臺灣博碩士論文加值系統

積分方程用於解外域聲場Helmholtz問題由來已久。如果單單使用奇異積分方程（UT式）或超強奇異積分方程（LM式）會產生虛擬頻率的非物理現象。CHIEF（Combined Helmholtz Interior integral Equation Formulation）法因易於使用而受到歡迎，可以克服外域虛擬頻率的問題；但是如果內域點配置不當則會導致失效，此為一般所謂節點的位置。CHIEF法結合SVD（Singular Value Decomposition）技巧可以更有效率的過濾虛擬頻率，唯SVD只能處理實數，在計算時必須先將複數係數矩陣轉化為實數係數矩陣以方便處理。在文章中先分析探討虛擬頻率發生的機制，而後以二維圓柱為數值範例，明確指出徑向、幅向節點線的所在位置，也就是失敗點所在的位置。

Integral equation has been used to solve exterior acoustic problems (radiation and scattering) for many years. It is well know that fictitious (irregular) frequencies stem from the numerical resonance instead of the physical resonance if integral representation for the solution is assumed. It is found that if the singular (UT) or the hypersingular (LM) equation is used alone results in fictitious frequencies for exterior problem. The CHIEF (Combined Helmholtz Integral Equation Formulation) method is a very popular technique that can overcome the nonunique problem but this method breaks down if the internal points are not properly chosen. The CHIEF method in conjunction with the SVD (Singular Value Decomposition) technique is easy and efficient to have a unique solution for the exterior problem. An analytical example for the fictitious frequencies of a circular cavity is studied. The optimum numbers and proper positions for the points in the interior domain are analytically studied and suggested in the numerical scheme. Also, numerical experiments are designed to verify the analytical results. Finally, we can get the exact positions of nodal lines; i.e. the additional failure interior points positions. Thus, if the additional points are properly chosen, there are no more than two points are needed.

中文摘要 Ⅰ
英文摘要 Ⅱ
目錄 Ⅲ
表目錄 Ⅳ
圖目錄 Ⅴ
第一章 緒論
1.1 研究動機 1
1.2 研究目的 1
1.3 研究方法 2
1.4 研究內容 2
第二章 虛擬頻率發生之原因與解決方法
2.1 虛擬頻率發生之原因 3
2.2 解決虛擬頻率的方法 3
2.3 對偶邊界積分方程之理論推導 4
2.4 CHIEF法之理論推導 10
第三章 CHIEF方法結合SVD技巧在二維外域
Helmholtz場聲學問題 13
第四章 CHIEF失敗點之分析研究
4.1 間接法奇異積分方程 16
4.2 超強奇異積分方程 22
第五章 數值範例 29
第六章 結論 31
參考文獻 52

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