眾所周知的，時域有限差分法用來解決電磁的問題是一個很好的分析工具。若在同一模擬空間內，有一結構的尺寸遠小於其他結構，為了描述此一結構，必需把空間網格切小，且由於穩定條件的關係，單位時間步階的設定亦需同時變小。由於空間網格與時間步階的縮小，相對的便需要使用較多記憶體及計算時間。因此便有一些數值方法的提出來克服這些問題，比如次網格法和非均勻網格法。

本論文提出一個有效率的方法來產生 FDTD次網格的方程式。我們建構一個二階的巨集模型來取代傳統FDTD的次網格區域，再利用模型階數減縮法(Model Order Reduction)將原始的矩陣減縮，並轉換成新的FDTD的方程式。此方法的另一個優點是，當同一物件重覆的出現在模擬空間中時，減縮模型只需計算一次，便可將得到的減縮模型使用於其他重覆出現的區域。

It is well known that the finite difference time domain (FDTD) method is a powerful numerical analysis tool for solving electromagnetic problems. In a simulated area, in order to discretize an object which is much smaller than the others, a very small space increment is needed and hence the time step should be decreased too for stability consideration in traditional FDTD. The small space and time increments will respectively increase the memory requirement and calculation time. To overcome these problems, some numerical methods were developed, such as the subcell and nonuniform grid method, to handle the small feature size.

This thesis describes an efficient method for generating FDTD subcell equations. We construct a second order macromodel system instead of the subcell region in conventional FDTD. The macromodel system can be reduced with model order reduction techniques (MOR) and then translated into new FDTD update equations. When the problem contains several objects of the same size and material properties, the MOR subcell has the advantage of reusability. This means that the reduce-order model of the object needs to be generated only once nonetheless can be applied to every position where the objects originally occupied.

目錄……………………………………………………………..……………………IV
圖表目錄…………………………………………………………….………………..V
第一章 序論…………………………………………………………………………..1
1-1 概述………………………………………………………………………….1
1-2 論文大綱……………………………………………………………………3
第二章 模型階數減縮法 ………………………………………………………..…4
2-1模型階數減縮法原理 ………………………………………..…………….5
2-1-1相似轉換及其不變性…………………………………………..…….7
2-1-2 Krylov 子空間……… ………………………..………..……………. 7
2-2-Lanczos 和Arnoldi演算法 ……………………………...………………10
2-2-1 Lanczos演算法 …………..……………………………………….10
2-2-2 Arnoldi演算法 ………..……..……………………………………. 12
2-3 Laguerre-SVD 演算法………………………………………...…………...13
2-4 二階 Krylov 子空間………………………………………...….………...15
2-4-1效率化節點階數減縮法(ENOR)…………………………...….……….16
2-4-2應用二階Arnoldi 法的被動式階數減縮(SAPOR) …………….……17
第三章 模型階數減縮法結合FDTD……...…………………..……………………20
3-1模型階數減縮法建立次網格 …………………………………………….21
3-2 模型階數減縮法的應用……………………………………...……………27
第四章 模型階數減縮法的改進…………...……………………..…………………35
4-1 第一次修改 ………………………………………………………………35
4-2 第二次修改 ………………………………………………………………44
第五章 結論………………………………………………………………………...50
參考文獻……………………………………………………………………………..51

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