On the constraints and consistency in implicit constitutive modelling using ANNs and indirect training

On the constraints and consistency in implicit constitutive modelling using ANNs and indirect training

LOURENÇO Rúben, CUETO Elías, GEORGIEVA Pétia, ANDRADE-CAMPOS António

Abstract. The training of an Artificial Neural Network (ANN) for implicit constitutive modelling mostly relies on labelled data pairs, however, some variables cannot be physically measured in real experiments. As such, the training should preferably be carried out indirectly, making use of experimentally measurable variables. The unconstrained training of an ANN’s parameters often leads to spurious responses that do not comply with the physics of the problem. Applying constraints during training ensures not only the physical meaning of the ANN predictions but also potentially increases the convergence to a global minimum, while improving the model’s performance. An ANN material model is trained using a novel indirect approach, where the local and global equilibrium conditions are ensured employing the Virtual Fields Method (VFM). An example of physical constraint is analyzed and applied during the training process.

Keywords
Constitutive Model, Elastoplasticity, Neural Networks, Indirect Training, Constrained Optimization

Published online 4/19/2023, 12 pages
Copyright © 2023 by the author(s)
Published under license by Materials Research Forum LLC., Millersville PA, USA

Citation: LOURENÇO Rúben, CUETO Elías, GEORGIEVA Pétia, ANDRADE-CAMPOS António, On the constraints and consistency in implicit constitutive modelling using ANNs and indirect training, Materials Research Proceedings, Vol. 28, pp 1143-1154, 2023

DOI: https://doi.org/10.21741/9781644902479-125

The article was published as article 125 of the book Material Forming

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

References
[1] Y. C. Lin, J. Zhang, J. Zhong, Application of neural networks to predict the elevated temperature flow behavior of a low alloy steel, Comput. Mater. Sci. 43 (2008) 752-758. https://doi.org/10.1016/j.commatsci.2008.01.039
[2] K. Hornik, M. Stinchcombe, H. White, Multilayer feedforward networks are universal approximators, Neural Networks 2 (1989) 359-366. https://doi.org/10.1016/0893-6080(89)90020-8
[3] K. Linka, E. Kuhl, A new family of Constitutive Artificial Neural Networks towards automated model discovery, Arxiv (2022). https://doi.org/10.48550/arxiv.2210.02202
[4] J. Ghaboussi, D.A. Pecknold, M. Zhang, R.M. Haj-Ali, Autoprogressive training of neural network constitutive models, Int. J. Numer. Methods Eng. 42 (1998) 105-126. https://doi.org/10.1002/(SICI)1097-0207(19980515)42:1<105::AID-NME356>3.0.CO;2-V
[5] A. Zhang, D. Mohr, Using neural networks to represent von Mises plasticity with isotropic hardening, Int. J. Plast. 132 (2020) 102732. https://doi.org/10.1016/j.ijplas.2020.102732
[6] F. Masi, I. Stefanou, P. Vannucci, V. Maffi-Berthier, Thermodynamics-based Artificial Neural Networks for constitutive modeling, J. Mech. Phys. Solids 147 (2021) 104277. https://doi.org/10.1016/j.jmps.2020.104277
[7] F. As’ad, P. Avery, C. Farhat, A mechanics-informed artificial neural network approach in data-driven constitutive modeling, Int. J. Nume.r Methods Eng. 123 (2022) 2738-2759. https://doi.org/10.1002/NME.6957
[8] L. Borkowski, C. Sorini, A. Chattopadhyay, Recurrent neural network-based multiaxial plasticity model with regularization for physics-informed constraints, Comput. Struct. 258 (2022) 106678. https://doi.org/10.1016/J.COMPSTRUC.2021.106678
[9] M. Raissi, P. Perdikaris, G.E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys. 378 (2019) 686-707. https://doi.org/10.1016/j.jcp.2018.10.045
[10] E. Cueto, F. Chinesta, Thermodynamics of learning physical phenomena, 2022.
[11] A. Danoun, E. Prulière, Y. Chemisky, Thermodynamically consistent Recurrent Neural Networks to predict non linear behaviors of dissipative materials subjected to non-proportional loading paths, Mech. Mater. (2022) 104436. https://doi.org/10.1016/j.mechmat.2022.104436
[12] A. Bower, Appl. Mech. Solids, CRC Press, 2009.
[13] K. Xu, D.Z. Huang, E. Darve, Learning constitutive relations using symmetric positive definite neural networks, J. Comput. Phys. 428 (2021) 110072. https://doi.org/10.1016/j.jcp.2020.110072
[14] 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. https://doi.org/10.1016/j.ijmecsci.2018.07.013
[15] X. Liu, F. Tao, H. Du, W. Yu, K. Xu, Learning nonlinear constitutive laws using neural network models based on indirectly measurable data, J. Appl. Mech. Trans. ASME 87 (2020) 1-8. https://doi.org/10.1115/1.4047036
[16] M. Grédiac, Principe des travaux virtuels et identification, Comptes Rendus de l’Académie des Sciences 309 (1989) 1–5.
[17] A. Marek, F.M. Davis, F. Pierron, Sensitivity-based virtual fields for the non-linear virtual fields method, Comput. Mech. 60 (2017) 409–431. https://doi.org/10.1007/s00466-017-1411-6
[18] F. Pierron, M. Grédiac, The virtual fields method: extracting constitutive mechanical parameters from full-field deformation measurements. Springer Science & Business Media, 2012.
[19] M. Grediac, F. Pierront, S. Avrilt, E. Toussaint, The Virtual Fields Method for Extracting Constitutive Parameters From Full-Field Measurements: a Review, Strain 42 (2006) 233-253. https://doi.org/10.1111/j.1475-1305.2006.tb01504.x
[20] A. Marek, F.M. Davis, M. Rossi, F. Pierron, Extension of the sensitivity-based virtual fields to large deformation anisotropic plasticity, Int. J. Mater. Forming 12 (2019) 457-476. https://doi.org/10.1007/S12289-018-1428-1/FIGURES/13
[21] T.B. Stoughton, J.W. Yoon, Review of Drucker’s postulate and the issue of plastic stability in metal forming, Int. J. Plast. 22 (2006) 391–433. https://doi.org/10.1016/J.IJPLAS.2005.03.002