Null-field integral equation approach for eigenproblems with circular boundaries

Abstract

In this paper, the eigenproblems with circular boundaries of multiply-connected domain are studied by using the null-field integral equations in conjunction with degenerate kernels and Fourier series to avoid calculating the Cauchy and Hadamard principal values. An adaptive observer system of polar coordinate is considered to fully employ the property of degenerate kernels. For the hypersingular equation, vector decomposition for the radial and tangential gradients is carefully considered in the polar coordinate system. Direct-searching scheme is employed to detect the eigenvalues by using the singular value decomposition (SVD) technique. Both the singular and hypersingular equations result in spurious eigenvalues which are the associated interior Dirichlet and Neumann problems of interior domain of inner circles, respectively. It is analytically verified that the spurious eigenvalue depends on the radius of any inner circle and numerical experiments support this point. Several examples are demonstrated to see the validity of the present formulation. More number of degrees of freedom of BEM is required to obtain the same accuracy of the present approach.