Technical Summary: Mask-Morph Graph U-Net (MMGUNet)

1. Problem Statement

Nonlinear finite element (FE) crash simulations are the standard for assessing vehicle crashworthiness but are computationally expensive, hindering their use in iterative design optimization. While machine learning surrogates based on Graph Neural Networks (GNNs) offer a faster alternative, existing approaches face a fundamental trade-off between predictive capacity and geometric generalisability:

• Shared-weight Message Passing: Models using shared update functions across all nodes/edges (e.g., MeshGraphNet) generalize well across different topologies but often lack the capacity to capture complex nonlinear relationships accurately.
• Edge-Specific Aggregation: Models using non-shareable, edge-specific weights (e.g., Multi-channel Aggregation Networks) achieve higher accuracy on fixed topologies but require fixed graph connectivity.
• The Generalisability Bottleneck: Hierarchical Graph U-Nets with edge-specific layers (like ReGUNet) rely on fixed coarse graph templates. Cross-graph edges are typically constructed based on spatial proximity between the input mesh and the fixed template. When geometric variations are large (e.g., significant shape changes, scale differences, or mesh density variations), spatial proximity fails to establish correct node correspondence. This leads to fine nodes connecting to inappropriate coarse regions, degrading prediction performance and limiting transferability to unseen geometries.

2. Methodology

The authors propose Mask-Morph Graph U-Net (MMGUNet), a framework designed to retain the high capacity of edge-specific layers while enabling generalisation across varying geometries. The methodology consists of three core components:

A. Feature-Aligned Barycentric Morphing

To resolve the correspondence issue without sacrificing the fixed topology required for edge-specific layers, MMGUNet morphs the coarse graph hierarchy to match each input fine mesh before constructing cross-graph edges.

• Process: Both the fixed-topology template and the input mesh are mapped to a shared 2D parameter domain (UV domain) using Tutte's barycentric embedding.
• Feature Alignment: Unlike standard circular boundaries, the method anchors the UV boundary to case-specific polygons (e.g., squares or octagons) defined by detected corner nodes. This ensures semantically similar regions (e.g., corners, flanges) maintain consistent UV locations across different shapes.
• Morphing: The target 3D nodal coordinates are interpolated from the UV domain back to the template, updating the coarse graph geometry to align with the input shape while preserving its fixed connectivity. Cross-graph edges are then constructed based on this morphed, aligned geometry.

B. Network Architecture

MMGUNet utilizes a hierarchical Graph U-Net backbone:

• Fine Levels: Uses shared-weight MLP-based operations for input encoding and in-graph message passing, allowing flexibility for arbitrary input topologies.
• Coarse Levels: Employs fixed-topology graphs with edge-specific downsampling and upsampling layers. These layers assign non-shareable weights per channel, maximizing representation capacity for nonlinear relationships.
• Cross-Graph Connections: Established after the morphing step, ensuring meaningful local correspondence between fine and coarse nodes.

C. Training Strategy: Masked Pretraining and Parameter-Efficient Fine-Tuning

To improve robustness and data efficiency, the model employs a two-stage training protocol:

1. Masked Supervised Pretraining: A subset of nodes and associated edges is randomly masked during training. The model predicts the full displacement field from the partially observed graph. Crucially, nodes with boundary conditions or contact constraints are protected from masking to preserve physical constraints. This acts as a stochastic regularizer.
2. Parameter-Efficient Fine-Tuning: For adaptation to specific target tasks, the high-parameter edge-specific layers (downsampling/upsampling) and coarse-level message passing layers are frozen. Only the encoder, first fine-level message passing, and downsampling message passing layers are updated. This reduces the trainable parameter count to approximately 0.48% of the total, preventing overfitting and accelerating adaptation.

3. Key Contributions

• Mask-Morph Graph U-Net: A multiscale mesh-based GNN surrogate that combines topology-preserving coarse-graph morphing with edge-specific aggregation layers for crashworthiness field prediction.
• Feature-Aligned Morphing: A novel procedure using barycentric parameterisation with landmark-anchored boundaries, enabling the reuse of fixed-topology, edge-specific operators across geometrically varying finite-element meshes.
• Transfer Learning Strategy: A supervised masked pretraining and parameter-efficient fine-tuning approach that significantly improves generalisability and reduces data requirements for target adaptation.
• Comprehensive Evaluation: A multi-geometry case study suite involving seven scenarios (six B-pillar variants and one U-channel case) with varying scales, mesh densities, and geometric complexities, evaluated under in-distribution, out-of-distribution, and cross-component transfer settings.

4. Experimental Results

The model was evaluated using Mean Euclidean Distance (MED) for displacement fields and Maximum Intrusion Percentage Error (MIPE) for peak intrusion.

• Ablation Studies:
  • Positional Encoding: Combining one-hot node-type encoding and distance-to-critical-regions (DTC) features yielded the best performance, outperforming Laplacian eigenvectors which introduced instability due to sign ambiguity.
  • Mask Ratio: A 20% mask ratio provided the optimal trade-off between generalisation and prediction quality.
• Baseline Comparisons:
  • MMGUNet outperformed external baselines (MeshGraphNet, MS-MGN, Multigrid) and ablated versions (GUNet-fix, MGUNet) on both B-pillar and U-channel cases.
  • Morphing Impact: MGUNet (with morphing but no masking) significantly reduced errors compared to GUNet-fix (fixed connectivity), confirming that geometric alignment is critical for large variations.
  • Pretraining Impact: MMGUNet (with morphing + masking) further reduced errors and narrowed the train-test gap, demonstrating improved robustness and reduced overfitting.
• Generalisability and Transfer:
  • Out-of-Distribution: Direct generalisation to unseen shapes (e.g., different scales or mesh densities) remained challenging but was improved by the proposed architecture.
  • Transfer Learning: Pretrained models fine-tuned on target tasks consistently outperformed models trained from scratch. Notably, in Trial B (shape generalisation), a pretrained model fine-tuned with only 50 samples outperformed a non-pretrained baseline trained with 300 samples.
  • Cross-Component Transfer: The model successfully transferred from B-pillar geometries to a U-channel component (Trial E), showing positive gains even across distinct component families, though with smaller relative improvements compared to intra-family transfers.

5. Significance and Claims

The paper claims that MMGUNet provides a practical route toward reusable, data-efficient mesh-based surrogate modelling for crashworthiness design exploration.

• Addressing the Trade-off: The method successfully resolves the tension between the need for fixed coarse topology (for high-capacity edge-specific layers) and the need for geometric flexibility (for generalisability).
• Design Support: The surrogate is positioned as a decision-support tool for early-stage design exploration, enabling rapid estimation of deformation fields and intrusion measures for many geometric variants before committing to expensive nonlinear FE simulations.
• Data Efficiency: The combination of morphing and parameter-efficient fine-tuning significantly reduces the amount of target data required to adapt the model to new geometries, making it viable for iterative workflows where generating large datasets for every new design is impractical.

The authors acknowledge limitations, noting the current scope is restricted to terminal displacement prediction under fixed loading and boundary conditions, and that the morphing strategy relies on identifiable geometric landmarks. Future work is proposed to extend the framework to time-dependent responses and variable loading conditions.