Approximations and full numerical simulations for the conductivity of three dimensional checkerboard geometries

This paper focuses on the problem of macroscopic effective permittivity (permeability, conductivity) of a regular three-dimensional two-phase checkerboard structure. There is no exact solution for the Laplace equation in this geometry. Various classical mixing rules and matrix methods exist in the literature to estimate the effective permittivity of mixtures and they are often used beyond their area of applicability. We compare the results from these methods with our full numerical solution of the problem for various contrasts between the phases. The limitations of the approximate models are pointed out