以自適性網格應用於晶格波茲曼之電磁學模擬研究

Abstract

晶格波茲曼法是一種傳統上用於模擬流體問題的演算法，在2008年時被應用於電磁學的模擬。波的色散現象是晶格波茲曼法的其中一個缺點，本研究以泰勒展開為基礎，推導出晶格波茲曼法演化中的第三階項，進而修正色散造成的誤差。經過修正後的晶格波茲曼法可以使用自適性網格點的架構使模擬的精度勝於均勻網格點的模擬。為了 提高模擬的效率，本研究也使用了顯示卡的平行運算作為輔助。本研究提出在模擬中加入完美導體網格點所需要的邊界條件，並利用此項邊界條件，模擬與探討偏振分向器、反射式望遠鏡與將細針置於拋物面鏡焦點所產生的近場效應等等問題。

Lattice Boltzmann method(LBM) is traditionally used in the simulation of hydrodynamic problems, and it has been developed to solve electromagnetic wave problems in 2008. The wave dispersion due to grid effects has been known to be a serious problem in the LBM approach. We devised a scheme which bases on third-order Taylor-expansion to correct the dispersion error. The new LBM scheme can naturally incorporate adaptive mesh refinement(AMR) to enhance the accuracy from the uniform grid simulation. We also used graphic processing units(GPU) as a tool to obtain speed-up in computation. This research provides a proper boundary condition for perfect conductor grids. By using this boundary condition, problems like wave polarizer, reflecting telescope and the near-field effect of a needle on the focus of a parabolic mirror are studied by this method.