Homogenization of thin 3D periodic structures in the time domain - effective boundary and jump conditions

This chapter presents a two-scale homogenization to encapsulate the effect of rigid inclusions in the vicinity of a rigid wall or forming a structured film in effective conditions. The chapter sets the different ingredients needed to conduct the asymptotic analysis. It provides the definition of a small parameter, which is used to differentiate between the rapid variations of the evanescent field and the slow variations of the propagating field. In the matched asymptotic expansion technique, two different expansions of the solution (termed outer and inner expansions) are sought which are valid far and close to the inclusions. The corresponding outer and inner problems are complemented using matching conditions. The chapter also argues that the effective boundary condition in equations requires the calculation of three parameters and the effective jump conditions require the calculation of six parameters. These parameters are deduced from static elementary problems.