On the spurious eigenvalues for a concentric sphere in BIEM

Abstract

Researchers have paid attention on spurious eigenvalues for multiply-connected domain (2D) eigenproblems by using BEM/BIEM. This paper employs the null-field integral equation method to study the occurring mechanism of spurious eigenvalues for 3D problems with an inner hole. By expanding the fundamental solution into degenerate kernels and expressing the boundary density in terms of spherical harmonics, all boundary integrals can be analytically determined. It is noted that our null-field integral formulation can locate the collocation point on the real boundary thanks to the degenerate kernel. In addition, the spurious eigenvalues are parasitized in the formulations, e.g. singular and hypersingular formulations in the dual BIEM while true eigensolutions are dependent on the boundary condition such as the Dirichlet or Neumann problem. By using the updating terms and updating document of singular value decomposition (SVD) technique, true and spurious eigenvalues can be extracted out, respectively. Besides, true and spurious boundary eigenvectors are obtained in the right and left unitary vectors in the SVD structure of the influence matrices. This finding agrees with that of 2D cases.