An accurate triangular spectral element method-based numerical simulation for acoustic problems in complex geometries

Abstract

An accurate triangular spectral element method (TSEM) is developed to simulate acoustic problems in complex computational domains. With Fekete points and Koornwinder-Dubiner polynomials introduced, triangular elements are used in the present method to substitute quadrilateral elements in traditional spectral element method (SEM). The efficiency of discretizing complex geometry is enhanced while high accuracy of SEM is remained. The weak form of the second-order governing equations derived from the linearized Euler equations (LEEs) are solved, and perfectly matched layer (PML) boundary condition is implemented. Three benchmark problems with analytical solutions are employed to testify the exponential convergence rate, convenient implementation of solid wall boundary condition and capable discretization in complex geometries of the present method respectively. An application on Helmholtz resonator (HR) is presented as well to demonstrate the possibility of using the present method in practical engineering. The numerical resonance frequency of HR reaches an excellent agreement with the theoretical result.

Learning Transient Partial Differential Equations with Local Neural Operators

“…For example, the finite element method (FEM) divides the computational domain into small elements and handles them in the same way [26,27]. Similar ideas are also adopted in the development from spectral method to the spectral element method (SEM) [28][29][30]. It provides FEM/SEM great flexibility to be independent of the computational domain and the boundary conditions, which enables a wide application of FEM.…”