Addition formulas and the Rayleigh identity for arrays of elliptical cylinders

Conductivity of a two-dimensional composite containing elliptical inclusions

We develop a method of functional equations to derive analytical approximate formulae for the effective conductivity tensor of the two-dimensional composites with elliptical inclusions. The sizes, the locations and the orientations of the ellipses can be arbitrary. The analytical formulae contain all the above geometrical parameters in symbolic form.

Duality relation for the Maxwell system

This paper is intended to establish a link between the vector Maxwell system for three-dimensional (3D) and 2D finite photonic crystals in the low-frequency limit. For this, we generalize the classical results of Keller and Dykhne (chessboard problem) to periodic media described by piecewise continuous permittivity profiles: our theorem enlights the result of Mendelson (polycrystalline and multiphase media) in the framework of homogenization theory of elliptic operators. In fine, we give illustrative examples by using both integral equation and variational approaches via the so-called method of fictitious charges and finite-element method.