High-Order Elasticity Interpolants for Microstructure Simulation

Abstract

We propose a novel formulation of elastic materials based on high-order interpolants, which fits accurately complex elastic behaviors, but remains conservative. The proposed high-order interpolants can be regarded as a high-dimensional extension of radial basis functions, and they allow the interpolation of derivatives of elastic energy, in particular stress and stiffness. Given the proposed parameterization of elasticity models, we devise an algorithm to find optimal model parameters based on training data. We have tested our methodology for the homogenization of 2D microstructures, and we show that it succeeds to match complex behaviors with high accuracy.

Citation

@article{HiOInterp,
	author    = {Chan-Lock, Antoine and Perez, Jesus and Otaduy, Miguel A.},
	title     = {High-Order Elasticity Interpolants for Microstructure Simulation},
	journal   = {Computer Graphics Forum (Proc. SCA)},
	number    = {8},
	volume    = {41},
	year      = {2022}
}

Description

In this paper, we develop a novel formulation of elastic materials based on high-order interpolants, which fits accurately complex elastic behaviors, but remains conservative. The contributions of our work are:

The design of tensor basis functions to interpolate derivatives of elastic energy. These basis functions can be regarded as a high-dimensional extension of radial basis functions (RBFs).
Based on the tensor interpolants, we design a parameterization of elasticity models. This paramterization provides suitable degrees of freedom to fit both the stress and stiffness behavior of complex materials.
An algorithm to optimize the parametric elasticity model based on training data, which finds the control points and coefficients of the elasticity interpolants.
The application of the methodology to homogenization of 2D microstructures. This includes the generation of representative training data and the application of the estimation algorithm mentioned above.

In the paper, we evaluate the accuracy of our method, we compare it to other variants, and we analyze the effect of various design choices. As a conclusion, the proposed methodology for the design of elasticity models succeeds at capturing complex behaviors. We have tested the methodology on 11 2D microstructures with different deformation behaviors, and we discuss the full results.