本論文利用加法定理及疊加技巧來求解集中力與螺旋差排問題的格林函數。此兩類問題均可利用疊加技巧分為兩部份。一部分為基本解的問題;另一部份為典型的邊界值問題。於基本解的部份，我們利用加法定理展開成退化核的型式。而在求解螺旋差排問題時，將角度型的基本解展成退化核的型式，這在作者的認知中文獻並未發現。我們引入疊加技巧及加法定理來決定典型邊界值問題之邊界條件。再利用NTOU/MSV在零場積分方程結合傅立葉級數求解典型邊界值問題之成功經驗，第二部份的解將能迎刃而解。將兩部分的場解做疊加即可得到完整的格林函數。另外，選取不同項數的傅立葉級數進行收斂性分析來測試本方法的收斂速率。最後，我們將利用含圓形邊界(洞及夾雜)之集中力與螺旋差排問題，來驗證此方法的準確性。本法最大特色可免除傳統邊界元素法中的五項缺點:(1)奇異積分的主值計算、(2)病態矩陣、(3)邊界層效應、(4)線性收斂及(5)網格切割。

In this thesis, we employ the addition theorem and superposition technique to derive the Green function of the concentrated forces and screw dislocation problems. By using the superposition technique, the problems can be decomposed into two parts. One is the problem of the fundamental solution and the other is a typical boundary value problem (BVP). The fundamental solution is expanded into the degenerate kernel by using the addition theorem. The angle-type fundamental solution of the screw dislocation problem has not been expanded into the degenerate form before to our best knowledge. Following the success of null-filed integral formulation for solving the typical BVP with Fourier boundary densities in the NTOU/MSV group, the second part boundary condition can be easily obtained by introducing the superposition technique and addition theorem. After superposing the two solutions, the Green function can be obtained. Convergence rate using various numbers of terms for Fourier series is also examined. Finally, some concentrated force and screw dislocation problems with circular boundaries, including holes and inclusions, were demonstrated to see the validity of present method. Five disadvantages, (1) calculation of principal value, (2) ill-posed model, (3) boundary-layer effect, (4) linear convergence and (5) mesh generation, can be avoided by using the present approach in comparison with the conventional BEM.

Contents

中文摘要 Ⅰ
Abstract Ⅱ
Contents Ⅲ
Table captions Ⅴ
Figure captions Ⅵ
Notations Ⅸ

Chapter 1 Introduction
1.1 Overview of BEM and motivation 1
1.2 Organization of the thesis 8
Chapter 2 Derivation of Green’s function for Laplace problems with circular boundaries using addition theorem and superposition technique
Summary 20
2.1 Introduction 20
2.2 Review of the null-field integral formulation for a typical boundary problem with Fourier boundary densities 21
2.2.1 Expansion of kernel function and boundary density 22
2.2.2 Adaptive observer system 23
2.2.3 Linear algebraic equation 23
2.3 Present approach for constructing the Green’s function 26
2.4 Numerical examples 28
2.5 Conclusions 31


Chapter 3 Derivation of screw dislocation solution for Laplace problems with circular boundaries using addition theorem and superposition technique
Summary 42
3.1 Introduction 42
3.2 Problem statements and mathematical formulation 43
3.3 Expansions of fundamental solutions and boundary densities 44
3.3.1 Degenerate (Separable) kernel for the angle-based fundamental solution 44
3.3.2 Fourier series expansion for boundary density 46
3.4 Matching of interface conditions and solution procedures 46
3.5 Illustrative examples and discussions 48
3.6 Conclusions 51
Chapter 4 Conclusions and further research
4.1 Conclusions 65
4.2 Further research 67
References 68
Appendix
Appendix 1 Equivalence between the solution using Green’s third identity and that using superposition technique 76
Appendix 2 Relation between the Smith solution and the present solution for screw dislocation by using the addition theorem 77

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