Degenerate-Scale Free Bie Formulation and Boundary Element Implementation

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簡歷

基本資料

Project title

Code/計畫編號

MOST103-2221-E019-012-MY3

Translated Name/計畫中文名

無退化尺度之邊界積分方程推導及邊界元素法之應用

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=11709332

Year

2016

Start date/計畫起

01-08-2016

Expected Completion/計畫迄

01-07-2017

Bugetid/研究經費

930千元

ResearchField/研究領域

土木水利工程

"邊界積分方程法(BIEM)的數學研究與邊界元素法(BEM)在工程的應用，發展至今已超過四十年。然而，邊界積分方程與偏微分方程，兩者解空間的等效性卻鮮有論述。而退化尺度、退化邊界、假根與虛擬頻率等問題只有在使用邊界元素法才會發生，說明此研究之重要性。此計畫擬將研究重心放在退化尺度。由於邊界元素法中影響係數矩陣的病態問題，邊界元素法會有無限多解(非充分邊界積分方程)或無解(非必要邊界積分方程)的狀況產生。所以此三年計畫擬以確保唯一解之充要邊界積分方程為研究主軸。在此三年期之提案計畫中，擬以數學與力學的觀點探討此議題為研究重心。首先，我們將回顧退化尺度的兩個檢測指標(影響係數矩陣的最小奇異值、複數函數中的對數容量)與確保唯一解的四種正規化方法(超奇異積分方程法、剛體模態法、邊界勢能通量平衡方程與CHEEF法)。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。並透過對 Fichera想法裡自由常數之了解，從而提出一個檢測退化尺度的新指標。最後，將推導出一個對所有尺度皆充分且必要的邊界積分方程式。因此，便無需使用正規化方法。此三年期計畫案之工作規劃架構如下：本計劃與先前研究的主要差異是在於本提案擬提供一個良態的邊界積分方程式(充要而無病)，取代傳統邊界元素法中病態系統(非充要需治療) (參見圖 1)。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)。並擬以圓形與橢圓形為例，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。此外，擬以 BEM進行數值驗證。於第二年計畫中，我們擬將第一年二維 Laplace問題(純量)之相關研究延伸至平面彈力問題(向量場)。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(圓與橢圓)與數值算例(一般外形)來驗證我們的想法。" "Mathematical study of the boundary integral equation method (BIEM) and engineering applications of the boundary element method (BEM) has been developed more than 40 years. However, the equivalence of the solution space using BIE and PDE was not noticed. Degenerate scale, degenerate boundary, spurious eigenvalues and fictitious frequencies etc. only occurring in the BEM indicate that the equivalence check is very important. Here, we will focus on the degenerate scale. Owing to the ill-conditioned influence matrices of BEM, BEM may have infinite solutions (not sufficient BIE) or no solution (not necessary BIE). Therefore, sufficient and necessary BIEs for ensuring a unique solution are our goal in the three-years project. In this three-year proposal, we will focus on this topic from the viewpoint of mathematics and mechanics. In the first, we will review the two indexes (the minimum singular value of the influence matrix and the logarithmic capacity in complex variables) and four regularization techniques (hypersingular formulation, method of adding a rigid body mode, rank promotion by adding the boundary flux equilibrium, CHEEF method) to ensure the unique solution. Then, we will introduce the Fichera’s technique and degenerate kernels to analytically study the rank deficiency mechanism. The BEM implementation will be accompanied with. In addition, we will give a new interpretation of Fredholm alternative theorem according to the structure of the singular value decomposition. A new index for detecting the degenerate scale will be proposed after understanding the free constant in the Fichera’s approach. Finally, we will propose a sufficient and necessary BIE to ensure a unique solution for all size of the domain. Therefore, the regularization techniques will not be required. The frame of our work is shown below: The main difference between this project and our previous researches is that this proposal will focus on providing a well-posed BIE model instead of the ill-conditioned system for the conventional BEM/BIEM (as shown in Fig. 1). In the first year, we will prove that the promoting rank two for the extra degree of freedom by adding only 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 circular and elliptic cases. Besides, numerical implementation will be done. In the second year, we will extend the Laplace problems (scalar field) to plane elasticity problems (vector field). We will focus on the two-dimensional biharmonic problems in the final year. Finally, two analytical examples (circle and ellipse) and numerical examples (general geometry) 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
奇異值分解
Fredholm二擇一定理
正規化技巧
BEM/BIEM
degenerate scale
singular value decomposition
Fredholm alternative theorem
regularization technique

DSpace CRIS

Degenerate-Scale Free Bie Formulation and Boundary Element Implementation

基本資料

Description