Efficient computational homogenization via tensor train format

TL;DR

This paper introduces a tensor train-based asymptotic homogenization method that significantly reduces computational costs in predicting macroscopic properties of complex heterogeneous materials.

Contribution

It develops a novel TT-format approach for efficient multiscale homogenization, improving computational speed and accuracy over traditional methods.

Findings

Effective in 2D and 3D thermal conductivity homogenization

Accurate elasticity property predictions

Reduces computational complexity significantly

Abstract

Real-world physical systems, like composite materials and porous media, exhibit complex heterogeneities and multiscale nature, posing significant computational challenges. Computational homogenization is useful for predicting macroscopic properties from the microscopic material constitution. It involves defining a representative volume element (RVE), solving governing equations, and evaluating its properties such as conductivity and elasticity. Despite its effectiveness, the approach can be computationally expensive. This study proposes a tensor-train (TT)-based asymptotic homogenization method to address these challenges. By deriving boundary value problems at the microscale and expressing them in the TT format, the proposed method estimates material properties efficiently. We demonstrate its validity and effectiveness through numerical experiments applying the proposed method for homogenization of thermal conductivity and elasticity in two- and three-dimensional materials, offering a promising solution for handling the multiscale nature of heterogeneous systems.