Energy-conserving equivariant GNN for elasticity of lattice architected metamaterials

Abstract

Lattices are architected metamaterials whose properties strongly depend on their geometrical design. The analogy between lattices and graphs enables the use of graph neural networks (GNNs) as a faster surrogate model compared to traditional methods such as finite element modelling. In this work, we generate a big dataset of structure-property relationships for strut-based lattices. The dataset is made available to the community which can fuel the development of methods anchored in physical principles for the fitting of fourth-order tensors. In addition, we present a higher-order GNN model trained on this dataset. The key features of the model are (i) SE(3) equivariance, and (ii) consistency with the thermodynamic law of conservation of energy. We compare the model to non-equivariant models based on a number of error metrics and demonstrate its benefits in terms of predictive performance and reduced training requirements. Finally, we demonstrate an example application of the model to an architected material design task. The methods which we developed are applicable to fourth-order tensors beyond elasticity such as piezo-optical tensor etc.

Figures (7)

• Figure 1: (a) X-ray CT scan of 3d-printed lattice. A computer model of the unit cell is shown as an inset. (b) Model schematic. The dimensionality of intermediary quantities is noted between layers using e3nn convention. We omit simple linear layers from the diagram for clarity.
• Figure 2: The evolution of (a) component loss, $L_\text{comp}$, (b) equivariance loss, $L_\text{equiv}$, and (c) percentage of lattices with a negative eigenvalue, $\lambda_{\%}^{-}$, during training.
• Figure 3: (a) Convergence with the amount of data for various model classes. (b) Sensitivity to the maximum 'frequency' $L_{max}$ and (c) correlation order $\nu$ for MACE model.
• Figure 4: (a) Unit cell of the lattice used as the starting point for optimization. (b) Original stiffness surface and (c) optimized stiffness surface. Inset shows projections into $x-y$ plane of the starting point, optimization target, and the result of ML-based optimization.
• Figure 5: (a) The assembly of millions of unit cells effectively behaves as continuum material, hence the name metamaterial. In this work we assume all unit cells have cylindrical struts with radius $r$. (b) Different types of unit cells could be used, such as triply periodic minimal surfaces (TPMS) which lead to shell-based lattices.