在三維色散介質中模擬電磁波的傳遞行為

Prediction of electromagnetic wave propagation in three-dimensional dispersive media

王豫潔(Yu-Chieh Wang)

指導教授 ： 許文翰

國立臺灣大學/工學院/工程科學及海洋工程學研究所/碩士(2014年)

https://doi.org/10.6342/NTU.2014.02127

摘要

本論文是在交錯網格上發展一三維時域有限差分法(FDTD)，以求解馬克斯威爾方程。本文的方法是在時域內，在滿足電場和磁場的零散度條件(亦即高斯定律)的架構下求解法拉第定律和安培定律。所提出的數值方法於時間離散方面使用具辛結構(Symplectic)之二級二階之Runge-Kutta方法，在經過長時間模擬後，解仍得以保持馬克思威爾方程的能量守恆性質；另透過法拉第及安培旋度方程空間微分項的推導，以期求得在色散關係上相當準確的解。

為了達到最佳數值色散性質，本文所提出的數值方法能在時間上和空間上保有相當好的收斂，且能有效地減少實解相速度與數值相速度的誤差，而得以顯著地降低了因時域有限差分所造成的明顯地數值色散誤差以及各向異性誤差。本研究證實了所提出的數值方法在具辛結構與色散關係上具有良好的保持性，尤其在針對經長時間馬克斯威爾方程的數值模擬後，其效果尤為顯著。本文亦在三種典型具代表性之色散介質Debye、Lornetz、Drude模型電磁波的模擬，透過計算，證明了文中所發展之數值方法於馬克斯威爾方程組在與頻率獨立和頻率相依上的有效性及在長時間模擬下的準確度。

關鍵字

馬克斯威爾方程 ； 交錯網格 ； 色散關係 ； 實解相速度和數值相速度 ； Debye介質 ； Lorentz介質 ； Drude介質

並列摘要

An explicit finite-difference scheme for solving the three-dimensional Maxwell's equations in staggered grids is presented in time domain. The aim of this thesis is to solve the Faraday's and Ampere's equations in time domain within the discrete zero-divergence context for the electric and magnetic fields (or Gauss's law). The local conservation laws in Maxwell's equations are also numerically preserved all the time using proposed the explicit second-order accurate symplectic partitioned Runge-Kutta temporal scheme.
Following the method of lines, the spatial derivative terms in the semi-discretized Faraday's and Ampere's equations are then properly discretized to get a dispersively very accurate solution. To achieve the goal of getting the best dispersive characteristics, this centered scheme minimizes the difference between the exact and numerical phase velocities with good rates of convergence are demonstrated for the problem. The significant dispersion and anisotropy errors manifested normally in finite difference time domain methods are therefore much reduced. The dual-preserving (symplecticity and dispersion relation equation) wave solver is numerically demonstrated to be efficient for use to get in particular long-term accurate Maxwell's solutions. The emphasis of this study is also placed on the accurate modelling of EM waves in the dispersive media of the Debye, Lorentz and Drude types. Through the computational exercises, the proposed dual-preserving solver is computationally demonstrated to be efficient for use to predict the long-term accurate Maxwell's solutions for the media of frequency independent and dependent types.

並列關鍵字

staggered grids ； dispersion relation equation ； exact and numerical phase velocities ； Debye ； Lorentz ； Drude

參考文獻

[1] K. S. Yee, Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media, IEEE Trans. Antennas Propagat. AP4 (1966) 302-307.

[2] G. Mur, Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-eld equations, IEEE Trans. Electromagnetic Compatibility 23 (1981) 377-382.

[3] J. P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Com. Physics 114 (1994) 185-200.

[4] J. P. Berenger, Perfectly matched layer for the FDTD solution of wavestructure interaction problems, IEEE Trans. Antennas Propagat. 44 (1996) 110-117.

[5] J. A. Roden and S. D. Gedney, Convolutional PML(CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media, Microwave Optical Tech. Lett. 27 (2000) 334-339.