Abstract

Classically, dynamic homogenization is understood as a low-frequency approximation to wave propagation in heterogeneous media. A particularly successful approach is the two-scale asymptotic expansion method and the notion of slow and fast variable1. The idea of high-frequency homogenization (HFH), introduced in2, is to use similar asymptotic methods to approximate how the dispersion relation and the media behave near a given point that satisfies the dispersion relation. This talk will be divided in two parts : in the first one, we extend the HFH method to the case of materials where imperfect contacts occur ; in the second one, we extend HFH to the case of dispersive media where the properties of the material depend on the frequency. Firstly, we consider an array of imperfect interfaces of the spring-mass type. HFH is applied to find an approximation of the wavefield and the dispersion diagram near a given point of the dispersion diagram. Especially, we consider edges of the Brillouin zone and three cases : when the eigenvalue solution of the dispersion relation is a single eigenvalue, when it is a double eigenvalue (the case of Dirac points where two branches intersect), or when eigenvalues are single but nearby. For these three cases, an approximation of the dispersion relation and an effective equation for the zeroth order wavefield are found. Furthermore, for the same example, the nearby case is observed to give much longer lived approximations than the single one as we get further from the edge.  This first part is a joint work with Raphael Assier, Bruno Lombard and Cédric Bellis. Secondly, we consider a doubly periodic structure on a square lattice. The physical properties (permittivity or permeability in electromagnetism, or effective elastic parameters arising from high-contrasts in elasticity, for example) may depend in some constituents of the unit cell on the frequency, following a Lorentz (or Drude) model. Far from the accumulation points that may occur at resonance in these cases, we get the approximation of the dispersion relation and an effective equation for the wavefield. This on-going work is a joint work with Benjamin Vial, Raphael Assier, Sébastien Guenneau and Richard Craster.  1 A. Bensoussan, J.-L. Lions, G. Papanicolaou, Asymptotic Analysis for Periodic Structures, AMS Chelsea Publishing (2011). 2 R. V. Craster, J. Kaplunov, A. V. Pichugin. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, The Royal Society, 466, 2341-2362 (2010).