cs.LGJul 5, 2025

Physics-Informed Graph Neural Networks to Reconstruct Local Fields Considering Finite Strain Hyperelasticity

Authors: Manuel Ricardo Guevara Garban, Yves Chemisky, Étienne Prulière, Michaël Clément

Organizations: Univ. Bordeaux, CNRS, Bordeaux INP, I2M, UMR 5295, Talence 33400 France · Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR 5800, Talence 33400 France · Arts et Metiers Institute of Technology, CNRS, Bordeaux INP, I2M, UMR 5295, Talence 33400 France

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Abstract

We propose a physics-informed machine learning framework called P-DivGNN to reconstruct local stress fields at the micro-scale, in the context of multi-scale simulation given a periodic micro-structure mesh and mean, macro-scale, stress values. This method is based in representing a periodic micro-structure as a graph, combined with a message passing graph neural network. We are able to retrieve local stress field distributions, providing average stress values produced by a mean field reduced order model (ROM) or Finite Element (FE) simulation at the macro-scale. The prediction of local stress fields are of utmost importance considering fracture analysis or the definition of local fatigue criteria. Our model incorporates physical constraints during training to constraint local stress field equilibrium state and employs a periodic graph representation to enforce periodic boundary conditions. The benefits of the proposed physics-informed GNN are evaluated considering linear and non linear hyperelastic responses applied to varying geometries. In the non-linear hyperelastic case, the proposed method achieves significant computational speed-ups compared to FE simulation, making it particularly attractive for large-scale applications.

Figures & tables

Figure 1: Overview of the proposed local stress reconstruction method. The mean microstructure stress response σˉ\bar{\sigma} , is uniformly assigned to all nodes of the input mesh. A Graph Neural Network (GNN) then acts as a localization operator, predicting the corresponding local stress field σi\sigma_{i} at each node.
Figure 2: Illustration of the graph representation of a mesh, node labeling in function of topological features and periodic edges added to reinforce message passing.
Figure 3: Schematics of the proposed model illustrating Encode-Message Passing-Decode architecture. Nodes vi{\bm{v}}_{i} and edges eij{\bm{e}}_{ij} of the input graph are embedded through the ϵV\epsilon^{V} and ϵE\epsilon^{E} encoders respectively. This embedded graph is then processed by a GNN which will update the nodes and edges embeddings through message-passing steps. Finally, the node embeddings are decoded with δV\delta^{V} to obtain the output field. Here, only a small subgraph is shown for visualisation purposes, but in practice the whole graph is processed by the architecture.
Figure 4: Top row: mesh examples of the generated database note that a mesh refinement is applied to every elements belonging to the hole surface, the mesh refinement and the global size of the triangles are different for each mesh. For this three examples the global triangles sizes corresponding to each mesh are (7.10,5.79,8.35)(7.10,5.79,8.35) and the triangle sizes belonging to the hole surface are (1.01,0.62,1.07)(1.01,0.62,1.07) Bottom row: Von Mises local stress fields (in MPa) corresponding to top row meshes.
Figure 5: Comparison of NMSE loss and divergence evolution during training of P-GNN model (in blue) and P-DivGNN model (in red).
Figure 6: Local stress reconstruction comparison between classical GNN (without periodic edges) in top row, P-GNN in middle row, and FEM in bottom row.
Figure 7: NSE between predicted local stress field and FEM stress field for both P-GNN and classical GNN (without periodic edges), error fields are plotted at node level.
Figure 8: Comparison of normalized local stress distributions of the entire database between P-GNN , P-DivGNN and FEM.
Figure 9: Comparison of local stress field reconstructions for P-GNN (top row), P-DivGNN (middle row), and FEM (bottom row) for the three stress components (σxx,σyy,σxy)(\sigma_{xx},\sigma_{yy},\sigma_{xy}) . Input mean stress values: σˉxx=−1154.90\bar{\sigma}_{xx}=-1154.90 MPa, σˉyy=−2024.47\bar{\sigma}_{yy}=-2024.47 MPa, σˉxy=−398.07\bar{\sigma}_{xy}=-398.07 MPa. The color bar scale is set to match the FE solution range.
Figure 10: NSE between predicted local stress field and FEM stress field for both P-GNN and P-DivGNN , error fields are plotted at node level.
Figure 11: Comparison of norm of divergence field of local stress field between P-GNN , P-DivGNN and FEM predictions.
 DivergenceNMSE
GNN1.13×10−31.13\times 10^{-3}2.33×10−22.33\times 10^{-2}
P-GNN1.11×10−31.11\times 10^{-3}1.13×10−21.13\times 10^{-2}
P-DivGNN3.23×10−4\mathbf{3.23\times 10^{-4}}1.08×10−2\mathbf{1.08\times 10^{-2}}
FEM9.18×10−49.18\times 10^{-4}–
Table 1: Comparison of divergence mean value and NMSE between GNN (without periodic edges), P-GNN , P-DivGNN , and FEM, evaluated on the whole linear elastic test dataset. The model P-DivGNN outperforms the baseline P-GNN and achieves lower divergence values than the FE simulation.
Figure 12: Performance comparison between FEM simulation P-DivGNN (red), P-DivGNN when adding periodic edges (orange) and without (blue). Tested using a CPU Intel Xeon W- 22552255 Processor, 1010 cores, 2020 threads for FEM simulation and GNN tests have been produced using a graphics card (GPU) Nvidia RTX A4000 with 16GB of VRAM where only 156MB were used.
Figure 13: Comparison of NMSE loss and divergence evolution during training of P-GNN model (in blue) and P-DivGNN model (in red) in the hyperelastic dataset.
Figure 14: Comparison of normalized local stress distributions of the entire database between P-GNN , P-DivGNN and FEM.
Figure 15: Comparison of local stress field reconstructions for P-GNN (top row), P-DivGNN (middle row), and FEM (bottom row) for the three stress components (σxx,σyy,σxy)(\sigma_{xx},\sigma_{yy},\sigma_{xy}) . Input mean stress values: σˉxx=0.59\bar{\sigma}_{xx}=0.59 MPa, σˉyy=−1.01\bar{\sigma}_{yy}=-1.01 MPa, σˉxy=0.33\bar{\sigma}_{xy}=0.33 MPa. The color bar scale is set to match the FE solution range.
Figure 16: NSE between predicted local stress field and FEM stress field for both P-GNN and P-DivGNN , error fields are plotted at node level.
Figure 17: Comparison of norm of divergence field of local stress field between P-GNN , P-DivGNN and FEM predictions for a non-linear hyperelastic case.
 DivergenceNMSE
P-GNN3.55×10−23.55\times 10^{-2}5.36×10−25.36\times 10^{-2}
P-DivGNN1.56×10−3\mathbf{1.56\times 10^{-3}}2.18×10−2\mathbf{2.18\times 10^{-2}}
FEM1.69×10−31.69\times 10^{-3}–
Table 2: Comparison of divergence mean value and NMSE between P-GNN , P-DivGNN , and FEM, evaluated on the whole non linear hyperelastic test dataset, the model P-DivGNN achieves lower divergence values than P-GNN and FE simulation.
Figure 18: Performance comparison between hyperelastic FEM simulation P-DivGNN (red), P-DivGNN when adding periodic edges (orange) and without (blue). Tested using a CPU Intel Xeon W- 22552255 Processor, 1010 cores, 2020 threads for FEM simulation and GNN tests have been produced using a graphics card (GPU) Nvidia RTX A4000 with 16GB of VRAM where only 156MB were used.
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