Fourier-based schemes for computing the mechanical response of composites with accurate local fields

TL;DR

This paper introduces a modified Fourier-based computational scheme for elasticity in composites, improving the accuracy of local stress and strain fields and accelerating convergence compared to previous methods.

Contribution

It develops a new Green operator based on centered differences on a rotated grid, enhancing accuracy and speed of FFT-based elasticity simulations.

Findings

More accurate local stress and strain fields.

Faster convergence of FFT schemes.

Improved estimates of effective elastic moduli.

Abstract

We modify the Green operator involved in Fourier-based computational schemes in elasticity, in 2D and 3D. The new operator is derived by expressing continuum mechanics in terms of centered differences on a rotated grid. Use of the modified Green operator leads, in all systems investigated, to more accurate strain and stress fields than using the discretizations proposed by Moulinec and Suquet (1994) or Willot and Pellegrini (2008). Moreover, we compared the convergence rates of the "direct" and "accelerated" FFT schemes with the different discretizations. The discretization method proposed in this work allows for much faster FFT schemes with respect to two criteria: stress equilibrium and effective elastic moduli.