Boundary element analysis for the Helmholtz eigenvalue problems with a multiply connected domain

For a Helmholtz eigenvalue problem with a multiply connected domain, the boundary integral equation approach as well as the boundary-element method is shown to yield spurious eigenvalues even if the complex-valued kernel is used. In such a case, it is found that spurious eigenvalues depend on the ge...

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Bibliographic Details
Published in:Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences Mathematical, physical, and engineering sciences, 2001-10, Vol.457 (2014), p.2521-2546
Main Authors: Chen, J. T., Lin, J. H., Kuo, S. R., Chyuan, S. W.
Format: Article
Language:English
Subjects:
Boundary conditions
Boundary element method
Boundary Elements
Burton-Miller Method
Circles
Circulant
Degenerate Kernel
Differential equations
Dirichlet problem
Eigenvalues
Matrices
Modal response
Multiply Connected Problem
Natural frequencies
Spurious Eigenvalue
Vibration
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Summary:For a Helmholtz eigenvalue problem with a multiply connected domain, the boundary integral equation approach as well as the boundary-element method is shown to yield spurious eigenvalues even if the complex-valued kernel is used. In such a case, it is found that spurious eigenvalues depend on the geometry of the inner boundary. Demonstrated as an analytical case, the spurious eigenvalue for a multiply connected problem with its inner boundary as a circle is studied analytically. By using the degenerate kernels and circulants, an annular case can be studied analytically in a discrete system and can be treated as a special case. The proof for the general boundary instead of the circular boundary is also derived. The Burton-Miller method is employed to eliminate spurious eigenvalues in the multiply connected case. Moreover, a modified method considering only the real-part formulation is provided. Five examples are shown to demonstrate that the spurious eigenvalues depend on the shape of the inner boundary. Good agreement between analytical prediction and numerical results are found.
ISSN:1364-5021
1471-2946
DOI:10.1098/rspa.2001.0806