Study of the Double-Degeneracy Problem in the Bem/Biem(1/3)

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Project title

Code/計畫編號

MOST109-2221-E019-001-MY3

Translated Name/計畫中文名

邊界元素法/邊界積分方程法中雙退化問題之探討(1/3)

Project Coordinator/計畫主持人

Jeng-Tzong Chen

Funding Organization/主管機關

National Science and Technology Council

Co-Investigator(s)/共同執行人

陳義麟
李家瑋

Department/Unit

Department of Harbor and River Engineering

Website

https://www.grb.gov.tw/search/planDetail?id=13534272

Year

2020

Start date/計畫起

01-08-2020

Expected Completion/計畫迄

31-07-2021

Bugetid/研究經費

1353千元

ResearchField/研究領域

土木水利工程

邊界積分方程法(BIEM) 與邊界元素法(BEM)發展至今已超過四十年。引入核函數與格林第三恆等式表示解空間，獲得降維度的好處。然而邊界元素法中所用積分方程將會有四種退化問題：退化尺度、退化邊界、假根與虛擬頻率。此計畫擬將研究重心放在退化尺度與退化邊界同時發生的雙退化問題。在求解外域Helmholtz方程時，內域的自然頻率竟會誘發外域數值虛擬共振。當橢圓形或長方形之輻射體或散射體退化成一條線時，參見表1，虛擬頻率(ωf)趨近於無窮，便可避開虛擬頻率，但卻有了退化邊界的解不唯一問題。對偶邊界元素法可提供解決方案。然而在靜力的退化尺度問題中，若橢圓形邊界退化成線裂縫或剛性線夾雜時，參見表1，退化尺度與退化邊界的問題將同時存在。然而雙退化問題，就申請人所知目前甚少相關學術研究提出探討，更何況如何解決此問題。此類問題為邊界元素法中影響係數矩陣的病態問題，因此可能有無窮多解或無解的狀況產生。所以此三年計畫擬提供一個良態的邊界積分方程式，取代傳統邊界元素法中病態系統為研究主軸，並企圖找到一個可同時解決退化尺度與退化邊界問題之方法。此三年期計畫案之工作規劃架構參見圖1。本計畫擬將先前科技部三年期計畫(MOST 103-2221-E-019-012-MY3)內域問題之成功經驗，拓展到外域問題，進而探討夾雜問題(參見圖2)。基於先前計畫成功經驗，我們擬引入Fichera方法與分離核之概念，針對矩陣秩降的原因進行解析研究。BEM之數值測試也將一併檢驗。我們擬以奇異值分解(SVD)之架構來詮釋Fredholm二擇一定理。而Fichera想法裡自由常數與束制條件在外域問題所扮演之角色與對應之物理意義也是本計畫之研究重點。於第一年計畫中，我們將證明在二維Laplace問題中使用Fichera方法－加入一個自由度與一個束制條件，使秩降的影響係數矩陣提升為滿秩矩陣(秩增加2)。並擬以圓形、橢圓形孔洞與裂縫邊界為例，解析推導與數值驗證該確保唯一解之充要邊界積分方程式。並擬以自由體圖的概念，結合內外域的充要邊界積分方程探討夾雜問題。此外，擬結合對偶邊界元素法探討雙退化問題也是本案之重點。於第二年計畫中，我們擬將第一年二維Laplace問題之相關研究延伸至平面彈力問題。第三年則擬致力於二維雙調和問題。最後，我們將藉由解析算例與數值算例來驗證我們的想法。 The boundary integral equation method (BIEM) and the boundary element method (BEM) have been developed more than 40 years. The main gain is the dimensional mesh reduction by introducing kernel functions and Green's third identity. However, the mathematical degeneracy in the BIEM/BEM appears in four aspects: degenerate boundary, degenerate scale, spurious eigenvalue and fictitious frequency. Here, we will focus on the double degeneracy of degenerate scale and the degenerate boundary. For the exterior Helmholtz equation, BIEM/BEM may result in a fictitious frequency, ωf, which is associated with the interior eigenvalues. When the scatter or radiator is shrunk to a thin-break down geometry as shown in Table 1, the fictitious frequency becomes infinite. It seems that the fictitious frequency disappears. However, it turns out to be the non-uniqueness due to the degenerate boundary. For the statics, there exists a degenerate (critical) scale resulting in the non-uniqueness solution. When the boundary is shrunk to a thin-break down as shown in Table 1, a degenerate scale still exists and the critical scale is finite. To the PI’s best knowledge, the simultaneous appearance of double degeneracy of the degenerate scale and boundary was not noticed, not to say how to circumvent. Owing to the ill-conditioned influence matrices of BEM, BEM may have infinite solutions or no solution. Therefore, derivation of sufficient and necessary BIEs for ensuring a unique solution of the double degeneracy problem is our goal in the three-years project. Finally, we will find a technique to overcome the problems of degenerate scale and degenerate boundary at the same time. The frame of our work is shown in Fig. 1.In this three-years proposal, we will extend the previous MOST project from interior to exterior problems and inclusions problems as shown in Fig. 2. Based on the successful experience of the previous MOST project, we will introduce the Fichera’s technique and degenerate kernels to analytically study the rank-deficiency mechanism. The BEM implementation will be accompanied. Based on the technique of free-body diagram, we will solve the inclusion problem by employing the necessary and sufficient BIEs for interior and exterior problems. In addition, we will give a new interpretation of Fredholm alternative theorem according to the structure of the singular value decomposition. In the first year, we will prove that the promoting rank two for the extra degree of freedom by adding a free constant and one constraint in Fichera’s method for two-dimensional Laplace problems. Then, we will examine the necessary and sufficient BIE to ensure the unique solution for the case of circular, elliptic and crack boundaries. In addition, it is also a focus of this three-year proposal to combine this necessary and sufficient BIE with the dual BEIM for dealing with the problem of the double degeneracy. In the second year, we will extend the Laplace problems to plane elasticity problems. We will focus on the two-dimensional biharmonic problems in the final year. Finally, analytical and numerical examples will be utilized to demonstrate our finding.

Keyword(s)

邊界元素法/邊界積分方程法
退化尺度
退化邊界
正規化技巧
雙退化
對偶邊界元素法
BEM/BIEM
degenerate scale
degenerate boundary
regularization technique
double degeneracy
Dual BEM

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Study of the Double-Degeneracy Problem in the Bem/Biem(1/3)

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