Interior three-dimensional acoustic modeling and modal analysis using wavelet-based finite-element approach

Abstract

This paper introduces two-dimensional (2D) and 3D acoustic modeling and modal analysis using the wavelet finite-element method (WFEM). Governed by the Helmholtz equation, the acoustic domain is parameterized and analyzed using the scaling functions of B-spline wavelets, which facilitates the construction of elements with varying numbers of nodes via multi-resolution analysis. The wavelet-based shape functions provide a semi-orthogonal basis that enables rapid searching for approximate solutions in Lebesgue spaces, thereby offering significantly reduced interpolation errors and computational burden. Numerical examples are considered using WFEM, comprising a 2D acoustic problem involving a tube for predicting acoustic pressure and eigenfrequency investigations, and 3D acoustic problems involving a cubic room and an L-shaped room for capturing acoustic characteristics. The results are compared with those of (i) standard FEM with the same mesh and (ii) analytical solutions. Importantly, WFEM demonstrates stability by being insensitive to internal mesh size variations, indicating that B-spline wavelet elements have minimal effects on the numerical results. Furthermore, B-spline wavelet elements effectively control the pollution (dispersion) error of numerical methods when imposing Neumann boundary conditions in the high-frequency range, and they reduce interpolation errors caused by polynomial interpolation in the low-frequency domain.