以時域有限差分方法求解三維馬克斯威爾方程組，探討手機于頭部之電磁波之比吸收率之分布

Abstract

本論文是在非交錯網格上發展一三維時域有限差分法(FDTD)，以求解馬克斯威爾方程。本文的方法是在時域內，在滿足高斯定律(即電場和磁場零散度條件) 的架構下求解法拉第定律和安培定律。本文所提出的數值方法能在時間上和空間上保有相當好的理論收斂斜率，且能有效地減少實解相速度與數值相速度之間的誤差，而得以顯著地降低了因時域有限差分所造成的數值色散誤差以及各向異性誤差。本研究證實了所提出的數值方法在具辛結構與色散關係上皆具有良好的保持性，尤其在針對經長時間馬克斯威爾方程的數值模擬後，其效果尤為顯著。 本文進而將此數值方法針對人體頭部進行預測及數值分析其暴露在手機輻射(RF) 下之特定比吸收率(Specific Absorbtion Rate) 的電磁場與SAR場的在頭部各器官組織的分布情形。人體在使用手機進行通話時，通常將手機聽筒貼置在左耳或右耳上，使得頭部將與手機直接貼觸，直接接受由手機天線發射出的低強度射頻電磁場(RF-EMF) 曝曬。然而，電磁曝曬的強度，將與手機種類以及手機輻射功率和作用頻段相關聯。本文選用複雜幾何之Apple iPhone4-like 模型，並與複雜幾何頭部組織進行電磁曝曬分析模擬，使用顯式非交錯(或稱並列) 網格方法進行模擬計算。此方法相當適合使用多圖形處理單元(GPUs) 平行計算，透過增加更多圖形處理單元減少計算時間，以換取計算空間之網格密度。 由於馬克斯威爾方程組屬於完全可積之方程，因此，我們採用具辛結構之Runge-Kutta方法來逼近時間導數項，並且保持馬克斯威爾方程組能量守恆的性質；同時透過最小化數值色散關係式與色散關係式之間的差，以減少數值色散誤差。結果顯示，所模擬行動電話的數值結果與實驗測量值相當接近，顯示本文所使用之數值方法，可以準確的預測出低頻射頻場對人腦的影響。

An explicit finite-difference scheme for solving the three-dimensional Maxwell’s equations in non-staggered grids is presented in time domain. Our aim is to solve the Faraday’s law and Ampère’s law in time domain under the constraint of Gauss’law. The numerical method presented in this paper can maintain a fairly good theoretical convergence slope in time and in space. It can effectively reduce the error between the actual solution phase velocity and the numerical phase velocity by dispersion relation analysis. with the concept of phase velocity preserving, this numerical method can significantly reduce the numerical dispersion error and anisotropy error. This study confirms that the proposed numerical method can retain on symplectic structure and dispersion relationship. Exposure to mobile (or cell) phone radiation will be numerically investigated in human head by solving the Maxwell’s equation. Our aim is to get the distribution of the electrical field in the calculation of Specific Absorption Rate (SAR). Cell phone handset is normally placed over left/ right ear. Exposure to low-intensity Radio Frequency–Electro/Magnetic Fields (RF-EMF) from cell phone is therefore a well localized issue. Moreover, the accompanied electrical field takes its highest magnitude in brain regions closest to cell phone antenna. The degree of exposure depends on the type of cell phone being used. As a result, Apple iPhone4 and a phantom head are chosen in this three-dimensional simulation of Maxwell’s equations. For performing a computationally effective simulation of Maxwell’s equations, calculation of Maxwell’s solution will be performed in non-staggered (or collocated) grids using the explicit finite difference scheme. As the result of the employed explicit discretization scheme in non-staggered grids, this simulation can be suitably executed in parallel on Graphic Process Units (GPUs), thereby reducing a dramatic amount of computing time. Maxwell’s equations belong to a class of completely integrable equations. Symplectic Runge-Kutta temporal scheme is therefore adopted to approximate time derivative terms so as to be able to preserve the embedded Hamiltonians and invariants embedded in Maxwell’s equations. In addition, the introduced numerical dispersion error is reduced by minimizing the difference between of numerical and exact dispersion relation equations. As a result, the emitted low-frequency radio frequency fields can be accurately predicted.