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研究生:陳義麟
研究生(外文):I-Lin Chen
論文名稱:邊界元素法求解赫姆茲方程秩降問題的處理及其應用
論文名稱(外文):Treatment of rank deficiency problems and its applications for the Helmholtz equation using boundary element method
指導教授:陳正宗教授,梁明德
指導教授(外文):J. T. ChenM. T. Liang
學位類別:博士
校院名稱:國立海洋大學
系所名稱:河海工程學系
學門:工程學門
學類:河海工程學類
論文種類:學術論文
論文出版年:2002
畢業學年度:90
語文別:英文
論文頁數:168
中文關鍵詞:rank deficiencyfictitious frequencyspurious eigenvaluesingular value decompositioncirculantsingular integral equationhypersingular integral equationFredholm alternative theorem
外文關鍵詞:秩降虛擬頻率假特徵值奇異值分解法循環矩陣奇異積分方程超奇異積分方程Fredholm 二則一定理
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本文針對邊界元素法分析聲場問題時所產生的退化問題進行深入的探討,並對此一退化問題提出一套統一的模式。當解內域特徵值問題時,採用實部或虛部核函數之邊界元素法會產生假特徵值。求解外域輻射或散射問題時在某些特殊頻率下會產生數值不穩定的現象,此頻率稱為虛擬頻率或不規則頻率。就數學本質而言,兩者均源於邊界元素法中之影響矩陣秩數不足。藉由採用退化核而取代閉合型的基本解,可對內域特徵值問題及外域虛擬頻率問題一併討論。本文利用圓形問題在離散系統予以解析推導。而這個推導不論對內外域聲場問題採用直接或間接邊界元素法所產生的退化問題皆可清楚的解釋,且對退化問題產生的機制可加以驗證。我們採用四個數學分析工具:退化核函數、循環矩陣、Fredholm二擇一定理及奇異值分解法(SVD)補充式的技巧,來協助我們瞭解退化問題發生的機制並提出解決方法。並利用這些方法來探討退化的機制及濾除假根,萃取造成數值不穩定的虛擬模態。同時經由解析的結果可知,真特徵模態及假特徵模態會分別藏身於左酉及右酉矩陣中,並分別對應於奇異值為零的奇異向量。對於外域聲場輻射及散射問題所產生的數值不穩定現象,我們則在連續系統及離散系統中同時提出了模態參與係數的觀念,同時發現虛擬模態亦藏在左酉向量中。我們並分別以CHIEF法及CHEEF法來克服虛擬頻率及假特徵值的問題。對於任意邊界的問題,我們則提出如何選用有效輔助點(CHIEF點或CHEEF點)的判別準則。本文同時對於奇異值分解法在外域輻射及散射問題上所具有的物理意義予以釐清。一個相對於邊界速度強度及場的聲壓有關的格林矩陣,在經由SVD分解後可得一組與輻射效率有關的奇異值,及兩組正交且分別與場點聲壓的模態及源點強度的模態有關的左酉及右酉向量。並將此二向量及奇異值與以映射法得到格林函數矩陣的基底函數,兩者之間予以聯繫。另外對於退化邊界及角點所造成秩降不足的問題,可瞭解到對偶邊界元素法為解此一類型問題有效的工具。最後,數值算例結果與解析的結果均相當吻合。
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.
[1-1] Motivation
[1-2] The structure and contribution of this thesis
[Chapter 2] A unified boundary integral formulation for
the Helmholtz equation
[2-1] Introduction
[2-2]A unified integral formulation for the Helmholtz
interior and exterior problems
[2-3] Analytical study for spurious and fictitious solutions
in BEM using degenerate kernels and circulants
[2-4] Fredholm alternative theorem\dotfill 13
[2-5] Fictitious values for exterior problem using BEM
[2-6] Spurious eigensolutions for interior problems using BEM
[2-7] Conclusions
[Chapter 3] Analytical study and numerical experiments for
radiation and scattering problems using BEM
[3-1] Introduction
[3-2] Statement for exterior boundary-value problems
of the Helmholtz equation
[3-3] The singular value decomposition technique in the CHIEF method
[3-4] Analytical study of the failure points
in the CHIEF method
[3-5] Modal participation factor for numerical instability
[3-6] Numerical examples
[3-7] Conclusions
[Chapter 4] Numerical experiments for radiation mode of a
circular cylinder using singular value decomposition
[4-1] Introduction
[4-2] The image method of acoustic field
[4-3] The singular value decomposition for the Green''s matrix
[4-4] The singular value expansion of the Green''s function matrix
[4-5] The singular value expansion and the singular value decomposition
[4-6] Numerical examples
[4-7] Conclusions
[Chapter 5] A new method for true and spurious
eigensolutions of arbitrary cavities using the CHEEF method
[5-1] Introduction
[5-2] Review of the real-part dual BEM for a two-dimensional
interior acoustic problem
[5-3] The CHEEF method for an interior two-dimensional acoustic problem in conjunction with SVD technique
[5-4] Analytical study of the failure points in the CHEEF method
[5-5] Numerical examples
[5-6] Conclusions
[Chapter 6] Applications of dual boundary integral formulation to degenerate boundary and corner problems
[6-1] Introduction
[6-2] Dual integral formulation for a membrane problem with degenerate boundaries of stringers
[6-3] Dual boundary element formulation using the constant
element scheme
[6-4] Eigenequation for the membrane with stringers
[6-5] Dynamic stress intensity factor for the membrane
with stringers
[6-6] Dual integral formulation for the Helmholtz equation with a corner
[6-7] Numerical examples
[6-8] Conclusions
[Chapter 7] Conclusions and further research
[7-1] Conclusions
[7-2] Further research
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