Geometric deep learning for computational mechanics Part I: Anisotropic Hyperelasticity

Abstract: This paper is the first attempt to use geometric deep learning and Sobolev training to incorporate non-Euclidean microstructural data such that anisotropic hyperelastic material machine learning models can be trained in the finite deformation range. While traditional hyperelasticity models often incorporate homogenized measures of microstructural attributes, such as porosity averaged orientation of constitutes, these measures cannot reflect the topological structures of the attributes. We fill this knowledge gap by introducing the concept of weighted graph as a new mean to store topological information, such as the connectivity of anisotropic grains in assembles. Then, by leveraging a graph convolutional deep neural network architecture in the spectral domain, we introduce a mechanism to incorporate these non-Euclidean weighted graph data directly as input for training and for predicting the elastic responses of materials with complex microstructures. To ensure smoothness and prevent non-convexity of the trained stored energy functional, we introduce a Sobolev training technique for neural networks such that stress measure is obtained implicitly from taking directional derivatives of the trained energy functional. By optimizing the neural network to approximate both the energy functional output and the stress measure, we introduce a training procedure the improves efficiency and generalize the learned energy functional for different microstructures. The trained hybrid neural network model is then used to generate new stored energy functional for unseen microstructures in a parametric study to predict the influence of elastic anisotropy on the nucleation and propagation of fracture in the brittle regime.