應用晶格波茲曼方法模擬電磁波傳遞和散射

Abstract

本研究基於三維晶格波茲曼方法，先證明虛擬粒子的分布函數可以重建馬克斯威方程組和電荷守恆方程式。然後應用晶格波茲曼方法，模擬不同極化方向且不同形狀金屬，其對於單一脈衝入射後的散射情形。 主要完成的工作有：基於三維波茲曼的晶格模型，以及電場、磁感應場和電流等宏觀量假設，用Chapman-Enskog展開晶格波茲曼BGK方程式，證明晶格波茲曼BGK方程式能滿足宏觀的電荷守恆定律、安培定律和法拉第定律。此外提出平均磁場的定義，改寫原來三維晶格波茲曼模型的安培定律，使安培定律符合SI單位系統下的常用定義。本研究者提出數學上簡單幾何形狀的表達形式，用雙曲正切函數加減、相乘、平移和放大等技巧，描述二維之長方形、圓形、菱形的物體，將可以應用於介電物質、金屬或導磁物質等建構。最後模擬TEz和TMz極化方向入射的脈衝波，探究電磁波打到圓形金屬、長方形金屬、菱形金屬後所造成的散射場差異，並且計算金屬物體之圓周方位接收到的回音寬強度。 研究目的主要藉由三維晶格波茲曼模型，模擬不同極化方向、不同幾何形狀金屬的電磁散射情形。首先以電磁波在一維介電質的傳播，測試晶格波茲曼模型。之後電磁波入射到一維金屬交界面，觀察電磁波的反彈情形及模型可靠性。最後討論因為極化方向不同，造成爬行波存在與否，對散射結果的不同。另外比較菱形金屬與其他兩種形狀的金屬，所產生的散射圖形差異。

This research adopts a D3Q13 lattice Boltzmann method and proves that Maxwell’s equations and continuity of charge can be reconstructed by distribution functions of virtual particles. Then the lattice Boltzmann method is applied to the simulation of electromagnetic scattering. Different polarized pulses and different shaped metals are considered here. The works are as followed: the lattice BGK equations are Chapman-Enskog expanded to get the continuity of charge, Ampere’s law and Faraday’s law. Besides, the researcher proposes the mean magnetic field, rewriting the Ampere’s law of this lattice model in a format of conventional unit system. Mathematical descriptions of simple geometries are presented: using tangential hyperbolic functions to construct cylinder, rectangle and diamond. In the end, TEz or TMz polarized’ incident pulse illuminates metallic cylinder, rectangle or diamond. Differences of scattering are discussed and echo widths with respect to bistatic angles are calculated. The main purpose is to simulate scattering of different polarized pulses by different shapes. Firstly, one-dimensional wave propagation in dielectric medium or conductive medium is performed to validate the lattice model. Then two-dimensional simulations of scattering are performed to investigate the effect of polarization and geometry. Relations between creeping wave and polarizations are discussed. Comparisons of metallic diamond with other shaped metals are also mentioned.