文獻中利用BEM/BIEM求解外域Helmholtz方程問題時，會遭遇虛擬頻率的問題。虛擬頻率產生的數值共振現象，並不存在於真實物理現象當中。由於解的積分表示式所產生的解不唯一之現象，在文獻中有直接法的Burton & Miller 法與CHIEF法。對應B&M法，間接法有混合勢能法。然因超奇異積分於計算上較為複雜，故CHIEF法普遍受到工程界之接受。相對於直接法中的CHIEF法，間接法苦無零場方程式所對應的束制條件。此問題已存在五十餘年仍未被解決，即是本研究將要探討的問題。本計畫擬發展一套屬於間接法之CHIEF法，藉此有效處理此虛擬頻率之病態問題。基於申請人先前提出之Fichera法與自救法計畫的成功經驗，本計畫擬延伸至解決以間接BEM與基本解法來解二維外域聲場問題時所產生之虛擬頻率問題，進而推展至三維聲場問題。本計畫首先探討直接法與間接法之間的等價關係，接著討論間接BEM(連續型)與基本解法(離散型)之間的關係。接著，為解決基本解法中所產生之虛擬頻率的問題，申請人擬引進類似CHIEF法之零場積分方程的概念，將CHIEF法中選取零場點改為選取額外源點 (場變源)，且結合自救法的束制方程式使加邊矩陣為方陣且滿秩，進而可逆並也可計算場解，建立完整且有效之解空間。在直接法的CHIEF法中，零場點用以增加束制條件。但屬於間接法的基本解法中選取額外源點是作為補充不足的基底，故直接法與間接法之間的關係亦將是本計畫的重點之一。最後，此法是否類似CHIEF法會有失效點將一併討論，透過數值算例來驗證此法之可行性。 In the literature, the numerical resonance occurs when boundary element method (BEM) / boundary integral equation method (BIEM) are employed to solve the exterior acoustics problems. The frequency does not exist in nature and results in the non-unique solution using the BEM and the method of fundamental solution (MFS). It is termed the fictitious frequency. For this issue, several regularization techniques, such as Burton & Miller approach (B&M) and the combined Helmholtz integral equation formulation (CHIEF) in the direct BEM, and mixed-potential method in the indirect BEM were studied. However, hypersingular integral equations are more complicated, the CHIEF method is popular in engineering community. Constraint is provided for the CHIEF null-field equation in the direct BEM, but it is not available in the indirect BEM. There is still no corresponding approach of the CHIEF idea in the indirect BEM since 1950s. We may fill the gap in this proposal.We will focus on a similar CHIEF method in the indirect BEM and the MFS to effectively deal with the problem of fictitious frequency. Based on the successful experience of the Fichera’s method (103-2221-E-019-012-MY3) and self-regularization technique (104-2221-E-019-007-MY3), we will alleviate the problem of fictitious frequency in the indirect BEM and the MFS by employing the proposed hybrid method. Two and three dimensional problem will be both considered. First, we will discuss the equivalence between the direct BEM and the indirect BEM. The relationship between the indirect BEM (continuous system) and the MFS (discrete system) will be addressed. Then, we will introduce the similar CHIEF idea in the MFS to solve the problem of fictitious frequencies. The corresponding CHIEF points in the indirect BEM and the MFS are the source points rather than the null-field points in the direct BEM. It means that the adding source points in the indirect BEM can provide the insufficient base of range rather than the constraint. Furthermore, we will also combine the constraint of the self-regularization technique to obtain the full-rank influence matrix. Then, the matrix is non-singular and the field solution can be obtained. In the CHIEF method of the direct BEM, the null-field points are used to add the constraint equation. But the extra source points in the complementary domain are used to provide the complete base in the range. The relationship between the direct BEM and the indirect BEM will be constructed in this project. Finally, the validity of the proposed method will be verified by several numerical cases. Possible failure source points will be analytically and numerically examined.
The indirect BEM/BIEM