Boundary element analysis for the Helmholtz eigenvalue problems with a multiply connected domain

For a Helmholtz eigenvalue problem with a multiply connected domain, the boundary integral equation approach as well as the boundary-element method is shown to yield spurious eigenvalues. even if the complex-valued kernel is used. In such a case, it is found that spurious eigenvalues depend on the geometry of the inner boundary. Demonstrated as an analytical case, the spurious eigenvalue for a multiply connected problem with its inner boundary as a circle is studied analytically. By using the degenerate kernels and circulants, an annular case can be studied analytically in a discrete system and can be treated as a special case. The proof for the general boundary instead of the circular boundary is also derived. The Burton-Miner method is employed to eliminate spurious eigenvalues in the multiply connected case. Moreover, a modified method considering only the real-part formulation is provided. Five examples are shown to demonstrate that the spurious eigenvalues depend on the shape of the inner boundary. Good agreement between analytical prediction and numerical results are found.