線性彈性力學在角點的奇異解的性質及精確解在理論和計算上是至關重要的。在這篇論文中，我們尋求固定（位移）邊界條件，無應力（外力）邊界條件和混合型邊界條件的各種角點奇異解，並詳細探討他們的角點奇異性及提供演算法和誤差分析。我們推導出線性彈性力學的奇異解，和一些在角點和裂縫帶奇異性的新模型。有效的數值方法，如collocation Trefftz methods（CTM），基本解法（MFS），特解法（MPS）及它們的混合型，在線性彈性力學問題中，這樣的解在研究其它奇異問題的數值方法是很有用處的。本論文由三部分組成，第一部分：基本方法，第二部分：進階研究，和第三部分：位移和外力條件的混合類型。第一和第二部分已發表在[47,82]。在第一部分，我們藉由CTM 獲取高精度的解，其中首項係數在雙精度計算下可達到14（或13）的有效數字。在第二部分中我們設計了對稱和反對稱的模型，並使用基本解結合了少量特解的方法。這種相結合的方法，在計算角點（即L 形區域）是一種重要的線性彈性力學，因為在近角點時連續平滑之基本解無法使角點數值再精確，況且奇異解只有透過數值計算尋求r^(ν_K)的次方項v_K來得到。因此，只用少數幾個奇異解就可大大簡化了數值算法的複雜度。第三部分延續第一和第二部分，發展出混合位移和無外力邊界條件的模型，據我們所知，這是線性彈性力學第一次在角點附近邊界條件是混合類型所提出的特解，並報告他們的數值計算結果。本論文探討在角點的奇異性及其數值方法，並且形成線性彈性力學基本理論和進階的計算的一個系統研究。

The singular solutions for linear elastostatics at corners are essential in both theory and computation. In this thesis, we seek new singular solutions for corners with the fixed (displacement), the free stress (traction) boundary conditions, and their mixed types, and to explore their corner singularity and provide the algorithms and error estimates in detail. The singular solutions of linear elastostatics are derived, and a number of new models of corner and crack singularity are proposed. Effective numerical methods, such as the collocation Trefftz methods (CTM), the method of fundamental solutions (MFS), the method of particular solutions (MPS) and their combinations: the so called combined method, are developed. Such solutions are useful to examine other numerical methods for singularity problems in linear elastostatics. This thesis consists of three parts, Part I: Basic approaches, Part II: Advanced topics, and Part III: Mixed types of displacement and traction conditions. Contents of Parts I and II have been published in [47,82]. In Part I, the collocation Trefftz methods are used to obtain highly accurate solutions, where the leading coefficient has 14 (or 13) significant digits by the computation with double precision. In part II, two more new models (symmetric and anti-symmetric) of interior crack singularities are proposed, for the corner and crack singularity problems, the combined methods by using many fundamental solutions, but by adding a few singular solutions are proposed. Such a kind of combined methods is significant for linear elastostatics with corners (i.e., the L-shaped domain), because the singular solutions can only be obtained by seeking the power νk of rνk numerically. Hence, only a few singular solutions used may greatly simplify the numerical algorithms; Part III is a continued study of Parts I and II, to explore mixed type of displacement and free traction boundary conditions. To our best knowledge, this is the first time to provide the particular solutions near the corner with mixed types of boundary conditions and to report their numerical computation with different boundary conditions on the same corner edge in linear elastostatics. This thesis explores corner singularity and its numerical methods, to form a systematic study of basic theory and advanced computation for linear elastostatics.

PART I. Basic Approaches 9
1 Linear Elastostatics Problems in 2D 11
1.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Traction Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 Particular Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Singular Solutions near Corners under Fixed Boundary Conditions 16
3 Model A of Crack Singularity and its Numerical Solutions 24
3.1 Model A of Crack Singularity . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 The Collocation Trefftz Method . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4 Singular Solutions near Corners under Free Stress Boundary Conditions
32
5 Model B of Crack Singularity and its Numerical Solutions 39
5.1 Model B of Crack Singularity . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6 Leading Powers ν_k of the Corner Solutions O(r^(ν_k)) 44
6.1 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.2 Numerical Results for Θ = π/4 , 3π/4 . . . . . . . . . . . . . . . . . . . . . . . 45
7 Concluding Remarks (I) 49
PART II. Advanced Topics 51
8 Complex Presentations of Solutions and Stress 53
8.1 Particular Solutions near Corners . . . . . . . . . . . . . . . . . . . . . . 57
9 Models of Crack Singularity 63
10 Symmetric Model C of Crack Singularity 65
10.1 The Collocation Trefftz Method . . . . . . . . . . . . . . . . . . . . . . . 66
11 Anti-Symmetric Model D of Crack Singularity 69
11.1 Equivalence to the Complex Solutions in Piltner [62] . . . . . . . . . . . 70
12 Fundamental Solutions 73
12.1 Equivalence to Complex FS . . . . . . . . . . . . . . . . . . . . . . . . . 75
12.2 Proof of Fundamental Solutions . . . . . . . . . . . . . . . . . . . . . . . 76
13 The Combined Trefftz Method with Many FS but a Few Singular PS 80
13.1 Combined Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
13.2 Numerical Results for Model C . . . . . . . . . . . . . . . . . . . . . . . 82
13.3 Numerical Results for Model D . . . . . . . . . . . . . . . . . . . . . . . 88
13.4 A Brief Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
14 Concluding Remarks (II) 93
PART III. Mixed Tpyes of Displacement and Traction Conditions. 95
15 Singularity near Corners with Mixed Types of Boundary Conditions 97
15.1 The Powers ν_k in r^(ν_k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
15.2 Special Cases with Θ = lπ/2 . . . . . . . . . . . . . . . . . . . . . . . . . . 103
16 Explicit Particular Solutions and Two Singularity Models 108
16.1 Explicit Particular Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 108
16.2 Models of Crack Singularity . . . . . . . . . . . . . . . . . . . . . . . . . 111
17 Numerical Experiments 113
17.1 Collocation Trefttz Method . . . . . . . . . . . . . . . . . . . . . . . . . 113
17.2 Computed Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
17.3 Conjugate Particular Solutions . . . . . . . . . . . . . . . . . . . . . . . . 122
17.4 Intensity Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
18 Mixed Types I and II 127
19 Mixed Types III and IV 140
20 Comparisons and New Discoveries 147
20.1 Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
20.2 New Discoveries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
21 Numerical Experiments 152
21.1 Collocation Trefttz Method . . . . . . . . . . . . . . . . . . . . . . . . . 152
21.2 Computed Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
22 Concluding Remarks (III) 165

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