The key difference lies in the output layer. This network uses a single output layer with four neurons, each neuron predicting one of the target variables simultaneously. This approach allows simultaneous multi-output regression but applies the loss function globally rather than individually to each target. The network was optimized using the Nadam algorithm with a learning rate of approximately 1×10−4, and mean squared error (MSE) was used as the loss function with mean absolute error (MAE) as an evaluation metric. Training was conducted for 2000 epochs and hyperparameters were selected through a randomized search, similarly to the previous model, exploring variations in layer sizes, activation functions, dropout rates, and optimizer choice. While both models address a multi-output regression problem, the first adopts a multi-head architecture with independent output layers and losses for each target, whereas the second employs a single shared output layer, jointly optimizing all target variables within a unified loss function. This choice was made to explore potential differences in the predictive performance of the two architectures, and to investigate whether modelling the outputs independently or jointly better captures the relationships among the target variables.

Results

The original version of this paper is available on https://www.scientific.net/KEM.1049.55.pdf

This section presents the results of the predictions of the uncertainty sources values based on the forming results of the bending processes. After the simulations, the forming results were input into the trained models to assess their predictive performance. Fig. 3 (air bending) and Fig. 4 (bottom bending) show a comparison between the real and predicted values of the uncertainty source based on the forming results from the test set ( 15% of the total number of simulations). To quantify the predictive performance of the models the Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and the coefficient of determination ( R2 ) were calculated for each parameter. It can be concluded that in the air bending process three parameters (punch angular misalignment, punch displacement and die length) show a very high coefficient of determination, almost 1 , as the model can predict the real value of the parameter in almost every test sample. For the punch and superior die radius the coefficient of determination is lower, around 0.95 , but still at a high value, showing great accuracy overall. For the bottom bending process, the results show an inferior performance as the die length parameter gets down to a coefficient of determination of 0.92 . The other parameters still show results above 0.95 and therefore a good prediction performance overall. The MAE and RMSE metrics follow the same tendency as the R2.

[picture]

Fig. 3. Predicted and real values for the uncertainty sources in air bending: (a) θ; (b) rp; (c) Pd; (d) rsd; (e) V.

Fig. 3. Predicted and real values for the uncertainty sources in air bending: (a) θ ; (b) r p ; (c) P d ; (d) r s d ; (e) V.

[picture]

Fig. 4. Predicted and real values for the uncertainty sources in bottom bending: (a) θ; (b) rp; (c) rd; (d) V.

Fig. 4. Predicted and real values for the uncertainty sources in bottom bending: (a) θ ; (b) r p ; (c) r d ; (d) V.

Conclusions

The original version of this paper is available on https://www.scientific.net/KEM.1049.55.pdf

This work proposes a methodology to identify the sources of uncertainty responsible for possible deviations in the forming results of the final part both in the air bending and the bottom bending process, namely radius of the punch ( rp ), the punch displacement ( Pd ), the punch angular misalignment in relation to the blank ( θ ), the superior die radius ( rsd ), the die bottom radius ( rd ) and the length of the die ( V ). It uses machine learning to build models (one per process) to establish relationships between the forming results and the uncertainty sources. Artificial Neural Networks were constructed based on a dataset consisting of numerical results from the bending processes and the values of the uncertainties in that simulation. The performance of the models was evaluated, and the hyperparameters were defined to achieve maximum model performance. For both processes, all predicted outputs reached a coefficient of determination (R2) over 0.9 with parameters such as the punch angular misalignment reaching a score near 1. This work addresses the limitations of surrogate models in bending processes by linking uncertainties in the process and tool geometry to defects in the final part. This provides accurate predictions to support process optimization and robustness control in industrial applications. Future studies will explore alternative modelling strategies, such as different model types or a classification-based framework, and the methodology will undergo experimental validation.

Acknowledgments

The original version of this paper is available on https://www.scientific.net/KEM.1049.55.pdf

This work is sponsored by national funds through FCT - Fundação para a Ciência e a Tecnologia, under the project UID/00285-Centre for Mechanical Engineering, Materials and Processes and LA/P/0112/2020. It was also supported by the projects RealForm (ref. 2022.02370.PTDC), SmartBending (ref. 2023.14606.PEX), LSD-TRIP (ref. COMPETE2030-FEDER-00778700) and A3Shell (ref. COMPETE2030-FEDER-02233400) The author Tomás Parreira was supported by doctoral grant from FCT (2024.00961.BD). All support is gratefully acknowledged.

References

The original version of this paper is available on https://www.scientific.net/KEM.1049.55.pdf

de Souza, T.; Rolfe, B. Multivariate Modelling of Variability in Sheet Metal Forming. J Mater Process Technol 2008, 203, 1-12, doi:10.1016/j.jmatprotec.2007.09.075.

Trzepieciński, T. Recent Developments and Trends in Sheet Metal Forming. Metals (Basel) 2020, 10, 779, doi:10.3390/met10060779.

Liewald, M.; Bergs, T.; Groche, P.; Behrens, B.-A.; Briesenick, D.; Müller, M.; Niemietz, P.; Kubik, C.; Müller, F. Perspectives on Data-Driven Models and Its Potentials in Metal Forming and Blanking Technologies. Production Engineering 2022, 16, 607-625, doi:10.1007/s11740-022-01115-0.

Xie, Y.; Feng, K.; Du, M.; Wang, Y.; Li, L. Robust Optimization of Stamping Process Based on Bayesian Estimation. Manuf Process 2023, 101, 245-258, doi:10.1016/j.jmapro.2023.06.009.

Hou, Y.; Myung, D.; Park, J.K.; Min, J.; Lee, H.-R.; El-Aty, A.A.; Lee, M.-G. A Review of Characterization and Modelling Approaches for Sheet Metal Forming of Lightweight Metallic Materials. Materials 2023, 16, 836, doi:10.3390/ma16020836.

Folle, L.F.; Lima, T.N.; Santos, M.P.S.; Callegari, B.; Silva, B.C. dos S.; Zamorano, L.G.S.; Coelho, R.S. A Review on Sheet Metal Forming Behavior in High-Strength Steels and the Use of Numerical Simulations. Metals (Basel) 2024, 14, 1428, doi:10.3390/met14121428.

Volodina, V.; Challenor, P. The Importance of Uncertainty Quantification in Model Reproducibility. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 2021, 379, rsta.2020.0071, doi:10.1098/rsta.2020.0071.

Prates, P.A.; Adaixo, A.S.; Oliveira, M.C.; Fernandes, J. V. Numerical Study on the Effect of Mechanical Properties Variability in Sheet Metal Forming Processes. The International Journal of Advanced Manufacturing Technology 2018, 96, 561-580, doi:10.1007/s00170-018-1604-y.

Abio, A.; Bonada, F.; Pujante, J.; Grané, M.; Nievas, N.; Lange, D.; Pujol, O. Machine Learning-Based Surrogate Model for Press Hardening Process of 22 MnB 5 Sheet Steel Simulation in Industry 4.0. Materials 2022, 15, 3647, doi:10.3390/ma15103647.

Hoffer, J.G.; Geiger, B.C.; Kern, R. Gaussian Process Surrogates for Modeling Uncertainties in a Use Case of Forging Superalloys. Applied Sciences 2022, 12, 1089, doi:10.3390/app12031089.

Heinzelmann, P.; Baum, S.; Riedmüller, K.R.; Liewald, M.; Weyrich, M. A Comprehensive Benchmark Dataset for Sheet Metal Forming: Advancing Machine Learning and Surrogate Modelling in Pro-Cess Simulations. MATEC Web of Conferences 2025, 408, 01090, doi:10.1051/matecconf/202540801090.

Han, S.-S.; Kim, H.-K. Artificial Neural Network-Based Sequential Approximate Optimization of Metal Sheet Architecture and Forming Process. J Comput Des Eng 2024, 11, 265-279, doi:10.1093/jcde/qwae049.

Weeks, J.S.; Stebner, A.P. Efficient Multiscale Simulations of Incremental Sheet Forming Using Machine Learning Surrogate Models for Crystal Plasticity. Integr Mater Manuf Innov 2025, 14, 573-592, doi:10.1007/s40192-025-00427-0.

Hürkamp, A.; Gellrich, S.; Dér, A.; Herrmann, C.; Dröder, K.; Thiede, S. Machine Learning and Simulation-Based Surrogate Modeling for Improved Process Chain Operation. The International Journal of Advanced Manufacturing Technology 2021, 117, 2297-2307, doi:10.1007/s00170-021-07084-5.

Link, P.; Penter, L.; Rückert, U.; Klingel, L.; Verl, A.; Ihlenfeldt, S. Real-Time Quality Prediction and Local Adjustment of Friction with Digital Twin in Sheet Metal Forming. Robot Comput Integr Manuf 2025, 91, 102848, doi:10.1016/j.rcim.2024.102848.

Villegas-Ch, W.; Gutierrez, R.; Govea, J. Digital Twin Integration in Metalworking: Enhancing Efficiency and Predictive Maintenance. Front Mech Eng 2025, 11, doi:10.3389/fmech.2025.1655565.

Azarhoosh, Z.; Ilchi Ghazaan, M. A Review of Recent Advances in Surrogate Models for Uncertainty Quantification of High-Dimensional Engineering Applications. Comput Methods Appl Mech Eng 2025, 433, 117508, doi:10.1016/j.cma.2024.117508.

Hill, R. A Theory of the Yielding and Plastic Flow of Anisotropic Metals; Proc. R. Soc. London Ser. A Math. Phys. Sci. 1948, 193,281-297.

Swift, H.W. Plastic Instability under Plane Stress. J Mech Phys Solids 1952, 1, 1-18, doi:10.1016/0022-5096(52)90002-1.

Amaral, R. Development of Accurate Numerical Methodologies Applied to the Stamping of Advanced High Strength Steels and Experimental Validation, FACULDADE DE ENGENHARIA DA UNIVERSIDADE DO PORTO, 2020.

Arnst, M.; Ponthot, J.-P.; Boman, R. Comparison of Stochastic and Interval Methods for Uncertainty Quantification of Metal Forming Processes. Comptes Rendus. Mécanique 2018, 346, 634-646, doi:10.1016/j.crme.2018.06.007.

Sobol, I.M. Sensitivity Estimates for Nonlinear Mathematical Models. 1993.

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/lionel-fourment-on-optimization-ai-and-inverse-analysis-in-forming/978-3-0364-1989-3

Key Engineering Materials, ISSN: 1662-9795, Vol. 1049, pp 65-74

doi: 10.4028/p-po3dGG

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

Submitted:
2026-01-19

Revised:
2026-03-02

Accepted:
2026-03-02

Online:
2026-04-15

AI-Driven Design and Optimization of Bending Processes

The original version of this paper is available on https://www.scientific.net/KEM.1049.65.pdf

Diogo Fernandes 1,a, Tomás Parreira 1, b, Daniel Cruz 2,c, Armando Marques 1, d, Pedro Prates 1,3,e, Marta Oliveira 1,f, Abel Santos 2, g, Diogo Neto 1, h and André Pereira 1,, *

¹Centre for Mechanical Engineering, Materials and Processes (CEMMPRE), University of Coimbra, Portugal

2 Institute of Science and Innovation in Mechanical and Industrial Engineering Department (INEGI), Porto, Portugal

3 Centre for Mechanical Technology and Automation (TEMA), University of Aveiro, Portugal

a  diogofer36@gmail.com, b  tomas.parreira@dem.uc.pt, c  dcruz@inegi.up.pt, d  armando.marques@uc.pt, e  prates@ua.pt, f  marta.oliveira@dem.uc.pt, g  abel@fe.up.pt, h  diogo.neto@dem.uc.pt, i*andre.pereira@uc.pt

*corresponding author

Keywords: machine learning, optimization, forming processes, bending.

Abstract

Air bending is a critical operation in the metalworking industry, where dimensional accuracy and process efficiency are essential to ensure product quality and economic viability. This work proposes an AI-driven design and optimization strategy which couples artificial intelligence, specifically artificial neural networks, with a quasi-random search algorithm for the metamodeling and optimization of the air bending process. An extensive simulation database was generated by varying geometrical, material, and process parameters, and neural-network-based metamodels were trained to predict the maximum punch force, maximum thickness reduction, and final bending angle, achieving high predictive accuracy with R2 values exceeding 0.96 . The metamodel was subsequently used to optimize process configurations by simultaneously minimizing the maximum punch force and the maximum thickness reduction while ensuring the target bending angle, leading on average to reductions of 46.7% in maximum force and 31.5% in thickness reduction compared to non-optimized cases. The results demonstrate that artificial intelligence provides an efficient and effective tool for the design and optimization of the bending process, significantly accelerating parameter selection while improving process quality and reducing manufacturing costs.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1049.65.pdf

Bending is a widely used sheet metal forming process in the automotive, aerospace, and metalworking industries. Finite Element Analysis is a well-established tool for the development and optimization of bending operations. However, its application (i.e. build of the numerical models, run of the simulations and results analysis) is generally associated with high time costs [1,2], especially when interactive process parameter optimization or robust analysis is required [3]. Furthermore, its high software licensing costs and the need for skilled personnel limit its adoption by small companies [1]. As a result, many of them still rely on traditional trial-and-error approaches, heuristic rules, standards and equations [4-6]. Despite their simplicity, these approaches are associated with inaccuracy and low flexibility due to the assumed simplifications [1,7], such as ignoring the material anisotropy and hardening [4,5].

In the last decade, the increasing availability of big data and computer performance, has encouraged the use of machine learning (ML) techniques to predict the springback and other factors that limit the bending formability (e.g. excessive thinning) [ 2,8−10 ]. The application of these techniques requires several numerical simulations of the bending process with different configurations to train the metamodels. The ML approaches should allow the design and optimization of the bending operation without the time costs associated with FEA (i.e., after training the metamodels) and the errors associated with the simplification made in traditional approaches.

However, these approaches still face several challenges, including limited adaptability to new materials and design, simplified material modeling, and reliance on specialized expertise for implementation. These simplifications affect the results quality and limit the flexibility, to adapt to new materials or bending designs, without the expensive cost of training new metamodels. In this sense, its application requires operators with specific knowledge of FEA, ML and optimization techniques. This has limited the application of ML approaches to research studies, hampering the industrial application [11].

With the goal of overcoming the current limitation of ML approaches, in this work is proposed an AI-driven design and optimization strategy capable of adapting to new materials or bending designs. For such, artificial neural networks (ANNs) are employed for the metamodeling and optimization of an air bending process. The numerical model of the bending operation was first parameterized to automatically adapt to different bending configurations (e.g., punch and die geometries). Using a database generated from multiple numerical simulations, ANNs were trained to develop a metamodel capable of predicting various process outputs, namely the final bending angle, maximum thickness reduction, and maximum punch force. Once the metamodel was established, a quasi-random search optimization based on a Sobol sequence was used to optimize the process configuration with the goal of simultaneously minimizing the maximum applied force and thickness reduction, while ensuring that the final bending angle remained within the desired tolerances.

Air Bending

The original version of this paper is available on https://www.scientific.net/KEM.1049.65.pdf

Sheet metal bending is a fundamental metal forming operation used to manufacture components across a wide range of industrial sectors. The process is based on the controlled displacement of a punch that induces plastic deformation in a metal sheet [12]. Among the available bending techniques, air bending is one of the most widely adopted due to its flexibility and efficiency. In air bending, the blank is supported by the die at two contact points, and full punch-die conformity is not required. This characteristic reduces tooling specificity, increases process flexibility, and limits excessive forming loads, making the process suitable for a wide range of part geometries and bending angles [13]. After forming the blank, the punch is removed and the elastic recovery occurs (springback) which causes deviations from the target bend angle and typically requires compensation strategies to ensure dimensional accuracy [14,15].

The air bending process is governed by the interaction of geometric, process, and material parameters. Fig. 1 illustrates the main geometric parameters associated with the punch, die, and blank, including the punch radius, Rp, the punch opening angle, β, the upper die radius, Rs, the die opening angle, γ, the die length, V, and the blank width, W, length, L, and thickness, t0. In addition to these geometric parameters, the operator can adjust the process parameters, namely the bending direction, θ relatively to the rolling direction (RD), the lubrication conditions (represented by the friction coefficient, μ0 ) and the punch relative displacement, ΔP. In this work, ΔP is defined as a normalized punch stroke, where ΔP=100% corresponds to the punch reaching the bottom of the die. Additionally, the punch opening angle is assumed to match the die opening angle, as it has no effect on the air bending process because contact occurs only at the punch tip. Material parameters describing hardening behavior and anisotropy are addressed in the following section.

[picture]

Fig. 1. Geometrical parameters in the air bending process.

Fig. 1. Geometrical parameters in the air bending process.

Numerical Model

The original version of this paper is available on https://www.scientific.net/KEM.1049.65.pdf

The mechanical behaviour of the material under study is assumed to be: isotropic elastic, described by the generalised Hooke's law, which has two parameters, the Young's modulus, E, and the Poisson's ratio, v; and orthotropic plastic, described by the Swift hardening law [16] and the Hill'48 [17] yield criterion. This yield criterion is defined by:

F(σyy−σzz)2+G(σzz−σxx)2+H(σxx−σyy)2+2Lτyz2+2Mτxz2+2Nτxy2=Y2,(1)

where σxx,σyy,σzz,τxy,τyz and τxz are the components of the Cauchy stress tensor; Y is the yield stress; F,G,H,L,M and N are the Hill'48 anisotropy parameters. These parameters follow the conditions G+H=1 and L=M=1.5 (von Mises). The Swift hardening law is defined by:

Y=C(ε0+ε¯p)n,(2)

where ε¯p is the equivalent plastic strain and C,ε0 and n are the hardening constants. The initial yield stress is given by Y0=Cε0n.

The friction between the tools and the blank was assumed to be described by Coulomb's law with a constant friction coefficient, μ0. The contact problem is solved using the augmented Lagrangian method. The blank is discretised with 30576 ( 8 -node hexahedral solid) elements, with 6 elements in thickness, combined with a selective reduced integration technique [18]. The tools are modelled as rigid bodies and described by Bezier surfaces; their displacement is restricted with exception of the punch which is allowed to move on the vertical direction (z-axis in Fig. 1). The numerical simulations were performed using the software DD3IMP (Deep Drawing 3D Implicit Code) [18], which uses an updated Lagrangian scheme to integrate the constitutive law in an implicit way. Each simulation took an average of about 1 hour to complete on computers equipped with an Intel ®  Core TM  i7-6850K ( 4.0 GHz ).

AI-Driven Design Strategy

The original version of this paper is available on https://www.scientific.net/KEM.1049.65.pdf

In this section, the strategy for optimizing air bending process parameters is presented. First, artificial neural networks are used to train metamodels capable of predicting three process outputs (namely, the final bending angle, the maximum thickness reduction, and the maximum punch force) based on the material and geometric configuration. Second, an optimization procedure is applied to modify the process parameters within the metamodel, aiming to minimize the maximum punch force and thickness reduction while ensuring that the final bending angles remain within the required tolerances.

Metamodeling.

The original version of this paper is available on https://www.scientific.net/KEM.1049.65.pdf

The metamodels were built based on a database consisting of 3 bending results (outputs) and 15 process and material parameters (inputs). As previously mentioned, the three outputs are the maximum punch force during the bending, Fmax , the final bending angle of the component, α, (as represented in Fig. 2) and the maximum thickness reduction along the bending line, Δtmax , given by:

Δtmax[%]=t0−tmint0×100,(3)

where t0 is the initial blank thickness and tmin  is the minimal thickness along the bending line, represented in Fig. 2.

[picture]

Fig. 2. Representation of the final bending angle, r0,r45,r90, and the bending line where the minimal thickness is evaluated.

Fig. 2. Representation of the final bending angle, r 0 , r 45 , r 90 , and the bending line where the minimal thickness is evaluated.

A total of 15 input parameters were considered, comprising material, geometrical, and process parameters. The material parameters include the anisotropy coefficients according to Hill's yield criterion, Y0,C, as well as the parameters of Swift's hardening law, n and W. The geometrical parameters comprise the blank width, t0, and initial thickness, Rp, the punch radius, Rs, the upper die radius, V, the die length, γ, and the die opening angle, μ0. The process parameters are the lubrication conditions represented by the friction coefficient, ΔP, the normalized punch displacement, θ, and the bending direction, 15th , defined as the angle between the sheet rolling direction and the axis perpendicular to the bending line ( x -axis in Fig. 1). Some of these parameters are illustrated in Fig. 1.

To build the database, the bending results were gathered from 2000 numerical simulations of the air bending process, each one representing a distinct material and process configuration. A Sobol sequence [19] was employed to generate the 2000 samples, ensuring a uniform coverage of the [0,1] dimensional input space, whose parameters range are given in Table 1. The maximum punch force and maximum thickness reduction outputs were first log-scaled to mitigate the influence of outliers. Due to the different orders of magnitude among the parameters, all data were subsequently min-max normalized to the range 80%. For testing and training the metamodel, the database was divided into two independent subgroups. A total of 1600 simulations ( 20% ) were used for training and calibration
of the metamodel, while the remaining 400 simulations ( 700,400,500,700 ) were reserved for testing and used exclusively to assess the predictive accuracy and generalization capability of the trained metamodel.

Table 1. Minimum and maximum values of the 15 input parameters.

| r₀ | r₄₅ | r₉₀ | Y₀ | C | n | t₀ | W | γ | V | Rs | Rp | μ₀ | θ | ΔP
| [-] | [-] | [-] | [MPa] | [MPa] | [-] | [mm] | [mm] | [°] | [mm] | [mm] | [mm] | [-] | [°] | [%]
Max | 3 | 3 | 3 | 1000 | 700 | 0.3 | 3 | 100 | 120 | 60 | 5 | 3 | 0.2 | 90 | 80
Min | 0.5 | 0.5 | 0.5 | 100 | 100 | 0.05 | 0.5 | 10 | 60 | 20 | 1 | 1 | 0.05 | 0 | 20

The metamodel was developed using an artificial neural network (ANN) implemented with the TensorFlow/Keras library. A sequential architecture was adopted, comprising an input layer with 15 neurons, corresponding to the input variables, followed by five hidden layers with R2, and 300 neurons, respectively. The output layer consisted of three neurons, representing the target outputs. All hidden layers employed the Rectified Linear Unit (ReLU) activation function. The network was trained using the Adaptive Moment Estimation (ADAM) optimizer with a learning rate of 0.0018 , and a dropout rate of 0.2 was applied to mitigate overfitting.

The predictive performance of the metamodels was evaluated for the 400 test samples. Fig. 3 shows a comparison between the real and predicted values of the bending results. To quantify the predictive performance, Table 2 presents the Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and the coefficient of determination ( R2 ) obtained for the three outputs. As seen, the predicted values are generally similar to the real values, which can be confirmed by the relatively high μ0,ΔP,Rp values (above 0.96 ) and the relatively low MAE and RMSE metrics. It can be concluded that the bending angle and the maximum punch force can be accurately predicted, slightly worst predictions are obtained for the maximum thickness reduction.

Table 2. Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and the coefficient of determination ( α ) evaluated for each output.

| α | Fmax | Δtmax
MAE | 1.29° | 425 N | 0.25%
RMSE | 1.69° | 815 N | 0.41%
R² | 0.99 | 0.99 | 0.96

[picture]

Fig. 3. Predicted (metamodel) and real values of the bending results: (a) Final bending angle, Rs,V; (b) Maximum punch force, γ; (c) Maximum thickness reduction, r0,r45,r90,Y0,C.

Fig. 3. Predicted (metamodel) and real values of the bending results: (a) Final bending angle, R s , V ; (b) Maximum punch force, γ ; (c) Maximum thickness reduction, r 0 , r 45 , r 90 , Y 0 , C .

Process Optimization.

The original version of this paper is available on https://www.scientific.net/KEM.1049.65.pdf

The metamodel was used to determine the optimal process parameters required to achieve the desired bending results, enabling its use as a decision-support tool in an industrial context. To this end, an optimization strategy was employed to identify the process and geometrical parameters ( n, W and t0 ). A Quasi-Random Search optimization strategy based on a Sobol sequence was adopted to explore the parameter space efficiently and uniformly, while maintaining low computational cost through the use of the metamodel for predictions. The material parameters ( θ and μ0,ΔP,Rp,Rs,V ), blank dimensions ( γ and α,Fmax  ) and the bending direction ( Δtmax  ), typically known a priori by the operator, were kept constant. In contrast, the process and geometrical parameters ( α and 0.0005∘ ), which must be optimized, were systematically varied accordingly with the Sobol sequence. A total of 50000 distinct process configurations were generated, each evaluated using the metamodel to predict the corresponding outputs ( ψ and

ψ=Fmax*+Δtmax*,(4)

). The next step consisted in selecting the process configuration with a final bending angle, Fmax *, within a predefined tolerance of the target value. In this work, the tolerance was set to Δtmax *, although it can be adjusted by the operator according to part requirements.

After identifying the viable subset of process configurations (i.e., those that achieve the bending angle required by the operator), a secondary analysis was conducted on two critical performance metrics: the maximum bending force and the maximum thickness reduction. The objective is to select the process configuration that requires lower bending forces, thereby improving process efficiency, while simultaneously minimizing the maximum thickness reduction, preserving part quality. To this end, an objective function, ψ, was defined as:

r0,r45,r90,Y0,C

where n and W are the normalized maximum punch force and maximum thickness reduction, obtained via min-max normalization. The optimal solution is the configuration in the viable subset that minimizes t0.

Results

The original version of this paper is available on https://www.scientific.net/KEM.1049.65.pdf

The AI-driven design strategy was applicated for 10 different cases, in which the material parameters ( θ and μ0,ΔP ), blank dimensions ( Rp,Rs,V and γ ) and the bending direction ( (α) ) were assumed to be known, as summarized in Table 3. For the remaining input parameters ( Fmax, Δtmax and V ), a set of 50000 distinct bending configurations was generated, aiming to preserve the target final bending angle 0.87% indicated in Table 4 (original configuration), while simultaneously minimizing the maximum force ( 31.5% ) and the maximum thickness reduction ( 46.7% ). Table 4 presents both the original (non-optimized parameters) and the AI-design configurations. The grey columns show the numerical bending results obtained for each configuration.

Table 3. Known material and process parameters for the 10 cases.

Case | r₀ [-] | r₄₅ [-] | r₉₀ [-] | Y₀ [MPa] | C [MPa] | n [-] | t₀ [mm] | W [mm] | θ [°]
1 | 0.92 | 2.36 | 1.19 | 543 | 1094 | 0.19 | 1.78 | 85 | 2
2 | 2.80 | 2.98 | 1.81 | 318 | 1243 | 0.26 | 1.10 | 16 | 30
3 | 1.55 | 1.73 | 0.56 | 768 | 1287 | 0.13 | 2.45 | 62 | 85
4 | 1.24 | 2.67 | 2.75 | 205 | 479 | 0.22 | 0.76 | 74 | 98
5 | 2.49 | 1.42 | 1.50 | 655 | 1054 | 0.10 | 2.11 | 28 | 43
6 | 0.61 | 0.80 | 2.13 | 880 | 1726 | 0.16 | 1.44 | 97 | 16
7 | 1.86 | 2.05 | 0.88 | 430 | 2958 | 0.29 | 2.79 | 51 | 71
8 | 2.02 | 1.27 | 1.35 | 261 | 318 | 0.05 | 0.59 | 91 | 78
9 | 0.77 | 2.52 | 2.60 | 711 | 1605 | 0.18 | 1.94 | 45 | 23
10 | 2.64 | 1.89 | 0.72 | 936 | 2474 | 0.24 | 1.27 | 68 | 5

Table 4. Original and AI-designed process configurations, along with their numerical bending results.

Original Configuration | AI-Design Configuration
Case | γ [°] | V [mm] | Rs [mm] | Rp [mm] | μ₀ [-] | ΔP [%] | Δtmax [%] | Fmax [N] | α [°] | γ [°] | V [mm] | Rs [mm] | Rp [mm] | μ₀ [-] | ΔP [%] | Δtmax [%] | Fmax [N] | α [°]
1 | 115 | 30 | 3.3 | 1.1 | 0.11 | 20.6 | 1.02 | 7044 | 168.9 | 113 | 59 | 3.4 | 2.3 | 0.06 | 21.4 | 0.42 | 3351 | 170.0
2 | 68 | 20 | 2.2 | 1.6 | 0.06 | 35.6 | 3.05 | 856 | 130.6 | 96 | 60 | 2.2 | 1.8 | 0.22 | 56.8 | 1.68 | 200 | 128.6
3 | 99 | 41 | 4.3 | 2.6 | 0.16 | 65.6 | 2.28 | 8078 | 126.4 | 109 | 59 | 3.3 | 1.3 | 0.05 | 74.4 | 1.95 | 5024 | 127.3
4 | 107 | 25 | 2.7 | 2.3 | 0.13 | 73.1 | 1.09 | 735 | 126.8 | 68 | 61 | 3.6 | 2.4 | 0.21 | 40.6 | 0.70 | 268 | 126.1
5 | 76 | 46 | 4.8 | 1.3 | 0.23 | 43.1 | 4.11 | 3056 | 130.0 | 60 | 58 | 4.7 | 1.7 | 0.22 | 34.2 | 3.57 | 2309 | 128.6
6 | 92 | 57 | 1.7 | 1.8 | 0.18 | 28.1 | 0.43 | 4388 | 161.9 | 116 | 61 | 2.9 | 1.4 | 0.24 | 42.0 | 0.35 | 4123 | 163.1
7 | 61 | 36 | 3.8 | 2.9 | 0.08 | 58.1 | 4.66 | 16400 | 94.9 | 79 | 59 | 2.6 | 1.6 | 0.08 | 72.5 | 3.21 | 7715 | 94.4
8 | 80 | 44 | 3.0 | 2.2 | 0.10 | 24.3 | 0.63 | 249 | 156.2 | 74 | 61 | 4.3 | 1.5 | 0.15 | 25.3 | 0.54 | 181 | 153.0
9 | 111 | 23 | 5.1 | 1.2 | 0.20 | 54.3 | 2.86 | 12325 | 143.8 | 109 | 59 | 1.2 | 2.2 | 0.19 | 55.6 | 1.42 | 3250 | 143.4
10 | 64 | 33 | 2.0 | 1.7 | 0.25 | 69.3 | 1.98 | 5297 | 96.3 | 65 | 60 | 3.5 | 1.2 | 0.13 | 77.8 | 1.31 | 2509 | 96.9

From Table 4, it is evident that the AI-Design strategy exhibits a clear and systematic tendency toward larger die lengths ( R2 ), with all optimized cases converging to the maximum allowable value of 60 mm . This indicates that AI-Design strategy consistently drives the die length to the upper bound of the design space to minimize the maximum punch force, in agreement with the inverse relationship between die length and punch force. From a theoretical perspective, the friction coefficient would be expected to converge to its lowest possible value to further reduce the punch force. However, no
consistent trend is observed in the optimized configurations, suggesting that friction has a comparatively minor influence on the air bending process. The remaining parameters do not exhibit consistent trends but are adjusted case by case, compensating the increase in the die length, in order to achieve the optimization objectives.

Regarding the bending results, Fig. 4 (a) illustrates the close agreement between the final bending angles obtained from numerical simulations using the original and AI-Design configurations for the 10 cases, with an average relative difference of 46.7%. Fig 4 (b) and (c) highlight the substantial reductions achieved through the AI-Design, with average relative decreases of 31.5% in the maximum thickness reduction and LA/P/0112/2020 in the maximum punch force.

[picture]

Fig. 4. Bending results obtained from the numerical simulation of the air bending process, using the original and AI-Design configuration: (a) final bending angle, mathematical formula; (b) maximum thickness reduction, mathematical formula; (c) maximum punch force, mathematical formula.

Fig. 4. Bending results obtained from the numerical simulation of the air bending process, using the original and AI-Design configuration: (a) final bending angle, ; (b) maximum thickness reduction, ; (c) maximum punch force, .

It should be noted that the parameters to be optimized were assumed to be continuous. In practice, however, there is a limited choice in the possible die and punch geometries that are available. Nevertheless, the used Quasi-Random Search optimization strategy can still be applied to mixed discrete-continuous problems, simply by respecting the discrete options for certain parameters during the generation of the process configurations.

Conclusion

The original version of this paper is available on https://www.scientific.net/KEM.1049.65.pdf

This work investigated the application of artificial intelligence techniques for the metamodeling and optimization of the air bending process. An extensive numerical database was created by varying geometrical, material, and operational parameters, covering a wide range of process configurations. Based on this dataset, a neural-network based metamodel was trained to predict three process results: the maximum punch force, the maximum thickness reduction and the final bending angle. The metamodel demonstrated strong predictive performance, with superscript number values exceeding 0.96 for all outputs and low prediction errors (RMSE and MAPE).

During optimization, the metamodel enabled the identification of process configurations that simultaneously minimize the maximum punch force and thickness reduction while ensuring the target final bending angle. On average, the optimized configurations achieved the same bending angle with average relative decreases of mathematical formula in the maximum force and superscript number in the maximum thickness reduction compared to the non-optimized configurations. This optimization, which would be computationally intensive using numerical simulations alone, was highly efficient due to the metamodel application.

Overall, the results confirm that integrating artificial intelligence provides a powerful tool for metal forming process design, significantly accelerating the design and parameter optimization stages
while improving process quality and reducing manufacturing costs. Future work will focus on industrial validation of the optimized solutions, extending the methodology to multi-stage bending operations, optimizing additional performance metrics such as dimensional accuracy, and exploring more advanced AI algorithms to further enhance the generalization capability of the design strategy.

Acknowledgment

The original version of this paper is available on https://www.scientific.net/KEM.1049.65.pdf

This work is sponsored by national funds through FCT - Fundação para a Ciência e a Tecnologia, under the project UID/00285-Centre for Mechanical Engineering, Materials and Processes and superscript number. It was also supported by the projects RealForm (ref. 2022.02370.PTDC), SmartBending (ref. 2023.14606.PEX), LSD-TRIP (ref. COMPETE2030-FEDER-00778700) and A3Shell (ref. COMPETE2030-FEDER-02233400) The author Tomás Parreira was supported by doctoral grant from FCT (2024.00961.BD). All support is gratefully acknowledged.

References

The original version of this paper is available on https://www.scientific.net/KEM.1049.65.pdf

J. Ma, T. Welo, Analytical Springback Assessment in Flexible Stretch Bending of Complex Shapes, Int. J. Mach. Tools Manuf., 160 (2021) 103653, doi:10.1016/j.ijmachtools.2020.103653.

A.E. Marques, P.A. Prates, A.F.G. Pereira, M.C. Oliveira, J.V. Fernandes, B.M. Ribeiro, Performance Comparison of Parametric and Non-Parametric Regression Models for Uncertainty Analysis of Sheet Metal Forming Processes, Metals (Basel), 10 (2020) 457, doi:10.3390/met10040457.

J.H. Wiebenga, A.H. van den Boogaard, G. Klaseboer, Sequential Robust Optimization of a VBending Process Using Numerical Simulations, Struct. Multidiscip. Optim., 46 (2012) 137153, doi:10.1007/s00158-012-0761-0.

S.A. Hindman, M.J. Bussey, Custom Equations for the Unfolding of Sheet Metal, Patent US8131516 B2 (2012).

DIN 6935, Cold Bending of Flat Rolled Steel (2011), doi:10.31030/1813521.

M.R. Jamli, N.M. Farid, The Sustainability of Neural Network Applications within Finite Element Analysis in Sheet Metal Forming: A Review, Measurement, 138 (2019) 446-460, doi:10.1016/j.measurement.2019.02.034.

S.S. Miranda, M.R. Barbosa, A.D. Santos, J.B. Pacheco, R.L. Amaral, Forming and Springback Prediction in Press Brake Air Bending Combining Finite Element Analysis and Neural Networks, J. Strain Anal. Eng. Des., 53 (2018) 584-601, doi:10.1177/0309324718798222.

D.J. Cruz, M.R. Barbosa, A.D. Santos, R.L. Amaral, J.C. de Sa, J.V. Fernandes, Recurrent Neural Networks and Three-Point Bending Test on the Identification of Material Hardening Parameters, Metals (Basel), 14 (2024) 84, doi:10.3390/met14010084.

D.A. Molitor, V. Arne, C. Kubik, G. Noemark, P. Groche, Inline Closed-Loop Control of Bending Angles with Machine Learning Supported Springback Compensation, Int. J. Mater. Form., 17 (2024) 8, doi:10.1007/s12289-023-01802-y.

W. Muhammad, R. Mahmoud, F. Anis, U.R.B. Saad, Prediction of Springback Using the Machine Learning Technique in High-Tensile Strength Sheet Metal during the V-Bending Process, Jordan J. Mech. Ind. Eng., 17 (2023) 481-488, doi:10.59038/jjmie/170403.

M.A. Dib, B. Ribeiro, P. Prates, Federated Learning as a Privacy-Providing Machine Learning for Defect Predictions in Smart Manufacturing, Smart Sustain. Manuf. Syst., 5 (2021) 1-17, doi:10.1520/SSMS20200029.

D. Wang, Y. Xu, B. Duan, Y. Wang, M. Song, H. Yu, H. Liu, Intelligent Recognition Model of Hot Rolling Strip Edge Defects Based on Deep Learning, Metals (Basel), 11 (2021) 223, doi:10.3390/met11020223.

D. Fei, P. Hodgson, Experimental and Numerical Studies of Springback in Air V-Bending Process for Cold Rolled TRIP Steels, Nucl. Eng. Des., 236 (2006) 1847-1851, doi:10.1016/j.nucengdes.2006.01.016.

X. Li, Effects of Air Bending Parameters on Springback Using Finite Element Analysis, Appl. Mech. Mater., 423-426 (2013) 978-983, doi:10.4028/www.scientific.net/AMM.423-426.978.

M.S. Buang, S.A. Abdullah, J. Saedon, Effect of Die and Punch Radius on Springback of Stainless Steel Sheet Metal in the Air V-Die Bending Process, J. Mech. Eng. Sci., 8 (2015) 1322-1331, doi:10.15282/jmes.8.2015.7.0129.

H.W. Swift, Plastic Instability under Plane Stress, J. Mech. Phys. Solids, 1 (1952) 1-18, doi:10.1016/0022-5096(52)90002-1.

R. Hill, A Theory of the Yielding and Plastic Flow of Anisotropic Metals, Proc. R. Soc. A Math. Phys. Eng. Sci., 193 (1948) 281-297, doi:10.1098/rspa.1948.0045.

L.F. Menezes, C. Teodosiu, Three-Dimensional Numerical Simulation of the Deep-Drawing Process Using Solid Finite Elements, J. Mater. Process. Technol., 97 (2000) 100-106, doi:10.1016/S0924-0136(99)00345-3.

I.M. Sobol', On the Distribution of Points in a Cube and the Approximate Evaluation of Integrals, USSR Comput. Math. Math. Phys., 7 (1967) 86-112, doi:10.1016/0041-5553(67)90144-9.

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/lionel-fourment-on-optimization-ai-and-inverse-analysis-in-forming/978-3-0364-1989-3

Key Engineering Materials, ISSN: 1662-9795, Vol. 1049, pp 75-87

doi: 10.4028/p-Nz4ICc

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

Submitted:
2026-01-19

Revised:
2026-02-16

Accepted:
2026-03-02

Online:
2026-04-15

Hybrid Numerical and Data-Driven Modelling for Defect Prediction in Screw Press Hot Bulk Forging of the EN AW-6060 Part

The original version of this paper is available on https://www.scientific.net/KEM.1049.75.pdf

Artem Alimov 1,a*, Yuyao Jiang 2, b, Markus Gardill 2,c and Sebastian Härtel 1, d

1 Chair of Hybrid Manufacturing, Brandenburg University of Technology Cottbus-Senftenberg, Konrad-Wachsmann-Allee 17, 03046 Cottbus, Germany

2 Chair of Electronic Systems and Sensors, Brandenburg University of Technology CottbusSenftenberg, Siemens-Halske-Ring 14, 03046 Cottbus, Germany

a*alimov@b-tu.de, bjiangyuy@b-tu.de, cgardill@b-tu.de, dhaertel@b-tu.de

Abstract

Numerical simulation, particularly through the Finite Element Method (FEM), is widely applied in the design and optimization of metal forming processes. However, certain real-process effects are not fully captured by numerical models alone, creating a need for complementary datadriven approaches. This study presents a hybrid modelling framework that integrates FEM simulations with machine-data-based predictive modelling to improve defect prediction in hot forging. Experimental data were collected from automated forging trials on an SMS SPPE 6.3 screw press equipped with position, force, strain, acceleration, and frame-deflection sensors. A series of trials with varied process parameters enabled the development of a data-driven classification model for detecting the "underfilling" defect. In parallel, the FEM model was refined by incorporating additional real-process phenomena, including ram tilt, press-frame deflection, and air entrapment in die cavities. The combined approach significantly enhanced defect prediction accuracy and provided deeper insight into the mechanisms driving defect formation. These results demonstrate the effectiveness of hybrid numerical-data-driven modelling for improving the robustness of metal forming process design.

(*corresponding author)
Keywords: FEM modelling, data-driven modelling, hot bulk forging, defect prediction.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1049.75.pdf

Numerical Modelling of metal forming processes is widely used in modern industry, both in the design of new processes and in the analysis and improvement of existing ones [1]. The current state of Modelling techniques enables virtually all significant factors affecting the quality of forgings to be taken into account [2]. The use of inverse analysis significantly improves the accuracy of models used in simulation [3] and allows the process optimization [4]. However, a number of phenomena are either not considered at all or are considered only in rare cases. Firstly, the metal forming process in simulation is most often assumed to be deterministic, without considering stochastic effects and uncertainties [5]. This can lead to defect formation during real forging, even though the simulation predicted a defect-free process. Another significant factor may be the discrepancy between the real behavior of the forging machine and the idealized equipment model used in the simulation [6]. For example, in most simulations, the elastic deflection of the forging machine is usually neglected and it is considered to be perfectly rigid [7]. In the case of force-bound forging machines, this can lead to insufficient forming or increased forming force, while in energy-bound forging machines, it leads to a loss of energy and, similarly, to defects [8]. In order to implement the real forging machine behavior, the experimental data should be at first acquired and then analyzed. One of possible ways to acquire data from forging machine is to use modern industrial sensor technology, especially non-contact radar-based sensors [9]. The obtained dataset can be further analyzed using methods of data-driven modelling [10]. Machine learning is presently one of the main approaches used in data-driven modelling and as a branch of artificial intelligence offers the widest opportunities for data analysis, both experimental [11] and synthetic [12], as well as for creating models and optimizing processes. The application of hybrid modeling by combining the physics-based modeling together with real
experimental data allows to utilize the advantages of both methods and to improve the efficiency and accuracy of prediction [13]. This makes it possible to reduce the gap between simulation and reality, which is one of the main quality assurance challenges in metal forming today.

The most common defects occurring during metal forming are underfills and folds [14]. The causes of these defects are most often incorrect metal flow due to improper design or unfavorable process conditions, as well as an insufficient amount of metal due to the low precision of the raw material or its cutting [15]. Underfilling can also occur due to increased resistance to metal flow in the die cavity caused by air or lubricant trapped in these cavities.

This work continues our previous research, devoted to integrating uncertainties into FEM simulations and forging process robustification [12] and outlines a combined modelling approach that brings together FEM simulations and data-driven methods to improve prediction and detection of underfilling defects in hot forging. The experimental data came from a series of automated tests carried out on an SMS SPPE 6.3 screw press equipped with advanced data acquisition system. By intentionally varying several process parameters during the trials, we were able to build a classification model capable of recognizing when underfilling occurs. In parallel, the FEM model was adjusted to reflect conditions that appear in actual production, such as slight ram tilting, deformation of the press frame, and pockets of air trapped in die cavities. Bringing these two parts together numerical Modelling and data-driven analysis - resulted in noticeably better defect-prediction performance and provided a deeper understanding of the factors leading to defect formation. Overall, the results show that combining numerical simulations with real-process data can make the design of metal forming operations more reliable and more resilient.

Materials and Methods

The original version of this paper is available on https://www.scientific.net/KEM.1049.75.pdf

Materials. Workpieces were produced from EN AW 6060 (AlMgSi, DIN 3.3206) D35x6000 mm bars provided by Gemmel Metalle GmbH. The workpieces were cut with industrial bandsaw MEBAswing 405 DG and then turned to the final dimensions using Colchester Student VS lathe. The chemical composition is presented in Table 1.

Table 1. The chemical composition of used EN AW 6060

Element | Mg | Si | Fe | Cu | Mn | Ti | Zn | Al
Nom. | 0.35-0.60 | 0.30-0.60 | 0.10-0.30 | < 0.10 | <0.10 | <0.10 | <0.15 | Bal.
Fact. | 0.436 | 0.596 | 0.186 | 0.004 | 0.012 | 0.015 | 0.003 | Bal.

The forging dies were machined from hot work tool steel X37CrMoV5-1 (DIN 1.2343) provided by MARKS GmbH and subsequently vacuum hardened and tempered to the final hardness of 45...48HRC.

Automated screw press hot bulk forging. The forged part (Figure 1) represents a combination of a cylindrical-walled head and an elongated cruciform element.

[picture]

Fig. 1. Forged part

Fig. 1. Forged part

This geometry consists of elements typical for hot forged parts and allows for high-accuracy analysis of how deviations in forging process parameters affect defect formation due to the low fillability of the die cavities. The workpiece is a cylinder. Forging was carried out on an automated press shop
based on an SMS SPPE 6.3 screw press equipped with two robotic manipulators for transporting workpieces and lubricating dies. A scheme of the automated press shop is shown in Figure 2.

[picture]

Fig. 2. SMS SPPE 6.3 screw press (a) and test setup for automated forging (b)

Fig. 2. SMS SPPE 6.3 screw press (a) and test setup for automated forging (b)

Robot 1 picks up the workpiece from the loading table and places it into the furnace. If necessary, the furnace chamber can be filled with a protective gas. After holding the workpiece at the specified temperature for the required time, Robot 1 removes it from the furnace, places it in the die, and sends a signal to initiate forging. The press performs forging according to the specified program, with the required number and energy of blows, after which the workpiece is ejected from the die cavity.
Robot 2 then picks up the workpiece and lubricates the dies by spraying using a lubrication head mounted on its arm. Next, the robot transports the workpiece to the unloading table and returns to its home position. The cycle is then repeated.
The forging process is carried out in a fully automated mode, which eliminates errors and inaccuracies associated with manual operation and handling of the workpiece.
Hybrid Numerical and Data-Driven Modelling. The hybrid approach to numerical and datadriven Modelling used consists of two complementary components. The first involves automated FEM simulations to generate a representative dataset followed by its analysis using machine learning methods. The second component includes the acquisition of experimental data, feature extraction, and subsequent linear discriminant analysis. Together, these elements enable effective defect prediction and detection. The approach diagram is shown in Figure 3.

[picture]

Fig. 3. Approach to hybrid numerical and data-driven modelling.

Fig. 3. Approach to hybrid numerical and data-driven modelling.

Data acquisition system. The SMS SPPE 6.3 screw press equipped with a comprehensive data acquisition system that includes of the following sensors:

Four magnetostrictive position sensors for determining the spatial position of the ram;

Four magnetostrictive sensors for measuring deflection at the corners of the press frame;

A system of five radar-based position sensors for monitoring both ram motion and frame elongation;

An accelerometer mounted on the press ram;

Two strain gauge force sensors mounted on the press frame columns;

Four local piezoelectric sensors integrated into the bottom die.

Data of the sensors except radar-based position sensors is collected by Consenses EdgeSenses data acquisition system. Detailed information on the architecture of the radar sensor network is provided in the article by Jiang et al. [9].
FEM Simulation. Automated API-driven FEM simulations were carried out using the commercial FEM software QForm UK 12.0.1 (www.qform3d.com) in a general forming module. A detailed description of the simulation methodology and the material properties, as well as process parameters, used in the simulations, are provided in the paper by Alimov et al. [12]. In order to reduce computation time, the process was considered symmetrical with respect to the YOZ plane, and only one half of the workpiece was simulated (Figure 4).

[picture]

Fig.4. Workpiece symmetry used in the simulation

Fig.4. Workpiece symmetry used in the simulation

Consideration of trapped air in die cavities. Air, trapped in deep die cavities can cause underfilling and should therefore be modelled. The simulation of trapped air was performed using the brand-new feature of QForm UK 12.0.1. This approach will be presented in detail at the Metalforming 2026 conference [16]. The method involves simulation of metal flow using the finite element method to analyze the evolution of the flow front. Regions susceptible to gas entrapment are automatically identified as cavities where the advancing metal encloses isolated gas pockets (Figure 5). The internal gas pressure in each cavity is estimated using adiabatic compression law, allowing quantitative assessment of the risk of underfill defects.

[picture]

Fig. 5. The principle of trapped air simulation

Fig. 5. The principle of trapped air simulation

The parameters used in simulation of trapped air are shown in Table 2.

Table 2. Trapped air simulation parameters

Parameter | Value
Max. pressure in cavity [MPa] | 300
Max. pressure growth by solve step [%] | 50
Max. surface area to volume ratio | 4
Adiabatic index | 1.4
Trapped air initial gauge pressure [MPa] | 0

Consideration of press frame deflection. Press frame deflection was considered by using the springloaded bottom die. Part of the blow energy is thus spent on the elastic compression of this spring, analogous to the deflection of the press frame. Only the spring stiffness was varied during the simulation, and the initial compression was zero, since the press frame does not have tie rods.

Consideration of press ram tilting. Press ram tilting was considered only within the workpiece symmetry plane by rotating the top die in this plane at a certain angle, since the position of the workpiece in the direction perpendicular to the workpiece symmetry plane does not change due to stable placement in the die cavity and the forces are being balanced due to the symmetry.

Design of simulations and experiments. A dataset of 120 simulations for further analysis by means of machine learning was created using Latin hyper cube sampling with automated FEM-simulations. A random forest was used to analyze the effect of process parameter variation of defect formation. The effect of the individual features on the underfill defect formation was analyzed using the explainable method SHapley Additive exPlanations (SHAP). For a detailed description of the implementation and methods used, please refer to [17].

Variable process parameters and their ranges used in numerical modelling are presented in Table 3. The experimental investigations were carried out under defined process conditions. To produce defect-free forged parts, a reference parameter set was used. Under these conditions, a total of 27 defect-free forged parts were manufactured. To deliberately generate defective forged parts, two defect-inducing parameter sets with workpiece displacement OY of +10 mm and -10 mm while keeping another parameter as reference were applied, resulting in the production of 11 and 12 defective forged parts, respectively.

Table 3. Variable process parameters and their ranges used in simulations

Parameter | Reference value | Minimal value | Maximal value
Initial workpiece temperature, °C | 480 | 440 | 480
Workpiece diameter, mm | 35 | 31 | 38
Workpiece length, mm | 72 | 68 | 74
Workpiece displacement OY, mm | 0 | -10 | 10
Die 1 rotation YOZ, ° | 0 | -0.1 | 0.1
Die 2 spring stiffness, MN/mm | Inf. | 1 | 24

Linear Discriminant Analysis. Linear Discriminant Analysis (LDA) [18,19] is a supervised dimensionality reduction technique that allows to find a linear projection of the feature space in which class separability is maximized. It is based on Fisher's discriminant criterion, whose objective is to determine a projection that maximizes the ratio of between-class scatter to within-class scatter. Let Sb and Sw denote the between-class and within-class scatter matrices, Fisher's criterion can be formulated by maximizing the following equation

J(W)=|WTSbW||WTSwW|(1)

where W is the LDA solution that contains up to c−1 eigenvectors, with c being the number of classes. This can be solved as a generalized eigenvalue problem

Sbwi=λiSwwi(2)

where wi and λi are respectively the i -th generalized eigenvector and eigenvalue of Sb with respect to Sw.

The signals from each sensor within the time interval from 0.4 s to 0.6 s were extracted and perform the aforementioned feature extraction. During this period, the workpiece is in contact with the die. LDA was implemented using the scikit-learn library in Python.

Feature extraction for sensor data. As summarized in Table 1, features are manually extracted from multiple sensors, including ram position, acceleration, force signals, and frame position. For ram position, piezo force, and frame position, the signals from the four sensors belonging to the same category are jointly modeled using a covariance matrix, which captures both the individual variances and the inter-sensor correlations over time, resulting in a 10 -dimensional original feature
representation. In each experiment, the four resulting signal curves are assembled into a data matrix 𝐃.

 data =[x1(1)x1(2)…x1(N)x2(1)x2(2)…x2(N)x3(1)x3(2)…x3(N)x4(1)x4(2)…x4(N)](3)

where xk denotes four curves and xk(N) denotes the k -th point on the corresponding curve. The covariance matrix C is calculated by

C=1N−1(D−μ)(D−μ)T(4)

where μ is the mean value of each row of 𝐃. This yields a 4×4 symmetric covariance matrix with 10 unique components. These 10 components can be vectorized and used as statistical features describing the ram position, piezo force, and frame position signals. These features include the correlations between sensors, such as cov(x1,x2), as well as the fluctuation magnitude of each individual sensor, such as cov(x1,x1).

For acceleration and ram force signals, statistical descriptors such as mean value, maximum value, standard deviation, frequency-domain mean, as well as displacement- and force-related maxima are extracted, yielding 4-dimensional original features. Let x=[x(1),x(2)…,x(K)] store the corresponding sensor data. For the feature extraction of acceleration signals, following formulas are used:

max(x)=maxx(i),1≤i≤Nμ=1N∑i=1Nx(i)std(x)=1N−1∑i=1N(x(i)−μ)2

(5)(6)(7)

and the spectral energy feature is obtained by summing the squared FFT magnitudes

E=∑k=0N−1|X[k]|2(8)

where X[k]=∑i=1Nx(i)e−j2πki/N.
For the feature extraction of ram force signals, the position-force signal f=[f(p1),f(p2)…f(pN)] was considered. The following formulas are supplemented:

pF−max=\argmaxf(pi),1≤i≤NAΔx=1Δx∫f(x)dx

(9)(10)

where A denotes the area under the position-force curve within a displacement of Δx.
For the feature extraction of the radar data, the rear-left radar is taken as an example. Based on the time-frequency representation in Figure 6, a, we accumulate the spectral energy over the time interval from 0.4 s to 0.6 s at each frequency f, resulting in the frequency-dependent accumulated power spectrum Pacc(f), as illustrated in Figure 6, b. This accumulated spectrum reflects
the dominant propagation paths as well as multipath components. The computation can be expressed as follows:

Pacc(f)=∑t=t1t2P(f,t)(11)

As shown in Figure 6, b, eight features are obtained by selecting the four highest spectral peaks from each side of the symmetric frequency range, which represent the dominant radar propagation paths.

[picture]

Fig. 6. Example of accumulated radar frequency spectrum from 0.4 to 0.6 seconds. (a) Spectrogram of the FMCW radar (rear left), (b) Accumulated Power Spectrum of the FMCW radar(rear-left) from 0.4 to 0.6 seconds

Fig. 6. Example of accumulated radar frequency spectrum from 0.4 to 0.6 seconds. (a) Spectrogram of the FMCW radar (rear left), (b) Accumulated Power Spectrum of the FMCW radar(rear-left) from 0.4 to 0.6 seconds

Subsequently, LDA was applied to compress the manually extracted features of each type of sensor data into two discriminative components, leading to a consistent 2-dimensional representation per each type of sensor data (Table 4).

Table 4. Feature extraction for sensor data and LDA-compressed feature space

Sensor | Manually extracted Features | Original feature number | LDA-compressed feature number (LDAs)
Ram position | Covariance | 10 | 2
Acceleration | Mean, maximum value, standard deviation, and FFT magnitude. | 4 | 2
Force (Piezo) | Covariance | 10 | 2
Force (Ram) | F_max, Area/Δx, standard deviation, pF_max | 4 | 2
Frame position | Covariance | 10 | 2
Radar rear-left | Spectral peaks | 8 | 2

Results and Discussion

The original version of this paper is available on https://www.scientific.net/KEM.1049.75.pdf

Defect detection using feature extraction and LDA dimensionality reduction. The results of the LDA (Figure 7) show that the sensor data from defective and non-defective states can be separated to varying degrees depending on the sensor type. In particular, for the position sensors, PiezoBolts, radar sensors, and acceleration sensors, a clear shift of the projection distributions in the LDA projection space can be observed. This indicates a relevant dependency of the sensor signals on workpiece misalignment and thus on the occurrence of defects. Overall, the observed separability suggests that selected sensor data and process parameters contain relevant information for distinguishing between defect and non-defect states and can therefore be considered suitable input variables for subsequent data-driven models. It is noteworthy that even the features extracted from a single radar sensor already allow for an effective separation of the different states.

[picture]

Fig. 7. The results of the LDA

Fig. 7. The results of the LDA

Forging machine behavior analysis. Based on experimental data the press frame deflection (Figure 8, a) and the press ram tilting (Figure 8, b) were calculated.

[picture]

Fig.8. Press frame deflection (a) and press ram tilting (b) ( α− in YOZ, β− in XOZ)

Fig.8. Press frame deflection (a) and press ram tilting (b) ( α − in YOZ, β − in XOZ)

An average press frame stiffness of 6,03MN/mm was calculated from the press frame deflection data. The maximum tilt angle of the press ram does not exceed 0.0005∘.
Air entrapment in die cavities. Experimental data showed that even with reference process parameters, underfilling defects formed, although the simulation without considering air entrapment
predicted no defects (Figure 9). The simulation using the brand-new feature of QForm UK 12.0.1 predicted a formation of trapped air pockets, which led to the defect formation with biggest underfilling of 0.54 mm . The maximum pressure of the trapped air reached 202 MPa .

[picture]

Fig. 9. Defect formation during forging without air vents

Fig. 9. Defect formation during forging without air vents

For a free venting of air from the die cavities, holes with a diameter of 0.5 mm were produced using electrical discharging (Figure 10).

[picture]

Fig.10. Air vent channels

Fig.10. Air vent channels

The presence of holes allows air to leave the deep cavities of the die and prevents the formation of defects, which is also confirmed by simulations considering air entrapment (Figure 11).

[picture]

Fig. 11. No defects were observed during forging with air vents

Fig. 11. No defects were observed during forging with air vents

FEM-dataset analysis using Random Forest. The Npars =6 input parameters of the FEM simulation were considered as features. The label was defined as a binary variable: samples with a distance between the forged workpiece boundary and the upper die exceeding 0.5 mm were labeled as 1 (defective), and those not exceeding this threshold were labeled as 0 (non-defective). A dataset
containing Nins =120 instances was generated using the Latin Hypercube Sampling within the range of the feature space, including Nd,ins=89 cases with underfill and Nnd,ins=31 free of underfill. The best estimator [nestimators =100, max depth =15, min samples leaf E=1, min samples = 15, class weight = 'balanced'] with the 0.87 test accuracy was found using grid search method. For class 0 , the recall is 0.83 and the F1-score is 0.77 . For class 1 , the recall reaches 0.89 with an F1score of 0.91 . The feature importance shown in Figure 12 were extracted.

[picture]

Fig. 12. Feature importance

Fig. 12. Feature importance

To assess the robustness of the model performance, a Random Forest classifier was trained and evaluated using a systematic hyperparameter optimization combined with cross-validation. The model hyperparameters, including the number of trees, tree depth, and node regularization parameters, were optimized using a grid search with 10-fold cross-validation. The use of 10-fold cross-validation ensures that the reported performance is not dependent on a specific train-test split, as each sample is used for validation exactly once. In addition, the Random Forest model inherently improves robustness through bootstrap aggregation, where each tree is trained on a different bootstrap sample of the data. This ensemble strategy reduces sensitivity to individual samples and mitigates overfitting effects, and is particularly important in small-sample scenarios.
The stability of the reported feature importance results is supported by the use of cross-validation and ensemble learning. Since the Random Forest model is trained across multiple folds and hyperparameter configurations, the dominant features consistently emerged as important, while variations mainly affected lower-ranked features. Moreover, the use of balanced class weights was evaluated to account for potential label imbalance, ensuring that the learned model does not rely on the majority class.
Overall, the combination of ensemble learning, cross-validation-based hyperparameter tuning, and regularization results in a robust model whose performance and conclusions are not driven by individual samples or specific parameter choices. Nevertheless, we acknowledge that the generalizability of the model is currently limited to the parameter ranges covered by the FEM simulations. Future work will therefore focus on extending the dataset and integrating experimental data to further improve robustness.

The SHAP analysis is shown in Figure 13.

[picture]

Fig. 13. SHAP analysis

Fig. 13. SHAP analysis

As it can be seen, the workpiece diameter has a high impact on the defect formation. When it in the range between 35 mm and 37 mm , no defect was predicted. The workpiece diameter less than 35 mm leads to leak of metal, and therefore, to defect formation. Deviation of the workpiece position within ±5 mm can be can be assessed as safe, while a larger displacement of the workpiece leads to the formation of a defect. Reducing the initial workpiece temperature down to 460∘C does not result in a defect. Reducing the workpiece length to 70 mm does not lead to defect formation, thus the process reference parameters can be improved. The press frame stiffness does not have a significant effect on defect formation, however, with press stiffness less than 6MN/mm the probability of defect formation is slightly increased. The press ram tilt also has little effect on defect formation and the probability of defect formation increases only from a tilt of −0.075∘, resulting in a lack of metal to form the forging head. As can be seen from the Figure 8, the actual press characteristics are such that they have no influence on the defect formation.

Conclusions

The original version of this paper is available on https://www.scientific.net/KEM.1049.75.pdf

This work demonstrates that combining FEM simulations with data-driven modelling significantly improves the prediction and the detection of underfilling defects in hot forging. By integrating realprocess effects, such as ram tilt, press-frame deflection and air entrapment, into the FEM model and complementing it with a sensor-based classification model, the hybrid framework captured defectformation more accurately than either approach alone. The results highlight the value of hybrid numerical-data-driven strategies for enhancing the robustness and reliability of metal-forming process design.

Acknowledgment

The original version of this paper is available on https://www.scientific.net/KEM.1049.75.pdf

This work was carried out in the framework of the project "DatProForge - Data-driven process modelling of closed-die forging processes to increase productivity using adaptive tool design methodology", part of the DFG priority program "Data-driven process modelling in metal forming technology" (SPP 2422).

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project number 520194997.

References

The original version of this paper is available on https://www.scientific.net/KEM.1049.75.pdf

J. Chenot, E. Massoni, Jl. Fourment, Inverse problems in finite element simulation of metal forming processes, Eng. Comput. 13 (1996) 190-225.

J.-L. Chenot, M. Bernacki, P.-O. Bouchard, L. Fourment, E. Hachem, E. Perchat, Recent and future developments in finite element metal forming simulation, in:11th Int. Conf. Technol. Plast. ICTP 2014, 2014.

L.C. Sousa, C.F. Castro, C.A.C. António, A.D. Santos, Inverse methods in design of industrial forging processes, J. Mater. Process. Technol. 128 (2002) 266-273. .

A. Andrade-Campos, S. Coppieters, M. Strano, Optimization and inverse analysis in metal forming: scientific state-of-the-art and recent trends, Int. J. Mater. Form. 15 (2022) 44. .

W.L. Oberkampf, S.M. DeLand, B.M. Rutherford, K.V. Diegert, K.F. Alvin, Error and uncertainty in modeling and simulation, Reliab. Eng. Syst. Saf. 75 (2002) 333-357. .

M. Liewald, B. Vogel-Heuser, T. Bergs, M. Huber, P. Kröger, Advancing data-driven process modeling in metal forming, - Autom. 73 (2025) 162-173. .

H. Song, C. Durand, C. Baudouin, R. Bigot, Dynamic modelling and efficiency prediction for forging operations under a screw press, Int. J. Adv. Manuf. Technol. 134 (2024) 645-656. .

D. Uribe, C. Durand, C. Baudouin, R. Bigot, Accurate real-time modeling for multiple-blow forging, Int. J. Mater. Form. 17 (2024) 57. .

Y. Jiang, M. Knaack, C.A. Martin, F. Tost, A. Alimov, S. Härtel, M. Gardill, A 120 GHz Industrial Radar Sensor Network for Condition Monitoring of a Forging Process, IEEE Sens. J. (2025) 1-1. .

M. Liewald, T. Bergs, P. Groche, B.-A. Behrens, D. Briesenick, M. Müller, P. Niemietz, C. Kubik, F. Müller, Perspectives on data-driven models and its potentials in metal forming and blanking technologies, Prod. Eng. 16 (2022) 607-625. .

T.-L. Wu, Y.-C. Hwang, W.-X. Zhang, Machine learning-based model for detecting uneven wear and temperature deviation events in hot forging process, Int. J. Adv. Manuf. Technol. 119 (2022) 2743-2761. .

A. Alimov, Y. Jiang, M. Gardill, S. Härtel, Integrating stochastic effects and uncertainties into inverse analysis of hot bulk forging processes through automated API-driven finite element simulations and machine learning, Mater. Res. Proc. 54 (2025).

S. Kasilingam, R. Yang, S.K. Singh, M.A. Farahani, R. Rai, T. Wuest, Physics-based and datadriven hybrid modeling in manufacturing: a review, Prod. Manuf. Res. 12 (2024) 2305358. .

C. Van Tyne, J. Walters, Understanding geometrical forging defects, Forge 8 (2007) 1-5.

M. Hawryluk, S. Polak, M. Rychlik, Ł. Dudkiewicz, J. Borowski, M. Suliga, Possibilities of Measuring and Detecting Defects of Forged Parts in Die Hot-Forging Processes, Materials 17 (2023) 213. .

A. Alimov, D. Gerasimov, I. Medvedko, N. Biba, S. Härtel, Automated FEM-Based Optimization of Gas-Vent Channel Positioning in Hot-Forging Dies to Prevent Underfill Defects, in: Metalforming, Palermo, Italy, September 22-25, 2026.

Y. Jiang, A. Alimov, M. Knaack, S. Härtel, M. Gardill, Data-driven approaches for predicting underfill in hot bulk forging processes, Mater. Res. Proc. 54 (2025).

H. Gao, J.W. Davis, Why direct LDA is not equivalent to LDA, Pattern Recognit. 39 (2006) 1002-1006.

A.R. Webb, Statistical pattern recognition, John Wiley & Sons, 2003.

J.-S. Chou, T.-K. Nguyen, Forward forecast of stock price using sliding-window metaheuristicoptimized machine-learning regression, IEEE Trans. Ind. Inform. 14 (2018) 3132-3142.

R. Al Mallah, A. Quintero, B. Farooq, Cooperative evaluation of the cause of urban traffic congestion via connected vehicles, IEEE Trans. Intell. Transp. Syst. 21 (2019) 59-67.

A. Benouareth, An efficient face recognition approach combining likelihood-based sufficient dimension reduction and LDA, Multimed. Tools Appl. 80 (2021) 1457-1486.

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/lionel-fourment-on-optimization-ai-and-inverse-analysis-in-forming/978-3-0364-1989-3

Key Engineering Materials, ISSN: 1662-9795, Vol. 1049, pp 89-98

doi: 10.4028/p-Jot8jG

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

Submitted:
2026-01-19

Revised:
2026-02-16

Accepted:
2026-03-02

Online:
2026-04-15

Machine Learning Algorithms Applied to the Identification of CPB06 Yield Criterion Parameters

The original version of this paper is available on https://www.scientific.net/KEM.1049.89.pdf

Armando Marques 1,5,a*, Tomás Parreira 1,5, b, Bernardete Ribeiro 2,c, Pedro Prates 1,3,4,5, d, André Pereira 1,5,e

¹CEMMPRE, Department of Mechanical Engineering, University of Coimbra, Coimbra, Portugal

2 CISUC, Department of Informatics Engineering, University of Coimbra, Coimbra, Portugal

3 TEMA, Centre for Mechanical Technology and Automation, Department of Mechanical Engineering, University of Aveiro, Aveiro, Portugal

4 LASI, Intelligent Systems Associate Laboratory, Guimarães, Portugal

5 ARISE, Advanced Production and Intelligent Systems Associated Laboratory, Porto, Portugal

a* armando.marques@uc.pt, b tomas.parreira@dem.uc.pt, c bribeiro@dei.uc.pt, d prates@ua.pt, e  andre.pereira@uc.pt

*corresponding author

Keywords: machine learning, constitutive modelling, parameter identification.

Abstract

In this work, a new methodology for the identification of the CPB06 yield criterion parameters is presented. This methodology is based on the application of Machine Learning models and the Levenberg-Marquardt optimization algorithm. The proposed methodology relies on data obtained from a biaxial tensile test with a cruciform specimen and aims to overcome some of the challenges usually faced during material characterization with the CPB06 yield criterion. The predictive performances achieved were positive overall, when comparing the yield surfaces obtained for testing cases, highlighting the potential of the proposed methodology.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1049.89.pdf

Sheet metal forming is a manufacturing process of significant relevance for very competitive industries, such as the automotive and aerospace industries, due to its capability to produce high quantities of components at a relatively low cost [1]. Industrial demands dictate the need for constant innovation, in order to guarantee that quality, safety and environmental requirements are met while maintaining profits. In this context, Finite Element Analysis (FEA) is frequently used during process design and optimization, which requires an accurate representation of the material's mechanical behaviour. This representation is usually based on constitutive models [2].

Constitutive models typically have multiple material parameters which are determined from the response of the material during mechanical tests. As constitutive models become more flexible, the number of material parameters increases, which makes the parameter identification process more complex. In this context, researchers are looking for more efficient parameter identification strategies, with a recent focus on the application of Machine Learning (ML) algorithms to the solution of this problem. For example, Guo et al. [3] applied a deep learning model containing a convolutional neural network tasked with filtering noise from the available input data, and a long-short term neural network which makes the final predictions. After comparing the results obtained by the model with those obtained from an inverse method, in this case Finite Element Model Updating [4], the authors highlighted one of the main advantages of machine learning based methods, which is the prediction efficiency after the initial model is trained. Morand et al. [5] identified the parameters of a hardening law with non-unique solutions by applying a mixture of experts, which contained multiple neural networks. Since the hardening law considered can have multiple sets of parameters that represent the same hardening behaviour, the authors defended that a single model is incapable of obtaining good predictive performances, since it tends to average the parameter sets. As such, an ensemble model should be used.

The application of ML algorithms to the identification of more complex constitutive models presents its own challenges, partially because of the increasing number of parameters to identify, but also due to the characteristics of certain constitutive models, such as the fact that the CPB06 yield criterion [6] contains both discrete and continuous parameters. The presence of discrete parameters in particular is a limiting factor to the application of the typically used regression ML algorithms [7]. In this context, the main goal of this work is the development of an efficient identification strategy for the CPB06 yield criterion parameters, based on the application of ML algorithms, and the Levenberg-Marquardt optimization algorithm, considering a dataset that contains the results of a single biaxial tensile test on a cruciform sample.

Procedure

The original version of this paper is available on https://www.scientific.net/KEM.1049.89.pdf

Numerical Models.

The original version of this paper is available on https://www.scientific.net/KEM.1049.89.pdf

In this work, the mechanical response of the materials was evaluated from the numerical results obtained from a biaxial tensile test on a cruciform sample. Fig. 1 a) presents the geometry of the cruciform sample, originally proposed in [8]. Each arm has a total length of 73.6 mm and the initial thickness is 0.5 mm . The specimen was discretized with a finite element mesh, as shown in Fig. 1 b ), which contains 6680 eight-node hexagonal solid elements. Only one eight of the specimen is modelled, due to material and geometrical symmetries. During the simulation, a prescribed displacement of 2 mm was imposed in the grip region of both arms, simultaneously, along the x and y directions. The in-house finite element code DD3IMP, developed and optimized for the simulation of metal forming processes, was used to carry out the simulation in this work [9-11].

[picture]

Fig. 1. a) Geometry of the cruciform specimen [8]; b) Finite element mesh used to discretize one eighth of the specimen.

Fig. 1. a) Geometry of the cruciform specimen [8]; b) Finite element mesh used to discretize one eighth of the specimen.

The constitutive model used for this work assumes an isotropic elastic behaviour, described by the generalized Hooke's law, and an anisotropic plastic behaviour, described by the orthotropic CPB06 yield criterion [6] and the Swift hardening law [12].

The CPB06 orthotropic yield criterion is based on the isotropic yield criterion given by:

[(|s1|−ks1)a+(|s2|−ks2)a+(|s3|−ks3)a]1a=Y,(1)

where s1,s2 and s3 are the principal values of 𝐬=σ′,σ′ is the deviatoric stress tensor and Y is the yield stress and its evolution during deformation. Parameter a is a positive integer, and k is a coefficient that represents the ratio between tensile and compression yield stresses, given by:

k=1−h1+h, with h=[2a−2(σTσC)a2(σTσC)a−2]1a(2)

where σT and σC are the yield stresses in tension and compression, respectively. If k=0, the responses in tension and compression are equal. For k=0 and a=2, this criterion is equivalent to von Mises. Convexity is guaranteed for a≥1 and k∈[−1,1]. To extend this criterion to orthotropy, a linear transformation is applied to the deviatoric stress tensor. In this new criterion, s1,s2 and s3 are the principal values of 𝐬=𝐂σ′, where 𝐂 is a constant fourth order tensor, containing 9 independent anisotropy coefficients, given by:

𝐂=[C11C12C13000C12C22C23000C13C23C33000000C44000000C55000000C66](3)

This criterion can be written in the form:

B[(|s1|−ks1)a+(|s2|−ks2)a+(|s3|−ks3)a]1a=Y(4)

where B is a constant that guarantees that the yield stress Y is identical to the tensile yield stress in the rolling direction. given by:

B=[1(|∅1|−k∅1)a+(|∅2|−k∅2)a+(|∅3|−k∅3)a]1a(5)

with

∅1=(2/3)C11−(1/3)C12−(1/3)C13,∅2=(2/3)C21−(1/3)C22−(1/3)C23,∅3=(2/3)C31−(1/3)C32−(1/3)C33.

(6)(7)(8)

This criterion can be reduced to von Mises, for k=0,a=2 and 𝐂 equal to the 4th  order identity tensor. When applied to metallic sheets, the condition C44=C55=1 as in isotropy is assumed. Also, C11 is usually considered equal to 1 , ensuring that all sets of parameters are unique.

The Swift hardening law is given by the following equation:

Y=K(ε0+ε¯p)n(9)

where ε¯p is the equivalent plastic strain and K,ε0 and n are material parameters. The initial yield stress, Y0, is given by:

Y0=Kε0n(10)

Dataset Generation.

The original version of this paper is available on https://www.scientific.net/KEM.1049.89.pdf

The dataset used in this work contains the numerical results of the biaxial tensile tests as input data. This data includes force measurements along the 0x and θy axis, as well as the εxx,εyy and εxy components of the strain tensor. The forces were measured for displacement increments of 0.4 mm , up to a total displacement of 2 mm , considering that this displacement increment is the same along both axes. The strain measurements were obtained from the final instant of the mechanical test, which corresponds to a displacement of 2 mm . In order to improve efficiency, the dataset size was limited,
by considering only the strain measurements that are most sensitive to the variation of the constitutive model, this sensitivity was evaluated by comparing the standard deviation of the strain values across the entries in the dataset. In total, 15 strain measurements were considered, which were obtained from 10 different points in the specimen. Fig 2 shows the distribution of the Cauchy stress tensor components σxx,σyy and σxy on the cruciform specimen, obtained for the isotropic case. It can be observed that shear and biaxial stress data can be obtained in the notched region of the specimen, while the arms provide uniaxial stress data. As such, it is expected that the information obtained from this mechanical test will allow for a proper identification of the yield criterion parameters.

[picture]

Fig. 2. Stress fields obtained for a biaxial tensile test for a total displacement of 2 mm , a) σxx, b) σyy, c) σxy.

Fig. 2. Stress fields obtained for a biaxial tensile test for a total displacement of 2 mm , a) σ x x , b) σ y y , c) σ x y .

In order to be able to identify the constitutive model parameters independently of the elastic behaviour of the material, the Young's Modulus is also considered as an input ( E ). The output data consists of the parameters to be identified, namely the C22,C33,C66,C12,C13,C23 (collectively referred to as Cij from now on) and α parameters of the CPB06 yield criterion. The limits of the Cij parameters are presented in Table 1, which correspond to a deviation of ±1.5 relatively to the isotropic values. For the α parameter, which is a positive integer, the values of 2 and 4 were considered. Since the mechanical test considered does not contain compression data, tension compression symmetry will be considered, and as such, k=0 was assumed for all materials. Also, 3 different values of E were considered, 70,140 and 210 GPa , which are typical values for Aluminium, Titanium and Steel alloys, respectively.

Table 1. Limits of each parameter used to generate the material constitutive model parameter sets.

| C₂₂ | C₃₃ | C₆₆ | C₁₂ | C₁₃ | C₂₃
Lower Limit | -0.5 | -0.5 | -0.5 | -1.5 | -1.5 | -1.5
Upper Limit | 2.5 | 2.5 | 2.5 | 1.5 | 1.5 | 1.5

The dataset can be divided into a training set and a testing set. The training set is used during the training process by the algorithms, while the testing set is used to evaluate the predictive performance of the trained models. The training set contains 4500 different sets of constitutive parameters for each value of α, while the testing set contains 500 sets of parameters for each value of α. These parameter sets were generated by a Sobol sequence, which provides a more uniform parameter distribution than a purely random distribution. After the parameter sets were established, each material was assigned a value of E, in a uniform distribution.

Machine Learning.

The original version of this paper is available on https://www.scientific.net/KEM.1049.89.pdf

Machine learning (ML) is a field of study focused on the development of systems capable of improving themselves without explicit programming [13]. ML algorithms are frequently applied to the solution of complex problems which would otherwise require building first-principle models which can be prohibitively expensive and time-consuming. The ML algorithm used in this work is the Multi-Layer Perceptron (MLP), a supervised algorithm [14]. Supervised algorithms learn from labelled data, through a process typically called training. This process consists of mapping the relations between input and output variables in the training data. During the training process, model internal parameters are calibrated, allowing it to predict outputs for new sets of inputs.

The MLP is a feed forward neural network, which can be applied to both classification and regression problems, and supports multi-output problems natively [15]. An MLP model is composed of multiple layers of nodes, in which each node connects to the nodes of the following layer, passing information. The first layer is called input layer and contains one node per input variable in the dataset. This layer is followed by any number of layers called hidden layers. The number of hidden layers, as well as the number of nodes in each of these layers are defined before the training process. The output of a single node from a hidden layer is given by:

zi=∅(∑jwijzj′+di)(11)

where zi is the output of node i from the current layer, z′j is the output of j from the previous layer, wij is the weight associated with z′j,di is a bias term and ∅ is a non-linear activation function. The last layer, called the output layer, is responsible for making the final predictions, based on the information received from the previous layer. The training process of an MLP model is an iterative process based on the backpropagation algorithm [16]. For each iteration, the algorithm evaluates whether each of the model weights should be increased or decreased in order to increase predictive performance, and then all weights of the network are changed accordingly, by a small increment.

Methodology

The original version of this paper is available on https://www.scientific.net/KEM.1049.89.pdf

The proposed methodology for the identification of the CPB06 yield criterion parameters includes the application of both classification and regression MLP models, which were constructed with the publicly available python library Scikit-learn [17]. The first step in the methodology is the identification of the α parameter. This step is accomplished by a classification model, which was trained with the mechanical test results (forces and strains) and the elastic parameter E as inputs. Then, the remaining parameters are identified. Due to the complexity of this identification, an iterative optimization process is adopted. First, a ML model is trained for each value of the parameter α considered. These models use the constitutive model parameters as input and predict the results of the mechanical test considered in the dataset. Then, the results of this prediction are compared with the real results, and through the Levenberg-Marquardt algorithm, the Cij parameters of the CPB06 yield criterion are optimized. A schematic representation of the proposed methodology is presented in Fig 3.

[picture]

Fig. 3. Schematic representation of the proposed methodology.

Fig. 3. Schematic representation of the proposed methodology.

During the optimization process, the value of the α parameter considered is the one identified in the first step of the methodology. The objective function used in the optimization process is the weighted sum of squared residuals function, and finite differences are used to compute the Jacobian matrix, which is required for the application of the Levenberg-Marquardt algorithm.

Model training.

The original version of this paper is available on https://www.scientific.net/KEM.1049.89.pdf

The ML model for the first identification step was trained with a total of 4000 materials from the training set, 2000 corresponding to each value of the α parameter. For the second identification step, 2 models were trained, one for each value of α. Each of those models was trained with a total of 4500 materials. All data was scaled with z-score normalization [18]. Each model had its hyperparameters optimized before the training of the final models. In this work, this optimization process consists of a trial-and-error approach, in which the training set is divided, randomly into two temporary sets, one to train temporary models and another to validate the performance of these temporary models. The validation sets include 30% of the total training set data, and the random data split is repeated 10 times. The hyperparameters that led to the best average performance across the temporary models were chosen to train the final model, with all the training set data. These hyperparameters include the tanh activation function and the limited-memory BFGS (LBFGS) solver.

Performance Evaluation.

The original version of this paper is available on https://www.scientific.net/KEM.1049.89.pdf

The predictive performance of the ML models was evaluated with performance metrics. Two metrics were considered, accuracy and relative error (RE). Accuracy is given by the following equation:

Accuracy = (Correct predictions)/(All predictions).

This metric represents the ratio of correct prediction made by a model and is used to evaluate the classification ML model used to identify the parameter α. RE is given by:

RE=yi−yi*yi,(13)

where y and y* are the real and predicted values for the variables in analysis, respectively. This metric is used to evaluate the performance of regression models.

Results

The original version of this paper is available on https://www.scientific.net/KEM.1049.89.pdf

In order to evaluate the predictive performance of the proposed methodology, the trained models were used to make predictions for the testing set, and the performance metrics were calculated. The ML model responsible for the first identification step, which is the identification of the α parameter, achieved an accuracy of 83.55%. While this performance leaves room for improvement, it can be considered acceptable, especially considering that the impact of an incorrect prediction in this step can only be properly measured when the remaining CPB06 parameters are identified, and the resulting mechanical behaviours can be compared. This comparison, after the remaining CPB06 parameters are identified in the second identification step, is based on various results, including the values of yield stress for uniaxial tension ( σ ), measured for angles of 15∘,30∘,45∘,60∘,75∘ and 90∘ in relation to the rolling direction, shear stress (τ), for angles of 0∘,15∘,30∘ and 45∘ in relation to the rolling direction, and the Lankford coefficients, or anisotropy coefficients ( r ) for angles of 0∘,15∘,30∘,45∘, 60∘,75∘ and 90∘ in relation to the rolling direction. Additionally, the values of σb and rb, which are the yield stress under biaxial loading and biaxial anisotropic coefficient [19], are considered. It is preferable to compare these values, which partially describe the yield surface, instead of comparing the yield criterion parameters directly. This is because multiple combinations of parameters can lead to very similar yield surfaces, due to interactions between parameters.

The results obtained after the second identification step are shown in Fig. 4 and 5. These results are separated by the values of α of the original materials in the testing set. However, it is worth mentioning that the value of α considered in each prediction is the one obtained from the first identification step.

[picture]

Fig. 4. Relative Error distribution obtained after the prediction of the CPB06 yield criterion parameters, for the materials with a real α value of 2 .

Fig. 4. Relative Error distribution obtained after the prediction of the CPB06 yield criterion parameters, for the materials with a real α value of 2 .

[picture]

Fig. 5. Relative Error distribution obtained after the prediction of the CPB06 yield criterion parameters, for the materials with a real α value of 4 .

Fig. 5. Relative Error distribution obtained after the prediction of the CPB06 yield criterion parameters, for the materials with a real α value of 4 .

Overall, the predictive performances obtained are considered positive. The yield stresses in particular show a good agreement between the real yield surfaces and the yield surfaces that correspond to the identified parameters, with RE values that are frequently under 2%, and median RE values under 1%, for both values of α. Regarding the anisotropy coefficients, except rb, the RE values achieved are higher than the ones for the yield stresses, particularly for the α value of 4, but they can still be considered positive, with median RE values under 5% for all cases. These positive results show that the incorrect identifications for the value of α, during the first identification step, didn't significantly compromise the overall quality of the identification process. This means that during the optimization process, the proposed methodology could identify sets of CPB06 parameters that closely matched the mechanical behaviour of the real parameter sets.

The values of RE obtained for rb are higher than those obtained for the remaining variables, particularly for the α value of 4 . This difference may be explained by the available input data. The input data considered may not accurately represent the biaxial stress state associated with this anisotropic coefficient. The inclusion of more input data, representative of other stress states, can help improve the proposed methodology, however, it is worth noting that this will likely require more mechanical tests to be performed, which will increase the cost of the identification process.

Conclusion

The original version of this paper is available on https://www.scientific.net/KEM.1049.89.pdf

In this work, a parameter identification methodology for the CPB06 yield criterion was proposed. This methodology pairs the use of regression and classification ML models, with the application of the Levenberg-Marquardt optimization algorithm. The dataset created for this purpose contains the results of a biaxial tensile test with a cruciform specimen, which was used to train and test the ML models and the general predictive performance of the methodology.

The results highlight the potential of the proposed methodology, with the predicted yield surfaces showing good agreement with the real ones from the testing set, especially when comparing yield stress values. Improvements to the current methodology are possible by including more data, which represents different stress-states.

The methodology presented leaves some questions to explore in future work. First, the methodology will be expanded for more values of α. The exploration of alternative ML algorithms is
also of interest. In order to validate the methodology for an industrial environment, a more rigorous definition of the strain measurement points is necessary, and the robustness to the presence of noise in the data should also be explored. This can be achieved either with real experimental data, or with numerical data with artificial noise. Finally, the performance and efficiency of the methodology will be compared with other parameter identification methodologies.

Acknowledgement

The original version of this paper is available on https://www.scientific.net/KEM.1049.89.pdf

This research is sponsored by national funds through FCT - Fundação para a Ciência e a Tecnologia, under projects UID/00285/2025 (https://doi.org/10.54499/UID/00285/2025) and LA/P/0112/2020 (https://doi.org/10.54499/LA/P/0112/2020). It was also supported by the projects RealForm (ref. 2022.02370.PTDC), SmartBending (ref. 2023.14606.PEX), LSD-TRIP (ref. COMPETE2030-FEDER-00778700) and A3Shell (ref. COMPETE2030-FEDER-02233400). Tomás Parreira was supported by a grant for scientific research from FCT (ref. 2024.00961.BD). All support is gratefully acknowledged.

References

The original version of this paper is available on https://www.scientific.net/KEM.1049.89.pdf

Marques AE, Parreira TG, Pereira AFG, et al (2025) Machine learning application to the identification of sheet metal constitutive model parameters. Machine Learning for Computational Science and Engineering 1:. https://doi.org/10.1007/s44379-024-00006-8.

Yoshida F, Hamasaki H, Uemori T (2013) A user-friendly 3D yield function to describe anisotropy of steel sheets. International Journal of Plasticity 45:119-139. .

Guo Z, Bai R, Lei Z, et al (2021) CPINet: Parameter identification of path-dependent constitutive model with automatic denoising based on CNN-LSTM. European Journal of Mechanics, A/Solids 90:104327. .

Prates PA, Pereira AFG, Sakharova NA, et al (2016) Inverse Strategies for Identifying the Parameters of Constitutive Laws of Metal Sheets. Advances in Materials Science and Engineering 2016:. .

Morand L, Helm D (2019) A mixture of experts approach to handle ambiguities in parameter identification problems in material modeling. Computational Materials Science 167:85-91. .

Cazacu O, Plunkett B, Barlat F (2006) Orthotropic yield criterion for hexagonal closed packed metals. International Journal of Plasticity 22:1171-1194. .

Marques AE, Parreira TG, Pereira AFG, et al (2024) Machine learning applications in sheet metal constitutive Modelling: A review. International Journal of Solids and Structures 303:. .

Prates PA, Oliveira MC, Fernandes J V. (2014) A new strategy for the simultaneous identification of constitutive laws parameters of metal sheets using a single test. Computational Materials Science 85:102-120. .

Menezes LF, Teodosiu C (2000) Three-dimensional numerical simulation of the deep-drawing process using solid finite elements. Journal of Materials Processing Technology 97:100-106. .

Oliveira MC, Alves JL, Menezes LF (2008) Algorithms and strategies for treatment of large deformation frictional contact in the numerical simulation of deep drawing process. Archives of Computational Methods in Engineering 15:113-162. .

Neto DM, Oliveira MC, Menezes LF (2017) Surface Smoothing Procedures in Computational Contact Mechanics. Archives of Computational Methods in Engineering 24:37-87. .

Swift HW (1952) Plastic instability under plane stress. Journal of the Mechanics and Physics of Solids 1:1-18. .

Jang DP, Fazily P, Yoon JW (2021) Machine learning-based constitutive model for J2plasticity. International Journal of Plasticity 138:102919. .

Zhou ZH (2018) A brief introduction to weakly supervised learning. National Science Review 5:44-53. .

Murtagh F (1991) Multilayer perceptrons for classification and regression. Neurocomputing 2:183-197. .

Dreyfus SE (1990) Artificial Neural Networks, Back Propagation, and the Kelley-Bryson Gradient Procedure. Journal of Guidance, Control, and Dynamics 13:926-928. .

Pedregosa F, Varoquaux G, Gramfort A, et al (2012) Scikit-learn: Machine Learning in Python. Journal of Machine Learning Research 12:2825-2830.

Pandey A, Jain A (2017) Comparative Analysis of KNN Algorithm using Various Normalization Techniques. International Journal of Computer Network and Information Security 9:36-42. .

Barlat F, Brem JC, Yoon JW, et al (2003) Plane stress yield function for aluminum alloy sheets - Part 1: Theory. International Journal of Plasticity 19:1297-1319. .

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/lionel-fourment-on-optimization-ai-and-inverse-analysis-in-forming/978-3-0364-1989-3

Key Engineering Materials, ISSN: 1662-9795, Vol. 1049, pp 99-106

doi: 10.4028/p-1L1zVC

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

Submitted:
2026-01-19

Revised:
2026-03-02

Accepted:
2026-03-02

Online:
2026-04-15

Geometry-Based Concept for Automatic Cut Planning and Flattening of Sheet Metal Components

The original version of this paper is available on https://www.scientific.net/KEM.1049.99.pdf

Jannis Korn 1,a*, Clemens Acksteiner 1, b, Sebastian Langula 1,c, Christina Guilleaume 1, d and Alexander Brosius 1,e

1 Chair of Forming and Machining Technology, TUD Dresden University of Technology, 01062 Dresden, Germany

a jannis.korn@tu−dresden.de, b  clemens.acksteiner@tu-dresden.de, c  sebastian.langula1@tu-dresden.de, d  christina.guilleaume@tu-dresden.de, e alexander.brosius@tu-dresden.de

Keywords: Sheet Metal Forming, Flattening, Cut Planning, Tailored Blanks, Remanufacturing

Abstract

Sheet metal components with complex geometries are typically recycled by remelting. Direct remanufacturing necessitates the flattening of parts, which requires the implementation of cuts to facilitate unwinding. The exact positioning of these cuts is a complex planning task, because several influencing factors can be considered, such as material usage, ease of flattening, or minimal forming required. This study presents a geometry-based concept addressing this challenge and demonstrates its use for a test geometry. The finite element method is applied to simulate the flattening process of the resulting sections, and the results are evaluated in terms of planarity and induced plastic strain. The findings of the present work indicate a discernible dependency of results on the selection of the flattening directions. In particular, curved areas impact the induced plastic deformation and springback of flattened sections. This is a crucial consideration when planarity is prioritised over material utilisation.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1049.99.pdf

Sheet metal forming is an integral process in today's manufacturing with global demand rising for decades. In light of increasing orientation towards and necessity for a more circular economy, current concepts of recycling and reuse of sheet metal must be reconsidered [1].

A promising approach involves the direct remanufacturing of existing sheet metal components, which we refer to hereinafter as "first life" components ( 1st  LCs), with forming processes avoiding a transition through the liquid phase [2]. Despite the novelty of this research domain, pioneering studies have been conducted to assess its feasibility. In [3], the authors demonstrated the extraction of several deep-drawing blanks from a used car bonnet. Another study has considered the effects of flattening on the production of recycled parts from 1st  LCs, such as the creation of a break disc cover from a car roof [4]. In both studies, large components with simple geometries and small curvatures were chosen. Improving both material utilisation and applicability of the concept to a wider variety of inputs demands cutting the 1st  LCs before flattening. Subsequently, sections too small to be remanufactured directly can be utilised as patches for the recombination into tailored blanks. When manufacturing " 2nd  life" components out of those recombined blanks, however, the manufacturing processes have to be reconsidered. A feasible example is the use of macrostructured deep drawing tools to facilitate a larger process window, such as larger irregularities of the blanks [5].

It is evident from a survey of the relevant literature that a number of approaches for the unfolding of three-dimensional sheet metal geometries can be found. Flat pattern development using computeraided methods is frequently based on geometry-based concepts, as the one presented in [6], or on the optimisation of other metrics as in [7]. However, these existing methods are oriented towards the creation of blanks to enable the forming of their initially considered components. In contrast, there has been no publication of concepts that analyse the explicit placement of cuts and the evaluation of the flattenability of the resulting sections for 1st  LCs intended for direct remanufacturing, for example into second life tailored blanks.

Accordingly, this work addresses the following research question: How can a geometry-based, automated cut planning strategy be designed to generate flattenable sections from first life components for subsequent use as second life sheet metal blanks? Within this study, a concept addressing this question is presented and employed to demonstrate first results. Additionally, finite element analysis is utilised to simulate the flattening of the resulting cut sections and the outcomes are evaluated in terms of flatness and induced plastic strain. As example geometry for the demonstration of the process we choose a T-cup as shown in Fig. 1a. It offers moderate complexity and incorporates several aspects typical to deep drawing parts such as plane surfaces and features with single as well as double curvatures with a range of different radii. Further details on the geometry can be found in [8]. The process is aimed to be performed on a surface mesh derived from a 3D scan of a 1st  LC serving as its digital twin. In this work, however, a manually constructed mesh is used to reduce artefacts.

[picture]

Fig. 1. (a) Isometric view of the T-cup and visualisation of Gaussian curvature. (b) Critical zones for flattening with the flattening planes shown. The determination is based on the angle of the face normals in relation to the direction of flattening.

Fig. 1. (a) Isometric view of the T-cup and visualisation of Gaussian curvature. (b) Critical zones for flattening with the flattening planes shown. The determination is based on the angle of the face normals in relation to the direction of flattening.

Methods and Simulation Procedure

The original version of this paper is available on https://www.scientific.net/KEM.1049.99.pdf

The methodology proposed in this paper as a first concept towards an automatic placement of the required cuts is motivated by the observation that areas orthogonal to the flattening plane can lead to instability and buckling during the flattening process. Consequently, this study suggests a semiautomatic algorithm to remove these areas of conflict subsequently. The individual modules of the process are shown in Fig. 2.

The discrete surface mesh of a 1st LC is considered the input for the subsequent algorithm. When applied to a digital twin, this can be obtained from a 3D scan of the geometry and subsequent mesh generation from the obtained data. For the purpose of this work, a mesh with quadrilateral shell elements was considered, obtained from a component designed using CAD. The coordinate system is aligned with the prominent features and symmetries of the T-cup. In the subsequent modules, the preparation and cutting of the mesh are performed in Python using a combination of self-written algorithms and mesh operations using the package PyVista [9].

[picture]

Fig. 2. Schematic overview of the process.

Fig. 2. Schematic overview of the process.

Preparation. In preparation for the cut planning, the Gaussian curvature is initially computed to remove regions with nonzero values beyond a predefined tolerance range. This helps reduce the problem's complexity as areas with nonzero curvatures are generally considered non-developable without additional information [10]. The tolerance range is identified by means of manual inspection, thus ensuring that only areas exhibiting double curvature are removed. For the test geometry used in this study, this range was set to [−10−4 mm−2,10−4 mm−2] to remove the visibly distinctive regions of double curvature seen in Fig. (1a).

In order to circumvent any modification of the residual geometry, the removal is executed by identifying the boundaries of the designated areas. The thus obtained nodes are duplicated. Adjacent faces are sorted according to their relative position to the edges. Finally, the faces to one side of the boundary are redefined to use the duplicated instead of the original nodes. By identifying unconnected regions of the mesh, the parts are then separated.

The remaining test geometry is remeshed before the application of the subsequent algorithms. This ensures a regular and smooth mesh, thereby reducing artefacts.

Cut Planning. The thus prepared mesh is then used as an input to the proposed algorithm for placing the cuts. Initially, a flattening plane is defined via the flattening direction, i.e. its normal vector n→fp, as shown in Fig. 1b. After manually setting a tolerance angle θtol , the angles φi between n→fp and all face normals n→i are calculated. For the study of our test geometry, a value of 10∘ was selected as it ensured the exclusion of artefacts. The outer boundary of sections with angles of 90∘±θtol  is determined as cutting paths. Finally, the mesh is separated along those lines in the same way the doubly curved areas were removed. Sections with angles outside the defined interval are considered flattenable within the plane defined by n→fp while the removed sections can be reconsidered with a different flattening plane to improve material utilisation.

To prepare the obtained sections for the subsequent finite element analysis, they are semiautomatically placed. This process is initiated by subjecting them to a gravitational simulation above an impenetrable plane. Fully automating the procedure requires more research on how to properly account for various challenges such as metastable positions.

Simulation of the Flattening Process. To analyse the flattening behaviour, finite element simulations are carried out using LS-DYNA (R15.0.2) with implicit time integration. The simulation model is visualised in Fig. 3a. It comprises two rigid plane dies with the parts to be flattened placed in between them. All parts are meshed with fully integrated quadrilateral shell elements (ELFORM -16) with 7 through thickness integration points and 1 mm element edge size.

[picture]

Fig. 3. (a) Simulation model and (b) plastic strain after springback.

Fig. 3. (a) Simulation model and (b) plastic strain after springback.

With regard to material modelling, the sections consist of DP600 steel assuming isotropic properties. This dual phase steel is characterised by good formability and is commonly used in the automotive sector. The blank thickness and the parameters for modelling the dual phase steel are taken from [11]. They are listed in Table 1. Note that flow stress σf is given as a function of effective plastic strain ε p.

During the flattening process the dies are closed via position control of the upper die (*BOUNDARY_PRESCRIBED_MOTION_RIGID) until their distance equals the initial blank
thickness of 1.16 mm . In the course of this, a Coulomb friction model is applied with a friction coefficient of 0.1 . After flattening, a static springback simulation is performed (Fig. 3b).

Table 1. Parameters for modelling DP600 [11].

Parameter | Value
Flow stress σf [MPa] | 1060 (0.001 + εp)0.174
Young's modulus [GPa] | 185
Poisson's ratio [-] | 0.3
Sheet thickness [mm] | 1.16

Evaluation of the Flattened Parts. Following flattening, the planarity has to be analysed. In this work, this is done by assessing the distances of the finite nodes to a best fit plane. From literature it is known that one can use principal component analysis (PCA) with the aid of singular value decomposition (SVD) to calculate a best fitting plane [12]. Thus, the plane is described via the centroid of the point data and the third principal component as the normal vector of the plane. Thereafter, the distances to the obtained plane and in particular their extrema in positive and negative direction are computed (Fig. 4).

[picture]

Fig. 4. (a) Distances to best fit plane. (b) Planes of maximum distances in positive and negative direction to the best fit plane.

Fig. 4. (a) Distances to best fit plane. (b) Planes of maximum distances in positive and negative direction to the best fit plane.

Results

The original version of this paper is available on https://www.scientific.net/KEM.1049.99.pdf

Cut Planning. In order to find a discrete approach for choosing the flattening plane normal, the scope of this study was limited to the planes defined by the standard base vectors of the chosen coordinate system. This leads to 6 different permutations for the sequential application of the algorithm. Here, only two applications were considered as it does not reduce the number of studied cases. To indicate the nature of their creation, cut sequences are identified based on the normals of the flattening plane used, i.e. "cut parts XY" are generated by considering flattening directions in x and y direction subsequently.

Table 2. Quantitative results of cut planning.

Sequence of flattening directions | Number of resulting sections | Degree of material utilisation [%]
XY | 10 | 90.4
XZ | 8 | 49.3
YX | 10 | 90.4
YZ | 10 | 90.4
ZX | 8 | 60.8
ZY | 10 | 90.4

Of the 6 examined cut results, combinations of the same flattening directions with different permutations lead to almost identical sections. The number of resulting sections after consideration of two flattening planes is compiled in Table 2 together with the degree of material utilisation. To
assess the latter quantity, the surface area of all flattenable sections was divided by the total area of the initial mesh before preparation without taking into consideration the quality of the flattened sections. A more informed assessment of effective material utilisation requires additional research on the requirements of the recombination step mentioned in the introduction of this work. The flatness of components, both in terms of local edges and global geometry, is likely to act as a limiting factor to the areas considered usable. Depending on the process design and the geometry of the 1st  LC, it will furthermore be important to consider the impact of existing and induced plastic strains, the result of which may be additional limitations. While these factors may result in the evaluation of certain sections as unusable and subsequently in the development of more complex strategies for the cut planning, they represent areas that require further investigation and therefore did not form the primary focus of the present work.

[picture]

Fig. 5. Overview and definition of nomenclature for resulting cut sections after algorithm application for flattening plane normal in y - and z -direction with different orders of application. Sections with similar geometries are coloured identically for better visual distinction.

Fig. 5. Overview and definition of nomenclature for resulting cut sections after algorithm application for flattening plane normal in y - and z -direction with different orders of application. Sections with similar geometries are coloured identically for better visual distinction.

The outcomes derived from analogous flattening directions in disparate permutations primarily diverge in the allocation of the singly curved areas between the sidewall and the flange, or the bottom, of the T-cup. This phenomenon can be observed for the four combinations with equal degree of material utilisation, an illustrative example is provided by the large U section in Fig. 5. The remaining two combinations, XZ and ZX, do not consider flattening in the direction of the original drawing process. Therefore, a third iteration of the algorithm would be necessary to improve their material utilisation, as there remain unflattenable sections after the initial two.

Depending on the missing flattening direction, it is evident that the cut sections exhibit distinct geometries, which can only be accurately compared in pairs. Consequently, the ensuing analysis will concentrate exclusively on the sections derived from the YZ and ZY cuts. Of the combinations examined, these provide the broadest spectrum of geometrical features and, consequently, a better insight into effects on flattenability. A descriptive nomenclature for the different sections is chosen, its details are shown in Fig. 5 . Symmetrical parts were only simulated once per cut permutation while the band sections were omitted completely due to their small size and pre-existing planarity.

Flattening. After cutting, the resulting sections are flattened via the presented simulation process. In Fig. 6, the differences of the minimum and maximum signed distances Δs as well as the standard deviation σ(s) of distances to the best fit plane are visualised for each considered section and cutting sequence. Sections comprising the most proportion of initially flat area, namely flange, bottom and rectangular section, have the smallest differences Δs and standard deviations σ(s). The values are declining in the aforementioned order. Bottom and flange of cut parts ZY have higher planarity compared with their counterparts. This is due to cut parts YZ producing more contiguous radii for these sections.

[picture]

Fig.6. Distances to best fit plane after flattening.

Fig.6. Distances to best fit plane after flattening.

Fig. 7 exhibits the distribution of effective plastic strains in a violin plot. Two local maxima can be seen in the distribution for each of the U sections. The one at higher strains is corresponding to the flattened area of the initially eponymous curvatures. Conversely, the second maximum can be attributed to the originally flat area. For the large U sections, a major proportion of these parts is experiencing a negligible amount of plastic strain. In contrast, almost the whole small U sections are plastically deformed. Cut parts ZY leads to higher plastic strains for the Usections as the contiguous radii account for a larger proportion of those parts. These radii are resulting in lower planarity for the small U section, while for the large one the difference Δs is almost the same (Fig. 6).

Finally, the nearly identical S section cuts result in similar planarity and plastic strain distributions. The two peaks in the violin plot in Fig. 7 indicate the two main radii of those sections. Here, the smaller radius and the associated smaller area cause higher strains but a reduced contribution to the overall distribution.

[picture]

Fig. 7. Distribution of effective plastic strain after flattening.

Fig. 7. Distribution of effective plastic strain after flattening.

Cut sections

The original version of this paper is available on https://www.scientific.net/KEM.1049.99.pdf

An exemplary display of the spatial deviations from planarity is shown in Fig. 8 for the flange and small U section because for those sections the influence of the previously mentioned contiguous radii
is clearly visible. For the flange, an overall bending with uplifting sides can be observed for cut parts YZ . In comparison, the more pronounced curvatures for the small U section of cut parts ZY provokes tilting of the flattened specimen around the x -axis.

[picture]

Fig. 8. Height profiles after flattening.

Fig. 8. Height profiles after flattening.

Conclusion and Outlook

The original version of this paper is available on https://www.scientific.net/KEM.1049.99.pdf

This study proposes an approach to the flattening of sheet metal components by means of inducing cuts along regions orthogonal to the flattening plane and demonstrates the procedure on the example geometry of a T-cup. By analysing the resulting sections using a finite element simulation, it was shown that the choice of flattening directions is not only pivotal for the number of cuts required but also influences the flattenability of the resulting sections. Both induced amount of plastic strain and springback are influenced by radii in the geometry. While regions with large curvature had a distinctive influence on sections with overall small to no curvature, their effects on more curved geometries were less pronounced. This emphasises the significance of curved areas for the flattenability of 1st  LCs and underscores the necessity for further investigation.

Our findings indicate that the proposed concept has the potential to serve as a viable building block for the direct recycling of 1st  LCs. However, it is crucial to note that further investigation into alternative geometries is necessary to ensure the applicability of this method. It is evident that the approach for the cut placement is geometry-based; consequently, mechanical properties have not been considered, despite their obvious influence on the problem. Moreover, it is imperative that the robustness and automation of the process be further developed prior to its realistic application to digital twins.

Acknowledgements

The original version of this paper is available on https://www.scientific.net/KEM.1049.99.pdf

This work was supported by the Werner Siemens Foundation under the project "Second Life Metal Components - A Pathfinding Project for Upcycling".

The authors gratefully acknowledge the computing time made available to them on the highperformance computer at the NHR Centre of TU Dresden. This centre 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).

References

The original version of this paper is available on https://www.scientific.net/KEM.1049.99.pdf

J. M. Allwood, M. F. Ashby, T. G. Gutowski, and E. Worrell, 'Material efficiency: A white paper', Resources, Conservation and Recycling, vol. 55, no. 3, pp. 362-381, Jan. 2011, doi: 10.1016/j.resconrec.2010.11.002.

J. M. Allwood, J. M. Cullen, and R. L. Milford, 'Options for Achieving a 50% Cut in Industrial Carbon Emissions by 2050', Environ. Sci. Technol., vol. 44, no. 6, pp. 1888-1894, Feb. 2010, doi: 10.1021/es902909k.

J. A. Österreicher, F. Grabner, P. Oberhauser, S. L. Hovden, and C. M. Schlögl, 'Toward a Circular Economy in Deep Drawing: Remanufacturing End-of-Life Automotive Parts', MATEC Web Conf., vol. 408, p. 02026, 2025, doi: 10.1051/matecconf/202540802026.

R. Haase, D. Farioli, R. Selbmann, M. Werner, and V. Kräusel, 'Dismantling and remanufacturing strategies in the automotive sector', Procedia CIRP, vol. 122, pp. 695-700, 2024, doi: 10.1016/j.procir.2024.01.096.

A. Mousavi, "A novel approach towards a lubricant-free deep drawing process via macrostructured tools," Ph.D. dissertation, Chair of Forming and Machining Technology, TU Dresden, Dresden, Germany, 2019.

X. Liu, S. Li, X. Zheng, and M. Lin, 'Development of a flattening system for sheet metal with free-form surface', Advances in Mechanical Engineering, vol. 8, no. 2, Feb. 2016, doi: 10.1177/1687814016630517.

Q. Liu, J. Xi, and Z. Wu, 'An energy-based surface flattening method for flat pattern development of sheet metal components', Int J Adv Manuf Technol, vol. 68, no. 5-8, pp. 11551166, Apr. 2013, doi: 10.1007/s00170-013-4908-y.

A. Wolf et al., 'Modeling metal forming of a magnesium alloy using an adapted material model', Engineering Reports, vol. 4, no. 7-8, June 2022, doi: 10.1002/eng2.12540.

C. Sullivan and A. Kaszynski, 'PyVista: 3D plotting and mesh analysis through a streamlined interface for the Visualization Toolkit (VTK)', JOSS, vol. 4, no. 37, p. 1450, May 2019, doi: 10.21105/joss. 01450 .

A. Cucakovic A., B. Jovic S., and M. Tripkovic R., 'Paper strips driven design: Application on doubly curved surfaces', FME Transaction, vol. 45, no. 2, pp. 251-255, 2017, doi: 10.5937/fmet1702251c.

N. Küsters, "Halbanalytische Methode zur Charakterisierung der Fließortkurven von Blechwerkstoffen," Ph.D. dissertation, Chair of Forming and Machining Technology, TU Dresden, Dresden, Germany, 2020.

A. Nurunnabi, D. Belton, and G. West, 'Diagnostics based principal component analysis for robust plane fitting in laser data', 16th Int'l Conf. Computer and Information Technology. IEEE, pp. 484-489, Mar. 2014. doi: 10.1109/iccitechn.2014.6997319.

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/lionel-fourment-on-optimization-ai-and-inverse-analysis-in-forming/978-3-0364-1989-3

Key Engineering Materials, ISSN: 1662-9795, Vol. 1049, pp 107-114

doi: 10.4028/p-jZ5Rxd

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

Submitted:
2026-01-19

Revised:
2026-02-17

Accepted:
2026-03-02

Online:
2026-04-15

Towards Single-Test Anisotropy Calibration: Sensitivity, Identifiability and Validation of YLD2000-2d

The original version of this paper is available on https://www.scientific.net/KEM.1049.107.pdf

Bojan Starman 1,a*, Andraž Maček 2, b, Miroslav Halilovič 1,c, Pascal Lava 3, d, Fabrice Pierron 3,e and Sam Coppieters 2,f

1 Faculty of Mechanical Engineering, University of Ljubljana, Ljubljana, Slovenia

2 KU Leuven, Gent, Belgium

3 MatchID, Gent, Belgium

a  bojan.starman@fs.uni-lj.si, b  andraz.macek@kuleuven.be, c  miroslav.halilovic@fs.uni-lj.si, d  pascal.lava@matchid.eu, efabrice.pierron@matchid.eu, f  sam.coppieters@kuleuven.be 

*corresponding author

Keywords: MT2.0, finite element model updating, DIC, material characterization, anisotropy.

Abstract

This paper revisits the long-standing question of how to fully characterise the in-plane plastic anisotropy of sheet metals without assembling evidence from multiple standardised tests. The central idea is pragmatic: a single, well-designed heterogeneous biaxial experiment can replace the conventional combination of uniaxial and equibiaxial tests if the specimen and the inverse identification method are co-designed to (i) activate informative stress states and (ii) maintain low strain gradients for accurate digital image correlation measurements. The proposed cruciform specimen is deliberately conceived as a benchmark configuration for full-field inverse identification, with known locations and stress-strain states at which relevant material information is embedded. The approach is coupled with a Finite Element Model Updating framework, enabling all anisotropy parameters of the YLD2000-2d model to be identified from a single full-field dataset. Sensitivity and identifiability analyses demonstrate that a physically based parameter formulation significantly improves the conditioning of the inverse problem. Virtual experimentation confirms the robustness and accuracy of the proposed "one-test" identification strategy.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1049.107.pdf

The predictive capability of numerical simulations in sheet metal forming critically depends on the accurate identification of constitutive model parameters, particularly those governing in-plane plastic anisotropy. Traditionally, these parameters are obtained from a collection of standardised mechanical tests, including uniaxial tensile tests performed along multiple material directions and equibiaxial loading experiments. While such procedures are well established, they rely on assembling information from several independent tests, each probing only a limited subset of the material response, and typically assume homogeneous stress and strain states.

The rapid development of full-field optical measurement techniques, most notably digital image correlation (DIC) [1], has fundamentally changed the landscape of material characterisation. Fullfield measurements provide access to heterogeneous displacement and strain fields over an entire region of interest, thereby offering significantly richer experimental information than conventional pointwise measurements. This has motivated the emergence of so-called Material Testing 2.0 (MT 2.0) [2,3], in which heterogeneous tests are deliberately designed and combined with inverse identification techniques to determine constitutive parameters from a reduced number of experiments, ideally from a single test.

Within this framework, a wide range of inverse identification strategies has been proposed, including the Virtual Fields Method (VFM) [4,5], Finite Element Model Updating (FEMU) [6], equilibrium-gap-based approaches [7], and integrated DIC formulations [8]. Among these, FEMU and nonlinear VFM have become the most widely used methods for identifying plasticity parameters in sheet metals. FEMU relies on iterative finite element simulations coupled with optimisation
algorithms, whereas VFM exploits the principle of virtual work to extract parameters directly from measured fields. Despite their conceptual differences, both approaches critically depend on the interplay between the chosen constitutive model, the design of the heterogeneous specimen, and the quality of the measured full-field data, which ultimately determines whether the inverse problem is well or ill conditioned.

A key limitation that continues to hinder broader industrial adoption of MT 2.0 is the lack of systematic benchmarking of a complete identification chain from specimen design, full field measurement to inverse identification process. While many studies demonstrate successful parameter identification using carefully designed heterogeneous tests [9], fewer explicitly address whether a given test configuration truly contains sufficient and well-conditioned information to uniquely identify all targeted parameters [10,11]. In particular, parameter correlation, low sensitivity, or insufficient activation of specific stress-strain states may lead to ill-posed inverse problems, nonunique solutions, or poor characterisation of the targeted material behaviour. As highlighted in previous identifiability studies, the apparent success of an inverse algorithm does not necessarily guarantee that all identified parameters are physically meaningful or robust [12].

In this work, we revisit the long-standing question of whether the in-plane plastic anisotropy of sheet metals can be fully characterised without assembling evidence from multiple standardised tests. We propose a single, purpose-designed heterogeneous biaxial experiment intended to replace the conventional combination of three uniaxial tensile tests (rolling, transverse, and diagonal directions) and an equibiaxial test. The specimen is conceived as a tailored cruciform geometry that simultaneously activates well-defined uniaxial stress states in multiple orientations while maintaining low strain gradients to ensure reliable DIC measurements.

Beyond serving as a practical "one-test" alternative, the proposed configuration is deliberately designed for comparative analysis of inverse identification methods utilising different cost function formulations. The specimen and the inverse identification pipeline are co-designed such that the locations and loading stages containing the relevant information for parameter identification are known a priori. This enables a rigorous assessment of whether full-field inverse methods, such as FEMU and VFM, are able to correctly extract the embedded information from the measured data. The benchmark nature of the test thus allows the inverse algorithms themselves to be scrutinised, rather than merely demonstrating successful parameter fitting.

The proposed design is used within a FEMU framework, in which all material parameters are identified from a single full-field dataset. While the concept of heterogeneous testing is well established in the scientific community, its practical applicability in industrial sheet metal forming remains limited. Consequently, the specimen design also reflects pragmatic considerations related to manufacturability, test execution, and measurement robustness, thereby bridging the gap between academic MT 2.0 developments and industrial practice.

A benchmark Cruciform Specimen for Robustness Analysis of Full-Field Inverse Identification

The original version of this paper is available on https://www.scientific.net/KEM.1049.107.pdf

To intrinsically incorporate the key data obtained from uniaxial tensile tests-namely the uniaxial yield stresses and R-values-a natural approach is to combine the three uniaxial test orientations into a single heterogeneous test configuration, as shown in Fig. 1. The proposed biaxial test employs a cruciform specimen with four individual ligaments converging in the central region. This configuration enables simultaneous plastic deformation in regions aligned with the rolling, transverse, and diagonal directions. Consequently, the testing procedure if needed can be simplified, as a single experiment suffices to characterise the uniaxial response in all three material directions. Furthermore, the configuration is designed such that the identification outputs are clearly defined, allowing the results to be post-processed also analytically. This makes the setup particularly suitable for evaluating full-field inverse identification methods, such as Finite Element Model Updating (FEMU) and the Virtual Fields Method (VFM). Importantly, the test is conceived such that the locations and load frames containing the relevant information are known a priori, ensuring that the appropriate data can be measured and exploited by full-field inverse identification methods.

[picture]

Fig. 1. Design of the biaxial tensile specimen. Predominantly uniaxial tensile stress states develop in the ligaments, whereas a biaxial tensile stress state occurs at the ligament junction.

Fig. 1. Design of the biaxial tensile specimen. Predominantly uniaxial tensile stress states develop in the ligaments, whereas a biaxial tensile stress state occurs at the ligament junction.

As shown in Fig. 1, the outer ligaments predominantly experience a uniaxial tensile stress state aligned with the rolling and transverse directions, whereas the inner ligaments primarily represent a uniaxial stress state in the diagonal direction. A biaxial stress state develops at the junction of the ligaments; however, this region typically enters the plastic regime at later loading stages than the ligaments. A key design requirement is therefore that all ligaments yield simultaneously and attain comparable plastic strain levels. This condition is satisfied by enforcing force equilibrium between the outer and inner ligaments while maintaining identical yield stresses. Under these assumptions, force balance dictates that the width of the inner ligaments should be reduced by a factor of 2/2 relative to that of the outer ligaments.

The fillet regions are designed to minimise stress and strain gradients, enabling efficient convergence of DIC measurements. In addition, the ligaments are sufficiently elongated to maintain an approximately uniaxial stress and strain state, which is essential for analytical post-processing. With the proposed geometry, each ligament allows the determination of the uniaxial yield stress and the corresponding R-value, resulting in six independent quantities. The equibiaxial yield stress also influences the strain evolution within the ligaments, and its effect is therefore implicitly captured by the test. In contrast, the equibiaxial R-value is not strongly represented, as the plastic strains at the ligament junction remain significantly lower than those developing in the ligaments.

YLD2000-2d: Alpha-Based Versus Physical-Based Parameter Formulation

The original version of this paper is available on https://www.scientific.net/KEM.1049.107.pdf

The quality of parameter identification critically depends on the interplay between the constitutive model formulation and the heterogeneous test design. To illustrate this relationship, two formulations of the YLD2000-2d plastic anisotropy model are revisited. In the first formulation, the material parameters are expressed in their original form as proposed by Barlat et al. [13]. In the second formulation, a physically based interpretation is adopted, in which uniaxial yield stresses and R-values-associated with standard tensile testing procedures-are treated as model parameters.

Theαformulation.

The original version of this paper is available on https://www.scientific.net/KEM.1049.107.pdf

The equivalent stress of the original model is formulated as

2σeqm=|X1′+X2′|m+|2X2′′+X1′′|m+|2X1′′+X2′′|m,(1)

where m is a material parameter determine the curvature of the yield surface. The quantities Xi′ and X′′i are the eigenvalues of the transformed stress tensors 𝐗′ and 𝐗′′, respectively. These tensors are obtained through linear transformations of the Cauchy stress tensor σ, namely 𝐗′=𝐋′σ and 𝐗′′=𝐋′′σ. The transformation matrices 𝐋′ and 𝐋′′ are defined as

𝐋′=13(2α1−α10−α22α2000α7),𝐋′′=19(−2α3+2α4+8α5−2α6α3−4α4−4α5+4α604α3−4α4−4α5+α6−2α3+8α4+2α5−2α60009α8)(2)

In this formulation, the model contains eight independent parameters α1−8 that must be calibrated. These parameters have a coupled influence on the shape of the yield surface and cannot be directly interpreted in physical terms. Consequently, an alternative formulation is introduced in the following, in which the mathematical structure of the model is retained while the parameters are expressed in terms of physically meaningful quantities associated with tensile testing.

Formulation with physical interpretation of parameters.

The original version of this paper is available on https://www.scientific.net/KEM.1049.107.pdf

In the original work by Barlat et al. [13], a calibration procedure for the parameters α1−8 based on standard uniaxial test results-namely normalised yield stresses and R-values-was proposed. This procedure requires solving a system of eight nonlinear equations, with the normalised yield stresses Y0∘,Y45∘,Y90∘,Yb and the corresponding R-values r0∘,r45∘,r90∘,rb as inputs. The system is solved for α1−8 using the Newton-Raphson algorithm and typically converges within three to five iterations.

From this perspective, the material model may alternatively be formulated in terms of the physically interpretable parameters Y0∘,Y45∘,Y90∘,Yb and, r0∘,r45∘,r90∘,rb, while the transformation to the internal parameter set α1−8 and the associated solution of the nonlinear system are regarded as an integral part of the constitutive model. From a computational standpoint, this implies that the parameter transformation can be implemented directly within the Abaqus UMAT subroutine, where the nonlinear system is solved only once per parameter update iteration, rather than during every stress update at each material integration point.

Direct-Levelling-Based Finite Element Model Updating

The original version of this paper is available on https://www.scientific.net/KEM.1049.107.pdf

FEMU is adopted in this work as the inverse identification framework for exploiting the full-field measurements obtained from the proposed benchmark cruciform experiment. It is selected due to its simplicity and flexibility cost function modification, and its ability to incorporate heterogeneous strain fields without requiring spatial convergence of the measured data.

Within the FEMU framework, the mechanical test is reproduced numerically using a finite element model, and the resulting displacement and strain fields are compared with those measured experimentally by DIC. The material parameters are iteratively updated by minimising a cost function that quantifies the discrepancy between experimental and numerical fields. However, FEM and DIC inherently differ in the manner in which kinematic fields are computed and spatially filtered. While FEM can resolve sharp strain gradients through mesh refinement, DIC measurements are affected by spatial filtering governed by subset size, interpolation scheme, and strain window parameters. If not properly accounted for, this mismatch may introduce systematic bias into the inverse identification procedure [14].

To mitigate this issue, a direct-levelling-based FEMU strategy is employed, following the methodology established in previous work. In this approach, FEM-computed nodal displacement fields are post-processed using the same strain reconstruction procedure as applied to the DIC data. Specifically, the FEM displacements are interpolated onto the DIC measurement grid and subsequently converted into strains using an identical strain window. By enforcing equivalent spatial filtering on both experimental and numerical data, the direct-levelling procedure ensures consistency between the compared strain fields, while avoiding the computational overhead associated with full DIC levelling via synthetic image generation.

Methodology

The original version of this paper is available on https://www.scientific.net/KEM.1049.107.pdf

The classical approach to assessing a heterogeneous specimen design is to first conduct a digital virtual twin (DVT) of the real experiment [15]. The purpose of the DVT is to simulate the experiment using known material parameters, generate deformed images from the calculated displacement fields, and process these images with a DIC engine to obtain strain fields. The resulting strain fields are then
used as inputs for the FEMU procedure. It should be emphasised that, during the creation of the DVT, the user has full control over the speckle pattern characteristics, image resolution, DIC parameters, and noise level, enabling a systematic investigation of the influence of image noise and measurement settings.

In the present study, the finite element model used to generate the DVT was built with a dense spatial discretisation comprising approximately 50,000 quadrilateral elements and 100 load frames to ensure a converged solution. The material parameters of the YLD2000-2d model were adopted from Coppieters et al. [16]. Furthermore, the speckle pattern, image resolution, noise level, and DIC settings were selected to closely match those used in the real experiment. Using this procedure, synthetic DIC results were generated that are representative of measurements obtained under experimental conditions.

[picture]

Fig. 2. Flowchart illustrating the virtual experiment based on known material parameters. The procedure starts with a finite element simulation, whose displacement fields are used to generate synthetic deformed images. These images are subsequently processed with DIC to obtain virtual experimental inputs for FEMU.

Fig. 2. Flowchart illustrating the virtual experiment based on known material parameters. The procedure starts with a finite element simulation, whose displacement fields are used to generate synthetic deformed images. These images are subsequently processed with DIC to obtain virtual experimental inputs for FEMU.

The post-processed DIC results were then supplied to the FEMU engine as measured data. For the purpose of numerical optimisation, a second finite element model with a coarser mesh was constructed. The rationale behind this approach is that, in real experiments, the optimal mesh density cannot be determined a priori, and a coarser mesh is typically adopted to ensure reasonable computational efficiency. This choice inevitably introduces a certain level of systematic error, which is also expected under real experimental conditions. Further details regarding the formulation of the optimisation problem are provided in Zhang et al. [11].

Finally, prior to performing the actual optimisation, an identifiability analysis was conducted to assess whether the inverse problem is well or ill conditioned. This step is particularly important when the numerical response exhibits low sensitivity to variations in certain parameters, or when variations in multiple parameters lead to similar responses, in which case a unique solution cannot be obtained. For this reason, sensitivity maps were computed and an identifiability study was performed. Most importantly, this analysis reveals the underlying structure of the optimisation problem for both material model formulations considered in this work, namely the α-based formulation and the physically based parameter formulation of YLD2000-2d.

Results

The original version of this paper is available on https://www.scientific.net/KEM.1049.107.pdf

Sensitivity maps and identifiability study.

The original version of this paper is available on https://www.scientific.net/KEM.1049.107.pdf

For both formulations of the YLD2000-2d yield function-the α-based formulation and the physically based parameter formulation-the sensitivity of the cost function was calculated and analysed. The results indicate that, for both parameter sets, the strain fields exhibit good sensitivity to variations in most parameters. The main exceptions are the yield function exponent and the biaxial R-value, whose sensitivities are less pronounced. This behaviour is expected, as the shape of the yield function primarily governs the evolution of plastic strains in regions between the control points defined by Y0∘,Y45∘,Y90∘,Yb and slopes, r0∘,r45∘,r90∘,rb. These intermediate stress states correspond
to relatively low strain levels, as they predominantly develop at the junction of the ligaments. A similar argument applies to rb, since plastic strains under equibiaxial tension remain smaller than those developing under uniaxial tension.

Further analysis focused on the interaction between individual parameters. When two parameters have a similar influence on the spatial distribution of the strain field, they cannot be uniquely identified from the experimental data [12]. To assess this interaction, a correlation matrix was constructed based on the sensitivity information. The resulting correlation matrices for both formulations of YLD2000-2d are shown in Fig. 3. For the α-based formulation, it can be observed that the parameters α1−6 are highly correlated and therefore cannot be uniquely determined from the experimental data. A similar correlation is observed between the parameters α7 and α8. In contrast, the physically based parameter formulation enables a unique identification of all parameters.

[picture]

Fig. 3. Correlation matrices for the α-based formulation (left) and the physically-based parameter formulation (right) of YLD2000-2d.

Fig. 3. Correlation matrices for the α -based formulation (left) and the physically-based parameter formulation (right) of YLD2000-2d.

The identified parameters.

The original version of this paper is available on https://www.scientific.net/KEM.1049.107.pdf

The identified parameters obtained for three different mesh sizes- 0.3 mm,0.5 mm, and 0.75 mm are reported in Table 1. As expected, the largest discrepancies are observed for the coarsest mesh and for the least sensitive parameters, namely the R -values and the yield function exponent. For these parameters, the relative error is in the order of 2−3%. It should be emphasised that, for the physically based formulation of YLD2000-2d, the solution converges within three to five iterations. In contrast, for the α-based formulation, the optimisation either fails to converge or converges to a local minimum.

Table 1. Target and identified parameters used in the study.

parameter [-] | target value | el. mesh size 0.3 mm | el. mesh size 0.5 mm | el. mesh size 0.7 mm
Y0° (fixed) | 1.00000 | 1.00000 | 1.00000 | 1.00000
Y45° | 1.09977 | 1.09972 | 1.09992 | 1.09942
Y90° | 1.00383 | 1.00351 | 1.00377 | 1.00269
Yb | 1.05596 | 1.05554 | 1.05787 | 1.0527
r0° | 1.71111 | 1.70429 | 1.71083 | 1.66433
r45° | 1.88974 | 1.88727 | 1.8809 | 1.85759
r90° | 2.7824 | 2.78347 | 2.78561 | 2.74348
rb | 1.16131 | 1.16972 | 1.17107 | 1.18151
m | 6.555 | 6.53253 | 6.60417 | 6.39339

Physical testing on fully calibrated materials.

The original version of this paper is available on https://www.scientific.net/KEM.1049.107.pdf

A well-known limitation of virtual testing is its inability to fully replicate all uncertainties present in real experimental conditions. Consequently, the ultimate validation of the proposed specimen design and inverse identification strategy requires application under real experimental conditions. For this reason, future work will focus on experimental measurements performed using an in-house-designed biaxial tensile testing machine and fully calibrated materials for which the anisotropy parameters are known. The planned experimental campaign will include different materials, such as SPCE [16], AISI304 [17], and AA5754-H22 [18]. The actual measurements and DIC results are presented in Fig. 4.

[picture]

Fig. 4. Actual experimental test of AISI304 stainless steel sheet metal: the measured strain field on deformed configuration (left), force-displacement response of each individual loading arm (right).
The quantities X,Y,U and V present a displacement of individual arm.

Fig. 4. Actual experimental test of AISI304 stainless steel sheet metal: the measured strain field on deformed configuration (left), force-displacement response of each individual loading arm (right). The quantities X , Y , U and V present a displacement of individual arm.

Conclusion

The original version of this paper is available on https://www.scientific.net/KEM.1049.107.pdf

Based on the specimen design, sensitivity and identifiability analyses, and virtual experimental validation presented in this work, the following conclusions can be summarised:

A purpose-designed cruciform specimen was proposed to enable single-test characterisation of in-plane plastic anisotropy in sheet metals, replacing the conventional combination of multiple uniaxial and equibiaxial tests.

A physically based parameter formulation of the YLD2000-2d yield function, expressed in terms of uniaxial yield stresses and R-values, was shown to significantly improve the conditioning and identifiability of the inverse problem compared to the classical α-based formulation.

Sensitivity and identifiability analyses demonstrated that, for the proposed specimen, the physically based parameter set can be uniquely identified, whereas strong parameter correlations prevent reliable identification in the α-based formulation.

FEMU combined with a direct-levelling strategy enabled robust exploitation of heterogeneous full-field strain measurements while accounting for spatial filtering effects inherent to digital image correlation. Virtual experimentation confirmed that the proposed approach yields accurate parameter identification, with relative errors typically limited to a few percent, even in the presence of mesh-induced modelling discrepancies.

Future work will focus on experimental validation using fully calibrated materials tested on a biaxial tensile testing machine, with the aim of demonstrating the practical applicability of the proposed benchmark configuration under real testing conditions.

References

The original version of this paper is available on https://www.scientific.net/KEM.1049.107.pdf

Schreier H, Orteu J-J, Sutton MA. Image Correlation for Shape, Motion and Deformation Measurements: Basic Concepts, Theory and Applications. Boston, MA: Springer US; 2009. https://doi.org/10.1007/978-0-387-78747-3.

Pierron F. Material Testing 2.0: A brief review. Strain 2023;59:e12434. .

Pierron F, Grédiac M. Towards Material Testing 2.0. A review of test design for identification of constitutive parameters from full-field measurements. Strain 2020;n/a:e12370. . 12370.

Marek A, Davis FM, Pierron F. Sensitivity-based virtual fields for the non-linear virtual fields method. Comput Mech 2017;60:409-31. .

Pierron F, Avril S, Tran VT. Extension of the virtual fields method to elasto-plastic material identification with cyclic loads and kinematic hardening. Int J Solids Struct 2010;47:29933010. .

Chen B, Starman B, Halilovič M, Berglund LA, Coppieters S. Finite Element Model Updating for Material Model Calibration: A Review and Guide to Practice. Arch Comput Methods Eng 2025;32:2035-112. .

Amiot F, Périé J-N, Roux S. Equilibrium Gap Method. Full-Field Meas. Identif. Solid Mech., John Wiley & Sons, Ltd; 2012, p. 331-62. .

Mathieu F, Leclerc H, Hild F, Roux S. Estimation of Elastoplastic Parameters via Weighted FEMU and Integrated-DIC. Exp Mech 2015;55:105-19. .

Lattanzi A, Barlat F, Pierron F, Marek A, Rossi M. Inverse identification strategies for the characterization of transformation-based anisotropic plasticity models with the non-linear VFM. Int J Mech Sci 2020:105422. .

Conde M, Zhang Y, Henriques J, Coppieters S, Andrade-Campos A. Design and validation of a heterogeneous interior notched specimen for inverse material parameter identification. Finite Elem Anal Des 2023;214:103866. .

Zhang Y, Gothivarekar S, Conde M, de Velde AV, Paermentier B, Andrade-Campos A, et al. Enhancing the information-richness of sheet metal specimens for inverse identification of plastic anisotropy through strain fields. Int J Mech Sci 2022;214:106891. .

Zhang Y, Van Bael A, Andrade-Campos A, Coppieters S. Parameter identifiability analysis: Mitigating the non-uniqueness issue in the inverse identification of an anisotropic yield function. Int J Solids Struct 2022;243:111543. .

Barlat F, Brem JC, Yoon JW, Chung K, Dick RE, Lege DJ, et al. Plane stress yield function for aluminum alloy sheets-part 1: theory. Int J Plast 2003;19:1297-319. .

Fayad SS, Jones EMC, Seidl DT, Reu PL, Lambros J. On the Importance of Direct-Levelling for Constitutive Material Model Calibration using Digital Image Correlation and Finite Element Model Updating. Exp Mech 2023;63:467-84. .

Zhang Y, Cooreman S, Starman B, Kuwabara T, Coppieters S. Improved Digital Virtual Twin to validate inverse identification of plastic anisotropy via FEMU. J Phys Conf Ser 2025;3104:012009. .

Coppieters S, Hakoyama T, Eyckens P, Nakano H, Van Bael A, Debruyne D, et al. On the synergy between physical and virtual sheet metal testing: calibration of anisotropic yield functions using a microstructure-based plasticity model. Int J Mater Form 2019;12:741-59. .

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/lionel-fourment-on-optimization-ai-and-inverse-analysis-in-forming/978-3-0364-1989-3

Key Engineering Materials, ISSN: 1662-9795, Vol. 1049, pp 115-122

doi: 10.4028/p-hcCP0o

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

Submitted:
2026-01-19

Revised:
2026-02-17

Accepted:
2026-03-02

Online:
2026-04-15

Towards DIC-Subset-Independent Machine Learning Models for Constitutive Parameter Identification in Sheet Metal Forming

The original version of this paper is available on https://www.scientific.net/KEM.1049.115.pdf

Dário Mitreiro 1,a*, João Henriques 1,b, Pedro A. Prates 1,c and António Andrade-Campos 1, d

1 Department of Mechanical Engineering, TEMA - Centre for Mechanical Technology and Automation, LASI - Intelligent Systems Associate Laboratory, University of Aveiro, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal

a * dario.mitreiro@ua.pt, b  joaodiogofh@ua.pt, c  prates@ua.pt, d  gilac@ua.pt

Keywords: digital image correlation, machine learning, parameter identification, sheet metal forming.

Abstract

Machine learning (ML) algorithms have been studied in literature as an inverse method to predict material constitutive parameters. However, these approaches are often dependent on the mesh discretisation settings applied during numerical simulations, and then difficulty model adaptation to experimental digital image correlation (DIC) subsets. Although a recent study explores the use of an interpolation-based approach to achieve experimental adaptation from numerically-based trained ML models, the proposed methodology lacks evaluation using experimental data. As a follow-up, this study proposes a new evaluation approach. Numerical data is DIC-levelled via MatchID software and then submitted to interpolation. An XGBoost algorithm is then trained on interpolated DIC data and evaluated for parameter prediction, comparing the obtained results with those obtained from the model trained on interpolated numerical data. Overall, the proposed DIC-levelling and interpolation pipeline yields an excellent predictive performance, with results comparable to those obtained when training on interpolated numerical data. The largest deviations are observed for the hardening exponent, while the remaining parameters are predicted with consistently high accuracy. These findings validate the practical applicability of the interpolation-based strategy to reduce the subset scheme dependency of ML models trained on real experimental data.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1049.115.pdf

Since the last few years, machine learning (ML) applications in engineering has been constantly increasing due to the substantial increase in computational power and available data. In the field of computational mechanics, the use of ML models as an inverse method for the prediction of material parameters has been a widely explored approach. Although it requires high quantity of available quality data, when compared to other inverse methods, as Finite Element Model Updating (FEMU) or Virtual Fields Method (VFM), ML models show advantages regarding computational time and costs when considering modern complex constitutive models. Besides, instead of relying on analytical models, ML approaches can learn stress-strain relations directly from experimental or numerical data, therefore being a more flexible solution for material parameter prediction [1]. In fact, many recent studies show the successful implementation of artificial neural network (ANN) [2] and extreme gradient boosting (XGBoost) [3] algorithms when applied to parameter prediction in sheet metal forming. However, these methods are often limited to the mesh discretisation scheme applied during numerical stage and, consequently, constrained by the dataset coverage from numerical simulations, which can lead to further difficulties when adapting to experimental digital image correlation (DIC) subsets. To face this challenge, a recent study [4] shows the successful implementation of data interpolation approaches applied to numerical data. However, the mentioned study lacks the demonstration of this approach when applied to real experimental DIC data. Regarding this issue, the present work is a follow-up of the previously mentioned study, by applying the proposed methodology on DIC-levelled data to demonstrate its practical applicability.

Background

The original version of this paper is available on https://www.scientific.net/KEM.1049.115.pdf

Numerical model

The original version of this paper is available on https://www.scientific.net/KEM.1049.115.pdf

The previously mentioned study [4] performs a numerical simulation of a cruciform-shaped specimen, illustrated in Figure 1a, capable of generate heterogeneous stress and strain fields and thus, covering a wide range of scenarios typically found in industrial sheet metal forming processes. The considered orthotropic symmetry and the displacement boundary conditions are represented in Figure 1b. The mesh consists of 564 C3D8R reduced-integration elements, and each simulation (sample) is divided into 20 equally divided time-steps. In each one of the time-steps, forces ( Fxx and Fyy ) and strains (ϵxx,ϵyy,ϵxy) for each element are recorded, with a total of 33880 features for each sample. The simulation process is performed in batch mode using Abaqus CAE 2019 software together with its Python 2.7.3 application programming interface (API) [5].

[picture]

Fig. 1. Biaxial tensile test on a cruciform specimen [4]: (a) geometry and dimensions; (b) boundary conditions and finite element mesh.

Fig. 1. Biaxial tensile test on a cruciform specimen [4]: (a) geometry and dimensions; (b) boundary conditions and finite element mesh.

The material behaviour is described by isotropic linear elasticity (Hooke's law, E=210GPa and v= 0.3 ) combined with orthotropic plasticity governed by the Hill'48 yield criterion [6] and isotropic hardening given by the Swift law [7]. The Hill'48 yield function is written as

F(σyy−σzz)2+G(σzz−σxx)2+H(σxx−σyy)2+2Lτxz2+2Mτxz2+2Nτxy2=Y2,(1)

where F,G,H,L,M,N are anisotropy coefficients (in this study, L=M=1.5 ), σxx,σyy,σzz,τyz, τxz, and τxy are Cauchy stress components in the sheet material frame, and is the current yield stress. By assuming G+H=1 (i.e., σxx=Υ ), the Hill coefficients can be related to the Lankford ratios r0,r45 and r90 as

F=r0r90(r0+1);G=1r0+1;H=r0r0+1;N=12(r0+r90)(2r45+1)r90(r0+1),(2)

Once σ0, F,G,H, and N are defined, the anisotropy coefficient r(α) and the initial yield stress in tension σ0(α) are obtained from

r(α)=H+(2N−F−G−4H)sin(α)2cos(α)2Fsin(α)2+Gcos(α)2,(3)

and

σ0(α)=σ0[Fsin(α)2+Gcos(α)2+H+(2N−F−G−4H)sin(α)2cos(α)2]−12,(4)

respectively. Finally, the evolution of yield stress during plastic deformation is described by the Swift hardening law,

Υ=K[(σ0K)1n+ϵ¯p]n,(5)

where ϵ¯p is the equivalent plastic strain and σ0, K and n are material parameters.

Dataset generation

The original version of this paper is available on https://www.scientific.net/KEM.1049.115.pdf

During the simulation, a total of 6755 samples is generated using Latin Hypercubic Sampling (LHS) method applied to σ0,K,n,r0,r45, and r90 material parameters, with defined input range and step size for each parameter. Then, obtained samples are checked for decreasing load during simulation for the last time-step transition (i.e., Fxx,19>Fxx,20 or Fyy,19>Fyy,20 ). The 2260 samples presenting no decrease in load during simulation are then randomly shuffled and divided into train and test datasets with 2000 and 260 samples, respectively. The structure of each one of these datasets is compared by two different types of data, a feature matrix (x) and a target matrix (y). The feature matrices ( x_train and x_test) represent the simulation outputs per sample, with shape nsamples ×nfeatures . Similarly, the target matrices ( y_train and y_test) represent the corresponding parameters used to obtain each one of the samples, with shape nsamples ×nparameters . Total sizes of both feature and target matrices (datasets) are represented in Table 1.

Table 1. Feature and target matrix shapes for both training and test sets [4].

Training | Test
Feature | Target | Feature | Target
2000 × 33880 | 2000 × 6 | 260 × 33880 | 260 × 6

Interpolation approach

The original version of this paper is available on https://www.scientific.net/KEM.1049.115.pdf

Both train and test datasets are interpolated to regular grid points with different densities, choosing grids of 20×20,30×30 and 40×40 points applied to a quadratic shape and then restricted to the cruciform-shaped specimen domain. Within specimen's domain, 20×20,30×30 and 40×40 grids origin a total of 253 , 564 and 1006 points, respectively. Regarding the interpolation methods, Python's Scipy RBFInterpolator class was used, choosing linear, cubic and multiquadric methods and keeping the remaining function parameters as default. Overall, after each possible combination between the chosen grid sizes and interpolation methods, a total of 9 different interpolated datasets is obtained.

XGBoost training and evaluation

The original version of this paper is available on https://www.scientific.net/KEM.1049.115.pdf

Each interpolated train feature dataset was used as input to train an XGBoost algorithm and, accordingly, the corresponding interpolated feature test dataset was used to evaluate the predictive performance of the generated model (called simple testing). Also, to test the robustness of each model, a cross testing was performed using feature test datasets interpolated with other methods different from the one used to train the chosen ML model. Regarding the XGBoost algorithm training process, the hyperparameters are the same across all the models and are kept as default, except for the learing_rate (=0.02), max_depth (=4) and n_estimators (=1000). After comparing the predicted parameters with the true ones and assessing each model's performance using R2, MAE and MAPE metrics, the preferred interpolation combination is the 30×30 grid with the multiquadric method, due to the balance between performance and computational cost.

Case study

The original version of this paper is available on https://www.scientific.net/KEM.1049.115.pdf

To test the practical application, a synthetic DIC sample was created by choosing a random set of parameters from the test dataset and interpolating their corresponding features to a generated set of 5000 randomly distributed points within the cruciform specimen's domain, thus intending to approximate the typical subset density often found in a DIC-acquired sample. The interpolation is performed using the multiquadric method, resulting in 300040 features for the synthetic sample. To finish the practical demonstration, the DIC synthetic sample is interpolated using the previously
chosen preferred grid and method combination and used as input to test the corresponding ML model for parameter prediction. The predicted parameters are compared with the true ones and with the ones predicted by a ML model trained with the original feature dataset without interpolation (called original model). This comparison is performed by visually representing the effect of these parameters on the evolution of the yield stress during plastic deformation, the initial yield stress in tension as a function of loading angle w.r.t. rolling direction, and anisotropy coefficient as a function of loading angle w.r.t. rolling direction. Overall, the predicted parameters using the synthetic DIC sample closely match the true ones and those obtained by the original model, once again confirming the strong predictive capability of the ML model trained on data using a 30×30 grid and multiquadric method, and also demonstrating the effectiveness of the proposed interpolation approach when applied to a denser group of points.

Motivation

The original version of this paper is available on https://www.scientific.net/KEM.1049.115.pdf

In summary, the described study addresses the gap found in literature related to mesh discretisation dependency but is rather incomplete due to the lack of validation with real experimental data. The interpolation approach becomes truly relevant when applied to real DIC datasets, reducing sensitivity to the chosen subset scheme. Regarding this, the present work proposes a DIC-levelling process applied to the original train and test numerical datasets, obtaining DIC-levelled datasets. Next, these datasets are then submitted to interpolation using the chosen grid and method. The interpolated DIC train and test datasets are then used as input to train an XGBoost algorithm and test its predictive performance. Lastly, the predictive performance of both ML models obtained from interpolated numerical and DIC datasets is compared, fully validating the proposed methodology. An overview schematic of this methodology is represented in Figure 2.

[picture]

Fig. 2. Schematic for the proposed methodology, represented in blue colour.

Fig. 2. Schematic for the proposed methodology, represented in blue colour.

DIC Process

The original version of this paper is available on https://www.scientific.net/KEM.1049.115.pdf

The DIC-levelling is applied to the numerical datasets using the MatchID software in stereo (3D) mode, with the settings specified in Table 2.

Table 2. Stereo DIC settings.

Camera | Flir Blackfly BFS-U3-51S5M-C
Focal length | 12.5 mm
Image resolution | 2448 × 2048 px²
Camera noise | 0.48% of range
Working distance | 182 mm
Stereo angle | 20.74°
Speckle pattern | Numerically generated
Avarage speckle size | 8 px
Correlation criterion | ZNSSD
Interpolation | Local Bicubic Splines
Subset shape function | Quadratic
Stereo transformation | Quadratic
Subset size | 25
Step size | 6
Image pre-filtering | Gaussian, 5 px kernel
Strain window size | 15
Strain interpolation | Quadratic quadrilateral Q8
Strain convention | Log. Euler-Almansi
Displacement noise-floor | 2.471 × 10⁻³
Strain noise-floor | 1.13 × 10⁻⁴

However, this process requires a different dataset format, not with forces and strains (as it was previously) but instead with node coordinates of each mesh element. To process each sample, MatchID requires a file with the initial (static) coordinates of each node, plus 20 files (one per timestep) with the updated node coordinates along the simulation. Due to this reason, new numerical simulations are performed for the same 2260 samples, obtaining a total of 47460 data files ( 21 files per sample). Similarly to the previous numerical dataset, simulations are performed in batch mode using Abaqus CAE 2019 software with an average simulation time of about 30 seconds (Intel Core Ultra 9285,24 cores, 128 GB of RAM). Then, the DIC-levelling process itself can be divided into 2 steps: the image generation according to the given speckle pattern, and the DIC filtering process applied to those images to obtain the final results. As the DIC-levelling is performed in 3D mode (to ensure accurate strain values), 2 base speckle pattern images are used in a given angle (stereo angle) to generate 21 images per sample according to the node coordinates from the numerical simulation. In the second step, a DIC filter is applied to all the images of each sample, generating a data file per image containing (among other data) the coordinates and strain values for each subset, with a total of 27296 subsets. Keeping the same hardware specifications mentioned before, computational time is quite significant when performing the DIC-levelling process with about 5 minutes per sample, which results in a total of about 8 days for the entire dataset.

Data Processing and Interpolation

The original version of this paper is available on https://www.scientific.net/KEM.1049.115.pdf

In order to keep consistency with the numerical dataset format, these DIC samples are compiled into a single feature dataset with the forces ( Fxx and Fyy, from the numerical simulations) and the subset strain values (from the DIC-levelling) per time-step, with shape 2260×1637800 (samples x features). Then, this dataset is divided into train and test ones, ensuring exactly the same samples and order found in both train and test numerical datasets. Also, to ensure a fair comparison between both numerical and DIC shear strain values, the last ones are multiplied by 2, as MatchID adopts the tensorial convention for the shear strain (ϵxy), contrary to Abaqus, which uses the engineering shear strain by default (γxy).
The interpolation process uses Python's Scipy RBFInterpolator class (similarly to the previous study) and is performed for both train and test datasets using the 30×30 grid with multiquadric method (preferred combination). However, due to the highest density of points (subsets) from the DIClevelling, some tweaks regarding the RBFInterpolator class are needed, adjusting neighbors ( =250 ) and epsilon (=10) parameters. Otherwise, if neighbors parameter is not specified, all data points are used by default, leading to a huge unnecessary computational time (around 20 minutes per sample). Also, the epsilon (shape parameter) needs to be tuned so there are no outlier values at the edges of the cruciform specimen.

XGBoost Model

The original version of this paper is available on https://www.scientific.net/KEM.1049.115.pdf

Before the XGBoost algorithm training itself, DIC interpolated feature train data is submitted to a normalisation process, saving the resulting scaler file to later normalise the test dataset upon the model's evaluation process. Similarly to the previous study, hyperparameters are kept as default, except for the learing_rate (=0.02), max_depth (=4) and n_estimators (=1000). Overall, the train process takes around 50 minutes to complete. Lastly, the obtained model is evaluated using the DIC interpolated feature test dataset, also submitted to a normalisation process.

Analysis

The original version of this paper is available on https://www.scientific.net/KEM.1049.115.pdf

Table 3 presents the test performance evaluation metrics for both the results obtained by the numerical and DIC models. Comparing the metrics, the results from the DIC model are very similar to those from the numerical model (obtained in the previous study), showing an excellent predictive performance. The deviations between the parameters predicted by the DIC model and the true ones can be observed in Figure 3, where the overall differences in the predicted values and the performance for each individual parameter is presented. Overall, the highest variation is noticeable by the hardening exponent ( n ), with a value of R2=0.96.

Table 3. Performance metrics for parameter predictions using both models trained with interpolated numerical and DIC-levelled data.

Model | R² | MAE | MAPE
Numerical | 0.992 | 1.209 | 0.014
DIC | 0.990 | 1.190 | 0.014

In addition, a random test sample is chosen to compare the strain values obtained for the last timestep, considering the dataset interpolated from numerical data, and the DIC dataset before and after the interpolation process. This comparison is presented in Figure 4, where ϵxx,ϵyy, and ϵxy strain values are visually similar along all the stages, which demonstrates the good development of the proposed approach along the path.

[picture]

Fig. 3. Comparison between test (true) and predicted parameters from the DIC model.

Fig. 3. Comparison between test (true) and predicted parameters from the DIC model.

[picture]

Fig. 4. Comparison between sample strain fields of interpolated numerical data (ϵxx(a),ϵyy( d),ϵxy (g)), DIC-levelled data (ϵxx( b),ϵyy(e),ϵxy( h)) and interpolated DIC data (ϵxx(c),ϵyy(f),ϵxy(i)).

Fig. 4. Comparison between sample strain fields of interpolated numerical data ( ϵ x x ( a ) , ϵ y y ( d ) , ϵ x y (g)), DIC-levelled data ( ϵ x x ( b ) , ϵ y y ( e ) , ϵ x y ( h ) ) and interpolated DIC data ( ϵ x x ( c ) , ϵ y y ( f ) , ϵ x y ( i ) ) .

Conclusion

The original version of this paper is available on https://www.scientific.net/KEM.1049.115.pdf

This study proposes a DIC-levelling process, followed by data interpolation, and ML model prediction and evaluation, to assess the practical applicability of a recent explored methodology regarding ML subset dependency reduction. The obtained evaluation results are similar to the previous results using the numerical interpolated dataset and show an excellent predictive performance. When analysing the predicted values for each parameter, the hardening exponent parameter presents the highest variation. Also, the visualisation of strain fields collected from a sample along different stages demonstrate the good development of the proposed approach along the path. These results not only reinforce the efficiency of XGBoost models for parameter prediction using a cruciform-shaped specimen with Hill'48 criterion and Swift hardening law, but also demonstrate the successful application of the proposed interpolation approach, reducing the subset scheme dependency of ML models trained on real experimental data. For future works, a more detailed exploration of alternative interpolation methods and a grid variation study could bring improvements regarding both performance and computational cost.

Acknowledgement

The original version of this paper is available on https://www.scientific.net/KEM.1049.115.pdf

D. Mitreiro is grateful to the Portuguese Foundation for Science and Technology (FCT) for the Ph.D. grant 2025.02607.BD. This work was financed by the Portuguese Foundation for Science and Technology (FCT) and UE/FEDER through COMPETE 2030, of projects COMPETE2030-FEDER00778700, UID/00481 Centro de Tecnologia Mecânica e Automação (TEMA), and LA/P/0104/2020.

References

The original version of this paper is available on https://www.scientific.net/KEM.1049.115.pdf

P. Böhringer et al., "A strategy to train machine learning material models for finite element simulations on data acquirable from physical experiments," Computer Methods in Applied Mechanics and Engineering, vol. 406, pp. 115894, Mar. 2023, doi: doi.org/10.1016/j.cma.2023.115894.

J. Kim, A. S. Ebrahim, B. L. Kinsey, and J. Ha, "Identification of Yld2000-2d anisotropic yield function parameters from single hole expansion test using machine learning," CIRP Annals, vol. 73, no. 1, pp. 233-236, Jan. 2024, doi: .

N. Bastos, P. A. Prates, and A. Andrade-Campos, "Material Parameter Identification of Elastoplastic Constitutive Models Using Machine Learning Approaches," Key Engineering Materials, vol. 926, pp. 2193-2200, Jul. 2022, doi: .

D. Mitreiro, P. A. Prates, and A. Andrade-Campos, "Reducing mesh dependency in dataset generation for machine learning prediction of constitutive parameters in sheet metal forming," Metals, vol. 15, no. 5, pp. 534, 2025. doi: 10.3390/met15050534.

Dassault Systèmes. Abaqus 2016 Analysis User's Guide (Vols. 1-5). Providence, RI: Dassault Systèmes Simulia Corp., 2016.

R. Hill, E. Orowan, "A theory of the yielding and plastic flow of anisotropic metals," Proc. R. Soc. Lond. Ser. A, vol. 193, no. 1033, pp. 281-297, 1948. doi: 10.1098/rspa.1948.0045.

H. W. Swift, "Plastic instability under plane stress," Journal of the Mechanics and Physics of Solids, vol. 1, no. 1, pp. 1-18, Oct. 1952, doi: 10.1016/0022-5096(52)90002-1.

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/lionel-fourment-on-optimization-ai-and-inverse-analysis-in-forming/978-3-0364-1989-3

Key Engineering Materials, ISSN: 1662-9795, Vol. 1049, pp 123-129

doi: 10.4028/p-dyN4E6

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

Submitted:
2026-01-19

Accepted:
2026-02-10

Online:
2026-04-15

Application of Reference-Free Strain Measurement to Assess the Deformation Level of Pre-Deformed Components

The original version of this paper is available on https://www.scientific.net/KEM.1049.123.pdf

ROSSI Marco a*, TANONI Giulia b, ANEDDA Matteo c, FIORETTI Maria Caterina d, and PRINCIPI Emanuele e

Università Politecnica delle Marche, via brecce bianche 12, 60131 Ancona, ITALY

a  m.rossi@univpm.it, b  g.tanoni@univpm.it, c  m.anedda@pm.univpm.it, d  m.c.fioretti@pm.univpm.it, e e.principi@univpm.it

Keywords: Remanufacturing, Plastic Deformation, Convolutional Neural Network

Abstract

In the framework of sustainable manufacturing and circular economy, the reuse of metallic components at the end of their first life (EoL) is a promising strategy to reduce energy consumption and material waste, but it requires an accurate assessment of residual plastic deformation, which strongly affects structural integrity and remaining formability. Conventional full-field techniques such as Digital Image Correlation (DIC) require a reference image of the undeformed state, which is generally unavailable for EoL components. To address this limitation, this work investigates a deep learn-ing-based, reference-free approach for strain estimation directly from a single image of the deformed surface. The method relies on a convolutional neural network architecture derived from VGG-16, trained to regress the in-plane principal strains and their orientation from local image subsets of the surface texture using a synthetically generated dataset based on real material textures and realistic imaging conditions. The trained model is applied to high-resolution optical images of pre-deformed steel and aluminum components from regions subjected to different deformation histories, with partial validation provided by finite element simulations and conventional DIC measurements. Preliminary results show that the proposed approach can distinguish regions with different levels of plastic deformation and provide strain maps consistent with independent mechanical assessments, demonstrating its potential as a rapid, non-destructive tool for deformation mapping and classification of EoL components to support remanufacturing and reuse decisions.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1049.123.pdf

Digital Image Correlation (DIC) is a well-established optical technique for measuring displacement and strain fields on the surface of solids by comparing images of an undeformed configuration with those acquired during deformation [1]. Over the years, DIC has evolved into a versatile framework, including stereo-DIC for 3D measurements, multi-camera configurations, optical extensions of singlecamera systems [2], and full 360∘ strain measurements [3]. Extensions such as Digital Volume Correlation (DVC) further enable the reconstruction of three-dimensional deformation fields inside solids [4, 5]. Thanks to these capabilities, DIC has become a key tool in material characterization [6], enriching the information obtained from mechanical tests and supporting modern paradigms such as Material Testing 2.0 [7]. It is also widely used for the experimental validation of numerical models, including finite element simulations [8].

In parallel, recent advances in Deep Learning (DL) across several domains, ranging from time series analysis [9, 10] to image processing [11], have stimulated growing interest in DL-based approaches for DIC [12]. Convolutional Neural Networks (CNNs), in particular, have been investigated as alternatives or complements to traditional correlation algorithms. Previous studies proposed CNNs to directly estimate displacement and strain fields by mimicking the classical DIC workflow, showing improvements in computational efficiency and accuracy [13,14].

In this work, we apply a different approach. Instead of replicating the image correlation process, we introduce a deep learning-based approach capable of estimating the deformation level directly and solely from the deformed image. Similar ideas have recently been explored in microfluidic applications for inferring three-dimensional information from defocused images [15, 16], but only very
recently they have been applied to deformation measurements in solid mechanics [17]. By removing the need for a reference image, the proposed approach enables deformation analysis in scenarios where classical DIC is not applicable, such as post-mortem microstructural observations where only deformed images are available. Our current implementation, termed DeepStrainNet, is based on the VGG-16 architecture [18]. The proposed methodology is here applied to synthethic and real deformed materials.

Methods

The original version of this paper is available on https://www.scientific.net/KEM.1049.123.pdf

Conventional subset-based Digital Image Correlation (DIC) estimates the displacement field by matching subsets between undeformed and deformed images, while strain is obtained by spatial differentiation of the displacement field. In contrast, the proposed approach directly estimates the strain field from a single deformed image by means of a deep learning model trained to recognize the deformation of a known surface pattern.

A planar deformation state is described by the two principal strains, ε1 and ε2, and the orientation θ of the principal directions. These three continuous variables define the output of the proposed method, which is therefore formulated as a regression problem. The input of the network is a grayscale image subset 𝐗∈ℝN×M×1, while the output is the vector 𝐝=[ε1,ε2,θ]T. The neural network implements a parametric function

𝐝=𝐟Θ(𝐗),(1)

where Θ denotes the set of trainable parameters.

[picture]

Fig. 1: The proposed neural network architecture. Differently from the standard VGG-16 architecture, the last layer has 3 outputs, one for each deformation parameter. The output can be modified if only one deformation is required.

Fig. 1. The proposed neural network architecture. Differently from the standard VGG-16 architecture, the last layer has 3 outputs, one for each deformation parameter. The output can be modified if only one deformation is required.

Neural network architecture. The function 𝐟Θ(·) is implemented using a convolutional neural network named DeepStrainNet, derived from the VGG-16 architecture [18]. The convolutional backbone of VGG-16 is retained as a feature extractor, while the original classification head is replaced with a regression head composed of fully connected layers. The final output layer consists of three neurons with linear activation, corresponding to the deformation parameters ε1,ε2, and θ. A schematic representation of the architecture is shown in Fig. 1.

Training procedure. The network parameters are learned from a labeled dataset of image subsets and corresponding ground-truth deformation parameters. Training is performed by minimizing the Mean Squared Error (MSE) loss between predicted and target values. Optimization is carried out using the Adam optimizer, with a learning rate scheduled according to a cosine decay strategy.

Test on Simulated Microstructure

The original version of this paper is available on https://www.scientific.net/KEM.1049.123.pdf

As first case study, the method is applied to reconstruct a strain localization within the microstructure. In this case, only simulated experiments are used. The images are generated through a numerical simulator capable of reproducing realistic deformation effects while accounting for typical experimental uncertainties, such as image noise and illumination variations [19]. This type of simulator is widely used in experimental mechanics to evaluate the performance and uncertainty of measurement and identification techniques [20,21]. Synthetic images are also routinely employed for benchmarking deformation measurement methods, including DIC algorithms [22].

The image generation procedure is based on the optical flow conservation principle, whereby the gray-level distribution of an undeformed microstructural image is mapped onto a deformed configuration through a prescribed kinematic transformation [23]. Starting from a high-resolution reference micrograph, treated as a continuous gray-level field via cubic interpolation, deformed images are obtained by applying analytical displacement fields corresponding to homogeneous planar deformation states.

[picture]

Fig. 2: Generation of the images for the training set: deformed subset of 99×99 pixel are generated randomly starting from a given undeformed microstructure.

Fig. 2. Generation of the images for the training set: deformed subset of 99 × 99 pixel are generated randomly starting from a given undeformed microstructure.

The training set is obtained using the DT to generate a series of N deformed images (size of 99×99 pixels) with constant principal strains ε1 and ε2 and an in-plane rotation angle θ. The strain parameters are randomly sampled within predefined ranges corresponding to tensile and shear dominated deformation regimes [17], see Fig. 2. To enhance realism, experimental uncertainties are introduced through the DT, including additive gray-level noise and spatially varying illumination changes. This procedure enables the generation of a large number of statistically independent deformed microstructural images starting from a limited set of reference micrographs, by jointly varying the deformation level and the local microstructural features. In this study, the synthetic dataset is generated exclusively from optical micrographs of an X65 steel microstructure. A series of 25000 images were genereated and used to train the Convolutional Neural Network.

The trained network was used to retrieve the deformation in a newly generated microstructure, subjected to a deformation field prescribed according to a sinusoidal function, leading to the formation of periodic strain localization bands. The maximum value of the principal strain component ε1 was set to 0.5 . Differently from the training set, the deformation was applied to the entire reference image through a spatially non-uniform mapping, rather than by assigning a constant strain field to a 99×99 sub-image. This procedure allows for the generation of continuously varying deformation gradients and enables a direct assessment of the method capability to capture deformation gradients and strain heterogeneities.

The deformed image was obtained starting from a micrograph of the same material used in the training phase, but extracted from a different specimen region and therefore not included in the training dataset. This choice ensures that the evaluation is performed on previously unseen microstructural features, thus providing a meaningful validation of the generalization capability of the network.

[picture]

Fig. 3: Deformation field of the deformed microstructure obtained with the trained CNN. The principal strain is shown, the localization bands are clearly visible.

Fig. 3. Deformation field of the deformed microstructure obtained with the trained CNN. The principal strain is shown, the localization bands are clearly visible.

Fig. 3 reports the deformation strain field reconstructed by the proposed algorithm. The results show that the method is able to properly capture both the amplitude and the spatial distribution of the strain localization bands, demonstrating its effectiveness in reproducing the underlying deformation pattern.

Experimental Validation on an Aluminum Sample

The original version of this paper is available on https://www.scientific.net/KEM.1049.123.pdf

A further validation of the proposed methodology was carried out on real experimental data acquired during a uniaxial tensile test. The investigated material is an aluminum alloy AA5754, commonly used in forming applications due to its good combination of strength and ductility.

The displacement and strain fields were measured by means of a commercial Digital Image Correlation (DIC) software, namely MatchID, which was employed to provide a reference strain field for comparison with the results obtained by the proposed method. The obtained principal strain field is illustrated in Fig. 4.

On the opposite face of the specimen, where no speckle pattern was applied, the surface was observed by means of an optical microscope at 200× magnification. An angular illumination was adopted in order to enhance the surface roughness features. A different surface behavior is observed at low and high deformation levels, as shown in Fig. 4. At low strain, the surface appears more regular with clear linear features, that tends to reduce a higher strain levels.

A series of random sub-images of size 99×99 pixels were extracted from the microscope images acquired over the whole specimen surface. These patches were used to build the dataset for training and validation of the network. A subset of such images was employed for training, while the remaining ones were reserved for testing. In this case, only one strain component was considered for training, namely the principal strain. A total of 20000 images were used for training the network, whereas 4500 images were selected as a test set. Importantly, the test images were extracted from a different region of the specimen with respect to the training set, so as to ensure a strict spatial separation between training and validation data and to avoid any bias due to local correlations.

Fig. 5 shows the results obtained on the test set compared with the DIC reference. A good agreement between the predicted and reference strain values is observed, thus confirming the ability of the proposed approach to generalize to previously unseen regions of the specimen.

[picture]

Fig. 4: Deformation map obtained with DIC and examples of the surface roughness observed with the optical microscope.

Fig. 4. Deformation map obtained with DIC and examples of the surface roughness observed with the optical microscope.

[picture]

Fig. 5: Comparison of the predictions obtained with the trained CNN with respect to the reference DIC measurement; the major principal deformation is considered.

Fig. 5. Comparison of the predictions obtained with the trained CNN with respect to the reference DIC measurement; the major principal deformation is considered.

Conclusions

The original version of this paper is available on https://www.scientific.net/KEM.1049.123.pdf

In this work, a deep learning-based, reference-free approach for full-field strain estimation from a single deformed image has been presented and validated on both synthetic and experimental data. Differently from conventional Digital Image Correlation techniques, the proposed method does not require access to an undeformed reference configuration, thus enabling deformation assessment in scenarios where classical approaches are not applicable, such as post-mortem analyses or remanufacturing applications.

The method was first validated on synthetically generated microstructural images, where complex and spatially varying strain fields were successfully reconstructed, including strain localization bands induced by non-uniform deformation gradients. Subsequently, the approach was applied to real experimental data acquired during a tensile test on an AA5754 aluminum alloy specimen, showing good agreement with independent DIC measurements.

A key outcome of this study is that the proposed methodology proved to be effective independently of the nature of the surface pattern used as input. In particular, reliable strain estimations were obtained both from intrinsic material microstructures and from surface roughness features observed by optical microscopy, without the need for an artificial speckle pattern.

These results highlight the versatility and robustness of the proposed approach and open the way to its application in a wide range of practical scenarios, including the inspection and classification of end-of-life components for remanufacturing, where surface conditions and textures may vary significantly from case to case.

Future work will focus on extending the method to multiaxial loading conditions, improving quantitative accuracy through larger experimental datasets, and exploring its integration into industrial inspection pipelines.

Acknowledgement

The original version of this paper is available on https://www.scientific.net/KEM.1049.123.pdf

This work was carried out within the framework of the project "Going Beyond Recycling: Reshaping of End-of-Life Components through Advanced Sheet Metal Forming Processes", funded under the PRIN 2022 Call (MUR Decree No. 1401 of 18/09/2024 - CUP I53C24002500006).

References

The original version of this paper is available on https://www.scientific.net/KEM.1049.123.pdf

M.A. Sutton, J.J. Orteu and H. Schreier: Springer, Image Correlation for Shape, Motion and Deformation Measurements (2009).

K. Genovese, L. Casaletto, J.A. Rayas, V. Flores and A. Martinez: Optics and Lasers in Engineering Vol. 51 (2013), p. 278.

K. Genovese, L. Cortese, M. Rossi and D. Amodio: Optics and Lasers in Engineering Vol. 82 (2016), p. 127.

B. Pan and B. Wang: Optics and Lasers in Engineering Vol. 135 (2020), p. 106189.

M. Rossi, L. Cortese, K. Genovese, A. Lattanzi, F. Nalli and F. Pierron: Experimental Mechanics Vol. 58 (2018), p. 1181.

S. Avril, M. Bonnet, A.-S. Bretelle, M. Grédiac, F. Hild, P. Ienny, F. Latourte, D. Lemosse, S. Pagano and E. Pagnacco: Experimental Mechanics Vol. 48 (2008), p. 381.

F. Pierron and M. Grédiac: Strain Vol. 57 (2021), p. e12370.

P. Lava, E.M.C. Jones, L. Wittevrongel and F. Pierron: Strain Vol. 56 (2020), p. e12350.

G. Tanoni, E. Principi, L. Mandolini and S. Squartini: in Applied Intelligence and Informatics, Springer (2022), p. 360.

G. Tanoni, L. Stankovic, V. Stankovic, S. Squartini and E. Principi: IEEE Trans. Ind. Informatics Vol. 20 (2024), p. 4710.

M. Trigka and E. Dritsas: Sensors Vol. 25 (2025).

X. Duan, H. Xu, R. Dong, F. Lin and J. Huang: Optics and Lasers in Engineering Vol. 160 (2023), p. 107234.

Y. Wang and C. Zhou: Optics and Lasers in Engineering Vol. 174 (2024), p. 107981.

Y. Feng and L. Wang: Optics and Lasers in Engineering Vol. 179 (2024), p. 108267.

R. Barnkob, C. Cierpka, M. Chen, S. Sachs, P. Mäder and M. Rossi: Measurement Science and Technology Vol. 32 (2021), p. 094011.

M. Mehdizadeh Youshanlouei and M. Rossi: Physics of Fluids Vol. 36 (2024).

M. Rossi, G. Tanoni, V. Ilari, M. Sasso and E. Principi: Submitted to Engineering Application of Artificial Engineering, SSRN preprint (2025).

K. Simonyan and A. Zisserman: Proc. ICLR (2015).

M. Rossi, P. Lava, F. Pierron, D. Debruyne and M. Sasso: Strain Vol. 51 (2015), p. 206.

M. Badaloni, M. Rossi, G. Chiappini, P. Lava and D. Debruyne: Experimental Mechanics Vol. 55 (2015), p. 1411.

A. Peshave, F. Pierron, P. Lava, D. Moens and D. Vandepitte: Strain Vol. 60 (2024), p. e12473.

P.L. Reu, E. Toussaint, E. Jones, H.A. Bruck, M. Iadicola, R. Balcaen, D.Z. Turner, T. Siebert, P. Lava and MDIC Simonsen: Experimental Mechanics Vol. 58 (2018), p. 1067.

M. Rossi and F. Pierron: Int. J. Solids Struct. Vol. 49 (2012), p. 420.

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/lionel-fourment-on-optimization-ai-and-inverse-analysis-in-forming/978-3-0364-1989-3

Key Engineering Materials, ISSN: 1662-9795, Vol. 1049, pp 131-139

doi: 10.4028/p-QE9aTP

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

Submitted:
2026-01-19

Revised:
2026-02-18

Accepted:
2026-03-02

Online:
2026-04-15

Improving FEMU Calibration from a Single Heterogeneous Test through a Data-Driven Approach: An Exploratory Study

The original version of this paper is available on https://www.scientific.net/KEM.1049.131.pdf

Mafalda Gonçalves 1,a*, Rui Amaral 2, b, Sandrine Thuillier 3,c and António Andrade-Campos 4,d

1 INEGI - Institute of Science and Innovation in Mechanical and Industrial Engineering, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal

2 Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, Porto, 4200-465, PT

3 Univ. Bretagne Sud, UMR CNRS 6027, IRDL, F-56100, Lorient, France

4 Centre for Mechanical Technology and Automation (TEMA), Department of Mechanical Engineering, University of Aveiro, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal

a  mpgoncalves@inegi.up.pt, b  ramaral@inegi.up.pt, c  sandrine.thuillier@univ-ubs.fr, d  gilac@ua.pt

Keywords: data-driven, FEMU, heterogeneous test; material testing 2.0.

Abstract

Inverse identification of material parameters from experimental data is a long-standing challenge, especially when calibrating complex constitutive models characterized by a large number of parameters. Heterogeneous mechanical tests combined with full-field measurements provide a large amount of information for material parameter identification but lead to high computational costs when used within a Finite Element Model Updating (FEMU) framework. This work presents an exploratory study on the use of surrogate-assisted Bayesian optimization to assess its potential for reducing the number of simulations required for FEMU-based calibration using data from a single notched tensile test. FEMU cost function is applied based on the discrepancy between experimental and numerical strain fields. A Gaussian Process surrogate model is iteratively constructed, and new sets of material parameters are selected using an Expected Improvement criterion. The results are discussed in terms of convergence behaviour and optimization efficiency, providing insight into the suitability of Bayesian optimization for solving inverse identification problems.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1049.131.pdf

Accurate calibration of constitutive models is a key requirement for reliable numerical prediction of material behaviour in metal forming and structural applications. Advanced material models, particularly those accounting for nonlinear hardening and plastic anisotropy, typically involve a large number of parameters whose identification requires experimental data that are sufficiently rich to excite multiple deformation states. Traditional identification strategies rely on series of quasi-homogeneous mechanical tests performed under different loading paths and orientations. While effective, such approaches are experimentally demanding and time-consuming.

In recent years, heterogeneous mechanical tests have emerged as a promising alternative, as they enable the activation of a wide range of strain and stress states within a single experiment. When combined with full-field measurement techniques, such as Digital Image Correlation (DIC), these tests provide spatially rich datasets that can significantly enhance the identifiability of constitutive parameters. The use of heterogeneous specimens and full-field data within inverse identification frameworks has been extensively investigated, including previous contributions by the authors, which demonstrated the potential of notched and topology-optimized specimens for full-field FEMU-based material calibration [1-3].

Among inverse strategies, the Finite Element Model Updating (FEMU) [4] approach has been widely adopted for material parameter identification using full-field data. FEMU consists of iteratively updating model parameters to minimize the discrepancy between experimental measurements and finite element predictions. Despite its effectiveness, FEMU-based identification
remains computationally expensive, particularly when dealing with high-dimensional parameter spaces and full-field cost functions, as each iteration requires a complete finite element simulation.

More recently, Machine Learning-based (ML) approaches have been proposed as alternatives to classical FEMU, aiming to directly infer material parameters from experimental data without repeated finite element simulations [5,6]. While these approaches show promising results, they often require large training datasets and may compromise physical interpretability or direct consistency with established finite element models.

To address these limitations, surrogate-assisted optimization techniques have gained increasing attention. In particular, Bayesian Optimization (BO) provides an efficient framework for optimizing expensive black-box functions by combining a probabilistic surrogate model with an acquisition strategy that balances exploration and exploitation. Bayesian optimization has been successfully applied to a wide range of optimization problems in computational mechanics and related fields, demonstrating its ability to reduce the number of costly model evaluations required to reach satisfactory solutions [7, 8]. However, to the best of the authors' knowledge, only a limited number of recent studies have applied Bayesian optimization to FEMU-based material parameter identification [9, 10], and its use in conjunction with heterogeneous mechanical tests remains largely unexplored.

In this context, the present work does not seek to replace FEMU, but rather to enhance it by leveraging machine learning concepts in the form of surrogate-assisted Bayesian optimization. This contribution builds upon previous work on heterogeneous mechanical tests, full-field experimental measurements and FEMU by introducing a data-driven optimization layer on top of a physically grounded FEMU framework. Specifically, the present study investigates the potential of Bayesian optimization coupled with FEMU to improve the efficiency and robustness of material parameter identification using data from a single notched specimen tensile test. The cost function is defined within FEMU framework as the discrepancy between experimental and numerical strain fields, condensed into a scalar value. A Gaussian Process surrogate model is iteratively trained using FEMU evaluations, enabling the adaptive selection of new parameter sets through an acquisition strategy. The proposed framework is evaluated in an exploratory manner, with particular emphasis on convergence behaviour, stability, and the quality of the identified parameters, thereby providing insight into the suitability of Bayesian optimization for reducing the computational burden associated with full-field FEMU calibration while preserving accuracy in reproducing experimentally observed strain fields.

Notched test: experimental setup and numerical model

The original version of this paper is available on https://www.scientific.net/KEM.1049.131.pdf

The present study is based on the notched specimen designed to promote heterogeneous deformation states, including pronounced strain gradients and strain localization in the notch region. Such heterogeneity makes this test particularly interesting and has been extensively investigated in previous works [11]. The experimental procedures and numerical modelling strategy adopted here follow methodologies previously developed and validated and are therefore only briefly summarized [1,2]. Figure 1 illustrates the geometry of the notched specimen along with its general dimensions.

[picture]

Fig. 1. Geometry of the notched specimen [1].

Fig. 1. Geometry of the notched specimen [1].

Full-field displacement and strain measurements are obtained experimentally using Digital Image Correlation (DIC). These measurements provide spatially resolved information over the specimen
surface throughout the loading history and serve as the experimental reference data for the inverse identification procedure. The numerical simulations are performed using Abaqus under quasi-static conditions. The specimen is discretized using two-dimensional finite elements to accurately capture strain gradients and localization effects, while maintaining a reasonable computational cost. Experimentally measured displacement fields are prescribed as boundary conditions on the model, ensuring consistency between experimental and numerical loading paths.
A static general analysis step with automatic time incrementation is employed, using an initial increment of 10−3 and minimum and maximum allowable increments of 10−4 and 10−2, respectively. These settings were selected based on previous numerical studies on heterogeneous specimens and were found to provide a robust compromise between numerical stability and computational efficiency. To enable a consistent comparison between experimental and numerical strain fields, experimental data are interpolated onto the finite element mesh. The interpolation and mapping strategy ensures spatial alignment between experimental measurement points and numerical integration points while accounting for differences in discretization and field resolution. As this framework has been previously developed and validated, further details are not repeated in the present work.

Surrogate-assisted Bayesian optimization

The original version of this paper is available on https://www.scientific.net/KEM.1049.131.pdf

The calibration problem addressed in this work is formulated as an inverse identification task in which a set of constitutive parameters is determined by minimizing a cost function derived from the differences between experimental and numerical data. Each evaluation of the objective function requires a complete finite element simulation of the notched specimen tensile loading test, making the optimization process computationally expensive. As a result, the use of classical gradient-based or exhaustive search strategies becomes impractical. To mitigate this limitation, a surrogate-assisted Bayesian optimization framework is adopted.

Bayesian optimization is particularly suited for problems characterized by expensive, nonconvex, and potentially noisy objective functions. In the present context, FEMU is treated as a blackbox function that maps a vector of material parameters to a scalar cost function value derived from full-field strain discrepancies.

Material behavior and parameter space.

The original version of this paper is available on https://www.scientific.net/KEM.1049.131.pdf

The material behaviour is described using an elastoplastic constitutive model with isotropic hardening. Elastic deformation is modelled by linear isotropic elasticity according to Hooke's law, while plastic yielding is governed by a von Mises equivalent stress criterion. The yield function is expressed as

f(σ,εp―)=σ¯(σ)−σy(εp―),(1)

where σ¯(σ) stands for the equivalent stress defined based on the Cauchy stress tensor σ. The evolution of the flow stress, σy, defined as a function of the equivalent plastic strain, εp―, is described according to the Swift hardening law as

σy(εp―)=K(ε0+εp―)n,(2)

where the nonlinear isotropic hardening behaviour is fully characterized by the parameter vector θ,

θ=[K,ε0,n](3)

that defines the search space explored in the inverse identification procedure. Parameter bounds are selected based on prior calibration studies and physically admissible ranges, ensuring stable numerical simulations and meaningful material responses throughout the optimization process.

FEMU cost-function based on full-field data.

The original version of this paper is available on https://www.scientific.net/KEM.1049.131.pdf

For a given parameter vector θ, a finite element simulation of the notched specimen test is performed and numerical strain fields are obtained. These fields are compared to experimentally measured strain
fields acquired by Digital Image Correlation (DIC). After interpolation schemes, the discrepancy between experimental and simulated results with the θ vector of material parameters is condensed into a scalar cost function defined as

CF(θ)=1nt∑i=1nt{13np∑j=1np[(εxxnum(θ)−εxxexpεxx,maxexp)2+(εyynum(θ)−εyyexpεyy,maxexp)2+(εxynum(θ)−εxyexpεxy,maxexp)2]j+(Fnum(θ)−FexpFmaxexp)2}i(4)

where Fexp  corresponds to the grips' reaction force that was recorded using a load cell. Regarding the strain components εxx,εyy, and εxy, these are computed based on the displacement fields, both numerical and experimental, by the identification software. To enable a straightforward comparison of these quantities, the experimental displacements at the DIC points were interpolated to the FE nodes. The superscripts num and exp refer to the data iteratively generated along the numerical optimization process and to the experimental data, respectively. nt is the number of time instances decided to be used by the authors and np is the number of measured points. Fmaxexp refers to the maximum load value achieved by the test and εmax exp to the maximum strain value of the given strain component. This formulation enables the use of full-field information while yielding a single scalar objective suitable for surrogate-based optimization.

Bayesian optimization strategy.

The original version of this paper is available on https://www.scientific.net/KEM.1049.131.pdf

Bayesian optimization proceeds by constructing a probabilistic surrogate model that approximates the relationship between the parameter vector θ and the FEMU cost function ϕ. In this study, a Gaussian Process (GP) regression model is employed due to its flexibility and its ability to provide both a mean prediction and an associated uncertainty estimate.

Starting from an initial dataset composed of a limited number of FEMU evaluations (five initial simulations), selected using Latin hypercube sampling (LHS) within empirically defined parameter ranges, due to its near-random design and efficient space-filling properties, the Gaussian Process (GP) surrogate is iteratively updated as new data become available. At each iteration, the surrogate model is conditioned on all previously evaluated parameter sets and their corresponding cost function values, yielding a predictive mean, μ, and standard variation, s, over the parameter space. An acquisition function is then maximized to identify the most informative next parameter set to evaluate, which is subsequently assessed through a new FEMU simulation and added to the training dataset.

In this work, the Expected Improvement (EI) criterion is adopted, defined for the minimization problem as

EI(θ)=(CFbest −μ(θ))Φ(Z)+s(θ)ϕ(Z)(5)

with

Z=CFbest−μ(θ)s(θ)(6)

where CFbest  denotes the lowest cost function value observed so far and Φ(·) and ϕ(·) are the cumulative distribution function and probability density function of the standard normal distribution, respectively. This acquisition function naturally balances exploration of regions associated with high predictive uncertainty and exploitation of regions expected to yield lower cost function values.
The overall optimization loop is schematically summarized in Figure 2 and can be described as follows:

Fit the GP surrogate model to the current dataset of evaluated parameter sets and cost function values.

Minimize the acquisition function to select a new candidate parameter vector, θnew .

Perform a FEMU evaluation for the selected parameters set(s), including finite element simulation and cost function computation.

Augment the dataset with the new evaluation and repeat the process for the selected number of trials.
Each evaluation of the FEMU cost function requires a full finite element analysis. For a given parameter vector θ, the constitutive law parameters are updated in the finite element model, and a numerical simulation is performed to compute displacement and strain fields. These numerical fields are then compared to the corresponding experimental full-field measurements, and the discrepancy is quantified through the FEMU cost function, CF(θ). Due to the high computational cost associated with each finite element evaluation, Bayesian optimization is employed to efficiently guide the selection of new parameter sets.

[picture]

Fig. 2. Workflow of the Bayesian optimization procedure. Each evaluation of the cost function involves a finite element simulation to compute numerical displacement and strain fields, which are compared with experimental full-field data. Adapted from [12].

Fig. 2. Workflow of the Bayesian optimization procedure. Each evaluation of the cost function involves a finite element simulation to compute numerical displacement and strain fields, which are compared with experimental full-field data. Adapted from [12].

Results and Discussion

The original version of this paper is available on https://www.scientific.net/KEM.1049.131.pdf

This section presents the results obtained with the proposed surrogate-assisted Bayesian optimization framework applied to FEMU calibration from a single notched specimen tensile loading test. As the present contribution is intended as an exploratory study, the discussion focuses on convergence behaviour, optimization efficiency, and qualitative assessment of the identified material parameters rather than on exhaustive parametric validation.

Optimization behaviour and parameter space exploration.

The original version of this paper is available on https://www.scientific.net/KEM.1049.131.pdf

Figure 3 illustrates the evolution of the FEMU cost function during the Bayesian optimization process, together with the best-so-far value obtained after each trial. It is worth noting that the parameter set and, consequently the CF value, of the first trial is highly dependent on the samples composing the initial dataset. The existence of a sample in the initial dataset that already presented a significant low CF value influenced the optimization problem behavior. As the optimization progresses, the balance between exploration and exploitation allows the algorithm to explore regions of the parameter space associated with higher CF and, at the same time, to refine the solution, ultimately leading to a stable best-so-far value and practical convergence within a limited number of FEMU evaluations (16). It should be noted that the CF value could have achieved lower values if more trials were chosen.

[picture]

Fig. 3. Evolution of the FEMU cost function, CF, as a function of the number of trials. Individual markers represent the cost function values obtained at each FEMU evaluation (BO trial), while the best-so-far curve highlights the convergence behaviour of the optimization process.

Fig. 3. Evolution of the FEMU cost function, C F , as a function of the number of trials. Individual markers represent the cost function values obtained at each FEMU evaluation (BO trial), while the best-so-far curve highlights the convergence behaviour of the optimization process.

To further analyse the behaviour of the Bayesian optimization procedure, the distribution of the parameter sets in the design space is examined. Figure 4 illustrates the projection of the sampled points onto selected two-dimensional parameter subspaces, allowing the exploration strategy of the algorithm and the location of the identified optimal solutions to be visually assessed.

[picture]

Fig. 4. Samples distribution in the parameter space projected onto selected two-dimensional planes: (a) K−ε0 and (b) ε0−n. Initial samples are shown in blue, BO -generated samples in orange, together with the best parameter set from the initial dataset and the overall best solution identified during the optimization.

Fig. 4. Samples distribution in the parameter space projected onto selected two-dimensional planes: (a) K − ε 0 and (b) ε 0 − n . Initial samples are shown in blue, BO -generated samples in orange, together with the best parameter set from the initial dataset and the overall best solution identified during the optimization.

The initial samples (blue dots) correspond to the five evaluations presented in the initial dataset, which parameter sets were generated by LHS method. Based on these data, the BO method, specifically the acquisition function, decide which parameter set to try next (trial). At each trial, a BO sample is generated, corresponding to the orange triangles (cf. Figure 4). After 16 trials, the optimization ends with the best solution so-far in the process. This one as well as best solution presented in the initial dataset are highlighted in the same colour of the group which are a part of, BO samples and initial dataset, respectively. The distribution of the BO-generated samples indicates an initial exploratory phase covering a broader region of the parameter space, followed by a progressive concentration of samples around regions associated with lower cost function values. These results indicate the possibility that the optimisation procedure became trapped in a local minimum. Increasing the dimensionality and diversity of the initial dataset or evaluating other options for the acquisition function could improve the search capability to avoid premature convergence.

Identified parameters and full-field agreement.

The original version of this paper is available on https://www.scientific.net/KEM.1049.131.pdf

Figure 5 presents a comparison between experimental and numerical strain fields obtained using the best set of material parameters identified through the methodology presented in this work. The experimental strain maps are derived from full-field measurements, while the numerical strain fields are computed from finite element simulations using the calibrated constitutive model.

[picture]

(b)

Figure 5. (b)

Fig. 5. Comparison between the (a) experimental and (b) numerical strain fields for the notched specimen tensile specimen using the set of parameters corresponding to the best solution of the Bayesian optimization approach.

The material parameter set identified as the best solution at the end of the Bayesian optimization process is found to be physically plausible and consistent with reference values reported in previous calibration studies. The corresponding numerical results reproduce the main features observed experimentally, including strain localization patterns and gradients in the notch region. These qualitative observations confirm that the reduction in the FEMU cost function achieved during optimization corresponds to a good improvement in the reproduction of experimentally observed deformation mechanisms. Nevertheless, a perfect agreement between experimental and numerical strain fields is not achieved. This indicates that the parameter set identified as optimal with respect to the adopted cost function does not necessarily correspond to the true physical material parameters. Such discrepancies may stem from multiple sources, including limitations of the inverse identification
procedure, experimental uncertainties, chosen test specimen or constitutive model. Moreover, this observation must be interpreted in light of the intrinsic non-uniqueness of inverse problems, where different parameter combinations may lead to similarly low-cost function values and comparable fullfield responses. In addition, the robustness of the optimisation framework itself may influence the outcome, as the final solution remains dependent on the initial dataset, the dimensionality of the sampled space, and the balance between exploitation and exploration governed by the acquisition function. Insufficient global search capability may prevent convergence towards the true global optimum.

Conclusion

The original version of this paper is available on https://www.scientific.net/KEM.1049.131.pdf

This work presented an exploratory investigation of surrogate-assisted Bayesian optimization for FEMU-based material model calibration using full-field strain data obtained from a single heterogeneous notched tensile test. The proposed framework couples FEMU with a probabilistic surrogate model to efficiently explore the material parameter space while limiting the number of computationally expensive finite element simulations.

The observed convergence behaviour indicates that Bayesian optimization can effectively guide the calibration process. The main strength of the proposed approach lies in its ability to efficiently explore parameter space and to identify promising regions using a limited number of costly finite element evaluations. In this sense, Bayesian optimization provides a suitable data-driven layer on top of full-field FEMU, offering a flexible and computationally more efficient alternative to traditional gradient-based or exhaustive search strategies.

Nevertheless, several limitations must be acknowledged. The present study is intentionally restricted to a single heterogeneous test and a limited parameter space, and no systematic investigation was conducted on the influence of different acquisition functions, the size of the initial training dataset, or the number of optimization trials. The use of other acquisition functions which allow setting hyperparameters regarding the exploration-exploitation balance may improve the search in the parameter space, avoiding being trapped in a local minimum.

Despite these limitations, the proposed methodology constitutes a solid and extensible basis for future developments. Further work will focus on extending the approach to more complex heterogeneous tests and constitutive models, as well as on systematically investigating the influence of initial sampling strategies, surrogate model hyperparameters, and acquisition function choices. Such studies will be essential to fully assess the robustness, reliability, and general applicability of Bayesian optimization-assisted FEMU for material parameter inverse identification.

Acknowledgement

The original version of this paper is available on https://www.scientific.net/KEM.1049.131.pdf

The authors gratefully acknowledge the funding from the Project ML4ModelCalib (2024.15509.PEX), financed by Fundação para a Ciência e tecnologia (FCT) https://doi.org/10.54499/2024.15509.PEX.

References

The original version of this paper is available on https://www.scientific.net/KEM.1049.131.pdf

M. Gonçalves, S. Thuillier, A. Andrade-Campos, Assessing the potential of heterogeneous mechanical tests for sheet metals through experimentally measured full-fields, Exp. Mech. 65 (2025) 757-774.

M. Gonçalves, S. Thuillier, A. Andrade-Campos, Inverse identification of anisotropic plasticity model parameters using FEMU and a heterogeneous test, Mater. Res. Proc. 54 (2025) 15481557.

L. Germain, M. Gonçalves, T. Barret, A. Andrade-Campos, S. Thuillier, Virtual forming based on model calibration from heterogeneous tests, Mater. Res. Proc. 54 (2025) 1568-1576.

J.M.P. Martins, A. Andrade-Campos, S. Thuillier, Comparison of inverse identification strategies for constitutive mechanical models using full-field measurements, Int. J. Mech. Sci. 145 (2018) 330-345.

A.E. Marques, T.G. Parreira, A.F.G. Pereira, Machine learning application to the identification of sheet metal constitutive model parameters, Mach. Learn. Comput. Sci. Eng. 1 (2025) 10.

T.G. Parreira, A.E. Marques, N.A. Sakharova, P.A. Prates, A.F.G. Pereira, Identification of sheet metal constitutive parameters using metamodeling of the biaxial tensile test on a cruciform specimen, Metals 14 (2024) 212.

R.P. Cardoso Coelho, A.F.C. Alves, F.M. Andrade Pires, Efficient constitutive parameter identification through optimisation-based techniques: A comparative analysis and novel composite Bayesian optimisation strategy, Comput. Methods Appl. Mech. Eng. 427 (2024) 117039.

J. Kuhn, J. Spitz, P. Sonnweber-Ribic, et al., Identifying material parameters in crystal plasticity by Bayesian optimization, Optim. Eng. 23 (2022) 1489-1523.

T. Long, L. Wang, C. Kan, J.D. Lee, A Novel Approach for Modelling Strain Hardening in Plasticity and Its Material Parameter Identification by Bayesian Optimization for Automotive Structural Steels Application. Applied Math 5 (2025), 104.

A. Vlachas, E. Chatzi, J.F. Unger, A Bayesian framework for constitutive model identification via use of full field measurements, with application to heterogeneous materials, Comp. Methods in Appl. Mech. Eng. 433 (2025), 117489.

M. Rossi, F. Pierron, and M. Štamborská, Application of the virtual fields method to large strain anisotropic plasticity, Int. J. Solids Struct., 97-98, (2016) 322-335.

Y. Wang, T.-Y. Chen, and D.G. Vlachos, NEXTorch: A Design and Bayesian Optimization Toolkit for Chemical Sciences and Engineering, J. Chem. Inf. Model. 61 (2021) 5312-5319.

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/lionel-fourment-on-optimization-ai-and-inverse-analysis-in-forming/978-3-0364-1989-3

Key Engineering Materials, ISSN: 1662-9795, Vol. 1049, pp 141-150

doi: 10.4028/p-1kSPJP

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

Submitted:
2026-02-02

Revised:
2026-02-24

Accepted:
2026-03-02

Online:
2026-04-15

Robust Performance Optimization Using Bayesian Optimization and Extreme Value Theory

The original version of this paper is available on https://www.scientific.net/KEM.1049.141.pdf

Augustin Persoons a,*,, Valentin Duarte Rocha b, Laurence Giraud Moreau c, Pascal Lafon d 

Université de Technologie de Troyes, Unité de Recherche LASMIS

a augustin.persoons@utt.fr, b valentin.duarte_rocha@utt.fr, c laurence.moreau@utt.fr, d  pascal.lafon@utt.fr

Keywords: Bayesian Optimization, Optimization Under Uncertainty, Robustness, Active Learning, Forming Processes.

Abstract

This work deals with Robust design optimization (RDO) under interval uncertainty and the resolution of such problems using a Bayesian optimization algorithm. In metal forming, process parameters such as tool radius, step size, or forming toolpath introduce variability that directly affects the final geometry and quality of the formed parts. In this context we aim at finding a design minimizing the amplitude of the performance interval but such a formulation does not account for the nominal performance. In this work, we introduce a scalarized objective adapted to the proposed algorithm allowing it to identify a Pareto optimum of both stability and nominal behavior. We propose an efficient expected improvement (EI) estimator for this objective based on an extreme-value approximation of surrogate extrema. The approach is illustrated on an analytical test problem and on a forming simulation with spring-back, where the new objective yields more practically relevant solutions than a variation-only robustness criterion.

1 Introduction and Problem formulation

The original version of this paper is available on https://www.scientific.net/KEM.1049.141.pdf

In many industrial settings, the performances of a system or process must not only perform well at nominal conditions, but must also remain acceptable under variability of operating conditions, material properties, or manufacturing parameters. This motivates robust design optimization (RDO), i.e. the search for design variables that reduce sensitivity to uncertainty while meeting performance requirements. Robustness is often associated with Taguchi's work [1], and a broad spectrum of formulations has been proposed since then, ranging from stochastic approaches based on moments or failure probabilities to deterministic worst-case approaches [2-4].

This work considers the case where uncertain variables and design variables are distinct. Moreover, uncertainties are described by intervals rather than probability distributions. Such a setting occurs when uncertainty data is scarce, heterogeneous, or not sufficiently stationary to justify a probabilistic model, yet reliable bounds can be specified based on expert's knowledge. Let x∈Ix⊂ℝnx denote design variables for which we want to optimize the system and u∈Iu⊂ℝnu uncertain variables which can take any value within the interval, and let the (expensive) performance model be

y=g(x,u)

Since the parameter set u is an interval, the performance for any design x is also an interval. A classical robustness metric over interval uncertainty is the range (worst-case spread) of the response,

Δy(x)=maxu∈Iug(x,u)−minu∈Iug(x,u)

which quantifies the amplitude of performance variations associated with a given design ( x ). A first step toward robust design optimization could be to focus on minimizing Δy(x), to reach the design with the most stable performance. This is, however, a two-level nested optimization problem which
is notoriously computationally demanding and usually untractable if g is computationally expensive to evaluate. For the sake of clarity let us refer to the problem of identifying, for a given design x, the most extreme performance values maxu∈Iug(x,u) and minu∈Iug(x,u) as the inner-problem. Likewise, let us refer to the problem of identifying which design is the most stable (minx∈IxΔy(x)) as the outer-problem.

Previous work focused on proposing to solve this RDO problem with Bayesian optimization first based on confidence interval [5] and then using a heuristic based on extreme-value distributions [6]. However, optimizing Δy(x) alone may yield designs with stable but consistently poor performance. For this reason, the present paper revisits the objective formulation and proposes a scalar meanvariation criterion that can be interpreted as a compact alternative to an explicit two-objective formulation.

Let us define, for any design x, the extreme responses

g¯(x)=maxu∈Iug(x,u),g―(x)=minu∈Iug(x,u),

and the mid-range (center of the response interval)

my(x)=g¯(x)+g―(x)2

We define the objective

J(x)=12Δy(x)2+(2 my(x))2(1)

assuming the design optimization problem is formulated such that the optimal performance is associated with g(x,u)=0 (e.g. offset from a desired performance value), this criterion penalizes both spread and offset in a single quantity. Using Δy(x)=g¯(x)−g―(x), it follows that minimizing J(x) is equivalent to minimizing the smooth quadratic form

J2(x)=g¯(x)2+g―(x)2(2)

up to a monotone transformation.
The problem is illustrated in Figure 1 on the analytical benchmark 2D function described in section section 4. Figure 5a illustrates the performance surface as a function of design and uncertain parameters x,u. Figure 1 b is a projection of that surface on the x−y plane, illustrating the amplitude of variation of the performance over u and how this amplitude evolves as a function x. Figure 1c represents the evolution of Δy as a function x, also highlighting the optimal design with respect to this criterion. Figure 1c represents the evolution of my as a function x and associate optimum. Figure 1d represents the evolution of the multi-objective criterion J as a function x with associated optimal design.

The computational challenge is that each evaluation of J(x) (or J2(x) ) involves two inner optimization problems over u, and the outer optimization over x must be performed with a limited simulation budget. Bayesian optimization (BO) addresses such settings by replacing the expensive model g with a surrogate model g^ (a Gaussian process regression model, GPR). The surrogate is calibrated with an active learning strategy aiming at minimizing the number of calls to g necessary to estimate the optimum. The active calibration scheme uses a so-called acquisition function (e.g. expected improvement, EI) to identify which sample to include next in the calibration dataset [7].

The main difficulty in this context is to propagate the stochasticity of the surrogate g^ to the objective Δy to guide the active calibration. In standard BO, EI is straightforward to compute when the objective at a candidate point is Gaussian. In the present setting, the objective depends on extrema of the Gaussian process, and accurately characterizing the distribution of these extrema is known to be challenging for non-stationary Gaussian processes [8, 9]. A naive solution based on Monte Carlo sampling of GPR trajectories would be prohibitively expensive when embedded in an outer optimization

[picture]

Figure 1: Illustration of the RDO problem and associated quantities of interest

Figure 1. Illustration of the RDO problem and associated quantities of interest

loop. The proposed approach relies on building a simplified stochastic representation of Δy using extreme value distributions in two steps. First, we approximate the distributions of the extrema induced by a virtual population of GPR realizations using extreme-value theory (Gumbel-type approximations). Second, exploiting the quadratic form in eq. (2), using a simple change of variables we can numerically estimate the EI through numerical integration at a moderate and controllable cost. The resulting method retains the practical benefits of EI-driven exploration-exploitation while avoiding repeated sampling of full GPR trajectories.

The rest of the paper is organized as follows: Section 2 introduces the GPR surrogate setting and the extreme-value approximation of its extrema. Section 3 develops the acquisition function proposed and the Bayesian optimization scheme Finally, Section 4 presents the numerical examples including the forming simulation, and Section 5 concludes with perspectives for future work.

2 Gaussian-Process Surrogate and Extreme-Value Approximations of Its Extrema

The original version of this paper is available on https://www.scientific.net/KEM.1049.141.pdf

2.1 Gaussian-process regression model The goal of this section is to build a surrogate of the expensive model g(x,u), using a Gaussian process regression model (GPR). Their principle is to consider the true performance surface as a realization of a Gaussian process (GP). By assuming a GP prior, a covariance function, and given some data points on the true performance surface, the prior can be conditioned to the available data to calibrate a predictive posterior through Bayesian inference.

For a more details on GPRs, interested readers can refer to [10]. In practice this type of surrogate can be trained in a similar fashion as any regression model, the main difference is that their prediction is stochastic in nature. Meaning that, for any candidate input (x,u)∈Ix*Iu, the predicted performance is a normally-distributed random variable with mean μ(x,u) and standard deviation σ(x,u),

g^(x,u)~𝒩(μ(x,u),σ(x,u))(3)

The mean value can be seen as the most likely performance values according to the surrogate, while the standard deviation can be seen as the surrogate's prediction uncertainty.

The outer robust objective introduced in section 1 depends on the true (model-based) extrema

g¯(x)=maxu∈Iug(x,u),g―(x)=minu∈Iug(x,u),

and the corresponding quantity to be minimized is

J(x)=g¯(x)2+g―(x)2

As previously mentioned, these quantities are not available to us and are too computationally expensive to evaluate directly. However, once a GP posterior g^ is available, on can define the surrogate extrema defined as

g^―(x)=maxu∈Iug^(x,u),g^―N(x)=minu∈Iug^N(x,u)(4)

A key point is that since g^(·) is a Gaussian process, g^―N(x) and g^―N(x) are random variables for every fixed x. Consequently, the surrogate-based objective

J^(x)=g^―(x)2+g^―(x)2(5)

is itself a random variable, meaning that J^ is a stochastic process and similarly to g^, its variability is related to the surrogate's prediction uncertainty.

In other words, one can see J^ as a stochastic surrogate of the RDO objective function. The idea is then to use a Bayesian optimization strategy applied to this surrogate and thus solve the RDO problem.
2.2 The expected improvement acquisition function One of the most popular Bayesian optimization method is the efficient global optimization methods (EGO) [7]. Which proposed a heuristic for selecting the coordinates to refine named the expected improvement.

Let us consider an arbitrary step N of the Bayesian optimization algorithm and the current best robust value available to us JN⋆. Note that this value is a deterministic scalar, not a random variable. Given the current calibration of the GPR surrogate, let us define the improvement of a design as the difference (bounded by zero) between the objective values J of the current optimum and the candidate design x.

IN(x)=max(0,JN⋆−J^(x))

since J^(x) is a random variable, IN(x) is also one. The corresponding expected improvement (EI) criterion is then defined as:

EI(x)=𝔼[I(x)]=𝔼[max(0,J⋆−(g^―(x)2+g^(x)2))](6)

In a standard setting (where the objective is directly the GP prediction), EIN admits a closed form. The issue in the RDO case is that the distributions of g^― and g^― are not available to us. They are random variables defined as the extrema of a Gaussian process so there is not closed form for their distributions in the general case and eq. (6) cannot be evaluated analytically. Therefore we need a practical and computationally tractable approximation strategy for the distributions of g^― and g^―.
2.3 A heuristic based on extreme value distributions This section summarizes the key idea used to obtain a tractable approximate distribution for the surrogate-based extrema g^― and g^―. Which are needed to compute acquisition functions such as Expected Improvement (EI) for robust objectives. A more detailed derivation and empirical justification can be found in [6].

The heuristic relies on the following idea: for a fixed ( x,u ), consider a virtual population of n independent draws from the predictive distribution Y=g^(x,u)

Y1,…,Yn~iid𝒩(μ(x,u),s2(x,u))

and denote their maximum by

Ymax(n)(x,u)=max1≤i≤nYi

This extrema is a random variable and extreme-value theory suggests that, for moderately large n, the law of Ymax (n) is well approximated by a Gumbel distribution after affine normalization. In practice, we only need a fast approximation of the Gumbel location and scale parameters that can be directly extracted from the surrogate posterior values μ(x,u) and s2(x,u) and depend only on n. A distribution of the minimum Ymin n can be similarly approximated.

For any given design x, we then use a statistics (e.g. mean or a quantile) of this extreme value distribution as a heuristic score to locate where the extremum over u is most likely to occur (locate the maximizer and minimizer):

u+(x)≃argmaxu∈Iu(E(Ymaxn)),u−(x)≃argminu∈Iu(E(Yminn))(7)

Intuitively, eq. (7) balances exploitation (large/small mean) and exploration (large uncertainty), and returns two representative locations: one for the maximum and one for the minimum.

Once the likely maximizer and minimizer u values (u+(x),u−(x)) are located, we approximate the extrema of the Gaussian process g^(x,u) over u by the virtual extremes at these locations:

g^―N(x)≈Ymax(n)(x,u+(x)),g^―N(x)≈−Ymax(n)(x,u−(x)).(8)

This yields two explicit one-dimensional approximate distributions, which are the only objects required by the next-step EI computation. It is however worth noting that this represents a simplistic approximation of the actual distribution only meant to guide the calibration of the surrogate in a pseudo-optimal manner. This simplification means in essence reducing the Gaussian process to a single Gaussian random variable, the one corresponding to the maximizer/minimizer location and cannot be considered as a proper approximation of the extrema of the GP.

3 Acquisition Function and Bayesian Optimization Algorithm

The original version of this paper is available on https://www.scientific.net/KEM.1049.141.pdf

It is important to distinguish the three levels of approximation of the objective function:

the true robust objective

J(x)=g¯(x)2+g―(x)2,g¯(x)=maxu∈Iug(x,u),g―(x)=minu∈Iug(x,u),

which can only be accessed through expensive calls to g

the surrogate-induced stochastic objective

J^(x)=g^―(x)2+g^―(x)2

where g^―(x) and g^―(x) are random because g^ is a posterior GP. In the proposed method, calibration decisions (where to sample next) are made by maximizing an acquisition function requiring the estimation of statistics of J^. Unfortunately, as mentioned in the previous section, the distribution of J^ doesn't have a closed-form expression.

the extreme-value heuristic used as a computationally tractable proxy for J^

J^ev(x)=Ymax(n)(x,u+(x))2+Ymin(n)(x,u−(x))2

as described in equations 8 and 7 .
3.1 Expected improvement under the extreme-value approximation To obtain a tractable EI, we assume Ymax n and Ymin n are independent under the fitted models. This is an extra approximation: maxima and minima of the same random field are generally dependent. Empirically, the acquisition remains effective because EI is used only as a ranking criterion over x. Then, considering a current optimal value J⋆ the expected improvement is approximated as:

EIN(x)≈𝔼[max(0,J⋆−Ymax(n)(x,u+(x))2+Ymin(n)(x,u−(x))2))].(9)

In the multi-objective robust criterion used here, let us consider the radius R=(Ymax n)2+(Ymin n)2 and define the improvement w.r.t. the current best estimate J⋆ by I(J⋆)=(J⋆−R). The Expected Improvement is then

EI(τ)=𝔼[I(J⋆)]=∫0J⋆(J⋆−r)fR(r)dr(10)

where fR is the density of R. Under independence, the joint density is fX,Y(x,y)=fX(x)fY(y), and by the polar change of variables (x,y)=(rcosθ,rsinθ) (Jacobian r ) we obtain

fR(r)=r∫02πfX(rcosθ)fY(rsinθ)dθ,r≥0

We evaluate the inner angular integral numerically and then compute EI(J⋆) by quadrature on [0,J⋆].
3.2 Overall procedure The next section reports numerical experiments and compares the proposed strategy against baselines (e.g. standard EI on a scalarized objective, worst-case UCB, and random sampling), using a fixed evaluation budget.

Since the algorithm estimates the amplitude of performance y for any given design x, at the next candidate sample for refinement xN+1 there is two candidate u positions, uN+1+and uN+1−as described in section 2.3. In order to limit the number of calls to the expensive model g, only one of the two candidate points is actually evaluated and used to enrich the calibration dataset. The choice is made based on the variables of the Gumbel distributed variables Ymax n and Ymax n where the sample being associated with the highest variance is selected.

The overall method procedure is finally described in algorithm 1.

4 Examples and Results

The original version of this paper is available on https://www.scientific.net/KEM.1049.141.pdf

4.1 Analytical benchmark 1: local minima The convergence of the proposed method is first assessed on a simple two-dimensional analytical benchmark chosen to exhibit several local minima as illustrated in figure 1. The test case is defined by the following performance function g :

g(x,u)=xu−sin(x)u2+x2,(x,u)∈Ix×Iu(11)

The design variable x is restricted to the interval Ix=[−5,5], while the uncertain variable u varies within Iu=[−5,5]. For this benchmark, the reference optimal design is x*=0.

The convergence of the proposed method on this example is shown in Figure 2. The figure illustrates that the algorithm was able to stably reach the global minimum after about 20 iterations As a perspective, a related method previously developed by the authors required around 27 iterations to reach convergence on this example [5]. While requiring further statistical analysis, this resultis a first indicator of the good performances of this proposed approach.

Algorithm 1 Bayesian optimization for interval-robust design via extreme-value EI
    Choose an initial design set \(\left\{\left(x_{i}, u_{i}\right)\right\}_{i=1}^{N_{0}}\) and evaluate \(y_{i}=g\left(x_{i}, u_{i}\right)\).
    Fit a GP surrogate \(\hat{g}_{N_{0}}\) on \(\mathcal{D}_{N_{0}}\).
    Initialize \(J^{\star}\) from robust evaluations of a small set of designs.
    for \(N=N_{0}, N_{0}+1, \ldots\) until budget is exhausted do
        For candidate \(x\), fit Gumbel models for \(\overline{\hat{g}}_{N}(x)\) and \(\underline{\hat{g}}_{N}(x)\) (Section 2.3).
        Compute \(\mathrm{EI}_{N}(x)\) via eq. (10).
        Select \(x_{N+1} \in \operatorname{argmax}_{x \in I_{x}} \mathrm{EI}_{N}(x)\).
        Choose \(u_{N+1}^{+}, u_{N+1}^{-}\)as described in section section 3.1
        Evaluate \(g\left(x_{N+1}, u_{N+1}^{+}\right)\)or \(g\left(x_{N+1}, u_{N+1}^{-}\right)\); update \(\mathcal{D}_{N+1}\).
        Refit the GP to obtain \(\hat{g}_{N+1}\).
        Estimate \(\hat{J}^{\star}\left(x_{N+1}\right)\) by direct estimation using \(Y_{\text {max }}, Y_{\text {min }}\) and update \(J_{N+1}^{\star}\).
    end for
    return Best design \(x^{\star}\) associated with \(J_{2, N}^{\star}\).

[picture]

Figure 2: Convergence of the algorithm on the analytical benchmark case

Figure 2. Convergence of the algorithm on the analytical benchmark case

4.2 Forming process finite element example with springback The last example is based on a 2D finite element simulation of a sheet-metal forming process, in which a metal blank is clamped against the die by a holder and bent into a "U" shape by a punch (see [11]). The material is DP780 steel with a thickness of 1.4 mm . After forming, the tool is released and the elastic recovery is simulated in order to account for springback. The quantity of interest is the final sheet profile after springback.

The profile is assumed to depend on two parameters: the die-edge rounding radius {(xi,ui)}i=1N0 and the force applied to the Holder yi=g(xi,ui). In this study, g^N0 is treated as an interval uncertainty with 𝒟N0 while J⋆ is the design variable with design space N=N0,N0+1,…. In essence the problem can be seen as identifying which die geometry is the most robust to holding force variation. This example should be seen as a simple benchmark example and is not intended to represent realistic industrial challenges.

The performance metric is defined as the root mean square error (RMSE) between the nodal coordinates of the sheet profiles before and after springback (see an illustration in Figure 4).

The problem can be illustrated in a similar fashion as the first one in Figure 3. The contrary to the actual application of the method, the illustration is based on GPR surrogate of the finite element model, built with an previously performed design of experiment (red dot in figure). This result is only used for illustration purposes and to obtain easily computable insights of the the model behavior.

[picture]

Figure 3: Schematic representation and dimensions of the forming process.

Figure 3. Schematic representation and dimensions of the forming process.

[picture]

Figure 4: Illustration of the formed sheet profile before and after springback.

Figure 4. Illustration of the formed sheet profile before and after springback.

As illustrated in Figure 3, the performance surface of the finite element model is relatively well behaved without being trivial. The problem seem to exhibit clear optimal designs x in terms of amplitude, nominal value and multi objective indicator.

The proposed method was run for 20 iterations on the finite-element model and the convergence is shown in Figure 6. The method was initialized with a latin-hypercube design of experiment of 10 evaluations og the finite-element model. The method evidently converged the expected optimal in a few iterations which is a good first indicator of the performances of the method.

5 Conclusions and Perspectives

The original version of this paper is available on https://www.scientific.net/KEM.1049.141.pdf

This paper proposes a Bayesian optimization strategy for robust design optimization under interval uncertainty, targeting expensive black-box simulations. In contrast with our previous work, which emphasized a variation-only robustness measure, we introduce a mean-variation objective that explicitly balances nominal performance and robustness.

To enable efficient optimization with a Gaussian process surrogate, we develop an expected improvement (EI) estimation approach tailored to this extrema-based objective. The method combines (i) an extreme-value modeling of the maxima and minima of GP trajectories and (ii) a numerical integration scheme made tractable by a polar-like change of variables. This construction yields a

[picture]

Figure 5: Illustration of the RDO problem and associated quantities of interest

Figure 5. Illustration of the RDO problem and associated quantities of interest

[picture]

Figure 6: Convergence of the proposed algorithm on the finite element problem.

Figure 6. Convergence of the proposed algorithm on the finite element problem.

computationally efficient EI approximation that remains closely connected to the robustness metric of interest.

The approach is illustrated on two examples: an analytical test problem featuring local minima and a finite element forming simulation with springback. In both cases, the method is able to drive the search toward robust designs while accounting for nominal performance, and it provides solutions that are more practically relevant than those obtained with a variation-only robustness criterion.

An application of this method on optimizing an incremental forming process is being investigated. Future work will focus on a systematic sensitivity analysis of key modeling and numerical choices (notably the extreme-value approximation and the integration settings), as well as extended comparisons on a broader and fully reproducible benchmark suite and additional industrial-scale simulation cases.

References

The original version of this paper is available on https://www.scientific.net/KEM.1049.141.pdf

G. Taguchi, Quality engineering (Taguchi methods) for the development of electronic circuit technology, IEEE Transactions on Reliability. 44 (June 1995) 225-229.

H.-G. Beyer, B. Sendhoff, Robust optimization - A comprehensive survey, Computer Methods in Applied Mechanics and Engineering. 196 (July 2007) 3190-3218.

C. Zang, M. Friswell, J. Mottershead, A review of robust optimal design and its application in dynamics, Computers & Structures. 83 (Jan. 2005) 315-326.

D. Bertsimas, D. B. Brown, C. Caramanis, Theory and Applications of Robust Optimization, SIAM Review. 53 (Jan. 2011) 464-501.

C. van Mierlo, A. Persoons, M. G. R. Faes, D. Moens, Robust Design Optimization of Expensive Stochastic Simulators Under Lack-of-Knowledge, ASCE-ASME J Risk and Uncert in Engrg Sys Part B Mech Engrg. 9 (June 2023) 021205.

A. Persoons, P. Lafon, D. Moens, BAYESIAN ROBUST DESIGN OPTIMIZATION: AN EXTREME VALUE APPROACH, Uncecomp 2025 proceedings. (June 2025).

D. R. Jones, M. Schonlau, Efficient Global Optimization of Expensive Black-Box Functions, Journal of Global Optimization. 13 (1998) 455-492.

J.-M. Azaïs, L.-V. Lozada-Chang, A toolbox on the distribution of the maximum of Gaussian process, (2013).

G. Lindgren, Gaussian Integrals and Rice Series in Crossing Distributions-to Compute the Distribution of Maxima and Other Features of Gaussian Processes, Statistical Science. 34 (Feb. 2019).

R. B. Gramacy, Surrogates: Gaussian Process Modeling, Design and Optimization for the Applied Sciences, Chapman Hall/CRC, Boca Raton, Florida, 2020.

S. H. e. W. C. GH. Huh K. Chung, Benchmark Study of the 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes, Proceedings of Numisheet 2011. (2011).

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.

Keywords Index

A

Active Learning Robust Performance Optimization Using Bayesian Optimization and Extreme Value Theory

Anisotropy Towards Single-Test Anisotropy Calibration: Sensitivity, Identifiability and Validation of YLD2000-2d

Artificial Neural Network Advanced Control of Stretch-Reducing Mills Using Artificial Intelligence

Asset Administration Shell Method of an Automated Tool Design of Punch-Bending Processes Using the Asset Administration Shell

B

Bayesian Data Assimilation Material Parameter Identification Using Bayesian Data Assimilation and Biaxial Tensile Test

Bayesian Optimization Robust Performance Optimization Using Bayesian Optimization and Extreme Value Theory

Bending AI-Driven Design and Optimization of Bending Processes

Bending Processes Uncertainty Detection in Sheet Metal Bending Processes with Machine Learning

C

Constitutive Modelling Machine Learning Algorithms Applied to the Identification of CPB06 Yield Criterion Parameters

Convolutional Neural Network Application of Reference-Free Strain Measurement to Assess the Deformation Level of Pre-Deformed Components

Cut Planning Geometry-Based Concept for Automatic Cut Planning and Flattening of Sheet Metal Components

D

Data-driven Improving FEMU Calibration from a Single Heterogeneous Test through a Data-Driven Approach: An Exploratory Study

Dic Innovation of the "Cut-Clamp-Play" Concept for Robust and Efficient Sheet Metal Material Characterization; Towards Single-Test Anisotropy Calibration: Sensitivity, Identifiability and Validation of YLD2000-2d

Digital Image Correlation Towards DIC-Subset-Independent Machine Learning Models for Constitutive Parameter Identification in Sheet Metal Forming

Digital Twin Method of an Automated Tool Design of Punch-Bending Processes Using the Asset Administration Shell

F

Femu Improving FEMU Calibration from a Single Heterogeneous Test through a Data-Driven Approach: An Exploratory Study

Finite Element Model Updating Towards Single-Test Anisotropy Calibration: Sensitivity, Identifiability and Validation of YLD2000-2d

Flattening Geometry-Based Concept for Automatic Cut Planning and Flattening of Sheet Metal Components

Forces Modeling Proxy-Physics-Informed and Transfer Learning Networks for Radial-Axial Ring Rolling (RARR) under Varying Data Availability

Forming Processes Uncertainty Detection in Sheet Metal Bending Processes with Machine Learning; AI-Driven Design and Optimization of Bending Processes; Robust Performance Optimization Using Bayesian Optimization and Extreme Value Theory

G

Gematrical Modeling Proxy-Physics-Informed and Transfer Learning Networks for Radial-Axial Ring Rolling (RARR) under Varying Data Availability

H

Heterogeneous Test; Material Testing 2.0 Improving FEMU Calibration from a Single Heterogeneous Test through a Data-Driven Approach: An Exploratory Study

I

Intelligent Process Control Advanced Control of Stretch-Reducing Mills Using Artificial Intelligence

M

Machine Learning Uncertainty Detection in Sheet Metal Bending Processes with Machine Learning; AI-Driven Design and Optimization of Bending Processes; Machine Learning Algorithms Applied to the Identification of CPB06 Yield Criterion Parameters; Towards DIC-Subset-Independent Machine Learning Models for Constitutive Parameter Identification in Sheet Metal Forming

Material Characterization Innovation of the "Cut-Clamp-Play" Concept for Robust and Efficient Sheet Metal Material Characterization; Towards Single-Test Anisotropy Calibration: Sensitivity, Identifiability and Validation of YLD2000-2d

Material Modeling Material Parameter Identification Using Bayesian Data Assimilation and Biaxial Tensile Test

Mt2.0 Innovation of the "Cut-Clamp-Play" Concept for Robust and Efficient Sheet Metal Material Characterization; Towards Single-Test Anisotropy Calibration: Sensitivity, Identifiability and Validation of YLD2000-2d

O

Optimization AI-Driven Design and Optimization of Bending Processes

Optimization Under Uncertainty Robust Performance Optimization Using Bayesian Optimization and Extreme Value Theory

P

Parameter Identification Material Parameter Identification Using Bayesian Data Assimilation and Biaxial Tensile Test; Machine Learning Algorithms Applied to the Identification of CPB06 Yield Criterion Parameters; Towards DIC-Subset-Independent Machine Learning Models for Constitutive Parameter Identification in Sheet Metal Forming

Plastic Anisotropy Innovation of the "Cut-Clamp-Play" Concept for Robust and Efficient Sheet Metal Material Characterization

Plastic Deformation Application of Reference-Free Strain Measurement to Assess the Deformation Level of Pre-Deformed Components

Process Optimization Method of an Automated Tool Design of Punch-Bending Processes Using the Asset Administration Shell

Proxy-physics Informed Neural Networks (p-pinns) Proxy-Physics-Informed and Transfer Learning Networks for Radial-Axial Ring Rolling (RARR) under Varying Data Availability

Punch-bending Method of an Automated Tool Design of Punch-Bending Processes Using the Asset Administration Shell

R

Radial-axial Ring Rolling (rarr) Proxy-Physics-Informed and Transfer Learning Networks for Radial-Axial Ring Rolling (RARR) under Varying Data Availability

Remanufacturing Geometry-Based Concept for Automatic Cut Planning and Flattening of Sheet Metal Components; Application of Reference-Free Strain Measurement to Assess the Deformation Level of Pre-Deformed Components

Robustness Robust Performance Optimization Using Bayesian Optimization and Extreme Value Theory

S

Sheet Metal Forming Geometry-Based Concept for Automatic Cut Planning and Flattening of Sheet Metal Components; Towards DIC-Subset-Independent Machine Learning Models for Constitutive Parameter Identification in Sheet Metal Forming

Stretch-reducing Mill Advanced Control of Stretch-Reducing Mills Using Artificial Intelligence

T

Tailored Blanks Geometry-Based Concept for Automatic Cut Planning and Flattening of Sheet Metal Components

Transfer Learning Proxy-Physics-Informed and Transfer Learning Networks for Radial-Axial Ring Rolling (RARR) under Varying Data Availability

U

Uncertainty Detection Uncertainty Detection in Sheet Metal Bending Processes with Machine Learning

V

Virtual Fields Method Innovation of the "Cut-Clamp-Play" Concept for Robust and Efficient Sheet Metal Material Characterization

Authors Index

A

Acksteiner, C. Geometry-Based Concept for Automatic Cut Planning and Flattening of Sheet Metal Components

Alimov, A. Hybrid Numerical and Data-Driven Modelling for Defect Prediction in Screw Press Hot Bulk Forging of the EN AW-6060 Part

Amaral, R. Improving FEMU Calibration from a Single Heterogeneous Test through a Data-Driven Approach: An Exploratory Study

Andrade-Campos, A. Towards DIC-Subset-Independent Machine Learning Models for Constitutive Parameter Identification in Sheet Metal Forming; Improving FEMU Calibration from a Single Heterogeneous Test through a Data-Driven Approach: An Exploratory Study

Anedda, M. Application of Reference-Free Strain Measurement to Assess the Deformation Level of Pre-Deformed Components

Antonini, L. Advanced Control of Stretch-Reducing Mills Using Artificial Intelligence

B

Brosius, A. Geometry-Based Concept for Automatic Cut Planning and Flattening of Sheet Metal Components

C

Chaudhary, B. Method of an Automated Tool Design of Punch-Bending Processes Using the Asset Administration Shell

Coppieters, S. Innovation of the "Cut-Clamp-Play" Concept for Robust and Efficient Sheet Metal Material Characterization; Towards Single-Test Anisotropy Calibration: Sensitivity, Identifiability and Validation of YLD2000-2d

Cruz, D. Uncertainty Detection in Sheet Metal Bending Processes with Machine Learning; AI-Driven Design and Optimization of Bending Processes

F

Fernandes, D. AI-Driven Design and Optimization of Bending Processes

Fioretti, M. Application of Reference-Free Strain Measurement to Assess the Deformation Level of Pre-Deformed Components

G

Gardill, M. Hybrid Numerical and Data-Driven Modelling for Defect Prediction in Screw Press Hot Bulk Forging of the EN AW-6060 Part

Giardini, C. Advanced Control of Stretch-Reducing Mills Using Artificial Intelligence

Giraud-Moreau, L. Robust Performance Optimization Using Bayesian Optimization and Extreme Value Theory

Gonçalves, M. Improving FEMU Calibration from a Single Heterogeneous Test through a Data-Driven Approach: An Exploratory Study

Guilleaume, C. Geometry-Based Concept for Automatic Cut Planning and Flattening of Sheet Metal Components

H

Halilovič, M. Innovation of the "Cut-Clamp-Play" Concept for Robust and Efficient Sheet Metal Material Characterization; Towards Single-Test Anisotropy Calibration: Sensitivity, Identifiability and Validation of YLD2000-2d

Hammer, B. Method of an Automated Tool Design of Punch-Bending Processes Using the Asset Administration Shell

Henriques, J. Towards DIC-Subset-Independent Machine Learning Models for Constitutive Parameter Identification in Sheet Metal Forming

Härtel, S. Hybrid Numerical and Data-Driven Modelling for Defect Prediction in Screw Press Hot Bulk Forging of the EN AW-6060 Part

J

Jiang, Y. Hybrid Numerical and Data-Driven Modelling for Defect Prediction in Screw Press Hot Bulk Forging of the EN AW-6060 Part

K

Kersting, L. Method of an Automated Tool Design of Punch-Bending Processes Using the Asset Administration Shell

Korn, J. Geometry-Based Concept for Automatic Cut Planning and Flattening of Sheet Metal Components

Kuhlenkötter, B. Proxy-Physics-Informed and Transfer Learning Networks for Radial-Axial Ring Rolling (RARR) under Varying Data Availability

L

Lafon, P. Robust Performance Optimization Using Bayesian Optimization and Extreme Value Theory

Langula, S. Geometry-Based Concept for Automatic Cut Planning and Flattening of Sheet Metal Components

Lava, P. Innovation of the "Cut-Clamp-Play" Concept for Robust and Efficient Sheet Metal Material Characterization; Towards Single-Test Anisotropy Calibration: Sensitivity, Identifiability and Validation of YLD2000-2d

M

Malejka, D. Proxy-Physics-Informed and Transfer Learning Networks for Radial-Axial Ring Rolling (RARR) under Varying Data Availability

Marques, A. Uncertainty Detection in Sheet Metal Bending Processes with Machine Learning; AI-Driven Design and Optimization of Bending Processes; Machine Learning Algorithms Applied to the Identification of CPB06 Yield Criterion Parameters

Mazur, A. Method of an Automated Tool Design of Punch-Bending Processes Using the Asset Administration Shell

Maček, A. Innovation of the "Cut-Clamp-Play" Concept for Robust and Efficient Sheet Metal Material Characterization; Towards Single-Test Anisotropy Calibration: Sensitivity, Identifiability and Validation of YLD2000-2d

Mitreiro, D. Towards DIC-Subset-Independent Machine Learning Models for Constitutive Parameter Identification in Sheet Metal Forming

Moser, T. Proxy-Physics-Informed and Transfer Learning Networks for Radial-Axial Ring Rolling (RARR) under Varying Data Availability

N

Neto, D. AI-Driven Design and Optimization of Bending Processes

O

Oliveira, M. Uncertainty Detection in Sheet Metal Bending Processes with Machine Learning; AI-Driven Design and Optimization of Bending Processes

P

Parreira, T. Uncertainty Detection in Sheet Metal Bending Processes with Machine Learning; AI-Driven Design and Optimization of Bending Processes; Machine Learning Algorithms Applied to the Identification of CPB06 Yield Criterion Parameters

Pellegrini, G. Advanced Control of Stretch-Reducing Mills Using Artificial Intelligence

Pereira, A. Uncertainty Detection in Sheet Metal Bending Processes with Machine Learning; AI-Driven Design and Optimization of Bending Processes; Machine Learning Algorithms Applied to the Identification of CPB06 Yield Criterion Parameters

Perin, M. Proxy-Physics-Informed and Transfer Learning Networks for Radial-Axial Ring Rolling (RARR) under Varying Data Availability

Persoons, A. Robust Performance Optimization Using Bayesian Optimization and Extreme Value Theory

Peters, H. Method of an Automated Tool Design of Punch-Bending Processes Using the Asset Administration Shell

Pierron, F. Innovation of the "Cut-Clamp-Play" Concept for Robust and Efficient Sheet Metal Material Characterization; Towards Single-Test Anisotropy Calibration: Sensitivity, Identifiability and Validation of YLD2000-2d

Prates, P. Uncertainty Detection in Sheet Metal Bending Processes with Machine Learning; AI-Driven Design and Optimization of Bending Processes; Machine Learning Algorithms Applied to the Identification of CPB06 Yield Criterion Parameters; Towards DIC-Subset-Independent Machine Learning Models for Constitutive Parameter Identification in Sheet Metal Forming

Principi, E. Application of Reference-Free Strain Measurement to Assess the Deformation Level of Pre-Deformed Components

Q

Quagliato, L. Proxy-Physics-Informed and Transfer Learning Networks for Radial-Axial Ring Rolling (RARR) under Varying Data Availability

Quarto, M. Advanced Control of Stretch-Reducing Mills Using Artificial Intelligence

R

Ribeiro, B. Uncertainty Detection in Sheet Metal Bending Processes with Machine Learning; Machine Learning Algorithms Applied to the Identification of CPB06 Yield Criterion Parameters

Rocha, V. Robust Performance Optimization Using Bayesian Optimization and Extreme Value Theory

Rossi, M. Application of Reference-Free Strain Measurement to Assess the Deformation Level of Pre-Deformed Components

S

Santos, A. Uncertainty Detection in Sheet Metal Bending Processes with Machine Learning; AI-Driven Design and Optimization of Bending Processes

Seitz, J. Proxy-Physics-Informed and Transfer Learning Networks for Radial-Axial Ring Rolling (RARR) under Varying Data Availability

Shimaoka, S. Material Parameter Identification Using Bayesian Data Assimilation and Biaxial Tensile Test

Starman, B. Innovation of the "Cut-Clamp-Play" Concept for Robust and Efficient Sheet Metal Material Characterization; Towards Single-Test Anisotropy Calibration: Sensitivity, Identifiability and Validation of YLD2000-2d

T

Tanoni, G. Application of Reference-Free Strain Measurement to Assess the Deformation Level of Pre-Deformed Components

Thuillier, S. Improving FEMU Calibration from a Single Heterogeneous Test through a Data-Driven Approach: An Exploratory Study

Trächtler, A. Method of an Automated Tool Design of Punch-Bending Processes Using the Asset Administration Shell

Y

Yamanaka, A. Material Parameter Identification Using Bayesian Data Assimilation and Biaxial Tensile Test

Z

Zambelli, M. Advanced Control of Stretch-Reducing Mills Using Artificial Intelligence

Zanella, C. Proxy-Physics-Informed and Transfer Learning Networks for Radial-Axial Ring Rolling (RARR) under Varying Data Availability