Rapidly convergent expansion method for calculating the effective conductivity of three-dimensional lattices of symmetric inclusions

Abstract

An exact method is introduced to determine the electric potential in an infinite rectangular lattice of particles described by curvilinear coordinates in which Laplace's equation separates. The potential is expanded in harmonic functions, and suitable auxiliary functions are used to obtain an infinite system of linear algebraic equations for the expansion coefficients. Special attention is paid to lattices of spheres and prolate spheroids. For these cases, the truncated system converges very rapidly as the number of terms in the truncation series increases. The method works well for calculating the effective conductivity for dense or sparse inclusions, and for highly conducting lattices or lattices of cavities. Numerical results for the effective conductivity are given and compared with data obtained by other methods.