Equivariant Machine Learning for Simulation of Buckling in Mechanical Metamaterials
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Mechanical metamaterials are materials with special properties such as tunable stiffness and negative Poisson’s ratio, resulting from their specially designed microstructure. Using structural instabilities like buckling, these materials can undergo controlled pattern transformations, making them suitable for applications in soft robotics and tunable sound attenuation. However, modelling, design, and optimization of such materials are limited by the high computational cost required to simulate these materials using conventional methods when deformations are large and there is buckling. To address these computational bottlenecks, this thesis splits into the following main four parts. 1. Accelerated Homogenization via SimEGNN First, SimEGNN (Similarity-Equivariant Graph Neural Network) is developed, which is a neural network that respects fundamental physical symmetries (Euclidean transformations, periodicity, and scaling). It predicts global quantities like strain energy density, stress and stiffness as well as the local deformation field in response to a macroscopic deformation gradient. It provides a high-fidelity alternative to traditional finite-element simulations. Quantitative results demonstrate that this architecture achieves superior data efficiency and provides order-of-magnitude speed-ups on a relevant test system (a microstructure with circular holes arranged in a square grid), making it a practical tool for rapid, iterative design. 2. A Dataset of Wallpaper Group-Based 2D Microstructures The development of robust surrogate models requires training data that reflects the full diversity of geometric possibilities. To this end, this thesis introduces a novel dataset of 1,020 high-quality 2D microstructures spanning all 17 wallpaper groups. These structures are generated via a periodic-graph "skeleton", used as a starting point to determine the shape and placement of holes parametrized by Bézier curves, ensuring connectivity. Each microstructure is simulated across multiple loading trajectories in the hyperelastic, finite-strain regime including buckling, providing a comprehensive resource for studying symmetry-property relationships such as auxeticity and mode multiplicity. 3. Equivariant Flow Matching for Bifurcation A fundamental challenge in modeling buckling metamaterials is multistability, where a single input can lead to multiple stable states. Because traditional deterministic machine learning models tend to average these outcomes into nonphysical predictions, this thesis introduces a probabilistic, equivariant flow-matching framework capable of capturing the full distribution of bifurcation outcomes. This framework is tested on various physical problems that show symmetry-breaking bifurcations. The framework ensures that the learned distributions respect the underlying physical symmetries of the system, even if individual outcomes break those symmetries. The models are trained more efficiently by utilizing symmetric coupling to find the optimal group-equivalent target for each sample. 4. An Integrated Pipeline for Deformation Distributions The final contribution of this thesis synthesizes the above three elements into a unified GNN and Flow-Matching pipeline. This integrated pipeline takes graph representations of metamaterials as input and generates entire distributions of deformation trajectories under mechanical loading as output. Temporal 1D U-Nets are incorporated to model long-range dependencies within these trajectories. The pipeline offers a powerful, end-to-end tool for predicting the complex, nonlinear behavior of mechanical metamaterials.
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