在曲線座標系統下發展一具最佳數值波數的馬克斯威爾方程數值方法

Development of an optimized numerical wavenumber Maxwell’s equation solver in curvilinear coordinates

梅嘉敏(Chia-Min Mei)

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(2011年)

https://doi.org/10.6342/NTU.2011.03078

摘要

本論文發展時域有限差分(Finite-difference time domain)方法在非正交曲線座標下求解馬克斯威爾方程式，
理論上，在時域和非交錯網格系統下，能維持零散度以及最佳波數保持特性。在時間離散上，因考慮馬斯威爾方程的漢米爾頓性質，
使用半隱式之具辛結構離散格式，在計算時間上能維持其能量守恆性質。此外針對複雜外型，使用面積比(Jacobian)保持緊緻格式來降低因座標轉換所造成之誤差。

由以上完整的時域有限差分方法，求解非正交曲線座標下之二維馬克斯威爾方程，
證實本論文所提出之求解程序的準確性及可行性。
並由測試問題得知本論文提出之格式能保持相當好的收斂斜率以及維持其能量守恆。

最後分析真實電磁波的問題，
本論文使用同軸性完美匹配層(UPML)來模擬無限域問題，
並且使用全場/散射場(TF/SF)和等位函數等數值方法求解非均勻介值之電磁問題
(二維TM模態米氏電磁波散射問題），
經由測試題目可知，本論文與前人所模擬的結果有相當好的一致性。
最後，再以相同的方法應用在其他外型之散射體問題上。

關鍵字

馬克斯威爾方程 ； 時域有限差分方法 ； 非交錯網格 ； 曲線座標 ； 零散度

並列摘要

In this thesis, a finite-difference time domain solver is presented for solving the Maxwell's equations in curvilinear coordinates.The scheme formulated in time domain and non-staggered grid system can
theoretically preserve zero-divergence condition and optimized numerical wavenumber characteristics.
To accommodate the Hamiltonian structure in the Maxwell's equations, the time integrator employed in the current semi-discretization falls into the symplectic category.
The inherent local conservation laws are also retained discretely all the time.To reduce the numerical error from the coordinate transformation in complex domain,
the Jacobian-preserving compact scheme is used in this thesis.

The integrity of the finite difference time domain method for solving the Maxwell's equations in two-dimensional curvilinear coordinates that are amenable to the exact solutions. The results with good rates of convergence are demonstrated for all the investigated problems.

For simulating wave problems on open domain, in this thesis, the Perfectly matched layer (PML),
Total-field-Scattered-field (TF/SF) and Level Set method are employed for solving scattering problems (2D (TM) Mie scattering problem). The results simulated from the proposed method agree well with other numerical and experimental results for the chosen problems.
Finally, this scheme to other scattered structure problems is applied.

並列關鍵字

finite-difference time domain ； non-staggered grids ； curvilinear coordinates ； divergence free

參考文獻

[33] J. X. Cai, Y. S. Wang, B. Jiang,

[39] C. M. Furse, S. P. Mathur, O. P. Gandi,

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equations in isotropic media, IEEE. Trans. Antenn. Propag. 1966; AP4: 302-307.

[3] B. Cockburn, F. Y. Li, C. W. Shu, Locally divergence-free discontinuous Galerkin methods for the Maxwell’s equations, J. Comput. Phys. 194, 588-610, 2004.