臺灣博碩士論文加值系統

本研究提出一個對應一維馬克斯威爾方程式的含特殊外力項晶格波茲曼方法，用來模擬電磁波在色散介質中的傳播行為。而隨時間變化的色散效果可由對頻域下的電容率作反傅立葉轉換來得到並藉由一等效外力項的形式展現在演化方程式中。藉由Chapman-Enskog多尺度分析，我們可以確保提出的外力項方法可以在數學上與目標的馬克斯威爾方程式連結起來。而數值的精確度則是透過將晶格波茲曼法求得的結果與時域有限差分法相比較來確立。結果顯示不論是空氣-水介面的反射係數或者真空-電漿介面的反射係數、穿透係數在各個頻率都與解析解相吻合。因此，此方法可以用來描述德拜、杜德甚至勞倫茲等模型的色散介質的電磁波傳播行為。

An extended lattice Boltzmann modeling with special forcing terms for one-dimensional Maxwell equations exerting on a dispersive medium is presented in this thesis. The time dependent dispersive effect is obtained by the inverse Fourier transform of the frequency-domain permittivity and is incorporated into the evolution equations of LBM via an equivalent forcing effect. The Chapman-Enskog multi-scale analysis is employed to make sure the proposed scheme is mathematically consistent with the targeted Maxwell’s equations. The numerical accuracy was then confirmed by comparing the LBM results with those from the FDTD. Results show that the numerical values for the frequency-dependent reflection coefficients at the air/water interface as well as the reflection and transmission coefficients at the vacuum/plasma interface obtained by these two methods were all in excellent agreement compared with the exact solutions. The present model can be used for dispersive media described by the Debye, Drude and Lorentz models.

誌謝 i
中文摘要 ii
Abstract iii
目錄 iv
圖目錄 vii
表目錄 ix
符號表 x
Chapter 1 緒論 1
1.1 研究背景與動機 1
1.2 論文架構 2
Chapter 2 電磁波理論與色散介質 3
2.1 古典電磁波理論 3
2.2 色散介質及其種類 4
2.2.1 色散介質基本介紹 4
2.2.2 色散介質的種類 5
2.3 色散介質之傳統數值方法回顧 6
2.3.1 時域有限差分法 7
2.3.2 時域有限元素法 7
2.4 晶格波茲曼方法應用於波動方程式及電磁波方程式 8
2.4.1 波動方程式 8
2.4.2 電磁波方程式 10
2.5 小結 13
Chapter 3 晶格波茲曼方法之電磁波於色散介質傳輸機制 14
3.1 波茲曼傳輸方程式 14
3.2 晶格波茲曼方法 15
3.3 電磁波晶格波茲曼法 17
3.4 特殊外力項 18
Chapter 4 數值模擬方法 21
4.1 初始條件、邊界條件 21
4.2 程式流程 22
4.3 程式驗證 24
Chapter 5 研究結果與討論 30
5.1 外力項之離散 30
5.2 電磁波在德拜介質中的傳播情形 33
5.3 模擬電磁波在杜德介質中的傳播情形 39
5.4 電磁波頻率對於脈衝與固體電漿作用結果之討論 44
Chapter 6 結論與未來工作 48
6.1 結論 48
6.2 未來工作 49
參考文獻 50
論文口試問題總整理 52

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