Rayleigh–Ritz Approximation of the Acoustic Vibrations of Clamped Superquadrics—Application to Free Core–Shell Objects

Abstract

A numerical approach based on the Rayleigh-Ritz method and using a modification of the so-called $xyz$ algorithm is introduced to calculate the acoustic vibrations of clamped objects whose shape is delimited by superquadrics. It is then used to improve the convergence for the free vibrations of core–shell objects. The issue in this case is first illustrated in the simpler one-dimensional case of the thickness breathing vibration of an infinite “core-shell” plate. Functions suitable for solving the clamped vibrations of the core are added to the original $xyz$ basis of functions to improve the convergence for core–shell superquadrics. The new basis obeys the same symmetry rules as the original one, which allows calculating vibrations for individual irreducible representations when the objects are made of cubic, tetragonal, or orthorhombic materials whose principal axes are aligned with those of the superquadrics. This method is validated for an isotropic spherical core–shell system for which analytic solutions exist.