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研究生: 陳俞廷
Yu-Ting Chen
論文名稱: 以周道積分解三維光子晶體之廣義特徵值問題
Solve three dimensional photonic crystal general eigenvalue problem by contour integral
指導教授: 黃聰明
Huang, Tsung-Ming
學位類別: 碩士
Master
系所名稱: 數學系
Department of Mathematics
論文出版年: 2014
畢業學年度: 102
語文別: 中文
論文頁數: 36
中文關鍵詞: 特徵值問題週道積分三維光子晶體平行運算
英文關鍵詞: general eigenvalue problem, contour integral, three dimensional photonic crystal, parallel computing
論文種類: 學術論文
相關次數: 點閱:87下載:8
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  • 在本文中,主要測試周道積分法(Contour integral)\cite {S2}的效能,測試並歸納出一些較好的使用原則。我們以此方法針對三維光子晶體的馬克斯威爾方程(Maxwell equation)取時諧波(time harmonic)後經過K. Yee的離散化程序\cite{T1}得到的廣義特徵值問題作效能的測試,觀察各參數間的交互作用,並且將其與E. Polizzi的FEAST\cite{E1,E2,E3}結合而提出了混合型的演算法。最後對這混合型的演算法與傳統的Lanczos作效能上的比較。文中的三維光子晶體馬克斯威爾方程包含兩種情形分別為簡易立方晶格(SC,Simple Cube)和面心立方晶格(FCC,Face Centered cube),並分為是否除去零空間(null space free)的兩種情況,於理論部分僅提供較簡易的簡易立方晶格的介紹,而我們的數值結果則著重於應用較廣的面心立方晶格。

    In this papper, we consider to solve general eigenvalue problem for three dimensional photonic crystal by contour integral, and focus on the solver's efficacy. At first, we take the time harmonic for three dimensional photonic crystal's Maxwell equation , and Discrete by Yee's scheme,then test the parameter for the sovler. We explain the implications of parameter for CIRR,and compare it with FEAST.After all, We propose a hybrid solver MLCIRR, it Combine CIRR and FEAST.

    導論 1 2 周道積分應用至瑞雷 -瑞茲投影特徵值解法 2 2.1 CIRR 的核心定理2 2.2 FEAST 演算法 4 2.3 CIRR 各項參數的意涵與作用6 2.3.1 M 7 2.3.2 L 8 2.3.3 N 9 2.3.4 積分路徑相關參數 9 2.4 混合型的 CIRR 10 3 馬克斯威爾方程與建構的矩陣 12 3.1 從馬克斯威爾方程到離散出的矩陣 14 3.2 以快速傅立葉變換得到的預處理矩陣 16 3.3 轉化為沒有零空間的問題 19 4 周道積分法與三為光子晶體的馬克斯威爾方程 22 4.1 Multi-level 的引入 23 4.2 線性系統與平移值之關聯 24 5 數值結果 25 5.1 線性系統迭代代價關係 25 5.2 移去零空間的必要性 27 5.3 參數 M 與 FEAST 的探討 27 5.4 參數 r 與 M 的交互影響 29 5.5 N 與 L 的比較 30 5.6 MLCIRR 與 Lancozs 的比較 31 6 結論 32 Reference 34

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