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

It has been found recently that the multiple reciprocity method (MRM) (Chen and Wong. Engng. Anal. Boundary Elements 1997; 20(1):25-33; Chen. Processi ngs of the Fourth World Congress on Computational Mechanics, Onate E, Idels ohn 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. Ci vil Hydraulics 1999; 11(2): 299-310 (in Chinese)). results in spurious eige nvalues for eigenproblems if only the singular (UT) or hypersingular (LM) i ntegral 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 dual BEM, the true and spurious eigenvalues can be separated by using singular value decomposition (SVD). To understand why spurious ei genvalues occur, analytical derivation by discretizing the circular boundar y into a finite degree-of-freedom system is employed, resulting in circulan ts for influence matrices. Based on the properties of the circulants, we fi nd that the singular integral equation of the real-part BEM for a circular domain results in spurious eigenvalues which are the zeros of the Bessel fu nctions of the second kind, Y-n(k rho), while the hypersingular integral eq uation of the real-part BEM results in spurious eigenvalues which are the z eros of the derivative of the Bessel functions of the second kind, Y-n(t)(k rho). It is found that spurious eigenvalues exist in the real-part BEM, an d that they depend on the integral representation one uses (singular or hyp ersingular; single layer or double layer) no matter what the given types of boundary conditions for the interior problem are. Furthermore, spurious mo des are proved to be trivial in the circular cavity through analytical deri vations. Numerically, they appear to have the same nodal lines of the true modes after normalization with respect to a very small nonzero value. Two e xamples with a circular domain, including the Neumann and Dirichlet problem s, are presented. The numerical results for true and spurious eigensolution s match very well with the theoretical prediction. Copyright (C) 2000 John Wiley & Sons, Ltd.