Finite element modeling of single periodic curved gratings

Abstract

The analysis of the scattering of a plane acoustic wave by an in-plane periodic, active, or passive structure has been previously performed with the help of the finite element method [A. C. Hladky-Hennion etal., J. Acoust. Soc. Am. 94, 621–635 (1993)]. In that case, the compliant tubes grating is a single periodic structure that is periodic in one direction and infinite in the other one. The method previously described has been extended to take into account the effect of the curvature of the grating, periodic along the symmetry axis. Within this approach, only a bidimensional mesh is needed because the structure is axisymmetrical. Moreover, only one unit cell of the periodic structure, including a small part of the surrounding fluid domain, has to be meshed, due to the use of a classical Block-type relation between the displacement components of points that are separated by the grating spacing. Finally, the effect of the external fluid domain is accounted for by matching the pressure field in the finite element domain to plane waves expressed in terms of a series of Bessel or Hankel functions. This paper describes the general mathematical model and then provides the results obtained for the scattering of a plane wave form simple axisymmetrical structures, this validating the method. Next, the method is applied to the case of curved compliant tubes gratings, thus showing the influence of the curvature.