Spurious and true eigensolutions of Helmholtz BIEs and BEMs for a multiply connected problem

Abstract

The spurious eigenvalues of an annular domain have been verified for the singular and hypersingular boundary‐element methods (BEMs) and circumvented by using the Burton‐Miller approach. Do they also occur in other formulations: continuous formulations such as the singular and hypersingular boundary integral equations (BIEs), the null‐field BIEs and the fictitious BIEs, or such discrete formulations as the null‐field BEMs and the fictitious BEMs? For the ten formulations of the multiply connected problem the study of otherwise the same issues is continued in the present paper. By using the degenerate kernels and the Fourier series, it is demonstrated analytically for the six continuous formulations of BIEs that spurious eigensolutions depend on the geometry of the inner boundary but not on that of the outer boundary. This conclusion can be extended to the six discrete formulations of BEMs. To filter out the spurious eigenvalues, the CHIEF (combined Helmholtz integral equation formulation) method is used here instead of the Burton‐Miller approach. The optimum number and appropriate positions of the CHIEF points are also addressed. It is then shown that, in the null‐field and fictitious BEMs, the spurious and true eigenvalues can be detected and distinguished by using the singular‐value‐decomposition‐updating techniques in conjunction with the Fredholm alternative theorem. Illustrative examples show the validity of the proposed methodologies.