Application of Graph Neural Networks to Accelerate Finite Element Simulations
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Finite Element (FE) simulations often involve unstructured mesh data, making training with traditional neural network models is impractical due to their reliance on structured data. Graph Neural Networks (GNNs), such as MeshGraphNet [1], offer a solution by effectively handling unstructured data, such as FE meshes, owing to their ability to capture node connectivity. We propose a hybrid GNN - FEM framework that alternates between GNN predictions for extended timesteps and conventional FE evaluations at sparse intervals, significantly reducing computational time while maintaining predictive accuracy. Generally, GNNs for time-dependent problems are trained in an autoregressive manner, which can lead to error accumulation and decreased accuracy over long-term predictions. To mitigate this, we adopt a hybrid approach [2]: for any two previous timesteps, the GNN captures spatial dependencies, while a 1D Convolution Network predicts the next k timesteps as a bundle. This combination enables stable sequential predictions by jointly considering spatial and temporal dependencies. The method is validated on a 3D torsion problem involving linear isotropic elastic-plastic deformation. The hybrid model accurately predicts the displacement, stress, and strain increments for unseen sequences, with relative errors within 0 – 0.01%, while reducing CPU time per increment from 52 s (FEM) to 5 s (GNN). In the case of torsion, as the applied load increases, torsional deformation causes a complex, curvilinear evolution of physical fields. In our work, we demonstrate the role of activation functions in GNN to capture the nonlinear, curvilinear evolution. These results demonstrate that the hybrid GNN-FEM approach can substantially accelerate the high-fidelity FE simulations by preserving near-perfect accuracy, enabling rapid structural analysis. REFERENCES [1] A. Sanchez-Gonzalez, J. Godwin, T. Pfaff, R. Ying, J. Leskovec, and P. W. Battaglia. Learning to Simulate Complex Physics with Graph Networks. Submitted Feb. 21, 2020; revised Sep. 14, 2020. Available: https://arxiv.org/abs/2002.09405 [2] J. Brandstetter, D. Worrall, and M. Welling. Message Passing Neural PDE Solvers. 2023, Available: https://arxiv.org/abs/2202.03376
