Helmholtz方程外问题的区域分解算法

发布时间：2018-02-23 22:24

本文关键词： 三维外问题 Helmholtz方程区域分解算法椭圆人工边界椭球人工边界自然边界归化　出处：《南京师范大学》2015年博士论文　论文类型：学位论文

【摘要】：由冯康先生首创的自然边界归化在各种边界归化中别具一格,它与有限元和辛几何算法一并构成了冯先生的三大学术贡献.后经余德浩等人的发展,除了直接将自然边界元方法用于求解某些特殊有界区域上的边值问题以外,基于自然边界归化的区域分解算法还是处理一般的无界区域及凹角断裂区域问题的有效手段之一.前人的研究工作在二维及三维领域里取得了许多重要的研究成果,人们往往以圆或球面作为人工边界,但对于某些特殊形状的内部区域,例如椭圆或长椭球区域,用椭圆或椭球面作人工边界,则可以大大减少计算区域,进而可以减少计算量和存储量.本文主要研究Helmholtz方程外问题的区域分解算法和椭圆外区域上依赖时间问题的Dirichlet-Neumann交替算法(D-N交替算法).第一章先介绍一些坐标变换、特殊函数和必需的Sobolev空间,这些内容都是后面各章节必要的理论工具.第二章研究三维Helmholtz方程外问题基于自然边界归化的D-N交替算法.首先给出D-N交替算法及其等价的预处理Richardson迭代算法,然后分析该算法的收敛性,给出等价的变分形式及其离散形式,并对其离散形式进行收敛性分析,最后给出数值例子以示该方法的可行性和有效性.第三章研究三维Helmholtz方程外问题基于自然边界归化的Schwarz交替算法.首先给出问题等价的变分形式,其次给出Schwarz交替算法并分析该算法的收敛性、收敛速度.最后给出数值例子验证算法的可行性和有效性.第四章研究长椭球外区域上三维Helmholtz方程基于自然边界归化的D-N交替算法.首先给出所研究的问题的D-N交替算法,然后分析了该算法的收敛性.第五章研究椭圆外区域上依赖于时间问题的D-N交替算法.首先对所研究的问题进行变量替换,得到相应的Helmhotz外问题,并得到椭圆外区域上问题的Poisson积分公式和自然积分方程,其次对时间离散化并用所得的自然边界归化的结果给出D-N交替算法,然后对算法的收敛性进行分析,最后给出数值算例说明该方法的可行性与有效性.
[Abstract]:Naturalization of natural boundary, initiated by Mr. Feng Kang, is unique among all kinds of boundary naturalization. It, together with finite element method and symplectic geometry algorithm, constitutes Mr. Feng's three major academic contributions. Later, through the development of Yu Dehao and others, In addition to using the natural boundary element method directly to solve boundary value problems in some special bounded regions, The domain decomposition algorithm based on natural boundary domestication is also one of the effective methods to deal with the general problem of unbounded region and concave fault region. Many important research results have been obtained in the field of 2D and 3D. A circle or sphere is often used as an artificial boundary, but for an inner region of a special shape, such as an ellipse or a long ellipsoid, the artificial boundary of an ellipse or an ellipsoid can greatly reduce the computational area. In this paper, we mainly study the domain decomposition algorithm for the Helmholtz equation and the Dirichlet-Neumann alternating algorithm for time-dependent problem in the outer elliptic region. In the first chapter, some coordinate transformations are introduced. Special functions and required Sobolev spaces, In the second chapter, the D-N alternating algorithm based on natural boundary normalization is studied. First, the D-N alternating algorithm and its equivalent preprocessing Richardson iterative algorithm are given. Then the convergence of the algorithm is analyzed, the equivalent variational form and its discrete form are given, and the convergence of the discrete form is analyzed. Finally, a numerical example is given to show the feasibility and effectiveness of the method. In chapter 3, the Schwarz alternating algorithm based on naturalization of natural boundary for three-dimensional Helmholtz equation is studied. At first, the equivalent variational form of the problem is given. Secondly, the Schwarz alternating algorithm is given and the convergence of the algorithm is analyzed. Finally, a numerical example is given to verify the feasibility and validity of the algorithm. In Chapter 4th, the D-N alternating algorithm for 3D Helmholtz equations based on natural boundary normalization is studied in the long ellipsoid region. First, the D-N alternating algorithm for the studied problem is given. Then the convergence of the algorithm is analyzed. Chapter 5th studies D-N alternating algorithm for time-dependent problems in the outer elliptic domain. The Poisson integral formula and the natural integral equation of the problem in the outer elliptic domain are obtained. Secondly, the D-N alternating algorithm is given for the discretization of time and the result of natural boundary normalization, and then the convergence of the algorithm is analyzed. Finally, a numerical example is given to illustrate the feasibility and effectiveness of the method.
【学位授予单位】：南京师范大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O175

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