Analytical study and numerical experiments for true and spurious eigensolutions of a circular cavity using the real‐part dual BEM
Analytical study and numerical experiments for true and spurious eigensolutions of a circular cavity using the real‐part dual BEM
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使用实部对偶边界元法对圆形腔的真实和虚假本征解进行分析研究和数值实验
DOI：
10.1002/1097-0207(20000730)48:9
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发表时间：
2000
期刊：
影响因子：
--
通讯作者：
C. X. Huang
中科院分区：
文献类型：
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作者：
S. Kuo;Jeng;C. X. Huang
最近发现多重互易法(MRM)(Chen和Wong)。英格。肛门。边界元1997；20(1)：25-33；陈。第四届世界计算力学大会的处理，Onate E，Idelsohn SR(编辑)。阿根廷，1998；106；陈和王。J.声音振动1998；217(1)：75-95。或实部BEM(Liou，Chen和Chen)。J.中文学院土木工程学报1999；11(2)：299-310)。如果仅使用奇异(UT)或超奇异(LM)积分方程，则会导致本征问题的伪本征值。本文以圆形空腔为例进行了分析研究。基于实部对偶边界元的框架，利用奇异值分解(SVD)分离真伪特征值。为了了解伪特征值产生的原因，采用了将圆形边界离散到有限自由度系统中的解析推导方法，从而产生了影响矩阵的循环。基于循环子的性质，我们发现圆域上实部边界元的奇异积分方程解的伪特征值是第二类贝塞尔函数Y(kρ)的零点，而实部边界元的超奇异积分方程解的伪特征值是第二类贝塞尔函数Yn‘(kρ)导数的零点。结果表明，在实部边界元中存在伪特征值，且伪本征值依赖于所使用的积分表示(奇异或超奇异；单层或双层)，而不管内部问题的边界条件是什么类型。此外，通过解析推导，证明了杂散模在圆腔中是微不足道的。在数值上，对于一个非常小的非零值，它们在归一化后看起来具有相同的真模式的节线。给出了两个圆形区域的算例，包括Neumann问题和Dirichlet问题。真本征解和伪本征解的数值结果与理论预测符合得很好。版权所有©2000 John Wiley&Sons，Ltd.
It has been found recently that the multiple reciprocity method (MRM) (Chen and Wong. Engng. Anal. Boundary Elements 1997; 20(1):25–33; Chen. Processings 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 dual 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 circulants, we find 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 functions of the second kind, Y (kρ), 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, Yn′(kρ). 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 or 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. Copyright © 2000 John Wiley & Sons, Ltd.