Historically, homogenization of periodic structures has been first investigated by the method of multiple scalings expansions. More recently, an interpretation has been given in terms of averages and effective moduli. Both approaches involve a slow variable at the macroscopic scale, and a fast one at the microscopic level. The periodicity of the solutions with respect to the second variable is a strong assumption made prior to any analysis. Although involving similar calculations, the two approaches differ and it is not so obvious to link them together. The matched asymptotic expansions presented here allow to give a common explanation to the two already mentioned approaches. The first one corresponds to an outer expansion while the second one describes the leading term of an inner expansion. Moreover, no a priori assumption is made, the periodicity of the solutions occurs as a consequence of the structure of the inner problems. The next term (involving a quadratic dependence on the local variable) of the inner expansion can be derived in the same way. The same matched asymptotics process can be used to define homogenized boundary conditions as well as boundary layers. These layers come from a mismatch between the general form of the solution within the domain and the boundary conditions which occur to be a perturbation of the periodicity. Indeed, it is not easy to give an exact definition of the boundary conditions in the original problem, the inner expansion defined on the enlarged domain allows one to give a precise framework for these conditions. They split into two parts, a macroscopic one defined on the smooth (homogenized) boundary and a microscopic periodic fluctuation taking into account the exact shape of the boundary.
A comparison between two-scale asymptotic expansions and Bloch wave expansions for the homogenization of periodic structures
