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

For a Helmholtz eigenvalue problem with a multiply connected domain, the bo undary 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 g eometry of the inner boundary. Demonstrated as an analytical case, the spur ious eigenvalue for a multiply connected problem with its inner boundary as a circle is studied analytically. By using the degenerate kernels and circ ulants, an annular case can be studied analytically in a discrete system an d can be treated as a special case. The proof for the general boundary inst ead of the circular boundary is also derived. The Burton-Miner method is em ployed to eliminate spurious eigenvalues in the multiply connected case. Mo reover, a modified method considering only the real-part formulation is pro vided. Five examples are shown to demonstrate that the spurious eigenvalues depend on the shape of the inner boundary. Good agreement between analytic al prediction and numerical results are found.