Equivariant graph convolutional neural networks for the representation of homogenized anisotropic microstructural mechanical response

TL;DR

This work provides neural network architectures that provide effective homogenization models of materials with anisotropic components that satisfy equivariance and material symmetry principles inherently through a combination of equivariant and tensor basis operations.

Abstract

Composite materials with different microstructural material symmetries are common in engineering applications where grain structure, alloying and particle/fiber packing are optimized via controlled manufacturing. In fact these microstructural tunings can be done throughout a part to achieve functional gradation and optimization at a structural level. To predict the performance of particular microstructural configuration and thereby overall performance, constitutive models of materials with microstructure are needed. In this work we provide neural network architectures that provide effective homogenization models of materials with anisotropic components. These models satisfy equivariance and material symmetry principles inherently through a combination of equivariant and tensor basis operations. We demonstrate them on datasets of stochastic volume elements with different textures and phases where the material undergoes elastic and plastic deformation, and show that the these network architectures provide significant performance improvements.

Figures (10)

• Figure 1: Polycrystal: relative deviation of the stress from the Hill estimate $\| {\bar{\mathbf{S}}-\bar{\mathbf{S}}_\text{Hill}} \| / \| \bar{\mathbf{S}} \|$ for the three elastic datasets.
• Figure 2: Polycrystal (left to right panels): first Euler angle of orientation vector field ${\boldsymbol{\phi}}$, stress $\mathbf{S}$ states: elastic (0.1%), transition (0.2%), plastic (0.4%) (colored by $\sigma_{11}$ with the same scale for all three panels [dark: $<$ 100 MPa, light: $>$ 400 MPa]).
• Figure 3: Neural network architecture for elastic response. Inputs are (a) the external strain loading of the sample $\bar{\mathbf{E}}(t)$, (b) the structure tensors for each cell $K$, and the graph ${\mathit{G}}$ that provides the cell-to-cell connectivity and cell volume fractions $\nu_K$. The output is the sample average stress $\bar{\mathbf{S}}$. After converting the scalar invariants and tensor basis generators of $\bar{\mathbf{E}}$ and $\mathbb{A}_K$, the local TBNN layer provides an embedding/reduction of these inputs at each cell $K$. This cell data is then processed by a stack of equivariant convolutions. The output of the stack is then pooled and this converted to the spatial output $\bar{\mathbf{S}}$.
• Figure 4: An equivariant RNN for inelastic response. Inputs are (a) the external strain loading of the sample $\bar{\mathbf{E}}(t)$, (b) the structure tensors for each cell $K$, and the graph ${\mathit{G}}$ that provides the cell-to-cell connectivity and cell volume fractions $\nu_K$. Output is the sample average stress $\mathbf{S}(t)$ over time $t$.
• Figure 5: Convex NODE potential GCNN for inelastic response. Inputs are (a) the external strain loading of the sample $\mathbf{E}(t)$, (b) the structure tensors for each cell $K$, and the graph ${\mathit{G}}$ that provides the cell-to-cell connectivity and cell volume fractions $\nu_K$. The additional latent variables $\mathsf{h}$ are initialized to zero and the NODE described in detail in Eq. (\ref{['eq:node']}). Output is the sample average stress $\mathbf{S}(t)$ over time $t$.

Theorems & Definitions (3)

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