臺灣博碩士論文加值系統

本論文旨在探討及改進傳統邊界元素法之數值奇異性。首先，利用高斯通量定理及等勢能理論推導拉普拉斯(卜易松)方程式、赫姆霍茲方程式及修正型赫姆霍茲方程式的非奇異性邊界積分方程式。其次，在圓形穴室流場及方形穴室流場中，以速度渦度法、必歐沙伐定理以及拉普拉斯(卜易松)方程式的非奇異性邊界積分方程式求解二維史托克斯流場，並與解析解及其他數值方法比較。在方形穴室流場中，利用速度渦度法以及修正型赫姆霍茲方程式的非奇異性邊界積分方程式求解低雷諾數流場，並與其他數值方法比較。利用赫姆霍茲方程式的非奇異性邊界積分方程式求解出方形導波管內的電磁場分佈，並與解析解及其他數值方法比較。本文發現利用非奇異性邊界積分方程式解析一些有趣的工程問題可以得到最有效率及最精準的結果。

In this study , the non-singular boundary integral equation method was used to circumvent the numerical singularity in traditional Boundary Element Method (BEM) . First of all , the non-singular boundary integral equations for the Laplace (Poisson) equation , the Helmholtz equation and the modified Helmholtz equation were derived by Gauss flux theorem and equal-potential theory . Then , the 2D Stokes flow of a circular cavity and a square cavity was computed by the velocity-vorticity formulation , the Biot-Savart law and the non-singular boundary integral equation for the Laplace equation . The velocity-vorticity formulation and the non-singular boundary integral equation for the modified Helmholtz equation are adopted to calculate the low Reynolds number flow field in a square cavity . In a square waveguide , the distributions of electromagnetic propagation for TE and TM modes were found by the non-singular boundary integral equation for the Helmholtz equation . All of the numerical simulations are compared with analytic solutions and other numerical results . The study shows that the non-singular boundary integral equations can get the most efficient and accurate results in solving some engineering problems associated with the governing equations of the Laplace , Poisson , Helmholtz , and modified Helmholtz equations .

1. Introduction................................................1
1.1 Motivations and Objectives..............................1
1.2 Literature Review.......................................2
1.3 Content of the Thesis...................................3
2. The Formulations of the Non-Singular Boundary Integral Equation.......................................................5
2.1 The Formulation of the NSBIE for the Laplace Equation...5
2.2 The Formulation of the NSBIE for the Helmholtz Equation.9
2.3 The Formulation of the NSBIE for the Modified Helmholtz Equation......................................................13
3. Analysis of Stokes Flow by the NSBIE for the Laplace Equation and the Biot-Savart Law..............................17
3.1 Governing Equations and Boundary Conditions............17
3.2 Numerical Procedures...................................20
3.3 Numerical Results and Comparisons......................21
4. Analysis of Low Reynolds Number Flow Field by the NSBIE for
the Modified Helmholtz Equation and the Laplace Equation......33
4.1 Governing Equations and Boundary Conditions............33
4.2 Numerical Procedures...................................35
4.3 Numerical Results and Comparisons .....................37
5. Analysis of the Electromagnetic Field by the NSBIE for the
Helmholtz Equation............................................50
5.1 Governing Equations and Boundary Conditions............50
5.2 Numerical Procedures...................................56
5.3 Numerical Results and Comparisons......................59
6. Conclusions and Recommendations............................86

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