三维Helmholtz方程外边值问题的虚边界元法

The Virtual Boundary Element Method for Exterior Problems of the 3-D Helmholtz Equation

基于位势的延拓,推导出三维虚边界积分方程.通过选择不同的虚边界,避免相应内问题的特征值与波数重合,从而保证解的唯一性.数值算例验证了该方法求解任意波数三维Helmholtz方程外边值问题的有效性.

A virtual boundary method for solving the Dirichlet and Neumann exterior problems of the 3-D Helmholtz equation, which is valid for all wave number is presented in this paper. When wave number is an eigenvalue of the interior Dirichlet or Neumann problem for the Laplacian, the solution of the boundary integral equation correspond to eigenvalue is not unique. Based on the extension of potential function, Virtual boundary integral equation is deduced. The uniqueness problem is granted by choosing different virtual boundary, with which the wave number do not coincide with the eigenvalue of the original interior Dirichlet or Neumann problem for the Laplacian. The results of numerical examples demonstrate that the scheme presented is practical and effective for the exterior problems of the 3-D Helmholtz equation with any Key wave numbers.