晶格波茲曼法應用完美匹配層吸收邊界於二維電磁波傳遞之研究

A perfectly Matched Layer for the Absorption of 2-D Electromagnetic wave with Lattice Boltzmann Method

李易騏(Yi-Chi Lee)

指導教授 ： 何正榮

國立中正大學/工學院/機械工程學系/碩士(2012年)

摘要

本研究引入Bérenger於時域有限差分法(Finite-Difference Time-domain method，簡稱FDTD)所提之「完美匹配層(Perfectly Matched Layer，簡稱PML)吸收邊界」，將之應用於晶格波茲曼法上，用以模擬二維電磁波在介質中的傳播行為，並分析其邊界之吸收效果。藉由Chapman-Enskog多尺度分析技巧，本研究於演化方程式中增添一組含有導電率及導磁率的特殊外力項，並透過於邊界上加入一層新的介質，以降低電磁波反射率。相較於傳統的輻射邊界，本文所提之PML-LBM其邊界晶格的吸收效果較佳，在使用8層邊界層之拋物線吸收函數時，可將反射率降至0.036%，除了滿足能量守恆外，更可維持二階準確度，其電場的誤差均小於十的負二次方
；與PML-FDTD方法相比，無論是單純的電磁波傳遞，或是針對電偶極子的模擬，PML-LBM在電場與磁場的計算上皆較PML-FDTD的計算時間快，以400x400的尺度搭配20層PML邊界，PML-LBM的運算時間較PML-FDTD快上46.38%。但礙於所採用的有限離散速度，在同樣邊界層數下，LBM的吸收率並沒有FDTD好。

關鍵字

晶格波茲曼法 ； 完美匹配層 ； 二維電磁波

並列摘要

Based on the Bérenger’s concept of perfectly matched layer (PML) absorbing boundary in the finite-difference time-domain (FDTD) method, this study proposes a lattice Boltzmann (LB) scheme that incorporates the PML technique for to simulating propagating electromagnetic waves in a two-dimensional medium. By introducing additional layers at boundary for reduction the reflection of electromagnetic waves, the PML effect is managed as a forcing term and incorporated into the LB evolution equations. The consistence between the proposed LB scheme and the macroscopic PML expressions is demonstrated using the technique of the Chapman-Enskog multi-scale analysis. Compared with the general LB method, the PML-LBM method has superior lattice absorption for the case of radiation boundary. When an eight PML boundary layers of the parabolic absorption function was used, the reflectivity could be reduced to 0.036%. This method does not only satisfy energy conservation, but also maintain a second order accuracy spatially. For performing both simulation cases of a simple transmission of an electromagnetic wave and the interactions of electric dipoles, the computational time of the PML-LBM is shorter than that of the PML-FDTD. For a 400  400 lattices with 20 PML boundary layers, the computation time for the PML-LBM is 46.38% shorter than that of for the PML-FDTD. The present PML-LBM scheme is based on the D2Q5 lattice, the limited discrete velocities, however, restricts its capability in reflection absorption. Results show that, under the same number of PML layers, the PML-FDTD demonstrations better ability in reflection absorption.

並列關鍵字

LBM ； PML ； Electromagnetic Wave

參考文獻

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