臺灣博碩士論文加值系統

摘 要
傳統複變數邊界元素法在處理彈性力學的平面問題時，僅取應力函數的實部項推求實變數型式的核函數，在求解含退化邊界問題，則必需建立奇異及超奇異積分式。本文是考慮應力函數的實部項及虛部項，由此拓展成複變數型式的核函數，經由傳統的功能互換原理，建立一套新的複變數邊界元素法，其離散後影響係數矩陣的積分式，能直接積分求得。在處理彈性力學平面問題，本文的複變數核函數是由實部及虛部二組獨立的函數，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來求解含退化邊界的問題。
本文所採用的複變數核函數，其虛部項具有分歧路徑的問題，由於分歧路徑通過求解域內時，無法直接由邊界元素法求解，因此本文引進相對夾角的概念並且配合應用軟體的輻角定義，建議可自行指定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分歧路徑的方法，以利分析凹形、多連通及外域的問題。最後，經由實例分析探討彈性力學的平面問題，以驗證本文提出複變數邊界元素法。

Abstract
When dealing with the plane of elastic mechanics, the traditional CVBEM can only derive the kernel functions of real variable form from the real part of stress functions, with establishing singular and hyper-singular equations to solve the problems of degenerated boundaries. But in this article, we consider the real and the imaginary part of the stress function at the same time to establish a new CVBEM. Through the traditionally reciprocal-work theorem, we can derive the integral equations of coefficient matrix after the divergence of boundary quantities directly by integrating the variables we considered. Therefore, the original problems in the plane of elastic mechanics and the degenerated boundaries will be resolved by the multi-variable kernel functions, including real and imaginary part, in the new CVBEM.
In our model, the imaginary part of the multi-variable kernel functions exists the problem of path diverging, which will cause the kernel functions unsolvable by BEM directly when them pass through the domain of objectives. Therefore, we use the method of relative angles, with the definitions of argument in computer software, to suggest some ways defined specifically to find the multi-variable kernel functions out of the objective domain. Finally, we extended this method with some numerical examples including the patch test, cantilever beams, simple beams, and two-dimensional crack problems in infinite region are considered to verify the validity of the proposed formulation.

目 錄
摘要………………………………………………………………………Ⅰ
Abstract…………………………………………………………………Ⅱ
目錄………………………………………………………………………Ⅲ
圖目錄……………………………………………………………………Ⅶ
表目錄……………………………………………………………………Ⅸ
第一章 緒論…………………………………………………………1
1.1 文獻回顧及研究動機………………………………………………1
1.2 本文架構與研究內容………………………………………………2
第二章 反平面問題…………………………………………………4
2.1 物理問題………………………………………………………4
2.2 功能互換定理…………………………………………………5
2.3 基本解…………………………………………………………6
2.4 複變數邊界元素法……………………………………………9
2.4.1 曳引力採用常元素………………………………………10
2.4.2 位移採用線性元素及常元素……………………………11
2.5 分歧路徑(Branch cut) ……………………………………13
2.5.1 簡介………………………………………………………13
2.5.2 無限長分歧路徑…………………………………………13
2.5.3 有限長分歧路徑…………………………………………15
2.6 複變理論與複變數邊界元素法的討論 ……………………17
第三章 平面彈性問題………………………………………………29
3.1 前言 …………………………………………………………29
3.2 位移、曳引力與應力函數的關係式 ………………………30
3.3 基本解及功能互換 …………………………………………35
3.4 複變數邊界元素法 …………………………………………39
3.4.1 邊界曳引力採用常元素…………………………………41
3.4.2 位移採用線性元素………………………………………42
3.5分歧路徑(Branch cut)………………………………………44
3.5.1 簡介 ……………………………………………………44
3.5.2 無限長分歧路徑 ………………………………………44
3.5.3 有限長度的分歧路徑 …………………………………45
3.6 複變數邊界元素法的比較 …………………………………46
第四章 實例分析 …………………………………………………52
4.1剛體運動測試與常應變測試 (Patch test) ………………52
4.2圓形平面問題…………………………………………………52
4.2a 內域圓形平面問題 ………………………………………53
4.2b 外域圓形平面問題 ………………………………………53
4.2c 多連通平面問題 …………………………………………53
4.3懸臂樑 (Cantilever beams) ………………………………54
4.4簡支樑 (Simple beams) ……………………………………55
4.5無限域裂縫問題………………………………………………56
第五章 結論與未來研究方向 ……………………………………66
5.1 結論 …………………………………………………………66
5.2 未來研究方向 ………………………………………………68
參考文獻 ………………………………………………………………69
附錄A……………………………………………………………………73
附錄B……………………………………………………………………75
附錄C……………………………………………………………………78
附錄D……………………………………………………………………79
附錄E……………………………………………………………………81
附錄F……………………………………………………………………82
附錄G……………………………………………………………………85
附錄H……………………………………………………………………87

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