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Please use this identifier to cite or link to this item: http://ntour.ntou.edu.tw:8080/ir/handle/987654321/36082

Title: 對偶邊界元素法與保角映射於二維拉普拉斯問題的退化尺度之研究
Study on degenerate scale for the two-dimensional Laplace problem by using the dual boundary element method and the conformal mapping
Authors: Yi-Wei Chen
陳逸維
Contributors: NTOU:Department of Mechanical and Mechatronic Engineering
國立臺灣海洋大學:機械與機電工程學系
Keywords: 邊界元素法;二維拉普拉斯問題;退化尺度;複變函數;單位對數容量;保角映射;對數核函數
boundary element method;two-dimensional Laplace problem;degenerate scale;complex variable;unit logarithmic capacity;conformal mapping;logarithmic kernel
Date: 2012
Issue Date: 2013-10-07T03:02:50Z
Abstract: 本文利用邊界元素法和保角映射來探討橢圓、正多邊形與半圓形的退化尺度問題。在邊界元素法中,若二維拉普拉斯問題之幾何外形為一特別尺寸時會導致矩陣的秩降,這不同於物理問題上的尺寸效應。此退化尺度可由兩個方向來做說明:一為邊界積分方程中因對數核函數所引起的解不唯一;另一為複變函數中單位對數容量所對應的保角映射。因此,我們利用保角映射和邊界元素法來得到退化尺度的解析推導與數值驗證。橢圓退化尺度的解析推導不僅可由保角映射搭配單位對數容量來得到,也可由退化核函數推導得到。對於弱奇異積分算子在橢圓領域的特徵值與特徵向量可由退化核推導得知,並且發現當特徵值為零時會導致退化尺度的發生。有趣的是藉由保角映射的技巧來找到對數容量絕對值等於1時為退化尺度。基於奇異值分解法,由退化尺度所造成的秩降(數學)模態是埋藏在弱奇異(U核函數)和強奇異(T核函數)積分算子的影響係數矩陣之左酉向量;在另一方面,強奇異(T核函數)和超強奇異(M核函數)算子的影響係數矩陣可以得到共同的右酉向量之剛體運動訊息。為了處理解不唯一的問題,我們利用四個正規化技巧(超強奇異邊界積分方程法、邊界勢能通量平衡式、剛體模態法與CHEEF法),來提升影響係數矩陣的秩數。並在正常尺度下解析推導與數值驗證外域零場與非零內場;在退化尺度時,邊界條件為齊次的Dirichlet邊界條件,得到內域零場與非零外場。也發現到利用邊界元素法得到退化尺度時非零外場的等高線分布是與保角映射所得結果一致。此外,可由非零外場來證實沒有失效的CHEEF點並且可用於解決退化尺度的問題。而退化尺度也可由一次錯誤嘗試即可計算得到數值解。最後,所有的數值算例解析和數值的結果是相互一致吻合。
Degenerate scales for elliptical, regular N-gon and half-disc domains are studied by using the boundary element method (BEM) and conformal mapping. For solving two-dimensional Laplace problems by using the BEM, the special size of geometry results in a rank-deficient matrix, and is different from the size effect problem in physics. Degenerate scale stems from either the non-uniqueness of BIE using the logarithmic kernel or the conformal mapping of unit logarithmic capacity in the complex variables. The degenerate scale can be analytically derived by using the conformal mapping as well as numerical detection by using the BEM in this thesis. Analytical formula of an ellipse for the degenerate scale can be derived not only from the conformal mapping in conjunction with unit logarithmic capacity, but also can be derived by using the degenerate kernel. Eigenvalues and eigenfunctions for the weakly singular integral operator in the elliptical domain are both derived by using the degenerate kernel. It is found that a zero eigenvalue results in a degenerate scale. By using the conformal mapping technique, it is interesting to find that the absolute value of the logarithmic capacity equals to 1 in the case of degenerate scale. Based on the singular value decomposition, the rank-deficiency (mathematical) mode due to the degenerate scale (mathematics) is imbedded in the left singular vector for the influence matrices of weakly singular (U kernel) and strongly singular (T kernel) integral operators. On the other hand, we obtain the common right singular vector in the dual integral formulation corresponding to a rigid body mode (physics) in the influence matrices of strongly singular (T kernel) and hypersingular (M kernel) operators. To deal with the problem of non-uniqueness solution, four regularization techniques, the hypersingular BIE, the constraint of boundary flux equilibrium, addition of a rigid body term in the fundamental solution and the Combined Helmholtz Exterior integral Equation Formulation (CHEEF) approach, are employed to promote the rank of influence matrices to be full rank. Null field for the exterior domain and interior nonzero field are analytically derived and numerically verified for the ordinary scale while the null field for the interior domain and nonzero exterior field are obtained for the homogeneous Dirichlet problem in the case of the degenerate scale. It is also found that the contour of nonzero exterior field for the degenerate scale using the BEM matches well with that of the conformal mapping. Besides, no failure CHEEF point outside the domain can be found due to the nonzero field of the complementary domain in the case of degenerate scale. Only one trial in the BEM is required to determine the degenerate scale. Both analytical and numerical results agree well in the demonstrative examples.
URI: http://ethesys.lib.ntou.edu.tw/cdrfb3/record/#G0019972011
http://ntour.ntou.edu.tw/handle/987654321/36082
Appears in Collections:[機械與機電工程學系] 博碩士論文

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