电磁场问题积分方程退化核的研究

发布时间：2018-12-18 07:43

【摘要】：求解大尺寸目标电磁散射问题是现阶段计算电磁学主要研究的问题之一,为了解决矩量法求解过程中存储量和计算量过大的问题,出现了一系列快速算法,其中快速多极子(FMM)方法可以看作一种具体的退化核逼近形式。退化核思想运用于求解电磁场积分方程的过程中,可以使核函数中的场点变量和源点变量分离,从而减少阻抗矩阵元素的存储量,同时也降低方程单次迭代求解的计算量。本文首先介绍了H矩阵方法,在理论上推导了H矩阵方法的运算量与存储量,二者的复杂度均接近O(Nlog N)数量级。然后基于退化核函数的思想,通过对快速多极子原理的分析和实现,从而进一步运用复数域上的最小二乘法对核函数进行拟合。拟合函数选取与快速多极子相同的指数形式,根据拟合点的数目和拟合过程中方向矢量的选取实现精度控制,得到一种新的指数退化核函数。将指数退化核与快速多极子退化核进行对比,验证了该退化核的正确性和逼近效率。最后,对导体球、导体立方体和NASA杏仁核三个模型问题的电磁散射特性进行分析,通过数值例算验证了H矩阵方法结合指数退化核的正确性与计算效率,结果表明H矩阵方法结合新的退化核逼近方法可以构建高效率的电磁场问题快速求解方法。
[Abstract]:Solving the electromagnetic scattering problem of large size target is one of the main research problems in computational electromagnetics at present. In order to solve the problem of excessive storage and computation in the process of solving the method of moments, a series of fast algorithms have appeared. The fast multipole (FMM) method can be regarded as a degenerate kernel approximation. Applying the degenerate kernel idea to solving the electromagnetic field integral equation, the field point variable and the source point variable in the kernel function can be separated, thus reducing the storage capacity of the impedance matrix element, and reducing the calculation amount of the single iteration solution of the equation. In this paper, the H matrix method is first introduced, and the computational complexity and storage capacity of the H matrix method are derived theoretically. The complexity of both methods is close to that of O (Nlog N). Then, based on the idea of degenerate kernel function, by analyzing and realizing the fast multipole principle, the kernel function is fitted by the least square method in complex field. The fitting function is chosen in the same exponential form as the fast multipole. According to the number of fitting points and the selection of the Chinese vector in the fitting process, a new exponential degenerate kernel function is obtained. By comparing the exponential degenerate kernel with the fast multipole degenerate kernel, the correctness and approximation efficiency of the degenerate kernel are verified. Finally, the electromagnetic scattering characteristics of three model problems, the conductor sphere, the conductor cube and the NASA amygdala, are analyzed, and the correctness and computational efficiency of H matrix method combined with exponential degenerate kernel are verified by numerical examples. The results show that the H-matrix method combined with the new degenerate kernel approximation method can be used to construct an efficient method for solving electromagnetic field problems.
【学位授予单位】：南京邮电大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：TN011

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