基于E-H的快速时域有限元算法研究
发布时间：2018-08-24 07:36
【摘要】：近几十年来,随着不断提高的工程电磁场应用需求,追求高效且精确的数值分析方法成为计算电磁学领域一直以来的重点研究工作。时域有限元法由于其能够对非均匀介质和复杂几何结构进行精确模拟的优势,越来越受到计算电磁学研究者们的关注。面对电磁工程分析越来越高的精度要求,本文研究的高阶叠层矢量基函数是一个有效的方法。通过算例计算结果表明,在同样的精度要求下,应用高阶基可以使网格的剖分尺寸更大,从而使未知量数目降低,计算效率能够得到有效提高。传统的时域有限元法在计算电大尺寸问题的时候,大型矩阵的求逆便会严重影响求解时间,甚至因为计算资源的影响无法求解。本文使用不连续伽辽金时域有限元法,通过在时间步进前对每个子域系统矩阵的分别处理,大大缩短了求逆时间。不连续伽辽金时域有限元法中,整个计算区域被分为多个互不重叠的子区域,每个子域中的电场和磁场单独求解,且只与相邻子域交界面相关。然而基函数在区域分界面处是不连续的,因此本文引入了中心数值通量,强加电磁场切向连续性。除此之外,本文还研究了另一种解决大型稀疏矩阵求解问题的方法——基于H-矩阵的高效有限元直接求解技术。在基于H-矩阵直接解法的研究中,首先介绍了H-矩阵的基本概念和运算法则,详细讨论了H-LU分解算法,最后仔细分析了该方法的计算复杂度和内存消耗。
[Abstract]:In recent decades, with the increasing demand for engineering electromagnetic applications, the pursuit of efficient and accurate numerical analysis methods has been the focus of research in the field of computational electromagnetics. Time domain finite element method (TFEM) has attracted more and more attention of computational electromagnetics because of its advantages of accurate simulation of inhomogeneous media and complex geometric structures. In the face of the higher and higher precision requirement of electromagnetic engineering analysis, the high order laminated vector basis function is an effective method studied in this paper. The numerical results show that the application of higher order basis can make the mesh size larger and reduce the number of unknowns under the same precision requirement. The computational efficiency can be improved effectively. When the traditional time domain finite element method is used to solve electrically large size problems, the inverse of large matrices will seriously affect the solution time, even because of the influence of computational resources can not be solved. In this paper, the discontinuous Galerkin time domain finite element method is used to deal with each subdomain system matrix separately before the time step, and the inverse time is greatly shortened. In the discontinuous Galerkin time domain finite element method, the whole computational region is divided into several non-overlapping subregions. The electric and magnetic fields in each subdomain are solved separately and are only related to the interface between adjacent subdomains. However, the basis function is discontinuous at the regional interface, so the central numerical flux is introduced and the tangential continuity of the electromagnetic field is imposed. In addition, another method to solve the problem of large sparse matrix is studied in this paper, which is an efficient finite element method based on H-matrix. In the research of direct solution based on H- matrix, the basic concepts and algorithms of H- matrix are introduced, and the H-LU decomposition algorithm is discussed in detail. Finally, the computational complexity and memory consumption of this method are analyzed in detail.
【学位授予单位】：南京邮电大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O441;O241.82
本文编号：2200087
[Abstract]:In recent decades, with the increasing demand for engineering electromagnetic applications, the pursuit of efficient and accurate numerical analysis methods has been the focus of research in the field of computational electromagnetics. Time domain finite element method (TFEM) has attracted more and more attention of computational electromagnetics because of its advantages of accurate simulation of inhomogeneous media and complex geometric structures. In the face of the higher and higher precision requirement of electromagnetic engineering analysis, the high order laminated vector basis function is an effective method studied in this paper. The numerical results show that the application of higher order basis can make the mesh size larger and reduce the number of unknowns under the same precision requirement. The computational efficiency can be improved effectively. When the traditional time domain finite element method is used to solve electrically large size problems, the inverse of large matrices will seriously affect the solution time, even because of the influence of computational resources can not be solved. In this paper, the discontinuous Galerkin time domain finite element method is used to deal with each subdomain system matrix separately before the time step, and the inverse time is greatly shortened. In the discontinuous Galerkin time domain finite element method, the whole computational region is divided into several non-overlapping subregions. The electric and magnetic fields in each subdomain are solved separately and are only related to the interface between adjacent subdomains. However, the basis function is discontinuous at the regional interface, so the central numerical flux is introduced and the tangential continuity of the electromagnetic field is imposed. In addition, another method to solve the problem of large sparse matrix is studied in this paper, which is an efficient finite element method based on H-matrix. In the research of direct solution based on H- matrix, the basic concepts and algorithms of H- matrix are introduced, and the H-LU decomposition algorithm is discussed in detail. Finally, the computational complexity and memory consumption of this method are analyzed in detail.
【学位授予单位】：南京邮电大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O441;O241.82
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