On the spurious eigenvalues for a concentric sphere in BIEM

Chen, Jeng‐Tzong; Kao, Shing‐Kai; Lee, Ying‐Te; Lin, Yi-Jhou

doi:10.1016/j.apacoust.2009.08.001

Cited by 8 publications

(4 citation statements)

References 19 publications

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“…These techniques have been successfully employed for rod (Chen et al, 2009a), membrane (Chen et al, 2003(Chen et al, , 2004b(Chen et al, , 2005(Chen et al, , 2007, plate (Lee and Chen, 2008a,b) and concentric sphere cavity (Chen et al, 2010a). Now, we will utilize the SVD updating technique to deal with the eigenproblems containing elliptical boundaries.…”

Section: Svd Updating Techniquementioning

confidence: 97%

“…This is the reason why spurious eigenvalues may appear in multiply-connected eigenproblems. In order to suppress the appearance of the spurious eigenvalues, three methods of extracting out true and spurious eigenvalues, including the CHIEF method (Chen et al, 2003), the SVD updating technique (Chen et al, 2003(Chen et al, , 2004b(Chen et al, , 2005(Chen et al, , 2007(Chen et al, , 2008b(Chen et al, , 2009a(Chen et al, , 2010a, and the Burton & Miller approach (Chen et al, 2001(Chen et al, , 2007) are adopted, respectively.…”

Section: Treatment Of the Spurious Eigenvalues For Multiplyconnected mentioning

confidence: 99%

“…Not only true eigenvalues but also spurious eigenvalues can be extracted out by using the SVD updating terms and SVD updating documents, respectively. This technique has been successfully employed for rod (Chen et al, 2009a), circular membrane (Chen et al, 2003(Chen et al, , 2005(Chen et al, , 2007, plate (Lee and Chen, 2008b) and concentric sphere cavity (Chen et al, 2010a). Chen et al (2001Chen et al ( , 2003 studied spurious eigenvalues of circular membranes in both the continuous and discrete systems by using the separable kernel and circulant, respectively.…”

Section: Introductionmentioning

confidence: 99%

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Analytical investigation for true and spurious eigensolutions of multiply-connected membranes containing elliptical boundaries using the dual BIEM

Chen

Lee

Leu

2011

International Journal of Solids and Structures

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a b s t r a c tIt is well known that the boundary element method may induce spurious eigenvalues while solving eigenvalue problems. The finding that spurious eigenvalues depend on the geometry of inner boundary and the approach utilized has been revealed analytically and numerically in the literature. However, all the related efforts were focused on eigenproblems involving circular boundaries. On the other hand, the extension to elliptical boundaries seems not straightforward and lacks of attention. Accordingly, this paper performs an analytical investigation of spurious eigenvalues for a confocal elliptical membrane using boundary integral equation methods (BIEM) in conjunction with separable kernels and eigenfunction expansion. To analytically study this eigenproblem, the elliptic coordinates and Mathieu functions are adopted. The fundamental solution is expanded into the separable kernel by using the elliptic coordinates and the boundary densities are expanded by using the eigenfunction expansion. The Jacobian terms may exist in the separable kernel, boundary density and boundary contour integration and they can cancel each other out. Therefore, the orthogonal relations are reserved in the boundary contour integration. In this way, a similar finding about the mechanism of spurious eigenvalues is found and agrees with those corresponding to the annular case. To verify this finding, the boundary element method and the commercial finite-element code ABAQUS are also utilized to provide eigensolutions, respectively, for comparisons. Good agreement is observed from comparisons. Based on the adaptive observer system, the present approach can deal with eigenproblems containing circular and elliptical boundaries at the same time in a semi-analytical manner. By using the BIEM, it is found that spurious eigenvalues are the zeros of the modified Mathieu functions which depend on the inner elliptical boundary and the integral formulation. Finally, several methods including the CHIEF method, the SVD updating technique and the Burton & Miller method are employed to filter out the spurious eigenvalues, respectively. In addition, the efficiency of the CHIEF method is better than those of the SVD updating technique and the Burton & Miller approach, since not only hypersingularity is avoided but also computation effort is saved.

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Section: Svd Updating Techniquementioning

confidence: 97%

Section: Treatment Of the Spurious Eigenvalues For Multiplyconnected mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Analytical investigation for true and spurious eigensolutions of multiply-connected membranes containing elliptical boundaries using the dual BIEM

Chen

Lee

Leu

2011

International Journal of Solids and Structures

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“…Several schemes, such as the real-part BEM [1], imaginary-part BEM [2], and complex-value BEM [3] have been proposed for this purpose, while the spurious eigenvalues may appear in such analysis [4][5][6]. To deal with this drawback, the dual multiple reciprocity method [7] and the real-part dual BEM [8] are suggested to extract out the spurious eigenvalues.…”

Section: Introductionmentioning

confidence: 97%

Singular boundary method for acoustic eigenanalysis

Chen

Pang

2016

Computers & Mathematics with Applications

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a b s t r a c tThis paper applies the singular boundary method (SBM) to two-(2D) and threedimensional (3D) acoustics eigenproblems in simply-and multiply-connected domains. The SBM is a strong-form boundary discretization numerical method and is meshless, integration-free, and easy-to-implement. By introducing the concept of the source intensity factors, the singularity of fundamental solutions can be isolated to avoid the singular numerical integrals in the boundary element method (BEM). Similar to the BEM, the spurious eigenvalues may arise in the SBM computation. In order to extract out spurious eigenvalues from SBM results, the singular value decomposition updating techniques and the Burton-Miller method are implemented. Several 2D and 3D benchmark examples subjected to Dirichlet and Neumann boundaries are tested to examine the accuracy and stability of the present SBM strategy.

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