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  • 學位論文

發展一具最佳數值色散關係方程式之顯式馬克斯威爾方程的時域有限差分方法

Development of an explicit FDTD scheme with optimized dispersion relation equation for solving Maxwell's equations

指導教授 : 許文翰

摘要


本論文是發展一具最佳數值色散關係方程式之顯式馬克斯威爾方程的時域有限差分法。在等向性、均勻、線性及非色散性的前提下,進行長時間之數值模擬。為了保持內涵的漢彌爾頓結構,使用一具守恆物理量的顯式具辛(explicitSymplectic)性質之方法,在非交錯網格系統內,配合空間方法開發具一維及二維色散關係方程式的方法,並以群速度觀念以保持色散關係式Dispersion-relation-equation-preserving (DREP),在空間內發展一準確且有效率得以保持光波傳遞色散關係的非交錯有限差分格式來離散一次微分項。並將此一explicit Symplectic DREP的差分方法運用於求解空間中包含散射體之電磁問題。 藉由求解二維光電電磁波方程,證實本論文所提出之求解程序的準確性及可行性,由測試問題可知本論文所提出之格式能保有相當好的收斂斜率及能量守恆性。為了模擬無限域的問題,本文中使用了軸向完美匹配層、全場/散射場與等位函數等數值技巧,以求解包含非均勻介質之電磁問題(包括正向入射波與斜向入射波於二維TM模態米氏散射問題和二維TM模態複雜非均勻介質光子晶體波導問題),經測試題目可知本論文所提出的方法可以得到相當好的準確性,且與前人所模擬之結果均呈現相當的吻合性。

並列摘要


In this thesis an explicit FDTD scheme was developed with the optimized dispersion relation equation for solving the Maxwell's equations. Consider an isotropic, homogeneous, linear and nondispersive medium, to accommodate the Hamiltonian structure in the Maxwell's equations, the time integrator employed in the current semi-discretization needs to fall into the explicit symplectic category. Discretization of Maxwell's equations using the explicit Symplectic time integrator in non-staggered grids, one- and two-dimensional dispersion relation equations (DRE) were developed first, and then using the DRE property to preserve wave propagation character by using the concept of group velocity. Application of the explicit Symplectic-DREP FDTD method to solve the Maxwell's equations involving scatters will be verified by solving the problem in two-dimensions that is amenable to exact solutions. Results with good rates of convergence are demonstrated for the problem. For the simulation of wave problems on an open region, in this thesis the uniaxial Perfectly matched layer (UPML), Total-Field-Scattered-Field (TF/SF) and level set methods are employed for solving the scattering problems, including the two-dimensional incident wave for TM-mode Mie scattering problem, and the PC-based L-shaped waveguide problem. The results simulated from the proposed method agree well with other numerical and experimental results for the chosen problems.

參考文獻


[1] K. S. Yee, Numerical solution of initial boundary value problems involving
[2] G. Mur, Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field 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] D. E. Merewether, R. Fisher, F. W. Smith, On implementing a numeric Huygen's source scheme in a finite difference program to illustrate scattering bodies, IEEE Trans. Nuclear Science 27 (1980) 1829-1833.

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