Fourier-based schemes with modified Green operator for computing the electrical response of heterogeneous media with accurate local fields

TL;DR

This paper introduces a modified Green operator for Fourier-based numerical schemes that improves convergence and local field accuracy in heterogeneous media with high property contrast, while also enabling efficient implementation.

Contribution

It proposes a new Green operator that enhances convergence and local field accuracy, and offers a memory-efficient implementation of the direct scheme.

Findings

Convergence tends to a finite value at infinite contrast.

Significantly improved local field accuracy near interfaces.

Efficient memory usage in the direct scheme implementation.

Abstract

A modified Green operator is proposed as an improvement of Fourier-based numerical schemes commonly used for computing the electrical or thermal response of heterogeneous media. Contrary to other methods, the number of iterations necessary to achieve convergence tends to a finite value when the contrast of properties between the phases becomes infinite. Furthermore, it is shown that the method produces much more accurate local fields inside highly-conducting and quasi-insulating phases, as well as in the vicinity of the phases interfaces. These good properties stem from the discretization of Green's function, which is consistent with the pixel grid while retaining the local nature of the operator that acts on the polarization field. Finally, a fast implementation of the "direct scheme" of Moulinec et al. (1994) that allows for parcimonious memory use is proposed.