电磁散射问题退化核快速解法的研究

发布时间：2018-03-27 08:33

本文选题：电磁散射　切入点：矩量法　出处：《南京邮电大学》2015年博士论文

【摘要】：采用数值方法可以灵活地对任意目标建模仿真求解,己成为获得目标散射特征的重要方法,是计算电磁学的重要研究领域。在各种数值方法中,矩量法由精度高,未知量规模小而得到了广泛应用,但是求解矩量法得到的线性方程组需要大量的存储空间和计算时间。为了克服上述问题发展出了一系列快速方法,其中远区作用矩阵块的高效表示是构建快速算法的重点和难点。退化核技术可以有效用于矩阵的低秩近似表示。本文以层次型矩阵(H矩阵)框架为基础,深入研究了格林函数的快速退化核构造方法,用来构建快速算法求解电磁散射问题。主要研究内容和贡献主要分为以下几个方面：1.提出一种线性计算复杂度的公共边生成方法,用于采用RWG基函数矩量法通用求解方法的前处理中。该算法基本思路为只通过比较邻近三角形面元的顶点编号来得到公共边信息,从而将计算复杂度由O(N2)减少为O(N),这里N为三角形面元数目。该算法可以集成到通用的仿真软件和算法测试平台中。此外,该算法己成功推广应用于快速生成四面体体元剖分的公共面。2.研究了采用Lagrange多项式和径向基函数(RBF)构造的插值退化核。格林函数为个震荡函数,基于距离的径向基函数插值的效率更高,只需要更少的插值点就能得到较高的插值精度。针对导体散射问题,提出一种近表面插值网格来减少插值点的数目。相比传统的均匀插值网格,近表面插值网格只选用靠近目标体表面的点,从而可以将插值插值点的数目由O(k3)下降为O(k2),其中k为每一维上的插值点数目。计算效率大幅提高,同时保持精度基本不变。3.基于分层基层次型矩阵(H2矩阵)框架,提出了一种用于求解导体电磁散射问题的混合插值网格的快速算法。该方法采用高斯径向基函数构造插值退化核函数,根据不同层划分得到的立方体块大小：对于较小的块,插值网格选用均匀网格,其它块选用近表面插值网格。该方法结合了两种网格的优点。相比于只采用均匀网格的方法,所提方法的计算量和存储量都有所下降。4.提出了一种新颖的指数退化核函数。指数退化核函数可以将格林函数源点与场点作用分离,得到的转移矩阵是对角矩阵,相比于多极子展开,达到相同精度时所需展开项数减少。从而可以有效地加快迭代求解算法中的矩阵向量乘法运算。在此基础上,提出一种新的方向选择方案,可以进一步减少展开项数至多极子展开的一半,且随着块尺寸变大比多极子展开增长更慢。数值算例验证了算法的有效性。
[Abstract]:Using the numerical method can flexibly to arbitrary target modeling simulation, has become an important method to obtain the target scattering characteristics, is an important research field of computational electromagnetics. In a variety of numerical methods, the method of moments by the high precision, the unknown quantity of small scale and has been widely used, but for the moment method of linear equations need a lot of computing time and memory space. In order to overcome the problems of the development of a series of rapid method, the far zone matrix block, that is important and difficult to construct a fast algorithm. The degradation of nuclear technology can be effectively used for the low rank matrix approximation. Based on the hierarchical matrix (H matrix) framework the rapid degradation of nuclear construction methods, in-depth study of the Green function, used to construct a fast algorithm for solving electromagnetic scattering problems. The main research contents and contributions are mainly divided into the following several aspects 1.: This paper proposes a linear computational complexity common edge generation method, used for pretreatment with RWG basis function moment method. The general method for solving the basic idea of the algorithm is to get the public information through the number of points is only edge adjacent triangle element, thus the computational complexity from O (N2) to reduce O (N), where N is the number of triangle element. This algorithm can be integrated into the simulation software and the algorithm of common test platform. In addition, the public face.2. the algorithm has been successfully applied to the rapid generation of tetrahedral element subdivision was studied by Lagrange polynomial and radial basis function (RBF) structure the interpolation degenerated kernel. The Green function as a shock function, efficiency based on radial basis function interpolation distance higher, requiring fewer interpolation points can get higher accuracy. According to the scattering problem, proposes a near surface interpolation grid To reduce the number of interpolation points. Compared with the traditional uniform grid interpolation, near surface interpolation grid use only near the target surface, which can be the number of interpolation points by O (K3) O (K2), decreased the K for the purpose of interpolation points for each dimension. At the same time, a substantial increase in the efficiency of calculation. Keep the accuracy invariable.3. layered base matrix (H2 matrix) based on the framework, put forward a fast algorithm for solving electromagnetic scattering problems of conductor mixed interpolation grid. This method adopts Gauss radial basis function interpolation degenerate kernel, according to cube block size in different layers were divided: for smaller blocks, interpolation the grid used the uniform grid, other blocks are made of near surface interpolation grid. This method combines the advantages of the two kinds of grid. Compared to the method using only uniform grid, the amount of calculation and storage of the proposed method are decreased .4. proposes a novel index index. Degenerate kernel degenerate kernel can separate the source points and field Green function, the transfer matrix is a diagonal matrix, compared to the multipole expansion, the expansion terms reduce to achieve the same precision. It can effectively accelerate the matrix vector multiplication in iterative algorithm. On this basis, put forward a new direction selection scheme can be further reduced by half approximation to multipole expansion, and the block size becomes larger than multipole expansion slower growth. Numerical examples show the validity of the algorithm.

【学位授予单位】：南京邮电大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：TN011

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