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博碩士論文 etd-0812111-171313 詳細資訊
Title page for etd-0812111-171313
論文名稱
Title
零場法與守恆型格式求解在狄利克雷與混合型邊界條件下的拉普拉斯方程
The Null-Field Methods and Conservative schemes of Laplace’s Equation for Dirichlet and Mixed Types Boundary Conditions
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
120
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
口試日期
Date of Exam
2011-06-09
繳交日期
Date of Submission
2011-08-12
關鍵字
Keywords
混合型邊界條件、諾伊曼條件、環形域、基本解、退化問題、誤差分析、狄利克雷條件、配置Trefftz方法、零場法
Neumann condition, Dirichlet condition, mixed type of boundary conditions, error analysis, degenerate kernel, fundament solutions, circular domains, collocation Trefftz method, Null field method
統計
Statistics
本論文已被瀏覽 5748 次,被下載 888
The thesis/dissertation has been browsed 5748 times, has been downloaded 888 times.
中文摘要
在論文中,定義邊界誤差來探討零場法的收斂速度,並以條件數來討論模型問題的數值穩定性。論文中也尋求應用在基本解展開當中較好的場點,發現場點的位置對收斂速度影響輕微,但對穩定性影響很大。若場點落在邊界上,則零場法得到最佳的穩定性;若場點遠離邊界,誤差會些微的變小,但條件數隨之增加,所以離邊界較近的場點是推薦選取的。
即使是拉普拉斯方程(Laplace’s equation)中的狄利克雷問題(Dirichlet problem),也會遭遇到解不存在或不唯一的情形。在零場法原始的顯式解當中由於不滿足守恆定律,可能會發生退化問題。論文中分析對零場法的退化問題,並藉由兩個特定未知數之間必須符合的關係式來得到新的守恆型演算法。守恆形式的零場法避開了退化問題,卻帶來不穩定的數值結果。此時使用超定矩陣系統(overdetermined system)和截斷奇異值分解法(truncated singular value decomposition)可以得到較好的穩定性,其中超定矩陣的演算法比截斷奇異值分解法簡單、誤差與穩定性也輕微的好一些。
對零場法提出誤差分析也是目標之一。論文中得到狄利克雷問題、諾伊曼問題(Neumann problem)的邊界誤差,並有數值實驗來支持理論分析的結束。
Abstract
In this thesis, the boundary errors are defined for the NFM to explore the convergence rates, and the condition numbers are derived for simple cases to explore numerical stability. The optimal convergence (or exponential) rates are discovered numerically. This thesis is also devoted to seek better choice of locations for the field nodes of the FS expansions. It is found that the location of field nodes Q does not affect much on convergence rates, but do have influence on stability. Let δ denote the distance of Q to ∂S. The larger δ is chosen, the worse the instability of the NFM occurs. As a result, δ = 0 (i.e., Q ∈ ∂S) is the best for stability. However, when δ > 0, the errors are slightly smaller. Therefore, small δ is a favorable choice for both high accuracy and good stability. This new discovery enhances the proper application of the NFM.
However, even for the Dirichlet problem of Laplace’s equation, when the logarithmic capacity (transfinite diameter) C_Γ = 1, the solutions may not exist, or not unique if existing, to cause a singularity of the discrete algebraic equations. The problem with C_Γ = 1 in the BEM is called the degenerate scale problems. The original explicit algebraic equations do not satisfy the conservative law, and may fall into the degenerate scale problem discussed in Chen et al. [15, 14, 16], Christiansen [35] and Tomlinson [42]. An analysis is explored in this thesis for the degenerate scale problem of the NFM. In this thesis, the new conservative schemes are derived, where an equation between two unknown variables must satisfy, so that one of them is removed from the unknowns, to yield the conservative schemes. The conservative schemes always bypasses the degenerate scale problem; but it causes a severe instability. To restore the good stability, the overdetermined system and truncated singular value decomposition (TSVD) are proposed. Moreover, the overdetermined system is more advantageous due to simpler algorithms and the slightly better performance in error and stability. More importantly, such numerical techniques can also be used, to deal with the degenerate scale problems of the original NFM in [15, 14, 16].
For the boundary integral equation (BIE) of the first kind, the trigonometric functions are used in Arnold [3], and error analysis is made for infinite smooth solutions, to derive the exponential convergence rates. In Cheng’s Ph. Dissertation [18], for BIE of the first kind the source nodes are located outside of the solution domain, the linear combination of fundamental solutions are used, error analysis is made only for circular domains. So far it seems to exist no error analysis for the new NFM of Chen, which is one of the goal of this thesis. First, the solution of the NFM is equivalent to that of the Galerkin method involving the trapezoidal rule, and the renovated analysis can be found from the finite element theory. In this thesis, the error boundary are derived for the Dirichlet, the Neumann problems and its mixed types. For certain regularity of the solutions, the optimal convergence rates are derived under certain circumstances. Numerical experiments are carried out, to support the error made.
目次 Table of Contents
Abstract i
1 The Null-Field Method 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Explicit Algorithms of Null-Field Methods . . . . . . . . . . . . . . . . . 2
1.2.1 Basic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Explicit Algebraic Equations . . . . . . . . . . . . . . . . . . . . 5
1.2.3 Explicit Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.4 Validity for the NFM as∈, ‾∈ → 0 . . . . . . . . . . . . . . . . . . 11
1.3 Stability Analysis of the NFM for Simple Cases . . . . . . . . . . . . . . 15
1.4 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4.1 Definitions of Boundary Errors . . . . . . . . . . . . . . . . . . . 22
1.4.2 Model Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.4.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2 Conservative Schemes and Degenerate Scale Problems 39
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2 Conservative Schemes of Null-Field Methods . . . . . . . . . . . . . . . . 40
2.3 Pitfalls of Degenerate Scale Problems . . . . . . . . . . . . . . . . . . . . 42
2.3.1 Interior Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.3.2 Exterior Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3.3 Circular Boundaries with the Same Origins . . . . . . . . . . . . . 45
2.3.4 Circular Boundaries with Different Origins . . . . . . . . . . . . . 46
2.3.5 Further Discussions for the Pitfall Nodes . . . . . . . . . . . . . . 48
2.4 Renovated Solution Methods to Retain Good Stability . . . . . . . . . . 59
2.4.1 Rationale of numerical Instability of the Conservative Schemes . . 59
2.4.2 Truncated Singular Value Decomposition . . . . . . . . . . . . . . 62
2.5 Numerical Results and Concluding Remarks . . . . . . . . . . . . . . . . 65
3 Neumann and Mixed Types of Boundary Conditions 77
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.2 Second Kinds of NF Equations . . . . . . . . . . . . . . . . . . . . . . . . 78
3.2.1 Explicit Algebraic Equations . . . . . . . . . . . . . . . . . . . . . 78
3.2.2 Validity of the Second Kind NF Equations for ∈ → 0 . . . . . . . 84
3.3 The Interior Field Method for Neumann Problems . . . . . . . . . . . . . 85
3.3.1 Explicit Interior Equations . . . . . . . . . . . . . . . . . . . . . . 85
3.3.2 Relations to the NFM . . . . . . . . . . . . . . . . . . . . . . . . 88
3.4 Collocation Trefftz Methods . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.5 Numerical Experiments by First Kind Field Equations . . . . . . . . . . 91
3.6 Numerical Experiments by Second Kind Field Equations . . . . . . . . . 94
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