Energy-conserving equivariant GNN for elasticity of lattice architected metamaterials

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 modeling. 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. A relatively new class of materials, architected (meta-)materials, emerged in the last century. As a subclass of architected materials, lattices are a collection of struts (edges) which are connected at nodes. See Figure 1a below and Figure 5 in the Appendix. Lattices are especially mechanically efficient, offering a very high specific stiffness (stiffness divided by density). For instance, it is possible to make materials with the density of water and the strength of steel.