Applications of Modified Green$S Functions for Solving Boundary Value Problems of 2d Laplace Equation

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簡歷

基本資料

Project title

Code/計畫編號

MOST105-2221-E019-004

Translated Name/計畫中文名

修正型格林函數於求解二維拉普拉斯方程邊界值問題之應用

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=11891204

Year

2016

Start date/計畫起

01-08-2016

Expected Completion/計畫迄

31-07-2017

Bugetid/研究經費

836千元

ResearchField/研究領域

土木水利工程

"零空間是線性代數系統中判斷矩陣是否奇異的重要指標。相對於連續系統，便是齊次方程式的解空間。如果有非無聊解，則其相對應之非齊次方程式便會解不唯一。透過 Fredholm 二擇一定理可得無限多解(不充分)與無解(非必要)之結果。邊界積分方程法 (BIEM)/邊界元素法(BEM)的退化尺度問題便是其中一種病態問題。此類病態問題之處理方式大致可分成：修正積分表示式(Fichera方法）、增補束制條件(CHEEF方法與邊界勢能通量平衡方程式）、結合不同的積分方程式(超奇異積分方程式)與修正核函數(剛體模態法）。本計晝擬將研究重心設定於修正核函數，即修正格林函數。藉由補解項的修正，改變其對應之解空間，讓因退化尺度而導致之值域缺損的問題得以解決。修正格林函數法有兩種方式，一種是直接增補其缺損之補解項於該函數中，另一種則是透過改變其定義，除了集中力之外透過對應的分布力以達修正格林函數之目的，而現行的剛體模態法則是屬於前者，本計晝則是後者(參見表1)。然而文獻中雖對修正格林函數有許多數學論證，但鮮有數值算例，亦是本計晝需執行之重點。本計晝擬提供一個良態的邊界積分方程式(充要而無病態），取代該方法僅是避開原退化尺度(非充要需治療）。因此，便無需使用正規化方法。不論是連續系統中值域缺損與離散系統的矩陣秩降問題都將一併進行討論。擬使用新的修正技巧使這些問題得以解決。本計晝與Fichera技巧最主要的差異在於格林函數的修正，兩者間的比較請參見表2。本計晝案之工作規晝架構如下 (參見圖1): 本計晝擬以修正自由空間之格林函數作為研究主軸。首先，我們將透過Fredholm 二擇一定理回顧退化尺度問題並引入分離核解析驗證此想法之機制與可行性。首先，我們擬以圓形與橢圓形為例探討Laplace問題，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，擬以邊界元素法進行數值驗證。最後，此法所導得之充要邊界積分方程式與先前計晝透過Fichera技巧所得之充要積分方程式的關係將進行比對探討。兩者間的等價性亦是本計晝之研究重點。最後，我們將藉由解析算例(圓與橢圓)與數值算例(一般外形)來驗證我們的方法。" "Null space is an important index to judge whether the matrix is singular in the linear algebraic system. In the continuous system, the null space is the solution space of a homogeneous equation. According to the Fredholm alternative theorem, the uniqueness of the solution exists while the solution is not trivial. Furthermore, the uniqueness of the solution is either infinite solutions from non-sufficient formulation or no solution from the non-necessary formulation. The degenerate scale problem occurring in the Boundary Iintegral Equation Method (BIEM) / Boundary Element Method (BEM) is an ill-posed problem. The ill-posed problem can be treated in several ways: using the modified integral formulation (Fichera’s method), adding the constraint equation (CHEFF method and rank promotion by adding the boundary flux equilibrium), introducing the hypersingular integral formulation and using the modified kernel function (the method of adding a rigid body mode). This project focuses on the modified kernel function namely the modified Green’s function. By adding the complementary solution to the unmodified Green’s function to change the corresponding solution space, the range deficiency due to the degenerate scale does not occur. There are two ways to modify Green’s function. The first one is adding the complementary solution to the kernel function which is range-deficient, and the other is adding the distributed force to the left-hand side of the governing equation for the unmodified Green’s function (lumped force). The method of adding a rigid body mode belongs to the former, and the present method belongs to the latter (refer to Table 1). The well-posed boundary integral equation (BIE) with the modified Green’s function (necessary and sufficient formulation) will be proposed to replace the previous way only avoiding the degenerate scale (not necessary and sufficient formulation and needs to be treated). Therefore, we don’t need to use the regularized technique. Both the range deficiency in the continuous system and rank deficiency in the discrete system are considered by using the present method. The difference between the present method and the Fichera’s method is the modification of the kernel function. The comparison of two methods is listed in Table 2. The frame of our work is also given below (Fig. 1): In this project, we will modify the free space Green’s function. First, we will revisit the degenerate scale problem according to the Fredholm alternative theorem, and introduce the degenerate (separate) kernel to verify the feasibility of this project. For problems of the circular and elliptical domain with the Laplace equation, analytical derivation and numerical implementation will be used to ensure the uniqueness of solution in the necessary and sufficient BIE. Besides, the BEM will be proposed for the numerical implementation. We will discuss the relationship between the Fichera’s method utilized in the previous project and the present method. Equivalence or non-equivalence of two methods will be considered in this project. Finally, several examples are utilized to demonstrate the validity of the present approach in the analytical and numerical implementation since only very few examples were found in the literature for the modified Green’s function method."

Keyword(s)

邊界元素法/邊界積分方程法
病態問題
Fredholm 二擇一定理
修正格林函數
退化尺度
BEM/BIEM
ill-posed
Fredholm alternative theorem
modified Green’s function
degenerate scale

DSpace CRIS

Applications of Modified Green$S Functions for Solving Boundary Value Problems of 2d Laplace Equation

基本資料

Description