A multiscale method for computing effective parameters of composite electromagnetic materials with memory effects

Abstract

We consider the problem of computing (macroscopic) effective properties of composite materials that are mixtures of complex dispersive dielectrics described by polarization and magnetization laws. We assume that the micro-structure of the composite material is described by spatially periodic and deterministic parameters. Mathematically, the problem is to homogenize Maxwell’s equations along with constitutive laws that describe the material response of the micro-structure comprising the mixture, to obtain an equivalent effective model for the composite material with constant effective parameters. The novel contribution of this paper is the homogenization of a hybrid model consisting of the Maxwell partial differential equations along with ordinary (auxiliary) differential equations modeling the evolution of the polarization and magnetization, as a model for the complex dielectric material. This is in contrast to our previous work (2006) in which we employed a convolution in time of a susceptibility kernel with the electric field to model the delayed polarization effects in the dispersive material. In this paper, we describe the auxiliary differential equation approach to modeling material responses in the composite material and use the periodic unfolding method to construct a homogenized model.