Narrow Results

In this paper, we proposed a unified formulation to explain the reason why spurious eigensolution occurs in the eigenproblem of interior acoustics using the real-part and imaginary-part BEMs and why fictitious frequency occurs in exterior acoustics using the complex-valued BEM. Both the two problems stem from the rank deficiency of the influence matrix. Based on the circulant properties and degenerate kernels, an analytical study in a discrete system for a circular cavity is conducted. The Fredholm alternative theorem is employed to study the rank-deficiency problems in conjunction with singular value decomposition updating technique. The spurious and fictitious boundary modes are found to locate in the column vectors of left unitary matrix. Also, the effects of different types of boundary condition on the spurious and fictitious solutions using direct and indirect methods are discussed. The mathematical and physical meanings for the nontrivial boundary solution in spurious eigensolution and fictitious frequency are explained. Numerical experiments are found to agree with the analytical predictions.

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.

A systematic approach of using the null-field integral equation in conjunction with the degenerate kernel and eigenfunction expansion is employed to solve three-dimensional (3D) Green’s functions of Laplace equation. The purpose of using degenerate kernels for interior and exterior expansions is to avoid calculating the principal values. The adaptive observer system is addressed to employ the property of degenerate kernels in the spherical coordinates and in the prolate spheroidal coordinates. After introducing the collocation points on each boundary and matching boundary conditions, a linear algebraic system is obtained without boundary discretization. Unknown coefficients can be easily determined. Finally, several examples are given to demonstrate the validity of the present approach.