Reconstructed Gaussian basis to characterize flexural wave collimation in plates with periodic arrays of annular acoustic black holes

While embedding acoustic black holes (ABHs) in plates has proved to be a lightweight and useful solution for noise and vibration reduction, their potential for wave manipulation is only in its beginnings. Recent studies seem to indicate that amazing properties can be attained with them. A plate with ABH indentations can be viewed as a single-phase metamaterial that can alter wave propagation direction thanks to the ABH power-law thickness profile. Some typical unconventional dispersion properties of multi-phase metamaterials, like collimation or negative refraction, can be recovered in a simple way. This work focuses on the collimation of flexural waves in plates by means of ABH arrangements. The first goal of the paper is to propose a predictive semi-analytical method to describe such phenomenon in an efficient manner. The method is also expected to allow one to perform quick parametric analyses for different ABH configurations. To that purpose, the Gaussian expansion method (GEM) is extended to deal with ABH phononic crystals on infinite plates. The Lagrangian for a single cell is built and a new basis of Gaussian functions is constructed satisfying the periodic boundary conditions of the problem. The Rayleigh-Ritz method is then adopted to compute the displacement field of annular and circular ABH arrays. The accuracy of the reconstructed GEM approach is validated by comparing dispersion curves and modal shapes with those obtained from finite element simulations. Equi-frequency contours are then used to determine the frequency bands prone to collimation. The focus is placed on annular ABH configurations which allow one to achieve collimation at lower frequencies than the circular ones. The second main contribution of this work therefore consists of a thorough analysis to characterize the influence of geometric parameters on the performance of ABH arrays. Moreover, new designs of annular ABH arrangements on plates are proposed for wave conduction, including curvature, and energy focusing in the long wavelength limit.