Similarity Equivariant Graph Neural Networks for Homogenization of Metamaterials

TL;DR

This work introduces SimEGNN, a similarity-equivariant graph neural network for fast, accurate homogenization surrogates of nonlinear, buckling-enabled metamaterials. By embedding $E(n)$-in-/equivariance, periodic RVE handling, and scale invariance, SimEGNN predicts deformation and homogenized tensors $\mathfrak{W}$, $\textbf{P}$, and ${}^4\textbf{D}$ directly from RVE geometry. Across extensive tests, SimEGNN achieves $\text{FVU} < 10^{-3}$ and $R^2 > 0.999$ for all targets, outperforms less symmetric baselines, and maintains robust generalization to unseen geometries while delivering significant speed-ups over FE simulations. The approach enables rapid design exploration and optimization of mechanical metamaterials, with potential extensions to more complex 3D geometries and bifurcation modeling.

Abstract

Soft, porous mechanical metamaterials exhibit pattern transformations that may have important applications in soft robotics, sound reduction and biomedicine. To design these innovative materials, it is important to be able to simulate them accurately and quickly, in order to tune their mechanical properties. Since conventional simulations using the finite element method entail a high computational cost, in this article we aim to develop a machine learning-based approach that scales favorably to serve as a surrogate model. To ensure that the model is also able to handle various microstructures, including those not encountered during training, we include the microstructure as part of the network input. Therefore, we introduce a graph neural network that predicts global quantities (energy, stress stiffness) as well as the pattern transformations that occur (the kinematics). To make our model as accurate and data-efficient as possible, various symmetries are incorporated into the model. The starting point is an E(n)-equivariant graph neural network (which respects translation, rotation and reflection) that has periodic boundary conditions (i.e., it is in-/equivariant with respect to the choice of RVE), is scale in-/equivariant, can simulate large deformations, and can predict scalars, vectors as well as second and fourth order tensors (specifically energy, stress and stiffness). The incorporation of scale equivariance makes the model equivariant with respect to the similarities group, of which the Euclidean group E(n) is a subgroup. We show that this network is more accurate and data-efficient than graph neural networks with fewer symmetries. To create an efficient graph representation of the finite element discretization, we use only the internal geometrical hole boundaries from the finite element mesh to achieve a better speed-up and scaling with the mesh size.

Figures (22)

• Figure 1: Schematic overview of the first-order computational homogenization procedure.
• Figure 2: (a) The RVE geometry, with four holes, slightly flattened by 1% into ellipses, with $2a$ the major axis and $2b$ the minor axis. The periodic boundaries are indicated by $\Gamma$, the control points by red dots, and the foci of the ellipses by black dots. The length and width of the RVE is $\ell$ and the domain is $\Omega$. The directions of the basis vectors are indicated by $\vec{e}_1$ and $\vec{e}_2$. (b) The finite element discretization of this geometry.
• Figure 3: The behavior of the metamaterial RVE under various loading conditions. In this plot, $U_{ij}$ are components of the right stretch tensor, see Section \ref{['subsec:sampling']}. The components $U_{11}$ and $U_{22}$ are varied, while $U_{12}$ is constrained at zero. The reference configuration (i.e., the unloaded state) is plotted in orange.
• Figure 4: Change in (a) stress component $P_{11}$ and (b) stiffness component $D_{1111}$ as a function of the biaxial compression.
• Figure 5: The sampled loading paths in $U_{11}$, $U_{22}$ and $U_{12}$ space. The lines indicate the loading paths, all starting from the black dot which indicates $\textbf{U} =\textbf{I}$. The colored dots represent the sampled values of final $\textbf{U}^{\textrm{fin}}$, with the blue dots indicating cases with $U_{11}^{\textrm{fin}}$ constant, orange dots indicating $U_{22}^{\textrm{fin}}$ constant and green dots indicating $U_{12}^{\textrm{fin}}$ constant.