在GPU上對於14種布拉菲晶格的三維馬克斯威爾方程的計算

Computing Maxwell's equations for 14 Bravais lattice on GPUs

陳子為(Tze-Wei Chen)

指導教授 ： 林文偉

國立交通大學/理學院/數學建模與科學計算研究所/碩士(2018年)

摘要

研究光子晶體的能代結構是現今很重要的課題，其中，尋找光子能隙是了解光子晶體特性很重要的關鍵。事實上，我們可以藉由解馬克斯威爾方程去算出光子晶體的能帶結構，進而找出光子能隙，依著Yee’s scheme和時間諧波的條件去離散化馬克斯威爾方程式，將之轉換成一個廣義的特徵值問題，再對旋度算子進行特徵分解，如此便可得到一個標準的特徵值問題。由於考慮的是在三維空間中的馬克斯威爾方程式，此問題的矩陣大小將會隨著離散網格大小的三次方成長，因此需要花費很多的時間去解這個特徵值問題，不過現今我們可以利用GPU來縮短解這些問題的時間：對於一個七百萬乘七百萬大小的特徵值問題，我們平均僅需要83.0915秒便可找到十個最靠近零的特徵值。本論文的重點就是介紹這些使用到GPU的CUDA程式用途以及整個程式執行的流程，及如何從選定的材料形狀開始結合MATLAB和CUDA程式算出能帶結構。此外，在三維空間中總共有14種布拉菲晶格，每一種材料都是按照其中某種晶格規則排列而成的，而我們這套程式包可以解決所有晶格材料的特徵值問題。最後，我們展現許多不同材料的能帶結構，並分析能帶結構是如何隨著形狀和介電係數的變化而改變。

關鍵字

能帶結構 ； 馬克斯威爾方程式 ； 圖形處理器 ； 統一計算架構

並列摘要

Studying the band structure is an important thing for photonic crystal, and finding the band gap is a key point to understand the feature of the photonic crystal. Solving Maxwell's equation can be used to plot the band structure of the photonic crystal and can observe the existence and the width of the band gap. With the Yee's scheme and the Bloch theorem assumption, we can change the Maxwell's equation into a generalized eigenvalue problem (GEP). Next, we do the singular value decomposition (SVD) of the single curl operator so that we can obtain a standard eigenvalue problem (SEP). Since we consider the Maxwell's equation in three dimensional(3D) space, the matrix will grow cubicly as the grid size increase. It may take lots time for solving the eigenvalue problem, also. Despite of that, we could use the graphics processing unit (GPU) to solve the problem and it could save lots of time. For example, we could find ten smallest eigenvalues for the standard eigenvalue problem with five million matrix size in only 83 seconds averagely. In this research, we focus on introducing the function of these CUDA codes and the whole progress from choosing the crystal shape to plotting the band structure using this package including Matlab codes and CUDA codes. On the other hand, there are 14 different Bloch lattice and each crystal shape will be formed in some lattice type. This package could solve the eigenvalue problem for all lattice types. The other thing is that we show a lot of band structures for different material shapes in different lattice types. Also, we have found some facts about the changing of band structures with the change of the material shape and the permittivity.

並列關鍵字

band structure ； Maxwell equation ； GPU ； CUDA

參考文獻

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