True and spurious eigensolutions for the problems with the mixed-type boundary conditions using BEMs

Abstract

Boundary integral equations and boundary element methods were employed semi-analytically and numerically to study the occurrence of spurious eigenvalues for the interior Helmholtz equations with the mixed-type boundary conditions. The degenerate kernels of fundamental function and Fourier series were utilized in the null-field integral equation to derive the true and spurious eigenfunctions semi-analytically. The complex-valued boundary element method (BEM), the real-part BEM, the imaginary-part BEM and the multiple reciprocity method were utilized to solve the eigenproblem. The results are compared with those of FEM. A simple case of one-dimensional eigenproblem was also addressed. Moreover, the SVD updating techniques in conjunction with the Fredholm's alternative theorem were adopted to sort out the spurious eigenvalues and extract the true ones. It is emphasized that the occurrence of spurious eigenvalues depends on the adopted method (singular or hypersingular formulation) no matter what the given type of boundary conditions for the problem is. The illustrative examples were verified successfully and the numerical results matched well with those of FEM.