Generalized finite difference method for three-dimensional eigenproblems of Helmholtz equation

Abstract

In this paper, a meshless numerical procedure, based on the generalized finite difference method (GFDM) is proposed to efficiently and accurately solve the three-dimensional eigenproblems of the Helmholtz equation. The eigenvalues and eigenvectors are very important to various engineering applications in three-dimensional acoustics, optics and electromagnetics, so it is essential to develop an efficient numerical model to analyze the three-dimensional eigenproblems in irregular domains. In the GFDM, the Taylor series and the moving-least squares method are used to derive the expressions at every node. By enforcing the satisfactions of governing equation at interior nodes and boundary conditions at boundary nodes, the resultant system of linear algebraic equations can be expressed as the eigenproblems of matrix and then the eigenvalues and eigenvectors can be efficiently acquired. In this paper, four numerical examples are provided to validate the accuracy and simplicity of the proposed numerical scheme. Furthermore, the numerical results are compared with analytical solutions and other numerical results to verify the merits of the proposed method.