Mohammad S. Khorrami

Sarthak Kapoor, Jaber Rezaei Mianroodi, Mohammad Khorrami et al.

The purpose of this work is the systematic comparison of the application of two artificial neural networks (ANNs) to the surrogate modeling of the stress field in materially heterogeneous periodic polycrystalline microstructures. The first ANN is a UNet-based convolutional neural network (CNN) for periodic data, and the second is based on Fourier neural operators (FNO). Both of these were trained, validated, and tested with results from the numerical solution of the boundary-value problem (BVP) for quasi-static mechanical equilibrium in periodic grain microstructures with square domains. More specifically, these ANNs were trained to correlate the spatial distribution of material properties with the equilibrium stress field under uniaxial tensile loading. The resulting trained ANNs (tANNs) calculate the stress field for a given microstructure on the order of 1000 (UNet) to 2500 (FNO) times faster than the numerical solution of the corresponding BVP. For microstructures in the test dataset, the FNO-based tANN, or simply FNO, is more accurate than its UNet-based counterpart; the normalized mean absolute error of different stress components for the former is 0.25-0.40% as compared to 1.41-2.15% for the latter. Errors in FNO are restricted to grain boundary regions, whereas the error in U-Net also comes from within the grain. In comparison to U-Net, errors in FNO are more robust to large variations in spatial resolution as well as small variations in grain density. On other hand, errors in U-Net are robust to variations in boundary box aspect ratio, whereas errors in FNO increase as the domain becomes rectangular. Both tANNs are however unable to reproduce strong stress gradients, especially around regions of stress concentration.

Mohammad S. Khorrami, Pawan Goyal, Jaber R. Mianroodi et al.

The purpose of the current work is the development of a so-called physics-encoded Fourier neural operator (PeFNO) for surrogate modeling of the quasi-static equilibrium stress field in solids. Rather than accounting for constraints from physics in the loss function as done in the (now standard) physics-informed approach, the physics-encoded approach incorporates or "encodes" such constraints directly into the network or operator architecture. As a result, in contrast to the physics-informed approach in which only training is physically constrained, both training and output are physically constrained in the physics-encoded approach. For the current constraint of divergence-free stress, a novel encoding approach based on a stress potential is proposed. As a "proof-of-concept" example application of the proposed PeFNO, a heterogeneous polycrystalline material consisting of isotropic elastic grains subject to uniaxial extension is considered. Stress field data for training are obtained from the numerical solution of a corresponding boundary-value problem for quasi-static mechanical equilibrium. This data is also employed to train an analogous physics-guided FNO (PgFNO) and physics-informed FNO (PiFNO) for comparison. As confirmed by this comparison and as expected on the basis of their differences, the output of the trained PeFNO is significantly more accurate in satisfying mechanical equilibrium than the output of either the trained PgFNO or the trained PiFNO.