Combination of the CHIEF and the self-regularization technique for solving 2D exterior Helmholtz problems with fictitious frequencies in the indirect BEM and the MFS

間接邊界元素法與基本解法結合CHIEF與自救法求解含虛擬頻率的二維外域聲場問題

李家瑋(Jia-Wei Lee) ； 粘祺烽(Chi-Feng Nien) ； 高聖凱(Shing-Kai Kao) ； 陳正宗(Jeng-Tzong Chen)

《中華民國振動與噪音工程學會論文集》 2018 (2018/06) Pp. 78-86

摘要

Regarding the treatment for the fictitious frequency as well as spurious resonance in using the indirect boundary element method (BEM) and the method of fundamental solutions (MFS), we propose an alternative approach in this paper. The present approach is different from the mixed potential approach in the indirect method as well as the Burton and Miller approach in the direct BEM. In the proposed approach, we add some fundamental solutions with unknown source strength in the representation of the field to complete the base of solution space. From the viewpoint of adding source, the present idea is similar to the combined Helmholtz interior integral equation formulation (CHIEF) method in the direct BEM. The difference between the added source points and the null-field point of CHIEF method is their role. The added source points supply the deficient base due to the fictitious frequency while the null-field point of CHIEF provides the extra constraint equations. Therefore, we examine the CHIEF constraint by employing the self-regularization technique for the influence matrix in the direct BEM. Based on this idea, the constraint equation in the present approach may be found by adding the right unitary vectors of zero singular value. Then, a square bordered matrix is obtained. The bordered matrix is invertible for the fictitious frequency if the extra source points do not locate at the failure position. This is the reason why the property is analogous to the idea of the CHIEF method in the direct BEM. Therefore, the proposed approach can fill in the gap that there is no CHIEF method in the indirect BEM and the MFS. Since the proposed approach only need using the single-layer potential, it has an advantage over the existing formulations. To demonstrate the validity of the present idea, the problem of an infinite plane containing a circular radiator or scatter is considered. In the real implementation, all fictitious frequencies in the certain range of the wavenumber are found first by the direct searching algorithm. Four cases with different boundary conditions and fictitious frequencies are considered. Finally, we also analytically derive the locations of possible failure source points by using the degenerate kernel.

關鍵字

indirect BEM ； MFS ； fictitious frequency ； CHIEF ； self-regularization ； degenerate kernel

並列摘要

關於間接邊界元素法與使用基本解法應用在解決虛擬頻率及虛假共振的問題上，我們在本篇文章中提出了另一個方法。本文的方法不同於間接法中的混和勢能法與直接法中的Burton和Miller法。在此法中，我們添加未知源點強度的基本解來補足基底空間的不足。從添加源點的觀念來看，此法和直接法中結合Helmholtz 內域積分方程公式（Combined Helmholtzinterior integral equation formulation, CHIEF）相似。兩者之間不同之處在於未知源點與CHIEF點的作用機制並不相同。額外添加的源點補足了因虛擬頻率導致而缺損的基底，CHIEF點則提供額外束制方程。基於這個想法，本法的束制條件可從影響係數矩陣做奇異值分解後對應所得之最小奇異值為零的右酉向量來獲得。若是額外添加的源點並不落於失效點位上，則此矩陣在虛擬頻率的波數依然可逆。這就是為何我們要提出一個性質類似於直接法中CHIEF法的原因。所提出的想法可以填補在間接法與基本解法中並無對應直接法的CHIEF法之缺憾。由於本法僅需使用單層勢能公式，故它有優於現行方法的優點。為了驗證本想法的有效性，我們考慮了含圓形散射體或輻射體的無限域平面問題。在實際計算中，首先以直接搜尋法找尋一定波數範圍內的虛擬頻率。我們考慮四個不同邊界條件與虛擬頻率的案例。最後，我們也應用分離核來解析推導可能發生失效點的所在位置。

並列關鍵字

基本解法 ； CHIEF法 ； 奇異值分解 ； 虛擬頻率 ； 自救法