Stress Predictions in Polycrystal Plasticity using Graph Neural Networks with Subgraph Training

TL;DR

This work tackles the computational bottleneck of crystal plasticity simulations by developing a GNN-based surrogate that learns the strain-to-stress map directly on FEM-derived mesh graphs. It introduces subgraph training and edge-aware message passing to preserve mesh connectivity and geometry while handling large, irregular graphs. The approach achieves near-FEM accuracy (coefficient of determination > 0.99) on training, testing, and unseen validation data and delivers substantial speedups (around 150x) over conventional FEM. These results enable fast, scalable surrogate predictions for polycrystal plasticity with strong generalization and practical implications for design, optimization, and digital twins in materials engineering.

Abstract

Numerical modeling of polycrystal plasticity is computationally intensive. We employ Graph Neural Networks (GNN) to predict stresses on complex geometries for polycrystal plasticity from Finite Element Method (FEM) simulations. We present a novel message-passing GNN that encodes nodal strain and edge distances between FEM mesh cells, and aggregates to obtain embeddings and combines the decoded embeddings with the nodal strains to predict stress tensors on graph nodes. The GNN is trained on subgraphs generated from FEM mesh graphs, in which the mesh cells are converted to nodes and edges are created between adjacent cells. We apply the trained GNN to periodic polycrystals with complex geometries and learn the strain-stress maps based on crystal plasticity theory. The GNN is accurately trained on FEM graphs, in which the $R^2$ for both training and testing sets are larger than 0.99. The proposed GNN approach speeds up more than 150 times compared with FEM on stress predictions. We also apply the trained GNN to unseen simulations for validations and the GNN generalizes well with an overall $R^2$ of 0.992. The GNN accurately predicts the von Mises stress on polycrystals. The proposed model does not overfit and generalizes well beyond the training data, as the error distributions demonstrate. This work outlooks surrogating crystal plasticity simulations using graph data.

Figures (26)

• Figure 1: Schematic diagram for the decomposition in different configurations in crystal plasticity formulation. The visualization is inspired by Refs. fepx_arxivhuajian_crystal_plas_jmps.
• Figure 2: Schematic illustration of the procedures for converting FEM meshes to graphs. FEM element cell centroids are treated as nodes, and neighboring cells (sharing 3 common nodes for 'tetra10' elements) share an edge.
• Figure 3: General schematic of the workflow for using GNN to learn polycrystal plasticity. Virtual polycrystals are generated using FEPX. It is converted to mesh graphs based on finite element cells. The subgraphs are extracted to train the GNN. The GNN is then deployed to surrogate polycrystal plasticity simulations.
• Figure 4: The general architecture for the GNN. The node-encoding layer takes the strains on neighboring nodes for the input edges, and the edge-encoding layer takes the mesh cell link length (i.e. Euclidean norm of mesh cells). The combined outputs are then fed input the embedding space ($\mathbb{R}^{2{\sf emb}}$). $\left\{h_{ij}^{2 \sf emb}\right\}^{(n)}$ and $\left\{h_{ij}^{2 \sf emb} \right\}^{(e)}$ are the decoded nodal and edge information in the embedding space. The output data is then fed input to the decoding layer that maps $\mathbb{R}^{2{\sf emb}}$ to $\mathbb{R}^4$. The output of the decoding layer is then passed to the message-passing operator (i.e. $\bigoplus$), where the decoded messages are passed on nodes. The edge information on nodes $\left\{\epsilon_{mn}\right\}_i$ are then combined to put into the equation layer, to predict the corresponding stress components $\left\{\sigma_{mn}\right\}_i$.
• Figure 5: Schematic illustration for the proposed subgraph training method using the subgraph extracted from the full graph. The subgraph $\mathcal{G}_{\sf sub}$ are extracted from the sampled nodes from the full graph $\mathcal{G}$, in which the nodes containing full-edge information (e.g., $i, j, \&\ k$) were considered in the loss function during training.