Self-Supervised Physics-Informed Surrogate Model for Elastic Local Fields in Polycrystals
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We propose a self-supervised physics-informed neural network for explicitly solving anisotropic linear elasticity on periodic heterogeneous media, with an application to polycrystalline materials. Our method relies on linking a convolutional neural network with residual connections to non-learnable layers that take inspiration from the mechanical problem addressed by Fast Fourier Transform (FFT) algorithms. Spatial differential operators are discretized via finite differences consistently with the Green operator used in the FFT computations and are treated as convolutions with fixed kernels. One of the layers informs the network of the known relation between crystalline orientation and stiffness tensor. A physical loss function derived from the stress field allows for updating weights and biases with no need of a loss term accounting for supervision on the ground truth given by an FFT solver. The network is trained on untextured synthetic microstructures realized via a stochastic generator based on point processes and tessellations. Once trained, the machine learning pipeline is able to predict the periodic part of displacement field from the unit quaternion field of an artificial representative elementary volume (REV), without iterating. The predictions are validated by comparison with the FFT solution. Furthermore, possible applications and extensions of the method are discussed, including arbitrary elasticity tensors and loading directions, and comparisons between machine learning predictions and numerical or analytical estimates in terms of effective properties.
