A Study on the Potential Problems with Degenerate Boundaries Using Complex-Variable BEM

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基本資料

Project title

Code/計畫編號

NSC89-2211-E019-020

Translated Name/計畫中文名

複數邊界元素法在含退化邊界勢能問題的研究

Project Coordinator/計畫主持人

Shyh-Rong Kuo

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=581400

Year

2000

Start date/計畫起

01-08-2000

Expected Completion/計畫迄

31-07-2001

Bugetid/研究經費

261千元

ResearchField/研究領域

土木水利工程

本計畫是將實變數的核函數直接推廣到複變數核函數，經由功能互換原理，建立一個新的複變數邊界元素法。由於傳統複變數邊界元素法在處理平面問題的拉普拉斯方程式時，是採用複變理論的柯西積分公式，必須引進一個虛變數，因而無法求解含退化邊界的問題。而本計畫推導的複變數邊界積分式無需多引進一個虛變數，且此複變數核函數是由實部及虛部二組獨立的解析函數組成，因此可直接利用複變數邊界元素法的實部及虛部影響矩陣，求解含退化邊界的問題。複變數的U核函數虛部具有分歧路徑的特性，為了避免分歧路徑通過求解域內，造成無法直接藉由邊界元素法求解，本計畫引進相對夾角的概念，並配合FORTRAN程式的輻角定義，建議一個可自行指定U 核函數分歧路徑在求解域外的方法；接著再推廣相對夾角的概念，提出一個有限長度分歧路徑的U 核函數，以利分析凹形及多連通邊界或外域的問題。最後，本計畫亦探討核函數虛部的反對稱性質，即元素切割具有特定的對稱性，將導致影響矩陣秩的不合理降低，而會有多出零的奇異值，本計畫建議簡單的採用不對稱的元素切割方式克服上述核函數虛部反對稱的問題。 Dual boundary integral equations in the real domain contain two parts, singular and hypersingular equations. Successful applications have been done for applied to crack and airfoil problems with degenerate boundaries. Complex variable BEM stems from the Cauchy integral formula and it can be applied to solve two-dimensional problems. In the Cauchy integral formula, real and imaginary parts are present. Although the Hadamard integral formula obtained by differentiating the Cauchy integral formula can solve the problems with degenerate boundaries, the Hadamard finite part must be determined. The project will build the dependent relationship for the real and imaginate part. An integral equation which one variable, either the real part or imaginate part, will be proposed to solve the potential problems with degenerate boundaries. Also, the relation between RVBEM and CVBEM will be discussed. Finally, some numerical examples including the potential flow with a sheet pile, torsion problem for a cracked bar and two-dimensional crack problems are considered to verify the validity of the proposed formulation. The results will be compared with analytical solutions, real dual boundary element method and finite element results.

Keyword(s)

邊界元素法
勢能分析
退化
複變數
分歧
Boundary element method
Potential analysis
Degeneration
Complex variable
Branch
BEM

DSpace CRIS

A Study on the Potential Problems with Degenerate Boundaries Using Complex-Variable BEM

基本資料

Description