當我們要解決電磁問題時, 時域有限差分法(Finite-Difference Time Domain, FDTD)是一種非常有效率的數值模擬方法用來解決諸類問題。然而傳統的FDTD法是屬於Explicit型式的差分方程式，因此它會被Courant-Friedrich-Levy(CFL)穩定準則給限制住，也就是最小的空間網格將限制住最大的時間步階。因此若要模擬細小結構時，相對所產生的最大時間步階將使的模擬時間大幅上揚。

Weighted Laguerre Polynomials(WLP-FDTD)法利用WLP法與FDTD法做結合，使WLP-FDTD成為Implicit型式的差分方程式，也因此它可以完全的跳脫CFL穩定準則的限制，進而選擇任意的時間步階來改善整個模擬時間。本論文中我們將非均勻網格法與WLP-FDTD做結合使用，用它來模擬細小結構上將可以使模擬時間減少和降低記憶體的使用，進一步的我們再將此法從二維擴展到三維並加入有損材質於原本公式中，使得這個方法的應用更為廣泛。

When we want to solve electromagnetic problems, the Finite Difference Time Domain (FDTD) method is a very useful numerical simulation technique to solve these problems. However, the traditional FDTD method is an explicit finite-difference scheme, so the method is limited by the Courant-Friedrich-Levy (CFL) stability condition. In other words, the minimum cell size will limit the maximum time-step size in a computational domain. Therefore, while simulating structures of fine scale dimensions, it will relatively result in a prohibitively high computation time generated by the maximum time-step size.

The WLP-FDTD is based on the Weighted Laguerre Polynomials technique and the traditional FDTD algorithm. It is an implicit finite-difference equations. Therefore, it can completely avoid the stability constraint, and then improve calculation time by choosing relatively large time-step. In this thesis, we incorporate non-uniform grid method into the WLP-FDTD. By using them to simulate the structures of fine scale dimensions can reduce the computation time and memory usage. Further, we extend this method from two-dimensional to three-dimensional and add loss media into original formulations that will make the application of this method more widely.

論文審定書………………………………………………………………………………………………………………i
誌謝…………………………………………………………………………………………………………………………ii
中文摘要………………………………………………………….…..………………………………………………...iii
英文摘要………………………………………..…………………….………………………………………………...iv
第一章 序論………………………………………………………………………………………………………….1
1.1 研究背景. ..……..…………………………………………………………………………………………1
1.2 論文大綱..……………..………………………………………………………………………………....2
第二章 FDTD演算法……………………………………………………………………………………………3
2.1 馬克斯威爾方程式………………...……..………..………………………………………………….3
2.2 三維方程式.……………………………….…..………………………………….……………………….3
2.3 二維方程式…………………………………………………………………………………….….…..…..6
2.3.1 TMz模態之中央差分法與網格配置…………………………………………........8
2.4 Courant穩定準則…………………………………………………………………………….……….....8
2.5 激發源……………………………………………………………………………………………….….…....9
2.5.1 取代源…………………………………………………………………………………………..….9
2.5.2 附加源………………………………………………………………………………………..…….9
2.5.3 阻抗性電壓源……………………………………………………………………………..….10
2.6 吸收邊界條件…………………………………………………………………………………….……..10
2.6.1 Mur一階吸收邊界條件……………………………………………………………..…...11
2.7 非均勻網格之時域有限差分法………………………………………………………….…….12
2.7.1 理論…………………………………………………………………………………………..…...12
第三章 WLP-FDTD演算法………………………………………………………………………………16
3.1 緣由與目的……………………………………………………..……………………………………..…16
3.2 Explicit法和Implicit WLP-FDTD法…………………………................................17
3.2.1 Explicit方法……………………………………………………..…………………………….17
3.2.2 Implicit WLP-FDTD方法………………………………..……………………………..20
3.3 二維TM模態之WLP-FDTD…………………………………….............................…23
3.3.1 TM模態公式與DBC吸收邊界之推導……………………..…………………….24
3.3.2 Uniform二維TM wave free space之模擬………..………………………………30
3.4 二維TE模態之Non-Uniform WLP-FDTD………............................………….32
3.4.1 Non-Uniform二維TE之模擬…………………………………..……………………. 34
第四章 三維 Non-Uniform WLP之應用………………………………………………………….. 42
4.1 Non-Uniform 3D WLP公式……………………………………………………………………..42
4.2 模擬結果………………………………………………………….………………………………….….47
4.3 結合有損介質於WLP法中……………………………………………….………............. 55
4.3.1 三維WLP法於有損介質之公式修正………………………..……………………. 55
4.3.2 模擬結果…………………………………………………………………..……………………...57
4.4 隱式WLP法與ADI法之比較……………………………………….………................ 60
4.4.1 模擬結果比較…………………………………………………………..……………………...60
第五章 低CPU需求之改良式WLP法………………….………………………………………….61
5.1 修正運作基底函數之方法……………………………………………………………………...61
5.2 修正WLP方法之缺點………………………….………............................................65
5.3 修正2D TE 模態之WLP方法………………………….………...............................68
第六章 結論………………………………………………………………………………………………………..77
參考文獻………………………………………………………………………………………………………………..78

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