Treatment of rank deficiency problems and its applications for the Helmholtz equation using boundary element method
| Autor: | I-Lin Chen, 陳義麟 |
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| Rok vydání: | 2002 |
| Druh dokumentu: | 學位論文 ; thesis |
| Popis: | 90 In this dissertation, we provide a perspective on the current status of the degenerate problems in the boundary element method for acoustics. A unified boundary integral formulation is proposed to study the spurious eigenvalue and the fictitious frequency for the interior and exterior acoustic problems, respectively. Mathematically speaking, both the two problems stem from the rank deficiency of the influence matrix. By using the degenerate kernels instead of the closed-form fundamental solution, we constructed the relationship between the interior and exterior problems. An analytical study in a discrete system for a circular cavity is conducted using the circulant. The occurring mechanism of the degenerate problem is examined. Four mathematical tools, the circulant properties, the degenerate kernels, the Fredholm alternative theorem and the singular value decomposition (SVD) updating technique, are employed to understand the mechanism of the degenerate problem. Based on the tools, we can filter out the spurious eigenvalue and extract out the fictitious modes. It is found that the true and spurious modes are imbedded in the right and left unitary vectors with respect to the zero singular values, respectively. Regarding to the numerical instability for radiation or scattering problems near the irregular frequency, we propose the concept of modal participation factor to explain the phenomenon. The modal participation factor for the fictitious mode which results in the numerical instability is derived for both the continuous system and discrete system. Also, the fictitious mode is imbedded in the left unitary vector. To promote the rank of the influence matrix, we propose the CHEEF method to overcome the spurious eigenvalue problems and adopt the CHIEF method to treat the fictitious frequency. For arbitrary cavities, a criterion for choosing the better CHIEF or CHEEF points is suggested. At the same time, the physical meaning of the SVD technique in analyzing sound radiation problem is examined. The Green''s matrix related the field of acoustic pressure to the strengths of sources on the surface of a body which radiates or scatters sound. The Green''s matrix can be decomposed by SVD result in a set of singular values and two sets of orthogonal singular vectors. The singular value relates to the radiation efficiencies and the two sets of orthogonal basis functions are found to describe the mode shapes, respectively. By using the image method and degenerate kernel, the Green''s function is obtained. The relationship between the components of the SVD with the basis function of the Green''s function matrix are connected. The dual boundary element method is one of the powerful tools to solve the rank-deficiency problem, such as the degenerate boundary and corner problems, which are also studied. Finally, numerical results of the illustrative examples are found to agree with the analytical predictions. |
| Databáze: | Networked Digital Library of Theses & Dissertations |
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