Analysis of a Fast Fourier Transform Based Method for Modeling of Heterogeneous Materials

TL;DR

This paper analyzes the convergence of a Conjugate Gradient method applied to a Fourier Transform-based homogenization problem for heterogeneous materials, demonstrating improved efficiency over traditional algorithms through theoretical proof and numerical validation.

Contribution

It provides a convergence proof for the CG method on a non-symmetric system from FFT-based homogenization and shows its practical advantages over the original algorithm.

Findings

Convergence of the CG method is proven for the problem.

Numerical results show significant improvement in efficiency.

The method better captures the physics of heterogeneous materials.

Abstract

The focus of this paper is on the analysis of the Conjugate Gradient method applied to a non-symmetric system of linear equations, arising from a Fast Fourier Transform-based homogenization method due to (Moulinec and Suquet, 1994). Convergence of the method is proven by exploiting a certain projection operator reflecting physics of the underlying problem. These results are supported by a numerical example, demonstrating significant improvement of the Conjugate Gradient-based scheme over the original Moulinec-Suquet algorithm.