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 本論文為了彌補基本解法在物理域外人工邊界的缺陷，引入兩種改良方法，即邊界配置法及正規化無網格法；這兩種數值方法不僅可以將觀測點及源點放置於真實邊界，也可以保有基本解法的精神。我們使用誤差評估技術評估兩種方法的數值誤差，藉由觀察其誤差曲線，我們可獲得最佳配置點的數目；於採取最佳配置點後，在無解析解的情況下，該方法收斂的數值解可以被獲得。此外，我們藉由多重座標系統的概念改良這個誤差評估技術，解決本誤差評估技術於求解內域及外域多孔洞問題。最後，我們舉一些數值算例來驗證誤差評估技術的準確性和穩定性；本論文將多個散射體的水波問題置於最後來驗證此誤差評估技術的應用性及可行性。
 In this thesis, two improved approach, called the boundary collocation method (BCM) and the regularized meshless method (RMM) were introduced to remedy the drawback of the controversial artificial boundary outside the physical domain of the method of fundamental solutions (MFS). Not only can both numerical methods retain the spirit of the MFS, but they can also distribute the observation and the source points on the coincident locations of the real boundary. Furthermore, we implemented a proposed error estimation technique to estimate the numerical error of the two methods. By observing the obtained error curve, we derived the optimal number of collocation points. After adopting the optimal number of collocation points, the convergent solution of problem can be obtained in unavailable analytical solution condition. In addition, we improved this error technique by the concept of the multiply coordinate systems when we treat with the interior and exterior problem with multiply holes. Finally, we took the numerical examples to verify the accuracy and stability of the proposed error estimation technique. The water wave problem with multiply scatters was taken in the final numerical example to verify the applicable and practicable of this error technique.
 Contents ITable captions IVFigure captions VNotation captions XVIAbstract XX摘要 XXIChapter 1 Introduction1.1 Motivation of the research 11.2 Organization of the thesis 3Chapter 2 The method of fundamental solutions2.1 Introduction 42.2 Problems statement 52.2.1 Boundary value problem 52.2.2 Green’s function problem 62.2.3 Water wave problem 72.3 Formulation 92.4 The proposed error estimation technique 122.4.1 Definition of an auxiliary problem 122.4.1.1 Defining the G.E., contour and B.C. type 122.4.1.2 Giving the exact solution 132.4.1.3 Specifying B.C.s 142.4.1.4 Difference between the auxiliary problem and the original problem 142.4.2 Error analysis in the auxiliary problem 152.4.3 Solving the original problem 162.5 Illustrative examples and discussions 162.6 Conclusions 23Chapter 3 The boundary collocation method3.1 Introduction 253.2 Problems statement 263.3 Formulation 263.3.1 Meshless formulation using radial basis function of the imaginary-part kernel 273.3.2 Calculation for the diagonal elements in the four influence matrices using the rule and invariant method 283.4 The proposed error estimation technique 293.5 Illustrative examples and discussions 293.6 Conclusions 33Chapter 4 The regularized meshless method4.1 Introduction 354.2 Problems statement 364.3 Formulation 364.3.1 Meshless formulation using radial basis functions 364.3.2 Derivation of diagonal coefficients of influence matrices 384.3.2.1 Diagonal coefficients of interior type 384.3.2.2 Diagonal coefficients of exterior type 394.4 The proposed error estimation technique 434.5 Illustrative examples and discussions 444.6 Conclusions 49Chapter 5 Conclusions and further research5.1 Conclusions 505.2 Further research 51ReferencesTable 2-1 The complementary solution for the Helmholtz equation 62Fig. 1-1 The frame of this thesis. 63Fig. 1-2 History of the proposed error estimation technique. 64Fig. 2-1 The problem statement of boundary value problem, (a) the interior domain with multiply holes, (b) the exterior domain with multiply holes. 65Fig. 2-2 The problem statement of Green’s function problem, (a) the interior domain with multiply holes, (b) the exterior domain with multiply holes. 66Fig. 2-3 The problem statement of water wave problem with multiply holes, (a) x-z plane, (b) x-y plane. 67Fig. 2-4 The sketch of the concept of the multiply coordinate systems. 68Fig. 2-5 The flowchart of the formulation in implementing the proposed error estimation technique for MFS. 69Fig. 2-6 Problem sketch of the case 1 of example 2-1. 70Fig. 2-7 The error analysis versus the parameter d, for case 1 of example 2-1, (a) Single-layer potential approach, (b) Double-layer potential approach. 71Fig. 2-8 R.M.S error versus the number of collocation points for case 1 of example 2-1, (a) Single-layer potential approach, (b) Double-layer potential approach. 72Fig. 2-9 The field solution of original problem for case 1 of example 2-1, (a) Single-layer potential approach, (b) Double-layer potential approach. 73Fig. 2-10 Problem sketch of the case 2 of example 2-1. 74Fig. 2-11 The error analysis versus the parameter d, for case 2 of example 2-1, (a) Single-layer potential approach, (b) Double-layer potential approach. 75Fig. 2-12 R.M.S error versus the number of collocation points for case 2 of example 2-1, (a) Single-layer potential approach, (b) Double-layer potential approach. 76Fig. 2-13 The field solution of original problem for case 2 of example 2-1, (a) Single-layer potential approach, (b) Double-layer potential approach. 77Fig. 2-14 Problem sketch of the example 2-2. 78Fig. 2-15 The error analysis versus the parameter d, for example 2-2, (a) Single-layer potential approach, (b) Double-layer potential approach. 79Fig. 2-16 R.M.S error versus the number of collocation points for example 2-2, (a) Single-layer potential approach, (b) Double-layer potential approach. 80Fig. 2-17 The field solution of original problem for example 2-2, (a) Single-layer potential approach, (b) Double-layer potential approach. 81Fig. 2-18 Problem sketch of the example 2-3. 82Fig. 2-19 The error analysis versus the parameter d, for example 2-3, (a) Single-layer potential approach, (b) Double-layer potential approach. 83Fig. 2-20 R.M.S error versus the number of collocation points for example 2-3, (a) Single-layer potential approach, (b) Double-layer potential approach. 84Fig. 2-21 The field solution of original problem for example 2-3, (a) Single-layer potential approach, (b) Double-layer potential approach. 85Fig. 2-22 Problem sketch of the example 2-4. 86Fig. 2-23 The error analysis versus the parameter d, for example 2-4, (a) Single-layer potential approach, (b) Double-layer potential approach. 87Fig. 2-24 R.M.S error versus the number of collocation points for example 2-4, (a) Single-layer potential approach, (b) Double-layer potential approach. 88Fig. 2-25 The field solution of original problem for example 2-4, (a) Single-layer potential approach, (b) Double-layer potential approach. 89Fig. 2-26 Problem sketch of the case 1 of example 2-5. (Lx=Ly=1, x0=y0=0.8) 90Fig. 2-27 The error analysis versus the parameter d, for case 1 of example 2-5, (a) Single-layer potential approach, (b) Double-layer potential approach. 91Fig. 2-28 R.M.S error versus the number of collocation points for case 1 of example 2-5, (a) Single-layer potential approach, (b) Double-layer potential approach. 92Fig. 2-29 The field solution of original problem for case 1 of example 2-5, (a) Actual solution, (b) Single-layer potential approach, (c) Double-layer potential approach. 93Fig. 2-30 Problem sketch of the case 2 of example 2-5. (Lx=Ly=1, x0=y0=0.8) 94Fig. 2-31 The error analysis versus the parameter d, for case 2 of example 2-5, (a) Single-layer potential approach, (b) Double-layer potential approach. 95Fig. 2-32 R.M.S error versus the number of collocation points for case 2 of example 2-5, (a) Single-layer potential approach, (b) Double-layer potential approach. 96Fig. 2-33 The field solution of original problem for case 2 of example 2-5, (a) FEM, (a) Single-layer potential approach, (b) Double-layer potential approach. 97Fig. 2-34 Problem sketch of case 1 of example 2-6. (r=1, θinc=0°) 98Fig. 2-35 The error analysis versus the parameter d, of case 1 of example 2-6, (a) Single-layer potential approach, (b) Double-layer potential approach. 99Fig. 2-36 R.M.S error versus the number of collocation points of case 1 of example 2-6, (a) Single-layer potential approach, (b) Double-layer potential approach. 100Fig. 2-37 The field solution of original problem of case 1 of example 2-6, (a) Single-layer potential approach, (b) Double-layer potential approach. 101Fig. 2-38 Free surface elevation around cylinder of case 1 of example 2-6. 102Fig. 2-39 Problem sketch of case 2 of example 2-6. (r/l=1/3, θinc=0°) 103Fig. 2-40 The error analysis versus the parameter d, of case 2 of example 2-6, (a) Single-layer potential approach, (b) Double-layer potential approach. 104Fig. 2-41 R.M.S error versus the number of collocation points of case 2 of example 2-6, (a) Single-layer potential approach, (b) Double-layer potential approach. 105Fig. 2-42 The field solution of original problem of case 2 of example 2-6, (a) Single-layer potential approach, (b) Double-layer potential approach. 106Fig. 2-43 Free surface elevation around cylinder (1) and (2) of case 2 of example 2-6. 107Fig. 2-44 The resultant forces on the corresponding cylinder for case 2 of example 2-6, (a) Single-layer potential approach, (b) Double-layer potential approach. 108Fig. 2-45 Problem sketch of the case 1 of case 3 of example 2-6. (r/l=0.8) 109Fig. 2-46 The error analysis versus the parameter d, for case 3(a) of example 2-6, (a) Single-layer potential approach, (b) Double-layer potential approach. 110Fig. 2-47 R.M.S error versus the number of collocation points for case 3(a) of example 2-6, (a) Single-layer potential approach, (b) Double-layer potential approach. 111Fig. 2-48 The field solution of original problem for case 3(a) of example 2-6, (a) Single-layer potential approach, (b) Double-layer potential approach. 112Fig. 2-49 Free surface elevation around cylinder (1), (2), (3) and (4) for case 3(a) of example 2-6. 113Fig. 2-50 The resultant force on the corresponding cylinder for case 3(a) of example 2-6, (a) D. V. Evans (1997), (b) Single-layer potential approach, (c) Double-layer potential approach. 114Fig. 2-51 The error analysis versus the parameter d, for case 3(b) of example 2-6, (a) Single-layer potential approach, (b) Double-layer potential approach. 115Fig. 2-52 R.M.S error versus the number of collocation points for case 3(b) of example 2-6, (a) Single-layer potential approach, (b) Double-layer potential approach. 116Fig. 2-53 The field solution of original problem for case 3(b) of example 2-6, (a) Single-layer potential approach, (b) Double-layer potential approach. 117Fig. 2-54 Free surface elevation around cylinder (1), (2), (3) and (4) for case 3(b) of example 2-6. 118Fig. 2-55 The resultant force on the corresponding cylinder for case 3(b) of example 2-6, (a) Single-layer potential approach, (b) Double-layer potential approach. 119Fig. 3-1 The flowchart of the formulation in implementing the proposed error estimation technique for BCM. 120Fig. 3-2 The error analysis versus the number of collocation point for case 1 of example 3-1. 121Fig. 3-3 The convergence rate of the error analysis versus and computational time, (a) Single-layer potential approach, (b) Double-layer potential approach. 122Fig. 3-4 The field solution of original problem for case 1 of example 3-1, (a) Single-layer potential approach, (b) Double-layer potential approach. 123Fig. 3-5 The error analysis versus the number of collocation point for case 2 of example 3-1, (a) Single-layer potential approach, (b) Double-layer potential approach. 124Fig. 3-6 The convergence rate of the error analysis versus and computational time, (a) Single-layer potential approach, (b) Double-layer potential approach. 125Fig. 3-7 The field solution of original problem for case 2 of example 3-1, (a) Single-layer potential approach, (b) Double-layer potential approach. 126Fig. 3-8 The error analysis versus the number of collocation point for example 3-2, (a) Single-layer potential approach, (b) Double-layer potential approach. 127Fig. 3-9 The convergence rate of the error analysis versus and computational time, (a) Single-layer potential approach, (b) Double-layer potential approach. 128Fig. 3-10 The field solution of original problem for example 3-2, (a) Single-layer potential approach, (b) Double-layer potential approach. 129Fig. 3-11 The error analysis versus the number of collocation point for example 3-3, (a) Single-layer potential approach, (b) Double-layer potential approach. 130Fig. 3-12 The convergence rate of the error analysis versus and computational time, (a) Single-layer potential approach, (b) Double-layer potential approach. 131Fig. 3-13 The field solution of original problem for example 3-3, (a) Single-layer potential approach, (b) Double-layer potential approach. 132Fig. 3-14 The error analysis versus the number of collocation point for example 3-4, (a) Single-layer potential approach, (b) Double-layer potential approach. 133Fig. 3-15 The convergence rate of the error analysis versus and computational time, (a) Single-layer potential approach, (b) Double-layer potential approach. 134Fig. 3-16 The field solution of original problem for example 3-4, (a) Single-layer potential approach, (b) Double-layer potential approach. 135Fig. 3-17 The error analysis versus the number of collocation point for case 1 of example 3-5, (a) Single-layer potential approach, (b) Double-layer potential approach. 136Fig. 3-18 The convergence rate of the error analysis versus and computational time, (a) Single-layer potential approach, (b) Double-layer potential approach. 137Fig. 3-19 The field solution of original problem for case 1 of example 3-5, (a) Actual solution, (b) Single-layer potential approach, (c) Double-layer potential approach. 138Fig. 3-20 The error analysis versus the number of collocation point for case 2 of example 3-5, (a) Single-layer potential approach, (b) Double-layer potential approach. 139Fig. 3-21 The convergence rate of the error analysis versus and computational time, (a) Single-layer potential approach, (b) Double-layer potential approach. 140Fig. 3-22 The field solution of original problem for case 2 of example 3-5, (a) FEM, (b) Single-layer potential approach, (c) Double-layer potential approach. 141Fig. 4-1 The flowchart of the formulation in implementing the proposed error estimation technique for RMM. 142Fig. 4-2 The convergence rate of the error analysis versus and computational time for case 1 of example 4-1. 143Fig. 4-3 The field solution of original problem for case 1 of example 4-1. 144Fig. 4-4 The convergence rate of the error analysis versus and computational time for case 2 of example 4-1. 145Fig. 4-5 The field solution of original problem for case 2 of example 4-1. 146Fig. 4-6 The convergence rate of the error analysis versus and computational time for example 4-2. 147Fig. 4-7 The field solution of original problem for example 4-2. 148Fig. 4-8 The convergence rate of the error analysis versus and computational time for example 4-3. 149Fig. 4-9 The field solution of original problem for example 4-3. 150Fig. 4-10 The convergence rate of the error analysis versus and computational time for example 4-4. 151Fig. 4-11 The field solution of original problem for example 4-4. 152Fig. 4-12 The convergence rate of the error analysis versus and computational time for case 1 of example 4-5. 153Fig. 4-13 The field solution of original problem for case 1 of example 4-5. 154Fig. 4-14 The convergence rate of the error analysis versus and computational time for case 2 of example 4-5. 155Fig. 4-15 The field solution of original problem for case 2 of example 4-5. 156Fig. 4-16 The convergence rate of the error analysis versus and computational time for case 1 of example 4-6. 157Fig. 4-17 The field solution of original problem for case 1 of example 4-6. 158Fig. 4-18 Free surface elevation around cylinder of case 1 of example 4-6. 159Fig. 4-19 The convergence rate of the error analysis versus and computational time for case 2 of example 4-6. 160Fig. 4-20 The field solution of original problem for case 2 of example 4-6. 161Fig. 4-21 Free surface elevation around cylinder (1) and (2) of case 2 of example 4-6. 162Fig. 4-22 The resultant forces on the corresponding cylinder for case 2 of example 4-6. 163Fig. 4-23 The convergence rate of the error analysis versus and computational time for case 3(a) of example 4-6. 164Fig. 4-24 Free surface elevation around cylinder (1), (2), (3) and (4) for case 3(a) of example 4-6. 165Fig. 4-25 Free surface elevation around cylinder (1), (2), (3) and (4) for case 3(a) of example 4-6. 166Fig. 4-26 The resultant force on the corresponding cylinder for case 3(a) of example 4-6, (a) D. V. Evans (1997), (b) RMM. 167Fig. 4-27 The convergence rate of the error analysis versus and computational time for case 3(b) of example 4-6. 168Fig. 4-28 The field solution of original problem for case 3(b) of example 4-6. 169Fig. 4-29 Free surface elevation around cylinder (1), (2), (3) and (4) for case 3(b) of example 4-6. 170Fig. 4-30 The resultant force on the corresponding cylinder for case 3(b) of example 4-6. 171
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 1 影響係數矩陣之對角線獲取技術應用於基本解法和正規化無網格法 2 以基本解法求解赫姆霍茲、擴散及柏格斯方程式 3 利用基本解法產生符合邊界的二維正交網格 4 Trefftz方法使用一般解求解3維Laplace方程 5 特解法結合基本解法中形狀參數變化的影響 6 正規化無網格法求解多連通邊界值問題 7 無網格法及邊界元素法於薄膜及板問題之退化尺度分析

 1 9. 翁順裕（2006），以交易成本經濟探討保險業之銀行通路－銀行保險，保險實務與制度，5（1），95-117。 2 7. 徐明松(1998)，「員工團體保險規劃實務」，空大學訊，第216期。 3 7. 徐明松(1998)，「員工團體保險規劃實務」，空大學訊，第216期。 4 5. 李瑞雲(1976)，團體保險基本型態的研究(一)，壽險季刊，第21期，頁9-15。 5 5. 李瑞雲(1976)，團體保險基本型態的研究(一)，壽險季刊，第21期，頁9-15。 6 9. 翁順裕（2006），以交易成本經濟探討保險業之銀行通路－銀行保險，保險實務與制度，5（1），95-117。 7 11. 陳武宗(1988)，「壽險行銷與管理的基本概念」，保險專刊 第14輯 8 11. 陳武宗(1988)，「壽險行銷與管理的基本概念」，保險專刊 第14輯

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