# 臺灣博碩士論文加值系統

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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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 1 採用部份浮體基礎隔震結構系統之動力反應分析 2 邊界元素法於板自由振動之數學分析與數值研究 3 利用邊界元素法求解赫姆霍茲方程式中波導管及散射問題之應用 4 邊界元素法之三維異向材料內部應變/應力分析 5 以非奇異性暇積分改善三維邊界積分法 6 邊界元素法於混合型邊界條問題之假根及虛擬頻率探討 7 以非奇異邊界積分方程求解電磁場問題 8 邊界元素法中退化問題之探討 9 退化尺度與多連通聲場問題分析 10 期後誤差估計之通用架構

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 1 邊界元素法中退化問題之探討 2 邊界元素法求解赫姆茲方程秩降問題的處理及其應用 3 含圓形邊界史托克斯流與板問題之半解析法 4 零場方程求解拉普拉斯含圓形邊界問題 5 邊界元素法於混合型邊界條問題之假根及虛擬頻率探討 6 半平面與多連通拉普拉斯問題之研究 7 二維彈力無因次基本解之退化尺度研究 8 退化尺度與虛擬頻率矩陣秩降問題的探討 9 二維彈力問題之充分且必要邊界積分方程法研究 10 對偶邊界元素法與保角映射於二維拉普拉斯問題的退化尺度之研究 11 零場邊界積分方程法求解含圓或橢圓邊界山谷或山峰之SH波散射問題 12 含圓形邊界拉普拉斯與雙諧和問題之格林函數解析解 13 零場積分方程求解含圓形邊界拉普拉斯與赫姆茲方程之格林函數 14 非零初始狀態下之訊號追蹤設計 15 零場積分方程求解含圓形邊界之應力集中問題

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