A Self-Regularized Boundary Element Method-Theory and Applications

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簡歷

基本資料

Project title

Code/計畫編號

MOST104-2221-E019-007-MY3

Translated Name/計畫中文名

自救式邊界元素法-理論與應用

Project Coordinator/計畫主持人

Jeng-Tzong Chen

Funding Organization/主管機關

National Science and Technology Council

Department/Unit

Department of Harbor and River Engineering

Website

https://www.grb.gov.tw/search/planDetail?id=12026498

Year

2017

Start date/計畫起

01-08-2017

Expected Completion/計畫迄

31-07-2018

Bugetid/研究經費

1029千元

ResearchField/研究領域

土木水利工程

"在工程實務中，工程師常需利用數學模式與相對應的數值方法處理工程問題。數值計算更是在所難免的。然而，數值方法可能因為其不完備的數學模式或不足的束制條件而產生病態問題，即秩降的奇異矩陣。去年申請人針對邊界積分方程法(BIEM)/邊界元素法(BEM)的病態問題提出一個研究計晝，將研究重心放在靜力問題的退化尺度。在 2013年，我們提出一個良態的邊界積分方程模式一Fichera方法，透過加入一個自由常數與一個束制條件確保唯一解。如此，可免除傳統積分方程式在二維Laplace問題中因退化尺度而產生的值域缺損問題。根據此計劃之成功經驗，申請人今年擬將此概念延伸，結合奇異值分解(SVD)與加邊矩陣之技巧，直接在離散的線性代數系統中將原本奇異矩陣轉換為一個非奇異之加邊矩陣。主要技巧係利用影響係數矩陣中零奇異值所對應的左右酉向量分別加邊至原影響矩陣之對應的行與列向量，並補上對應的鬆弛變數 (slack variable)。本計劃擬從數學與物理的角度，對BIEM/BEM所產生的病態問題之機制進行研究，並發展出利用加邊矩陣的新治療方法。因透過矩陣本身之有效訊息(左右酉向量)來加邊，故稱此法為自救方法。不僅是靜力問題，時間諧和之動力問題也將一併討論。此三年期計晝案之工作規劃架構如下(參見圖1): 此三年計晝擬以加邊矩陣將病態矩陣轉換成良態作為研究主軸。由物理(無束制（自由)結構之剛體模態)與數學(退化尺度、假根與虛擬頻率等)所導致之矩陣秩降問題都將一併進行討論。擬使用新的自救技巧使這些問題同時得以解決。首先，我們將回顧SVD 與加邊矩陣技巧。擬引入加邊矩陣的想法與增加之鬆弛變數，從物理與數學的角度再探 Fredholm 二擇一定理。順帶一提，雖然文獻提及直接法與間接法等價，但本計劃擬探討兩者在解空間之不等價關係。此外，也將探討本法與偽逆(pseudo-inverse)、廣義逆 (generalized inverse)和Moore-Penrose逆間的關係。前兩年計晝擬使用新的自救技巧針對靜力問題(Laplace, Navier與雙諧和方程式)之退化尺度進行探討。最後一年計晝則將延伸此方法，解決BEM在處理内外域Helmholtz問題時所分別產生的假根或虛擬頻率問題。並擬以幾個算例來驗證此法的準確性。最後將連結線性代數系統的此法與邊界積分方程的Fichera方法，擬延伸Fichera方法去處理Helmholtz的動力問題。此外，地震波問題裡近陷阱模態與聚焦現象的一些應用也將會被研究。並擬將此加邊矩陣的想法應用於無束制（自由)結構之剛體模態的結構解析上。並延伸到解退化邊界之正問題，另反問題之病態系統也是本法可能應用的課題。" "In engineering practice, engineers often employ mathematical models and corresponding numerical methods to solve engineering problems. The numerical implementation is always required. However, the incomplete mathematical model or insufficient constraint in numerical methods may result in an ill-posed model, i.e. a singular matrix of rank deficiency. In the last year, the PI proposed a project for solving problems of the ill-posed model when using the boundary integral equation method (BIEM)/ boundary element method (BEM) and focused on the degenerate scale in static problems. In 2013, we proposed a well-posed BIE model, Fichera’s method, by adding a free constant and a constraint to ensure a unique solution. Therefore, we can alleviate the problem of the range deficiency in the integral operator of the conventional BIE due to the degenerate scale for 2D Laplace problems. Following the successful experiences in the past project, we will extend this idea to transform a singular matrix to a non-singular matrix by employing the technique of singular value decomposition (SVD) and the bordered matrix in the linear algebraic system of a discrete version. According to the bordered technique, we will border the singular influence matrix by adding its left and right singular vectors with respect to the zero singular value and corresponding slack variables. Furthermore, we will study what causes ill-conditioned problems in the BIEM/BEM from the mathematical and physical viewpoint and will provide this new approach of a bordered matrix to solve these problems. Since the bordered matrix uses its own left and right singular vectors of zero singular values, we call this approach of a bordered matrix as a self-regularization method. Not only statics but also time-harmonic problems will be addressed. The frame of our work is given below (Fig. 1): In this three-year project, we will transform the ill-posed matrix to a well-posed one by using bordered matrices. Rank deficiency due to physics (a rigid body mode of a free-free structure) and mathematics (degenerate scale, spurious eigenvalues and fictitious frequencies etc.) will be addressed. The new self-regularization technique will be employed to deal with both problems at the same time. First, we will review the SVD approach and the technique of bordered matrices. We will revisit the Fredholm alternative theorem by introducing the idea of bordered matrices and corresponding slack variables from the mathematical and physical viewpoint. By the way, the nonequivalence for the solutions space between direct and indirect BEMs will be reexamined although the equivalence was mentioned in the literature. Besides, the relation among this approach, the pseudo-inverse, generalized inverse and Moore-Penrose inverse will be linked. In the first and second year, we will focus on the issue of the degenerate scale for 2D static problems (Laplace, Navier and biharmonic equation) by using this new self-regularization technique. In the last year, we will extend this approach to deal with the spurious eigenvalues and fictitious frequencies in the BEM for the interior and exterior Helmholtz problems, respectively. Several examples will be considered to demonstrate the validity of the present approach. Finally, we will make the linkage between this new approach of the linear algebraic system and Fichera’s method in the BIE and will extend Fichera’s method to deal with the Helmholtz problem. Besides, some applications to the near-trapped mode and focusing of wave problems will be studied. Structural analysis of a free-free structure containing the rigid body mode will be done by using the idea of bordered matrices. Possible extension for solving degenerate-boundary direct problems and ill-posed inverse problems is also our concern."

Keyword(s)

邊界元素法/邊界積分方程法
病態問題
奇異值分解
加邊矩陣
Fredholm 二擇一定理
奇異矩陣
BEM/BIEM
ill-posed
SVD
bordered matrix
Fredholm alternative theorem
singular matrix

DSpace CRIS

A Self-Regularized Boundary Element Method-Theory and Applications

基本資料

Description