## 1. Introduction

[2] Many engineering problems contain materials which are heterogeneous on a length scale (microscopic) that is small compared with the typical length scales (macroscopic) of the problem. Nevertheless, the underlying heterogeneous microscopic structure affects the macroscopic properties of the material. Accurate methods to model these microscopic effects are therefore important to develop.

[3] In electromagnetic problems, when the wavelength is long compared to the periodicity of the microstructure, the homogenized values of the electric or magnetic material parameters are pertinent macroscopic quantities. Several homogenization procedures have been suggested in the literature. Some of these are based upon physical arguments, e.g., the Maxwell Garnett formula, the Böttcher mixture rule or Bruggeman formula, and the coherent potential (CP) formula [*Sihvola*, 1999]. They have proven useful in many situations, e.g., low volume fraction of homogeneous spherical or ellipsoidal inclusions in a homogeneous host material, but they fail if the volume fraction is too high or if the inclusions are not spheres or ellipsoids. Then a more accurate homogenization procedure has to be used, which includes all contributions of the interaction between the inclusions.

[4] In this paper we review a general mathematical procedure to obtain the homogenized (or effective) material parameters, which includes all interactions effects between the inclusions. Specifically, we employ recent advances in the mathematics of two-scale convergence, which was introduced in 1989 by *Nguetseng* [1989]. Specifically, we review some of the more important results of this convergence concept, but for the mathematical details we refer to the existing literature on this subject.

[5] A typical homogenization situation is depicted in Figure 1, where we, in several steps, shrink the length scale ε, which is the periodicity of the material. The two-scaled convergence predicts the limit of this process and gives a procedure of how to compute the effective material parameters of the material with microstructure. The results are not restricted to low volume fractions of the inclusions or to a two-phase composition of materials. In fact, the results are quite general and can be applied to a large variety of engineering situations.

[6] The results obtained using two-scale convergence are self-contained in contrast to similar approaches that assume an asymptotic expansion of the solution in terms of the microscopic scale [e.g., see *Sanchez-Palencia*, 1980]. The theory of two-scale convergence does not rely on such an expansion. The periodic variations in the material parameters generate the same type of variations in the two-scale converged solution, which characterizes the microscopic solution completely.

[7] This paper is organized in the following way. In section 2 the basic assumptions of the homogenization problem are stated, and in section 3 the fundamental properties of the two-scaled convergence are reviewed. The local problem is stated in section 4, and a comparison between the classical mixture formulae and the exact homogenization procedure is presented in section 5. Some conclusions of the theory are discussed in section 6.