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研究生:徐傳硯
研究生(外文):Chuan-Yan Hsu
論文名稱:以修正型特雷夫茨法求解具非線性邊界條件之拉普拉斯方程式
論文名稱(外文):Modified collocation Trefftz method for solving Laplace equations with nonlinear boundary conditions Modified collocation Trefftz method for solving Laplace equations with nonlinear boundary conditions
指導教授:范佳銘
指導教授(外文):Chia-Ming Fan
學位類別:碩士
校院名稱:國立臺灣海洋大學
系所名稱:河海工程學系
學門:工程學門
學類:河海工程學類
論文種類:學術論文
論文出版年:2012
畢業學年度:100
語文別:中文
論文頁數:148
中文關鍵詞:修正型特雷夫茨法指數收斂純量同倫演算法非線性邊界條件拉普拉斯方程式西尼奧里尼問題熱傳導陰極保護
外文關鍵詞:modified collocation Trefftz methodexponentially convergent scalar homotopy algorithmnon-linear boundary conditionLaplace equationSignorini problemheat conductioncathode protection problem
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本論文以修正型特雷夫茨法結合指數收歛純量同倫演算法求解具非線性邊界條件之拉普拉斯方程式,此類型的物理問題包括西尼奧里尼問題、非線性材質之熱傳導問題以及陰極保護系統問題。
修正型特雷夫茨法是無網格法的其中一種,無網格法的優點在於空間離散時不需要進行網格產生和數值積分,因此能夠減少電腦運算時所需要的資源,進而有效增加電腦計算之效率;另一方面,拉普拉斯方程式的數值解能以完備基底之線性累加所表示,相較於傳統的數值方法,修正型特雷夫茨法在數值計算操作上較為簡單,且用很少點數就能夠得到良好的準確性,因此本論文採用修正型特雷夫茨法進行物理問題之空間離散。修正型特雷夫茨法透過在計算域之邊界上配點進行空間離散,使非線性邊界條件上的每一個邊界點都會形成一條非線性代數方程式,進而形成非線性代數方程組。本研究中,搭配指數收歛純量同倫演算法來求解非線性代數方程組。藉由引入虛擬時間以及純量同倫函數之概念,可以推導出指數收歛純量同倫演算法,並增加求解時的方便性與提高效率。指數收歛純量同倫演算法的使用範圍廣泛,能夠求解過定或欠定系統;對於初始猜值較不敏感並且有著全域收斂的特性;並且在求解的過程中不用計算雅可比的反矩陣;另外具有著指數收斂的特性,能夠快速降低殘差得到答案。基於以上的優點,故本論文採用此方法來求解非線性代數方程式組。
在本論文中,以數個數值算例來驗證所提出數值方法的效能及準確度。並測試數值算例中不同的參數,藉以證明方法的穩定性。在結果與比較圖中,我們可以證明本研究所提出之方法可以快速的求解具非線性邊界條件之拉普拉斯方程式,並具有工程應用之潛力。

In this thesis, the combination of modified collocation Trefftz method (MCTM) and exponentially convergent scalar homotopy algorithm (ECSHA) is proposed to analyze Laplace problems with non-linear boundary conditions. These types of physical problems include Signorini problem, heat conduction problem with material nonlinearity and cathode protection problem.
MCTM is one kind of boundary-type meshless methods and the numerical solution can be expressed by linear combination of the T-complete functions of the Laplace operator. MCTM is free from mesh and integral-free for spatial discretization. Hence, MCTM can efficiently analyze problem by using few computer resources. On the other hand, MCTM is easy operation and highly accurate by few collocation points. The spatial discretization of MCTM will result in a system of non-linear algebraic equations (NAEs). We used ECSHA to solve NAEs formed by MCTM. ECSHA is derived by concepts of fictitious time and scalar homotopy function and can solve over-determined system or under-determined system. Besides, ECSHA is insensitive to initial guess and can avoid calculating the inverse of Jacobian matrix. Finally, ECSHA is exponentially convergent.
In this thesis, we will verify the efficiency and accuracy of the combination of MCTM and ECSHA by some numerical examples. In addition, numerical experiments on testing different parameters are used to verify the stability of the proposed scheme.

誌謝 I
摘要 III
Abstract V
Chapter 1 Introduction 1
1.2 Literature Review 2
1.2.1 Mesh-dependent Methods 2
1.2.2 Meshless Methods 3
1.2.3 Exponentially Convergent Scalar Homotopy Algorithm (ECSHA) 5
1.3 Organization of the dissertation 6
1.4 Reference 7
Chapter 2 Modified Collocation Trefftz Method for Solving Signorini Problem 11
2.1 Introduction 11
2.2 Signorini Problem 14
2.2.1 Shallow Dam Problem 14
2.2.2 Electropainting Problem 16
2.3 Numerical Method 18
2.3.1 Modified Collocation Trefftz Method (MCTM) 19
2.3.2 Exponentially Convergent Scalar Homotopy Algorithm (ECSHA) 20
2.4 Numerical Results and Comparisons 22
2.4.1 Example 2.1 22
2.4.2 Example 2.2 24
2.5 Conclusions 25
2.6 References 27
Chapter 3 Modified Collocation Trefftz Method for Heat Conduction Problem with Material Nonlinearity 52
3.1 Introduction 52
3.2 Governing Equation 54
3.2.1 Quadrate Computation Domain 54
3.2.2 L-shaped Computation Domain 56
3.3 Numerical Method 57
3.3.1 Modified Collocation Trefftz Method (MCTM) 58
3.3.2 Exponentially Convergent Scalar Homotopy Algorithm (ECSHA) 58
3.4 Numerical Results and Comparisons 59
3.4.1 Example 3.1 60
3.4.2 Example 3.2 61
3.4.3 Example 3.3 62
3.5 Conclusion 63
3.6 Reference 64
Chapter 4 Modified Collocation Trefftz Method for Cathodic Protection Problem 83
4.1 Introduction 83
4.2 Cathodic Protection Problem 85
4.3 Numerical Method 87
4.3.1 Modified Collocation Trefftz Method (MCTM) 87
4.3.2 Exponentially Convergent Scalar Homotopy Algorithm (ECSHA) 88
4.4 Numerical Results and Comparisons 89
4.4.1 Example 4.1 90
4.4.2 Example 4.2 92
4.5 Conclusions 94
4.6 References 95
Chapter 5 Conclusion and Discussion 126
5.1 Conclusion 126
5.2 Discussion 129
Personal Information 130

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