Computational parameter retrieval approach to the dynamic homogenization of a periodic array of rigid rectangular blocks

Abstract

We propose to homogenize a periodic (along one direction) structure, first in order to verify the quasi-static prediction of its response to an acoustic wave arising from mixing theory, then to address the question of what becomes of this prediction at higher frequencies. This homogenization is treated as an inverse (parameter retrieval) problem, i.e., by which we: (1) generate far-field (i.e., specular reflection and transmission coefficients) response data for the given periodic structure, (2) replace (initially by thought) this (inhomgoeneous) structure by a homogeneous (surrogate) layer, (3) compute the response of the surrogate layer response for various trial constitutive properties, (4) search for the global minimum of the discrepancy between the response data of the given structure and the various trial parameter responses (5) attribute the homogenized properties of the surrogate layer for which the minimum of the discrepancy is attained. We carry out this five-step procedure for a host of given structures and solicitation parameters, notably the frequency. The result is that: (i) at low frequencies and/or large filling factors, the effective constitutive properties are close to their static equivalents, i.e., the effective mass density is the product of a factor related to the given structure filling factor with the mass density of a generic substructure of the given structure and the effective velocity is equal to the velocity in the said generic substructure, 2) at higher frequencies and/or smaller city filling factors, the effective constitutive properties are dispersive and do not take on a simple mathematical form, with this dispersion compensating for the discordance between the ways the inhomogeneous given structure and the homogeneous surrogate layer respond to the acoustic wave. At low frequencies, the given structure, represented as a layer with static homogenized properties, is quite adequate to account for the principal features of the far-field response (even for the phenomenon of total transmission when the spaces between blocks are very narrow). The model of the layer with dispersive homogenized properties is more suitable to account for such features as resonances due to the excitation of surface wave modes.

Dynamical homogenization of a transmission grating

“…However, the second peak, which is linked to the Wood anomaly, is not accounted-for in the layer model. In fact, the existence of the Wood anomalies (which occur for frequencies and incident angles at which an inhomogeneous scattered wave becomes homogeneous) is so tightly linked with the d-periodic nature of the scattering configuration (the inhomogeneous waves being absent in the layer modeel) that they cannot make their appearance in the response of a homogeneous layer unless the effective mass density and/or velocity of the surrogate layer are dispersive (this possibility was ruled out a priori herein, but was taken into accout in studies such as [14,57]).…”

“…In spite of this shortcoming, the layer model deriving from our low-frequency homogenization scheme, appears to give meaningful predictions of the response of the transmission grating well beyond the static limit and can therefore be qualified as 'dynamical'. Moreover, these predictions can be improved either by the technique outlined in [56,57] or by taking into account higher-order iterates in the scheme presented herein. Finally, our homogenization scheme may provide a useful alternative to traditional multiscale and field averaging approaches to homogenization of periodic structures as regards their response to dynamic solicitations.…”

“…), but, on the contrary, he wants to find out how its properties depend on the characteristics of the solicitation. Moreover, the requirement of normal incidence (proper to the NRW technique) can be relaxed provided one accepts to search for the solution of the inverse problem in an optimization, rather than explicit, manner [40,55,57].…”