Analysis of the propagation of plane acoustic waves in passive periodic materials using the finite element method

Abstract

The finite element approach has been previously used, with the help of the ATILA code, to model the scattering of acoustic waves by single or doubly periodic passive structures ͓A. C. Hladky-Hennion et al., J. Acoust. Soc. Am. 90, 3356 -3367 ͑1991͔͒. This paper presents a new extension of this technique to the analysis of the propagation of plane acoustic waves in passive periodic materials without losses and describes with particular emphasis its application to doubly periodic materials containing different types of inclusions. In the proposed approach, only the unit cell of the periodic material has to be meshed, thanks to Bloch-Floquet relations. The modeling of these materials provides dispersion curves from which results of physical interest can be easily extracted: identification of propagation modes, cutoff frequencies, passbands, stopbands, as well as effective homogeneous properties. In this paper, the general method is first described, and particularly the aspects related to the periodicity. Then a test example is given for which analytical results exist. This example is followed by detailed presentations of finite element results, in the case of periodic materials containing inclusions or cylindrical pores. The homogenized properties of porous materials are determined with the help of an anisotropic model, in the large wavelength limit. A validation has been carried out with periodically perforated plates, the resonance frequencies of which have been measured. The efficiency and the versatility of the method is thus clearly demonstrated.

Microscale inverse acoustic band gap structure in aluminum nitride

“…This approach had been demonstrated and previously validated by Gorishnyy et al 8 The acoustic wave traveling inside the periodic media is described by the elastic wave equation and the Bloch-Floquet theorem. 9 In the FEM structural analysis, the eigenfrequencies of the ABG are found by solving the elastic wave equation with periodic boundary conditions applied on the unit cell. Furthermore, the frequency band gap is revealed by sweeping the eigenfrequency solver over the symmetric directions of the first Brillouin zone ͑⌫-X-M-⌫͒ of the reciprocal lattice.…”

Ultra-high frequency, high Q/volume micromechanical resonators in a planar AlN phononic crystal

“…Then, the eigenfrequencies of the structure were found by solving the elastic wave equation that is described by the Bloch-Floquet Theorem. 23 In the simulation, all possible modes were considered, i.e., inplane, out-of-plane transverse, and longitudinal modes. Furthermore, the frequency bandgap is determined by sweeping the eigenfrequency solver over the symmetric directions of the irreducible Brillouin zone (C-X-M-C) for the reciprocal lattice to generate the band diagram shown in Fig.…”

Propagation number in periodic structures considering losses

“…Analytical solutions for longitudinal waves and transverse waves are developed in Langlet's work [6]. Dispersion relation is cos(ak) = cos( Here the wave number gets an imaginary part, contrary to the modal analysis by FEM that only gives the real part [7]. It helps the dispersion curve reading by following the modes branches.…”

“…Here the finite element method (FEM) provided by the ATILA software [4,5] is used. Thanks to this tool, the modal analysis of a meshed structure gives real solutions for the dispersion relation [6,7]. This computation considers materials without losses.…”