Growth-Inspired Graph Generation and Inverse Design of Mechanical Lattices via Dot Matrices Database Augmentation and GCNN

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.

Explicit dynamic finite element (FE) simulations are widely used for large deformation engineering analysis, but repeated simulations remain costly during design space exploration and optimisation. In explicit FE analysis, nodal kinematics and element level deformation measures evolve through coupled node element updates. This motivates graph learned simulators that approximate one step FE state transitions and roll them out autoregressively. However, many mesh based graph surrogates are node centred, which makes element level variables and native nodal elemental exchange less direct to represent. This work proposes CAttBiGNN, a cross attention based bipartite graph neural network for coupled nodal elemental learning. The graph represents FE mesh nodes and elements as distinct entities linked by directed node element edges, enabling nodal displacement increments and element level deformation states to be predicted on their native discretisation domains. An edge aware cross attention processor uses geometric edge embeddings to modulate directional node element message passing. For larger graphs, CAttBiUGNN combines the bipartite processor with graph downsampling and upsampling to improve long-range information propagation. The method is evaluated on dome shaped cold forming and corner shaped hot forming benchmarks. Comparisons with node centred baselines and bipartite and attention ablations show improved accuracy and balance in nodal displacement and elemental thinning prediction during autoregressive rollout. The results indicate that the proposed finite element inspired learned simulator can support manufacturability oriented field prediction and efficient design space exploration in large deformation sheet material forming.

Finite element analysis (FEA) is essential for structural design but remains computationally expensive, particularly when evaluating multiple design iterations or load scenarios. Machine learning surrogate models offer a promising alternative, yet most approaches struggle with a critical limitation: generalizing across varying geometries. This work presents a mesh graph network (MGN) for predicting von Mises stress fields in 2D structural components with arbitrary hole geometries. Unlike traditional machine learning approaches that use absolute node coordinates as features, the proposed model builds on existing MGN frameworks that encode node types (e.g., fixed boundary, free surface, hole edge), relative edge features (distance between neighbors), and global features (applied load). This architecture is inherently translation- and rotation-invariant, enabling generalization to unseen geometries without retraining. The MGN was trained on 11 plate geometries under 20 load conditions and evaluated on 7 unseen geometries and 3 unseen loads. In the most favorable case, the model achieves $R^2 \geq 0.97$ on an unseen geometry and unseen load, compared to $R^2 \approx 0.01$--$0.86$ for conventional models (Random Forest, Gradient Boosting , K-Nearest Neighbors) trained on identical data. However, even in less favorable cases, the MGN model still outperforms conventional models. This work extends the mesh-based simulation framework of Pfaff et al. (arXiv:2010.03409) to structural mechanics, demonstrating that graph neural networks can serve as efficient surrogates for finite element analysis across varying geometries.