Linkage between the Degenerate Scale in the Bem/Biems and Unit Logarithmic Capacity in the Theory of Complex Variables

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Project title

Code/計畫編號

NSC102-2221-E019-034

Translated Name/計畫中文名

邊界元素法中的退化尺度與複變理論中單位對數容量關聯之研究

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

郭世榮

Department/Unit

Department of Harbor and River Engineering

Website

https://www.grb.gov.tw/search/planDetail?id=3117583

Year

2013

Start date/計畫起

01-08-2013

Expected Completion/計畫迄

01-07-2014

Co-Investigator(s)

Shyh-Rong Kuo

Bugetid/研究經費

1032千元

ResearchField/研究領域

土木水利工程

"邊界元素法或邊界積分方程法在處理含 Dirichlet 邊界條件的二維拉普拉斯內域問題時會產生退化尺度的問題。此時，即使數學上邊界勢能給定為零而其邊界法向通量可為非零的情形發生。本計劃擬證明單位圓透過黎曼保角映射時，在其對數容量等於 1 時會有退化尺度發生。其中所謂對數容量是指黎曼保角映射的線性項之領導係數。首先，本計劃擬將邊界元素法中實數邊界積分方程法轉換成複數邊界積分方程法，以便與複變函數理論接軌。在黎曼保角映射中，將探討尺度擴縮與線性係數中兩個參數變換時對幾何外型與退化尺度發生和奇異影響係數矩陣的影響，並透過解析領域與分支切割線來證明單位對數容量將導致退化尺度發生。而對數容量等於 1 時，內場與外場勢能是否為零場，在本計劃中也將作一檢驗。本計劃將透過參數研究探討尺度擴縮與黎曼保角映射中 z 項的領導係數對幾何外形、影響係數矩陣奇異與退化尺度的影響。接著我們也會嘗試將此研究延伸到彈力問題。如何將彈力問題中的兩個退化尺度與對數容量做連結，也是本計劃所關注的議題。最後，也將嘗試在任意外型所發生退化尺度，以數值保角寫像決定出其對應的黎曼保角映射。齊次 Dirichlet 邊界條件下的非零邊界通量將以複變函數與邊界元素法分別予以解析證實與數值檢驗。此外，多連通領域的退化尺度與對數容量的關係亦是本計劃之重點。最後，將透過幾個數值算例來驗證本計劃之想法。 " "It is well known that BEM/BIEM results in a degenerate scale for a two-dimensional Laplace interior problem subject to the Dirichlet boundary condition. In such a case, there is a nontrivial boundary normal flux even the trivial boundary potential is specified. The goal of this project will prove that the unit logarithmic capacity in the Riemann conformal mapping with respect to the unit circle results in a degenerate scale. The logarithmic capacity is defined as the leading coefficient of the linear term in the Riemann conformal mapping. First, the real-variable boundary integral equation will be transformed to the complex-variable boundary integral equation in order to connect the theory of complex variables. In the Riemann conformal mapping, the expansion and shrinkage of the size and changing the linear coefficient will be considered as two parameters to detect the shape of geometry, the degenerate scale and the singular influence matrix. By considering the analytical field and taking care of the path of branch cut, we will try to prove that unit logarithmic capacity in the Riemann conformal mapping results in a degenerate scale. When the logarithmic capacity is equal to one, whether the interior field and exterior field are null (trivial) or not will also be examined at the same time. For the parameter study, the scaling constant and the leading coefficient of z term in the Riemann conformal mapping will be tested and the corresponding geometry will be discussed in this project. The extension work to elasticity problems will be implemented. How to link the two degenerate scales and the logarithmic capacity in elasticity problems is also our main concern. Finally, the Riemann mapping function for an arbitrary geometry of the degenerate scale will be calculated by using the numerical conformal mapping. Nonzero boundary flux for the homogeneous Dirichlet boundary condition will also be proved analytically and numerically detected by using complex variables and BEM, respectively. In addition, the relation between the degenerate scale of multiply-connected domain and the logarithmic capacity will be addressed. Several examples will be utilized to demonstrate our finding. "

Keyword(s)

邊界元素法/邊界積分方程法
對數容量
黎曼保角映射
零場
退化尺度
奇異影響係數函數
複數函數
BEM/BIEM
logarithmic capacity
Riemann conformal mapping
null field
degenerate scale
singular influence matrix
complex variable

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Linkage between the Degenerate Scale in the Bem/Biems and Unit Logarithmic Capacity in the Theory of Complex Variables

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