基于谱方法求解器的非均匀电介质下电势分布特性的研究

Investigation on the Distribution Characteristics of the Electric Potential in Inhomogeneous Dielectric Using the Spectral-Method-Based Poisson Equation Solver

以往研究中求静电场电势时,主要是求解均匀电介质的泊松方程,缺少对介电系数非均匀效应的分析.本文针对非均匀电介质的修正泊松方程进行求解,并研究非均匀介电系数对电势分布的影响.首先采用高效准确的谱方法求解器对极坐标系下的泊松方程进行求解,在较少的网格点下数值解能与理论解吻合良好.随后,对均匀介质泊松方程求解非均匀介质的电势分布所引入的误差进行研究,发现介电系数存在径向和周向梯度时均会产生误差,且周向梯度影响更明显,同时会破坏精确解原始的分布特性.最后,准确求解了非均匀电介质的电势分布,发现周向梯度对电势分布的影响更为显著,并发现电势分布在介电系数梯度趋于无穷时的渐进解.

Previous studies mainly solved the Poisson equation for homogeneous dielectric when solving the electrostatic potential field and lack the analysis on the inhomogeneous effect of the dielectric coefficient.In this paper,we turn to solve the modified Poisson equation for inhomogeneous dielectric and investigate the effect of the inhomogeneous dielectric coefficient on the electric potential field.First,we develop an efficient and accurate spectral-method-based solver for the Poisson equation in the system of polar coordinates,and get excellent agreement between numerical and analytic solutions with small grid number points.Next,we focus on the error brought by solving the electric potential field with inhomogeneous dielectric when using the homogeneous-dielectric-based Poisson equation.We find that this error exists with the gradient of the dielectric coefficient in either radial or circumferential direction.Furthermore,this error is more contributed by the circumferential gradient than the radial gradient of the dielectric coefficient,which would break the distribution characteristics of the original accurate solution meanwhile.Finally,we accurately solve the electric potential field with inhomogeneous dielectric,find that the circumferential gradient is more significant than the radial gradient and obtain the asymptotic solution with the gradient of the dielectric coefficient approaching infinity.