A physics-informed 3D surrogate model for elastic fields in polycrystals

We develop a physics-informed neural network pipeline for solving linear elastic micromechanics in three dimensions, on a statistical volume element (SVE) of a polycrystalline material with periodic geometry. The presented approach combines a convolutional neural network containing residual connections with physics-informed non-trainable layers. The latter are introduced to enforce the strain field admissibility and the constitutive law in a way consistent with so-called fast Fourier transform (FFT) algorithms. More precisely, differential operators are discretized by finite differences in accordance with the Green operator used in FFT computations and treated as convolutions with fixed kernels. The deterministic relationship between crystalline orientations and stiffness tensors is transferred to the network by an additional non-trainable layer. A loss function dependent on the divergence of the predicted stress field allows for updating the neural network's parameters without further supervision from ground truth data. The surrogate model is trained on untextured synthetic polycrystalline SVEs with periodic boundary conditions, realized from a stochastic 3D microstructure model based on random tessellations. Once trained, the network is able to predict the periodic part of the displacement field from the crystalline orientation field (represented as unit quaternions) of an SVE. The proposed self-supervised pipeline is compared to a similar one trained with a data-driven loss function instead. Further, the accuracy of both models is analyzed by applying them to microstructures larger than the training inputs, as well as to SVEs generated by the stochastic 3D microstructure model, utilizing various different parameters. We find that the self-supervised pipeline yields more accurate predictions than the data-driven one, at the expense of a longer training. Finally, we discuss how the trained surrogate model can be used to solve certain inverse problems on polycrystalline domains by gradient descent.