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研究生:劉立偉
論文名稱:多連通赫姆茲問題邊界積分推導與邊界元素法分析
論文名稱(外文):Boundary integral formulation and boundary element analysis for multiply-connected Helmholtz problems
指導教授:陳正宗陳正宗引用關係
指導教授(外文):J. T. Chen
學位類別:碩士
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:81
中文關鍵詞:邊界積分方程邊界元素法假根奇異值分解法
外文關鍵詞:boundary integral equationboundary element methodspurious eigenvaluesingular value decompositionFredholm alternative theoremCHIEF methoddegenerate kernal
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本論文針對使用邊界積分方程與邊界元素法求解多連通 Helmohltz 方程所產生的假根問題作深入的研究。
文中對於產生假根的機制,分別使用連續系統與離散系統的觀點來探討。
在連續系統中,運用 Fourier 級數與退化核函數配合零場邊界積分方程可解析出假根的發生和
其出現的的位置(波數)與內部邊界的尺度有關
;在離散系統下,邊界元素法結合奇異值分解(SVD)法與 Fredholm 二擇一定理,也可以得到相同的結果。
為克服假根問題,本文提出三種解決方法,分別為 CHIEF 法、SVD 補充列與補充行、與 Burtion \& Miller 法。
文中對於使用 CHIEF 法時,如何選取有效的 CHIEF 點;與 SVD 補充列與補充行在直接法與間接法的適用性也作詳細的剖析。
最後本文藉由不同的數值算例,來驗證上述的理論推導,並且針對邊界元素法與各種數值方法之優缺點作比較。

Spurious eigensolution for boundary element methods (BEMs) and boundary integral equations (BIEs)
in the Helmholtz equation with a multiply-connected domain were investigated in this thesis thoroughly.
The dual BIEs and BEMs, the null-field BIEs and BEMs and the fictitious BIEs and BEMs were addressed.
The occurring mechanism of spurious eigenvalue with a annular domain was studied analytically in the continuous
system by using the degenerate kernels and the Fourier series expansion.
Then, the occurring mechanism with multiply-connected domain was demonstrated in the numerical experiments
by using the singular value decomposition (SVD) in conjunction with the Fredholm alternative theorem.
To overcome the problem of spurious eigensolution, CHIEF method and SVD updating techniques, were employed.
Besides, the criteria for selecting the optimum CHIEF points were discussed in depth and the SVD updating techniques for
direct and indirect BEMs were developed successfully.
In addition, the root of the multiplicity two can be distinguished by using BEM in conjunction with SVD as illustrated in
numerical examples.

Contents I
Table captions III
Figure captions IV
Notations VII
摘要 X
Abstract XI
Chapter 1 Introduction 1
1-1. Motivation of the research 1
1-2. Organization of the thesis 3
Chapter 2 Spurious and True Eigensolutions of Helmholtz Boundary intergal equations for multiply-connected problems 5
2-1. Boundary integral equations of spurious and true eigenvalues for a multiply connected problem 5
2-2. Derivation of true and spurious modes by using degenerate kernels and the Fourier series expansions 11
2-3. Null-field formulations of BIEs 12
2-4. Fictitious boundary formulations of BIEs 14
2-5. CHIEF treatment for spurious eigensolutions 17
2-6. Review of the Burton \& Miller method 19
2-7. Concluding remarks 19
Chapter 3 Eigenanalysis for multiply-connected problems using the Boundary element methods 21
3-1. Detection of spurious and true eigenvalues in BEM using SVD updating techniques and Fredholm's alternative theorem 21
3-1-1. Detection of spurious eigenvalues in direct BEMs by SVD updating documents 22
3-1-2. Detection of true eigenvalues in direct BEMs by SVD updating terms 24
3-1-3. Detection of true eigenvalues in indirect BEMs by SVD updating documents 25
3-1-4. Detection of spurious eigenvalues in indirect BEMs by SVD updating terms 27
3-2. CHIEF method for arbitrary domain 28
3-3. Numerical experiments for doubly-connected problems 29
3-3-1. Annular domain 29
3-3-2. Polygonal domain 32
3-3-3. Eccentric domain 32
3-4. Numerical experiments for multiply-connected problems 33
3-4-1. A circular domain with two unequal holes 33
3-4-2. A circular domain with two equal holes 35
3-4-3. A circular domain with four equal holes 36
3-4-4. A circular domain with seven equal holes 37
3-5. Concluding remarks 37
Chapter 4 Conclusions and further research} 39
4-1. Conclusions 39
4-2. Further research 40
References 41

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