額外源點法在三維外域聲場虛擬頻率問題之數值分析

Numerical analysis of extra source points approach for solving fictitious frequency problems in three dimensional exterior acoustics

許書瑋(Shu-Wei Hsu)

指導教授 ： 李家瑋

淡江大學/工學院/土木工程學系碩士班/碩士(2020年)

https://doi.org/10.6846/TKU.2020.00062

摘要

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。

關鍵字

基本解法 ； 外域聲場問題 ； 虛擬頻率 ； 額外雙層勢能 ； 額外混合勢能 ； 退化核

並列摘要

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.

並列關鍵字

Method of fundamental solutions ； Exterior acoustic problems ； Fictitious frequency ； Extra double-layer potential ； Extra mixed potential ； Degenerate kernel

參考文獻

參考文獻

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