Eigenanalysis for a confocal prolate spheroidal resonator using the null-field BIEM in conjunction with degenerate kernels

Abstract

In this paper, we employ the null-field boundary integral equation method (BIEM) in conjunction with degenerate kernels to solve eigenproblems of the prolate spheroidal resonators. To detect the spurious eigenvalues and the corresponding occurring mechanisms which are common issues while utilizing the boundary element method or the BIEM, we use angular prolate spheroidal wave functions and triangular functions to expand boundary densities. In this way, the boundary integral of a prolate spheroidal surface is exactly determined, and eigenequations are analytically derived. It is revealed that the spurious eigenvalues depend on the integral, representations and the shape of the inner boundary. Furthermore, it is interesting to find that some roots of the confocal prolate spheroidal resonator are double roots no matter that they are true or spurious eigenvalues. Illustrative examples include confocal prolate spheroidal resonators of various boundary conditions. To validate these findings and accuracy of the present approach, the commercial finite-element software ABAQUS is also applied to perform acoustic analyses. Good agreement is obtained between the acoustic results obtained by the null-field BIEM and those provided by the commercial finite-element software ABAQUS.