本文提出了一種求解滿足拉普拉斯(Laplace)方程二維邊界問題的無網格邊界積分方程方法(Meshfree Boundary Integral Equation Method)來解決內域勢能問題和反平面力場外域問題，此方法與需要生成元素網格的傳統邊界元方法不同，本方法只需要邊界節點。在處理柯西主值奇異積分和固體角的計算時，引入了邊界點局部正確解的邊界積分方程。該局部正確解必須滿足三個條件，其中包括1.場解要滿足拉普拉斯方程式 2.邊界點的原始場量 3.其對邊界點場量的法向導數，因此，局部正確解是邊界物理量與滿足二維拉普拉斯方程對應形狀函數的線性組合。透過減去原問題的邊界積分方程式和局部正確解的邊界積分方程式，可技巧性地計算柯西主值奇異積分，也可以免去計算邊界點上的固體角。計算上述的邊界積分方程式只使用了一次高斯積分，所以邊界積分方程式只不過是一個代數方程式，這些邊界節點就是高斯積分點，這就是為什麼在本方法中只需要邊界節點的原因，同時它們也是獲得聯立方程式的配置點，這個想法還可以保留數值方法的靈活性，因此它適用於任何幾何形狀。總之，本方法有兩個優點，一種是無網格化，另一種是不用主值計算奇異積分。最後，本文考慮了一些例子，如壩基滲流問題、穩態熱傳導問題和受反平面剪切應力中的包含孔洞/剛性夾雜物的無限域問題，以檢驗此無網格邊界積分方程方法的有效性。

In this thesis, a meshfree boundary integral equation method is proposed to solve 2D boundary value problems for the Laplace equation. Both interior potential problems and exterior problems under anti-plane shear are considered. Only boundary nodes are required for the present method different from the conventional boundary element method that needs to generate the mesh. To technically deal with the Cauchy principal value and the solid angle, a boundary integral equation of a local exact solution for a boundary point is introduced. This local exact solution must satisfy three conditions, including Laplace equation for the domain point, original boundary datum and its normal derivative on a boundary point. Therefore, a local exact solution is a linear combination of boundary data with corresponding shape functions which satisfy the 2D Laplace equation. By subtracting the boundary integral equation of original problem and boundary integral equation of a local exact solution, the singular integral in the sense of the Cauchy principal value can be technically determined. Free of calculating the solid angle for the boundary point is also gained. Then, the Gaussian quadrature is employed only once for the above boundary integral equation, the boundary integral equation is nothing but an algebraic equation. This is the reason why only boundary nodes are required in the present method. Those boundary nodes are just the Gaussian quadrature points. Simultaneously, they are also the collocation points to obtain the simultaneous equation. This idea can also preserve the flexibility of numerical method, hence it is suitable for any geometry shape. In a word, there are two advantages in the present method. One is meshfree. The other is free of calculating singular integral by using the sense of principal value. Finally, some examples such as a seepage problem with a dam foundation, steady heat conduction problems and infinite plane problems containing a hole/rigid inclusion in anti-plane shear are considered to examine the validity of the present meshfree boundary integral equation method.

目錄
目錄	I
圖目錄	III
表目錄	VII
第一章 緒論	1
1.1	研究動機與目的	1
1.2	文獻回顧	1
1.2.1	內域勢能問題	1
1.2.2	外域反平面力場問題	2
1.2.3	零場邊界積分方程法	4
1.2.4	局部正確解	5
1.3 論文架構	5
第二章 勢能場問題	7
2.1 問題描述	7
2.2邊界積分方程式	10
2.3 解析計算奇異積分	13
2.3.1局部正確解	13
2.3.2 積分方程式轉換代數方程	16
2.3.3 局部正確解處理近乎奇異積分	18
2.4 數值例題	19
2.4.1 矩形領域的流場	19
2.4.2 三角形領域	20
2.4.3 圓形領域	21
2.4.3 三葉草形狀領域	22
2.4.4 大壩滲流問題	23
2.5 數值結果探討	24
2.5.1 矩形問題	25
2.5.2 三角形問題	25
2.5.3 圓形問題	26
2.5.4 三葉草圖形問題	27
2.5.5 大壩滲流問題	27
第三章 反平面力場	40
3.1 問題描述	40
3.2 無網格邊界積分方程法求解外域問題	42
3.3 應力集中因子	46
3.4 數值結果與討論	47
3.4.1 圓形邊界問題	48
3.4.2 橢圓形邊界問題	53
3.4.3 正三角形問題	57
3.4.4 正方形邊界問題	62
3.4.5 五角星形邊界問題	66
3.4.6 雙圓孔洞問題	68
第四章 結論與未來展望	85
4.1 結論	85
4.2 未來展望	86
參考文獻	87
  
圖目錄
圖1.1 論文架構圖	6
圖2.1 流體控制體積示意圖	7
圖2.2 積分路徑的分解	11
圖2.3 矩形流場示意圖	19
圖2.4 三角形流場示意圖	20
圖2.5 圓形流場示意圖	21
圖2.6 三葉草圖形數值問題示意圖	22
圖2.7 滲流問題水頭差示意圖	23
圖2.8 高斯積分點位分佈圖	28
圖2.9 矩形積分點位分佈圖	28
圖2.10 矩形邊界的高斯積分分佈圖	29
圖2.11 矩形邊界的矩形積分分佈圖	29
圖2.12 三角形邊界的高斯積分分佈圖	30
圖2.13 三角形邊界的矩形積分分佈圖	30
圖2.14 矩形問題使用高斯積分法的勢能等高線圖	31
圖2.15矩形問題使用矩形積分法的勢能等高線圖	31
圖2.16 矩形問題正確解的勢能等高線圖	32
圖2.17 三角形問題使用高斯積分法的勢能等高線圖	32
圖2.18 三角形問題使用矩形積分法勢能等高線圖	33
圖2.19 三角形問題的有限元素法結果	33
圖2.20 三角形問題的有限元素法網格分佈(元素數32256，節點數16369)	33
圖2.21 圓形問題使用高斯積分法的勢能等高線圖	34
圖2.22 圓形問題使用矩形積分法的勢能等高線圖	34
圖2.23 圓形問題使用高斯積分佈點的平均相對誤差收斂分析圖	35
圖2.24圓形問題使用矩形積分佈點的平均相對誤差收斂分析圖	35
圖2.25 圓形問題使用高斯積分佈點的最大絕對誤差收斂分析圖	35
圖2.26圓形問題使用矩形積分佈點的最大絕對誤差收斂分析圖	35
圖2.27 不同數值積分法的平均誤差比較圖	35
圖2.28 圓形問題其邊界t(s)的絕對誤差比較圖(採一段路徑)	36
圖2.29圓形問題其邊界t(s)的絕對誤差比較圖(採上下段路徑)	36
圖2.30 三葉草圖形問題使用高斯積分法的勢能等高線圖	37
圖2.31三葉草圖形問題使用矩形積分法的勢能等高線圖	37
圖2.32三葉草圖形問題的勢能等高線圖	37
圖2.33 未處理近乎奇異積分的勢能等高線圖	38
圖2.34 已處理近乎奇異積分的勢能等高線圖	38
圖2.35 大壩滲流問題使用有限元素法的結果	38
圖2.36 大壩滲流問題文獻結果[2]	39
圖3.1 無限域中含有單一孔洞的反平面力場問題示意圖	40
圖3.2 圓形孔洞問題疊加示意圖	41
圖3.3 外域邊界值問題示意圖	43
圖3.4 含圓形孔洞/剛性夾雜的無限平面示意圖	48
圖3.5含圓形邊界反平面力場問題的位移等高線圖	49
圖3.6含圓形邊界反平面力場問題的主應力場等高線圖	50
圖3.7含圓形邊界反平面力場問題的邊界位移分佈圖(孔洞，剪切應力在x方向時)	51
圖3.8含圓形邊界反平面力場問題的邊界位移分佈圖(孔洞，剪切應力在y方向時)	51
圖3.9 圓形邊界反平面力場問題的應力集中因子分佈圖(孔洞，剪切應力在x方向時)	51
圖3.10圓形邊界反平面力場問題的應力集中因子分佈圖(孔洞，剪切應力在y方向時)	52
圖3.11 圓形邊界反平面力場問題的應力集中因子分佈圖(剛性夾雜，剪切應力在x方向時)	52
圖3.12圓形邊界反平面力場問題的應力集中因子分佈圖(剛性夾雜，剪切應力在y方向時)	52
圖3.13 含橢圓形孔洞/剛性夾雜的無限平面示意圖	53
圖3.14 含橢圓形邊界反平面力場問題的位移等高線圖	54
圖3.15含橢圓形邊界反平面力場問題的主應力場等高線圖	55
圖3.16含橢圓形邊界反平面力場問題的邊界位移分佈圖	56
圖3.18 含正三角形形孔洞/剛性夾雜的無限平面示意圖	57
圖3.19 含正三角形邊界反平面力場問題的位移等高線圖	58
圖3.20 含正三角形邊界反平面力場問題的主應力場等高線圖	59
圖3.21 含正三角形邊界反平面力場問題的剪應力場τxz等高線圖	60
圖3.22 含正三角形邊界反平面力場問題的剪應力場τyz等高線圖	60
圖3.23 正三角形孔洞例題的邊界切向應力分佈圖	61
圖3.24 含正方形孔洞/剛性夾雜的無限平面示意圖	62
圖3.25含正方形邊界反平面力場問題的位移等高線圖	63
圖3.26 含正方形邊界反平面力場問題的主應力場等高線圖	64
圖3.27 正方形孔洞例題的邊界切向應力分佈圖	65
圖3.28 含五角星形孔洞/剛性夾雜的無限平面示意圖	66
圖3.29含五角星形邊界反平面力場問題的位移等高線圖	67
圖3.30 含五角星形邊界反平面力場問題的主應力場等高線圖	67
圖3.31 無限域中含有雙圓孔洞的反平面力場問題示意圖	68
圖3.32 無限域中含有雙圓孔洞之座標幾何關係[30]	68
圖3.33無限域中含有任意雙圓孔洞之座標幾何關係	69
圖3.34 含雙圓邊界反平面問題的位移等高線圖(d=2)	71
圖3.35含雙圓邊界反平面問題的位移等高線圖(d=0.1)	71
圖3.36 含雙圓邊界反平面問題的位移等高線圖(d=0.01)	72
圖3.37含雙圓邊界反平面問題的位移等高線圖(d=0)	72
圖3.38含雙圓邊界反平面力場問題的主應力場等高線圖(d=2)	73
圖3.39含雙圓邊界反平面力場問題的主應力場等高線圖(d=0.1)	73
圖3.40含雙圓邊界反平面力場問題的主應力場等高線圖(d=0.01)	74
圖3.41含雙圓邊界反平面力場問題的主應力場等高線圖(d=0)	74
圖3.42孔洞1的應力集中因子分佈圖(剪切應力在x方向)	75
圖3.43使用更多點數時孔洞1的應力集中因子分佈圖(剪切應力在x方向)	76
圖3.44孔洞2的應力集中因子分佈圖(剪切應力在x方向)	77
圖3.45使用更多點數時孔洞2的應力集中因子分佈圖(剪切應力在x方向)	78
圖3.46孔洞1的應力集中因子(剪切應力在y方向)	79
圖3.47使用更多點數時孔洞1的應力集中因子分佈圖(剪切應力在y方向)	80
圖3.48孔洞2的應力集中因子(剪切應力在y方向)	81
圖3.49使用更多點數時孔洞2的應力集中因子分佈圖(剪切應力在y方向)	82
圖3.50 x1,y1=(-1.5,3)及x2,y2=(0,0)時的位移等高線圖	83
圖3.51 x1,y1=(-1.5,3)及x2,y2=(0,0)時的主應力等高線圖	84

表目錄
表3-1 正方形孔洞例題的邊界最大主應力與積分總點數比較表	64

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