Numerical analysis of several FFT-based schemes for computational homogenization

TL;DR

This paper analyzes the convergence behaviors of various FFT-based computational homogenization schemes, proving their effectiveness in approximating theoretical coefficients and providing convergence rate estimates for the FEM scheme.

Contribution

It offers a rigorous convergence analysis of multiple FFT-based schemes, including Moulinec-Suquent's, Willot's, and FEM, with new rate estimates for the FEM scheme.

Findings

All schemes' effective coefficients converge to theoretical values.

Convergence is proven under reasonable assumptions.

FEM scheme has established convergence rate estimates.

Abstract

We study the convergences of several FFT-based schemes that are widely applied in computational homogenization for deriving effective coefficients, and the term "convergence" here means the limiting behaviors as spatial resolutions going to infinity. Those schemes include Moulinec-Suquent's scheme [Comput Method Appl M, 157 (1998), pp. 69-94], Willot's scheme [Comptes Rendus M\'{e}canique, 343 (2015), pp. 232-245], and the FEM scheme [Int J Numer Meth Eng, 109 (2017), pp. 1461-1489]. Under some reasonable assumptions, we prove that the effective coefficients obtained by those schemes are all convergent to the theoretical ones. Moreover, for the FEM scheme, we can present several convergence rate estimates under additional regularity assumptions.