本文主旨為利用邊界元素法來模擬電磁場之分佈。電磁場的控制方程式為馬克斯威爾方程式，經由時間諧和場假設其可推導成赫姆霍茲方程式。此即為本文的主要控制方程式。經由邊界元素法的離散後，其應用在實際問題上，首先考慮一內插金屬的波導管，因為其外型上下左右對稱，此計算區域可簡化為四分之一。未簡化和簡化後的已知邊界條件均為零，本文將利用奇異值分解法來求解其未知的邊界條件。此外應用在外場方面，本文考慮平面波射至完全導體的散射波問題，由於邊界元素法非常適合求解外域問題，其可直接求得散射遠場的值，此為邊界元素法之一大特點。最後將計算結果與解析解和其他數值模式比較，可驗證出本文的數值模式的精確度非常良好。

In this study, we use the Boundary Element Method to simulate the electromagnetic fields. The corresponding governing equations are the Maxwell’s equations. By separation of time variable, it can be transformed to the Helmholtz equation. This is the main equation of the discretization in the numerical simulation. After the discretization is determined, it would be applied in some electromagnetic problems. First, the rectangular waveguide with a metal insert will be considered. Because the rectangular waveguide with a metal insert is symmetric, we can consider only the quarter domain. And the known boundary conditions of the waveguide are all zero, we then will use the Singular Value Decomposition (SVD) to find out the non-trivial solution. Furthermore, the homogeneous scattering problems also are considered. When the incident plane wave passes through the perfect electrical conductor, it will produce a scattering field. And the perfect electrical conductor will be considered as a cylinder in 2D and sphere in 3D. Finally, comparisons of our numerical simulations with the analytical solutions and the other numerical methods are made, and we find that our numerical model is a good tool for the analysis in the field of electromagnetic wave propagation.

CONTENTS
誌謝
摘要
Abstract
Contents
Figure Lists
Table Lists
Chapter 1 Introduction 1
1.1 Motivations and Objectives 1
1.2 Literature Review 2
1.3 Content of the Thesis 3
Chapter 2 Theoretical analysis 5
2.1 Governing Equations 5
2.2 Boundary Condition 8
2.3 Polarization 9
2.4 Vector Potentials 10
Chapter 3 Boundary element method 12
3.1 The Fundamental Solution 12
3.2 Boundary Integral Equation 13
3.3 Discretization of Boundary Integral Equation 15
3.4 Evaluation of and Matrix 17
3.5 Evaluation of the Domain Values 22
Chapter 4 Waveguide problems 23
4.1 Waveguide 23
4.2 Governing Equations and Boundary Conditions 24
4.3 Numerical Result 26
Chapter 5 Homogeneous Scattering problems 56
5.1 Governing Equation and Boundary Condition 56
5.2 Two Dimensional Scattering Problems 57
5.3 Three Dimensional Scattering Problems 75
Chapter 6 Conclusions and Recommendations 84
6.1 Conclusions 84
6.2 Recommendations 85
References 86

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