The effective conductivity of arrays of squares: large random unit cells and extreme contrast ratios

TL;DR

This paper introduces an advanced integral equation scheme for efficiently computing the effective conductivity of complex, high-contrast composite materials, achieving high accuracy even for large random unit cells with over a million squares.

Contribution

It extends existing methods by simplifying preconditioning, employing a banded solver, and optimizing quadrature placement, enabling high-precision calculations for large, complex composites.

Findings

Achieved nine-digit accuracy for random checkerboards with over a million squares.

Successfully computed effective conductivities at contrast ratios up to 10^6.

Extended the method to handle complex and negative conductivities using a homotopy approach.

Abstract

An integral equation based scheme is presented for the fast and accurate computation of effective conductivities of two-component checkerboard-like composites with complicated unit cells at very high contrast ratios. The scheme extends recent work on multi-component checkerboards at medium contrast ratios. General improvement include the simplification of a long-range preconditioner, the use of a banded solver, and a more efficient placement of quadrature points. This, together with a reduction in the number of unknowns, allows for a substantial increase in achievable accuracy as well as in tractable system size. Results, accurate to at least nine digits, are obtained for random checkerboards with over a million squares in the unit cell at contrast ratio 10^6. Furthermore, the scheme is flexible enough to handle complex valued conductivities and, using a homotopy method, purely negative contrast ratios. Examples of the accurate computation of resonant spectra are given.