Derivation of higher-order terms in FFT-based numerical homogenization

TL;DR

This paper extends FFT-based numerical homogenization methods to include higher-order terms using asymptotic expansions, providing a hierarchy of related linear problems and numerical results for the first two orders.

Contribution

It introduces a novel extension of the Basic Scheme to derive and solve higher-order homogenization problems via asymptotic expansion.

Findings

Hierarchy of linear problems derived from asymptotic expansion

Extension of FFT-based homogenization scheme to higher orders

Numerical results demonstrating the first two problem orders

Abstract

In this paper, we first introduce the reader to the Basic Scheme of Moulinec and Suquet in the setting of quasi-static linear elasticity, which takes advantage of the fast Fourier transform on homogenized microstructures to accelerate otherwise time-consuming computations. By means of an asymptotic expansion, a hierarchy of linear problems is derived, whose solutions are looked at in detail. It is highlighted how these generalized homogenization problems depend on each other. We extend the Basic Scheme to fit this new problem class and give some numerical results for the first two problem orders.