關於使用基本解法應用在外域聲場問題而引致的虛擬頻率問題，其搭配使用額外源點法可以有效地克服虛擬頻率造成的無解情況，進而求出準確解。從添加額外源點的觀點來看，某些特性與結合Helmholtz內域積分方程公式(combined Helmholtz interior integral equation formulation,CHIEF方法)非常相似，因此可以補近十多年來間接法中沒有CHIEF法的空白。然而此法目前只應用在二維情況下並且存在失效點的風險，因此，本論文有兩個延伸方向，其一是將增加額外源點法推廣至三維外域聲場問題，其二是使用額外雙層勢能與額外混合勢能來取代額外源點的單層勢能。我們將採用退化核函數來推導出虛擬頻率發生的機制，同時也將一併探討其額外源點為單層勢能時可能的失效點位置。為了驗證本想法的有效性，我們考慮了含球體輻射體的外域聲場問題。在球體輻射體的問題下我們考慮四個不同的案例，分別是單根與三重根的情況，並在各自分為無解與無限多解的情況，此外對於額外單層勢能的徑向失效點位，改換使用額外雙層勢能，可有效地解決。最後則再考慮扁長橢球體的情況，來驗證本法的正確性。

Regarding the problem of the fictitious frequency caused by the exterior acoustic problems by using the method of fundamental solutions (MFS), using the extra source points approach can effectively overcome the problem of non-unique solution caused by the fictitious frequency. In this way, the accurate solution can be determined. From the viewpoint of adding extra source points, some properties are very similar to the combined Helmholtz interior integral equation formulation (CHIEF method). Therefore it can fill in the blank area that there is no CHIEF method in the indirect method on the recent decade. However, this method is only used to solve two-dimensional problems and there is a risk of failure points in the current state. Therefore, there are two extensions in this thesis. One is to extend the method of adding extra source points to the three-dimensional exterior acoustic problems. The second is using the extra double-layer potential and the mixed potential to replace the single-layer potential of the extra source. We use the degenerate kernel to derive the occurring mechanism of fictitious frequency. We also derive the locations of possible failure source points when the extra single-layer potential is used to demonstrate the validity of the present way. We consider an exterior acoustic problem with a spherical radiator. In the problem of spherical radiator, we consider four different cases which are the cases of single root and triple root with the cases of no solution and infinite solutions. In addition, for the failure point on the radial nodal line of the extra single-layer potential, using the extra double-layer potential can be effectively solved. Finally, we also consider the case of prolate spheroid radiator to verify the correctness of present method.

目錄
目錄	IV
表目錄	VI
圖目錄	VII
第一章 介紹	1
1-1研究背景與動機	1
1-2論文架構	4
第二章 問題與方法	7
2-1問題描述	7
2-2間接邊界元法	8
2-3基本解法	9
2-4額外源點法	10
2-4-1間接邊界元法	10
2-4-2基本解法	10
2-5虛擬頻率發生的機制	13
2-5-1球型案例	13
2-5-2扁長型橢球案例	16
2-6利用退化核來解析推導可能的失效點位置	18
第三章 數值例題	25
3-1說明例題	25
3-1-1球型案例	25
3-1-2扁長型橢球案例	27
第四章 結論與未來展望	56
4-1結論	56
4-2未來展望	56
參考文獻	58

表目錄
表1-1 比較解決虛擬頻率問題的方法[30]	5
表3-1 球型案例的四種情況	29
表3-2 扁長型橢球案例的四種情況	30
表3-3在0＜k＜10的範圍內，可能的虛擬頻率(球型)	31
表3-4 使用基本解法得到的Fredholm二擇一定理結果	31
表3-5 j1(ka)的前兩個零根	31
表3-6 球型諧合函數在特定平面的投影等高線圖	32
表3-7 在0＜k＜10的範圍內，可能的虛擬頻率(扁長型橢球)	33

圖目錄
圖1-1 論文架構	6
圖2-1 球型與扁長型橢球外域聲場問題	23
圖2-2閉合型基本解與退化核的絕對值等高線圖比較(XY平面)24
圖3-1 源點與邊界點的分布(球型,72個點)	34
圖3-2 球型案例的最小奇異值對k作圖	34
圖3-3 球型案例一的等高線圖(額外源點法)	35
圖3-4 球型案例二的等高線圖(額外源點法)	36
圖3-5 球型案例三的等高線圖(額外源點法)	37
圖3-6 球型案例四的等高線圖(額外源點法)	38
圖3-7 式2-58的最小奇異值分析	39
圖3-8 額外源點的失效點分析(球型案例四)	40
圖3-9 徑向失效點分析(球型案例四)	41
圖3-10 球型案例一的等高線圖(雙層與混合)	42
圖3-11 球型案例二的等高線圖(雙層與混合)	43
圖3-12 球型案例三的等高線圖(雙層與混合)	44
圖3-13 球型案例四的等高線圖(雙層與混合)	45
圖3-14 球型案例四混合勢能實數與虛數	46
圖3-15 源點與邊界點的分布(扁長型橢球,72個點)	47
圖3-16 扁長型橢球案例的最小奇異值對k作圖	47
圖3-17 扁長型橢球案例一的等高線圖(額外源點法)	48
圖3-18 扁長型橢球案例二的等高線圖(額外源點法)	49
圖3-19 扁長型橢球案例三的等高線圖(額外源點法)	50
圖3-20 扁長型橢球案例四的等高線圖(額外源點法)	51
圖3-21 扁長型橢球案例一的等高線圖(雙層與混合)	52
圖3-22 扁長型橢球案例二的等高線圖(雙層與混合)	53
圖3-23 扁長型橢球案例三的等高線圖(雙層與混合)	54
圖3-24 扁長型橢球案例四的等高線圖(雙層與混合)	55

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