臺灣博碩士論文加值系統

時域有限差分法 (finite difference time domain method, FDTD) 是利用具有二階精確度的中央差分公式離散Maxwell’s 旋度方程式，可分析電磁波在空間中傳播。由於此數值方法在應用上若遇到待解空間中有尺寸很小的物件，則必須以小尺寸的網格離散待解空間；其時間增量的大小則受限於 Courant-Friedrich-Levy (CFL) 穩定性條件。以上兩種情況會耗費大量計算基記憶體與計算時間，本論文即針對時域有限差分法中此兩項缺陷提出改善之道。
對於待解空間中有小尺寸物件的情況，常見的解決方法是利用子格子技巧以小尺寸的網格離散小尺寸物件周圍的空間、大尺寸的網格離散剩餘的空間，但此方法皆具有不穩定性而使其數值精確度的提昇有限。本論文藉由有限差分法之拉普拉斯內插技巧結合時域有限差分法，再以子格子技巧處理空間中網格的分布，提出了具有高度穩定性的子格子技巧與多層子格子技巧，並藉由分析散射體問題以證實此方法可提升數值精確度與效率。
雖然交換方向隱式時域有限差分法(alternating-direction implicit FDTD, ADI-FDTD)不受CFL穩定度條件的限制並可用非均勻網格離散空間，此方法的數值精確度仍隨時間增量的增大而降低。藉由波封技巧(wave-envelope technique)之觀念的結合而成為波封交換方向隱式時域有限差分法(envelope ADI-FDTD)，已被證實在二維與三維空間中可在大的時間增量下仍具有精確的數值解。本論文即分別探討二維與三維波封交換方向隱式時域有限差分法的數值特性，並藉由分析二維波導管與三維共振腔以顯示此方法可提升數值精確度與效率。

The finite difference time domain (FDTD) method is useful to analyze the propagation of the electromagnetic wave. In FDTD method, Maxwell’s curl equations are discretized by utilizing central-difference equations with second-order accuracy, and the electric and magnetic field components are located at the suitable positions on the Yee cell. Since the cell size is restricted by the size of the smallest object in the computation domain and the maximum temporal increment is limited by the Courant-Friedrich-Levy (CFL) stability condition, the simulation of the model with fine components can cost enormous memory and computation resources. In this research, several techniques are proposed to improve the numerical accuracy and efficiency of the FDTD method.
The subgridding scheme is a useful technique which deals with models with fine components, but the instability limits the numerical resolution. By utilizing the finite-difference Laplacian interpolation scheme (FDLIS) to the FDTD method and coupling with the subgridding scheme, the new subgridding and multilevel subgridding schemes for the FDTD and FDTD(2, 4) methods were proposed, and the stability and efficiency of the method were validated by solving a scattering problem.
Although the temporal increment is free of the CFL stability condition and the computational domain can be discretized in non-uniform mesh distribution using the alternating-direction implicit FDTD (ADI-FDTD) method, the numerical error grows with the increase of the temporal increment. By coupling wave-envelope technique with the ADI-FDTD method, it is found that the envelope ADI-FDTD method in two- and three-dimensional domains maintain good numerical accuracy for a large temporal increment. In this thesis, the numerical performances of the envelope ADI-FDTD and ADI-FDTD methods are discussed in two- and three-dimensional domains, respectively. A two-dimensional waveguide problem and a three-dimensional cavity problem were also solved by the envelope ADI-FDTD, ADI-FDTD, and FDTD methods, and the solutions showed the good performance of the envelope ADI-FDTD method in this thesis.

中文摘要 i
Abstract ii
List of Contents iv
List of Tables vi
List of Figures vii
Chapter 1 Introduction 1
1.1 Research Background 1
1.2 Improved Technique for the Finite Difference Time Domain Method 2
1.2.1 Subgridding Scheme 2
1.2.2 Alternating-Direction-Implicit Technique 3
1.2.3 Wave Envelope Technique 4
1.3 Organization of the Thesis 5
Chapter 2 Finite Difference Time Domain Method 6
2.1 Introduction to FDTD and FDTD(2,4) Methods 6
2.2 Stability and Dispersion 8
2.3 Absorbing Boundary Condition 12
2.3.1 Introduction 12
2.3.2 Concept to Berenger Perfectly Matched Layer 13
2.3.3 Split-Field Formulations for the FDTD Method 18
Chapter 3 Subgridding Scheme 22
3.1 Introduction 22
3.1.1 FDTD Subgridding and Multilevel Subgridding Schemes 22
3.1.2 FDTD(2,4) Subgridding Scheme 24
3.2 Finite-Difference Laplacian-Interpolation Scheme 25
3.3 Numerical Results 26
Chapter 4 Envelope Alternating-Direction-Implicit Finite Difference Time
Domain Method 33
4.1 ADI-FDTD Method 33
4.2 Envelope ADI-FDTD Method 36
4.2.1 Envelope ADI-FDTD Method in Two-Dimensional Domain 36
4.2.2 Envelope ADI-FDTD Method in Three-Dimensional Domain 40
4.3 Numerical Results 44
4.3.1 Study of the Phase Velocity in Two-Dimensional Domain 44
4.3.2 Simulation of a Two-Dimensional Waveguide Problem 47
4.3.3 Study of the Phase Velocity in Three-Dimensional Domain 48
4.3.4 Simulation of a Three-Dimensional Cavity Problem 50
Chapter 5 Conclusions and Original Contributions 65
5.1 Thesis Conclusions 65
5.2 Original Contributions 66
5.3 Further Research 66
References 68
Curriculum Vitae 73

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