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

Reconstructing local stress fields in heterogeneous microstructures under non-linear, history-dependent loading remains a major computational bottleneck in multi-scale simulations. We propose a coupled LSTM-GNN framework that links the temporal and spatial aspects of local stress field reconstruction. A Long Short-Term Memory network encodes macroscopic stress-strain sequences into a compact hidden state that captures the path-dependent constitutive response, while a physics-informed Graph Neural Network reconstructs the spatially-resolved stress field at each time step. We introduce a relative weighting strategy with linear warm-up to balance the data-driven reconstruction loss and a discrete divergence-based equilibrium penalty. This resolves the scale mismatch that prevents fixed-weight formulations from converging in the elasto-plastic regime. The model is trained on 10,000 non-proportional loading paths applied to a periodic plate-with-a-hole microstructure and von Mises elasto-plasticity. The model achieves three orders of magnitude speedup over finite element simulations and generalizes to loading sequences twice the training length, with 1.9% cumulative error. Because the graph relies on mesh connectivity instead of the specific element type, one trained surrogate can be applied directly without retraining to meshes with different element types and to both coarser and finer resolutions, while in all cases reproducing the high-fidelity quad-element FE field used during training. Indeed, the message passing characteristics inherent to GNN and MeshGraphNet architecture render the model mesh-agnostic. Analysis of the LSTM hidden states suggests a low-dimensional structure related to the internal state variables of the constitutive model.

Stress localization in heterogeneous solids is governed by the bimaterial interface, where the displacement field remains $C^0$-continuous, while in-plane stresses jump due to the stiffness mismatch. Coordinate-based physics-informed neural networks (PINNs) represent this jump via a prescribed regularization width or a weighted interface penalty, making their accuracy sensitive to how phase-contrast changes are handled. This work presents a variational, label-free physics-informed graph neural network (PI-GNN) in which the heterogeneity is carried by the discretization rather than by the trial field. The solver operates on a conforming adaptive mesh graph, assigns constitutive behavior per element, and minimizes the discrete total potential energy as a single unweighted objective in which only first derivatives appear. The discrete energy on piecewise-linear elements coincides with the finite element (FE) Ritz functional. Dirichlet conditions are enforced by construction, with no penalty term, no interface weight, and no prescribed transition width. Using one fixed architecture, optimizer, and loss across small-strain elasticity and finite-strain Neo-Hookean hyperelasticity in two and three dimensions, the von Mises error remains below $3.58\%$ across a stiffness-contrast sweep spanning $(E_{\mathrm{inc}}/E_{\mathrm{mat}}\in[10^{-2},10^{2}])$, where a strong-form PINN degrades to $5.58\%$, and its displacement error reaches $7.66\%$ against $0.49\%$ for the PI-GNN. A trained network halves the ($σ_{xx}$) error of an energy-based PINN ($5.01\%$ versus $10.94\%$). Training cost exceeds a single FE solve by more than an order of magnitude, so the construction is a variationally consistent, penalty-free interface representation for parametric surrogates and inverse identification rather than a replacement for a one-off FE analysis.

Iterative design and optimization of large, complex structures require fast and accurate prediction of stress, displacement, and other fields. Finite element analysis (FEA) is computationally expensive for this task. Existing neural network surrogates often struggle with varying topologies and complex boundary conditions. This study proposes the novel Dual-Stream Heterogeneous Graph Neural Network (DS-HGNN) for full-field stress and displacement prediction in thin-walled structures, demonstrated on box beams made of stiffened panels. DS-HGNN operates on panel-level heterogeneous graph representations and introduces physics-guided edge states initialized from edge types, spatial information, and boundary kinematics. These states are updated through dual-stream message passing that separates longitudinal and transverse structural information while allowing cross-stream exchange. Geometry and loading effects are incorporated through Feature-wise Linear Modulation (FiLM)-conditioned 1-D spectral convolutions, and physical fields are reconstructed using a spectral-bypass low-rank readout. The model is evaluated on stiffened panel datasets with different geometries, boundary kinematics, loading conditions, and material nonlinear responses. DS-HGNN achieves the lowest stress and displacement RMSE compared with six benchmark heterogeneous graph neural network models. It also reaches comparable accuracy to the strongest benchmark models using 19%-38% fewer training samples. A targeted evaluation further shows that DS-HGNN captures yield and post-yield stress features.