Partition of Unity Finite Element Method for the modelling of Acoustic Black Hole wedges

Abstract

The Acoustic Black Hole (ABH) phenomenon can be exploited to manipulate and mitigate flexural wave propagation in thin-walled structures. ABH structures feature unique space-dependent wavenumber variation and wave celerity reduction in the tapered ABH area, thus posing challenges to the existing modelling techniques. In this work, the Partition of Unity Finite Element Method (PUFEM) is revamped to simulate the structural response of an ABH wedge subject to a harmonic loading. This method allows the incorporation of auxiliary enrichment functions into the finite element framework in order to cope with the ABH-induced wave oscillating behaviour, exemplified by the varying wavenumber and amplitude in space. The PUFEM tapered Timoshenko beam elements are constructed by employing wave enrichment functions with the Wentzel-Kramers-Brillouin (WKB) approximation method. A wavelet enrichment is also investigated as hierarchic refinement. Using these enriched elements, the frequency responses of an ABH wedge and the convergence of numerical solutions are computed and compared with the classical linear FEM and the elements enriched with ‘local’ wave solutions. An adaptive meshing scheme is designed and implemented to further accelerate the solution convergence. It is shown that the PUFEM offers a good computational accuracy and drastic reduction of degrees of freedom for solving the broadband ABH problems, outperforming the classical FEM.