Applications of the generalized singular-value decomposition method on the eigenproblem using the incomplete boundary element formulation

Abstract

In this paper, four incomplete boundary element formulations, including the real-part singular boundary element, the real-part hypersingular boundary element, the imaginary-part boundary element and the plane-wave element methods, are used to solve the free vibration problem. Among these incomplete boundary element formulations, the real-part singular and the hypersingular boundary elements are of the singular type and the other two are of the regular type. When the incomplete formulation is used, the spurious eigensolution may be encountered. An auxiliary system, whose boundary conditions are linearly independent of those of the original system, is required in the proposed method in order to eliminate the spurious eigensolution. A mathematical proof is given to show that the spurious eigensolution will appear in both the original and auxiliary systems. As a result, one can eliminate spurious eigensolution by means of the generalized singular-value decomposition method. In addition to the spurious eigensolution problem, the regular boundary element formulation further suffers ill-conditioned behaviors in the problem solver. It is explained analytically using a circular domain why ill-conditioned behaviors exist. Two main ill-conditioned problems are the numerical instability of solution and the nonexistence of solution. To solve the numerical instability of solution problem existing in these proposed regular formulations, the Tikhonov's regularization technique is used to improve the condition number of the leading coefficient matrix; further, the generalized singular-value decomposition method is used to eliminate spurious eigensolutions. Numerical examples are given to verify the performance of these proposed methods, and the results match the analytical values well.