Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Efficient computational homogenization via tensor train format

Yuki Sato, Yuto Lewis Terashima, Ruho Kondo

TL;DR

The paper tackles the high computational cost of computational homogenization for multiscale, heterogeneous materials. It introduces a tensor-train (TT) based asymptotic homogenization framework that encodes microscale boundary-value problems and material fields in TT/QTT format and solves them with the MALS TT solver. The authors formulate cell problems for both thermal conductivity and linear elasticity on RVEs, demonstrate TT-based discretization and rank-adaptive solution in 2D and 3D, and analyze how TT ranks scale with microstructure volume fraction. The work shows that TT-based homogenization can significantly reduce computational cost for high-resolution RVEs, though 3D elasticity can incur higher TT overhead, motivating exploration of alternative tensor networks for further efficiency.

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.

Efficient computational homogenization via tensor train format

TL;DR

The paper tackles the high computational cost of computational homogenization for multiscale, heterogeneous materials. It introduces a tensor-train (TT) based asymptotic homogenization framework that encodes microscale boundary-value problems and material fields in TT/QTT format and solves them with the MALS TT solver. The authors formulate cell problems for both thermal conductivity and linear elasticity on RVEs, demonstrate TT-based discretization and rank-adaptive solution in 2D and 3D, and analyze how TT ranks scale with microstructure volume fraction. The work shows that TT-based homogenization can significantly reduce computational cost for high-resolution RVEs, though 3D elasticity can incur higher TT overhead, motivating exploration of alternative tensor networks for further efficiency.

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.
Paper Structure (2 sections, 37 equations, 4 figures)

This paper contains 2 sections, 37 equations, 4 figures.

Table of Contents

  1. Thermal conductivity
  2. Linear elastic material

Figures (4)

  • Figure 1: A conceptual diagram of computational homogenization.
  • Figure 2: The lowest value of the maximum rank of TT to estimate the homogenized tensors of ten randomly generated RVEs within the error of 0.01 for various values of the number of degrees of freedom. Lines represent the medians and shaded areas show the first and third quantiles for 10 RVEs.
  • Figure 3: The computational time required to estimate the homogenized tensors of ten randomly generated RVEs using the number of ranks achieving an error lower than 0.01 for various values of the number of degrees of freedom. Lines represent the medians and shaded areas show the first and third quantiles for 10 RVEs.
  • Figure 4: The lattice $\tilde{Y}$ in two dimensions ($d=2$) when $N=4$.