Analysis of the scattering of a plane acoustic wave by a doubly periodic structure using the finite element method: Application to Alberich anechoic coatings

The finite element approach has been previously used, with the help of the ATILA code, to model the scattering of acoustic waves by single periodic structures, such as compliant tube gratings [A. C. Hennion et al., J. Acoust. Soc. Am. 87, 1861–1870 (1990)]. In this paper, the same approach is extended to doubly periodic structures, such as Alberich anechoic coatings. To do this, only the unit cell of the periodic structure, including a small part of the surrounding fluid domain, has to be meshed, due to the use of classical Bloch type relations. Then, the effects of the remaining fluid domain are accounted for by matching the pressure field in the finite element mesh with simple plane-wave expansions of the ingoing and outgoing waves. After an outline of the method, the paper describes the results obtained for the scattering of a plane wave by different periodic structures. Internal losses are taken into account and the incident plane wave impinges at normal or oblique incidence. Numerical results obtained for Alberich anechoic coatings are first analyzed to check the convergence and then compared to previous numerical results or to experimental results, demonstrating that the finite element approach is accurate and well suited to predict the behavior of these gratings. Moreover, careful attention is devoted to the analysis of the inclusion vibrations, to identify the origin of the resonance mechanisms.