Analytical study and numerical experiments for true and spurious eigensolutions of a circular cavity using the real-part dual BEM

摘要

It has been found recently that the multiple reciprocity method (MRM) (Chen and Wong. Engng Anal. Boundary Elements 1997; 20 (1): 25-33; Chen. Processing of the Fourth World Congress on Computational Mechanics, Onate E, Idelsohn SR (eds). Argentina, 1998; 106; Chen and Wong. J. Sound Vibration 1998; 217 (1): 75-95.) or real-part BEM (Liou, Chen and Chen. J.Chinese Inst. Civil Hydraulics 1999; 11 (2): 299-310 (in Chinese)). results in spurious eigenvalues for eigenproblems if only the singular (UT) or hypersingular (LM) integral equation is used. In this paper, a circular cavity is considered as a demonstrative example for an analytical study. Based on the framework of the real-part deal BEM, the true and spurious eigenvalues can be separated by using singular value decomposition (SVD). To understand why spurious eigenvalues occur, analytical derivation by discretizing the circular boundary into a finite degree-of-freedom system is employed, resulting in circulants for influence matrices. Based on the properties of the circulant, we find that the singular integral equation of the real-part BEM for a circular domain results is spurious eigenvalues which are the zeros of the Bessel function of the second kind, Y_n (kp), while the hypersingular integral equation of the real-part BEM results in spurious eigenvalues which are the zeros of the derivative of the Bessel functions of the second kind, Y_n' (kp). It is found that spurious eigenvalues exist in the real-part BEM, and that they depend on the integral representation one uses (singular or hypersingular, single layer of double layer) no matter what the given types of boundary conditions for the interior problem are. Furthermore, spurious modes are proved to be trivial in the circular cavity through analytical derivations. Numerically, they appear to have the same nodal lines of the true modes after normalization with respect to a very small nonzero value. Two examples with a circular domain, including the Neumann and Dirichlet problems, are presented. The numerical results for true and spurious eigensolutions match very well with the theoretical prediction.