Learning constitutive relations using symmetric positive definite neural networks

JOURNAL OF COMPUTATIONAL PHYSICS

卷:428

Article

Xu, Kailai¹  Huang, Daniel Z.¹  Darve, Eric^1,2 

[1] Stanford Univ, Inst Computat & Math Engn, Stanford, CA 94305 USA

[2] Stanford Univ, Mech Engn, Stanford, CA 94305 USA

关键词: Neural networks;    Plasticity;    Hyperelasticity;    Finite element method;    Multiscale homogenization;   

DOI  :  10.1016/j.jcp.2020.110072

来源: Elsevier

【 摘 要 】

We present a new neural-network architecture, called the Cholesky-factored symmetric positive definite neural network (SPD-NN), for modeling constitutive relations in computational mechanics. Instead of directly predicting the stress of the material, the SPD-NN trains a neural network to predict the Cholesky factor of the tangent stiffness matrix, based on which the stress is calculated in incremental form. As a result of this special structure, SPD-NN weakly imposes convexity on the strain energy function, satisfies the second order work criterion (Hill's criterion) and time consistency for path-dependent materials, and therefore improves numerical stability, especially when the SPD-NN is used in finite element simulations. Depending on the types of available data, we propose two training methods, namely direct training for strain and stress pairs and indirect training for loads and displacement pairs. We demonstrate the effectiveness of SPD-NN on hyper elastic, elasto-plastic, and multiscale fiber-reinforced plate problems from solid mechanics. The generality and robustness of SPD-NN make it a promising tool for a wide range of constitutive modeling applications. (c) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

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