Influence of Fiber Orientation Clustering Methods in Virtual Process Chains of Discontinuous-Fiber Composite Components

Abstract:

In virtual process chains for discontinuous fiber-reinforced polymers, clustering of fiber orientation tensors reduces the number of macroscopic material cards required for downstream structural and warpage simulations. However, it remains unclear whether including the additional information provided by the fourth-order fiber orientation tensor improves clustering quality. This study investigates the influence of second-order vs. fourth-order informed clustering on clustering outcomes and the resulting orientation-averaged mechanical properties. Using parameterizations based on harmonic decomposition, rotation-invariant clustering is performed in both the second-order and fourth-order parameter spaces. Results from injection molding simulation data indicate that the level of tensorial information has limited effect when the fourth-order tensor is computed via a closure approximation, as the deviatoric parameters are nonlinearly dependent on the second-order parameters. In contrast, the choice of clustering algorithm -- KMeans vs. Birch -- has a more pronounced influence on cluster shapes and allocations. Furthermore, we demonstrate that clustering affects orientation-averaged stiffness properties, with deviations most pronounced near cluster boundaries and rarely occurring tensor shapes.

• Periodical: Solid State Phenomena (Volume 387)
• Pages: 33-41
• DOI: https://doi.org/10.4028/p-fNrp4N
• Online since: April 2026
• Authors: Louis Schreyer*, Luca Salvatore Pusineri Yaluff, Constantin Krauß, Luise Kärger
• Keywords: Clustering, Fiber Orientation, Molding Simulation, Structural Simulation

References

[1] F. Wittemann, R. Maertens, L. Kärger, and F. Henning, "Injection molding simulation of short fiber reinforced thermosets with anisotropic and non-Newtonian flow behavior," Composites Part A: Applied Science and Manufacturing, vol. 124, p.105476, Sept. 2019.

DOI: 10.1016/j.compositesa.2019.105476

[2] J. Görthofer, N. Meyer, T. D. Pallicity, L. Schöttl, A. Trauth, M. Schemmann, M. Hohberg, P. Pinter, P. Elsner, F. Henning, A. N. Hrymak, T. Seelig, K. Weidenmann, L. Kärger, and T. Böhlke, "Virtual process chain of sheet molding compound: Development, validation and perspectives," Composites Part B: Engineering, vol. 169, p.133–147, July 2019.

DOI: 10.1016/j.compositesb.2019.04.001

[3] P. Kennedy and R. Zheng, Flow analysis of injection molds. Carl Hanser Verlag GmbH & Co. KG, Apr. 2013.

[4] C. Krauß and L. Kärger, "Tensor interpolation in virtual manufacturing chains for fiber rein forced composites," International Journal of Mechanical Sciences, vol. 226, p.107378, July 2022.

DOI: 10.1016/j.ijmecsci.2022.107378

[5] K.-I. Kanatani, "Distribution of directional data and fabric tensors," International Journal of Engineering Science, vol. 22, p.149–164, Jan. 1984.

[6] S. G. Advani and C. L. Tucker, "The use of tensors to describe and predict fiber orientation in short fiber composites," Journal of Rheology, vol. 31, p.751–784, Nov. 1987.

DOI: 10.1122/1.549945

[7] M. Hohberg, Experimental investigation and process simulation of the compression molding process of sheet molding compound (SMC) with local reinforcements. PhD thesis, Karlsruhe Institute of Technology (KIT), Germany, 2022.

[8] V. Romanenko,Materialcharakterisierungunddurchgängige3D-Prozesssimulationfürkohlen stofffaserverstärktes Sheet Molding Compound. PhD thesis, RPTU University Kaiserslautern Landau, Kaiserslautern, 2021.

[9] J. Köbler, M. Schneider, F. Ospald, H. Andrä,andR. Müller,"Fiberorientation interpolation for the multiscale analysis of short fiber reinforced composite parts," Computational Mechanics, vol. 61, no. 6, p.729–750, 2018.

DOI: 10.1007/s00466-017-1478-0

[10] C. Krauß, J. K. Bauer, J. Mitsch, T. Böhlke, and L. Kärger, "On the averaging and closure of fiber orientation tensors in virtual process chains," Journal of Elasticity, vol. 156, p.279–306, Aug. 2024.

DOI: 10.1007/s10659-024-10050-3

[11] J. K. Bauer and T. Böhlke, "Variety of fiber orientation tensors," Mathematics and Mechanics of Solids, vol. 27, no. 7, p.1185–1211, 2022.

DOI: 10.1177/10812865211057602

[12] S. Forte and M. Vianello, "Symmetry classes for elasticity tensors," Journal of Elasticity, vol. 43, p.81–108, May 1996.

DOI: 10.1007/bf00042505

[13] J. Rychlewski, "A qualitative approach to Hooke's tensors. Part I," Archives of Mechanics, vol. 52, p.737–759, Jan. 2000.

[14] D. H. Chung and T. H. Kwon, "Invariant-based optimal fitting closure approximation for the numerical prediction of flow-induced fiber orientation," Journal of Rheology, vol. 46, no. 1, p.169–194, 2002.

DOI: 10.1122/1.1423312

[15] N. Meyer and J. Greif, "Injection molding simulations," May 2024.

[16] J. MacQueen, "Multivariate observations," in Proceedings of the 5th Berkeley symposium on mathematical statistics and probability, vol. 1, p.281–297, 1967.

[17] S. Lloyd, "Least squares quantization in PCM," IEEE transactions on information theory, vol. 28, no. 2, p.129–137, 1982.

DOI: 10.1109/tit.1982.1056489

[18] T. Zhang, R. Ramakrishnan, and M. Livny, "BIRCH: An efficient data clustering method for very large databases," SIGMOD Rec., vol. 25, p.103–114, June 1996.

DOI: 10.1145/235968.233324

[19] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and É. Duchesnay, "Scikit-learn: Machine learning in Python," Journal of Machine Learning Research, vol. 12, no. 85, p.2825–2830, 2011.

DOI: 10.3389/fninf.2014.00014

[20] G. P. Tandon and G. J. Weng, "The effect of aspect ratio of inclusions on the elastic properties of unidirectionally aligned composites," Polymer Composites, vol. 5, p.327–333, Oct. 1984.

DOI: 10.1002/pc.750050413

[21] T. Böhlke and C. Brüggemann, "Graphical representation of the generalized Hooke's law," Technische Mechanik, vol. 21, no. 2, p.145, (2001)