多连通区域数值保角变换计算的精度改善法

发布时间：2018-04-22 11:19

本文选题：保角变换 + 模拟电荷法　；参考：《昆明理工大学》2017年硕士论文

【摘要】：保角变换是复变函数的一个基本问题,广泛应用于物理学与工学.保角变换的主要求解方法有解析法和数值计算法.解析法的基础是Riemann存在与唯一性定理,指出保角变换函数的存在性,但是并不能给出函数的具体形式.因此,对于许多复杂的区域必须通过数值计算方法来求变换函数.目前的保角变换分为两类:一类是单连通区域的保角变换,另一类是多连通区域的保角变换.本文研究基于模拟电荷法多连通区域的数值保角变换计算法,主要工作有:(1)对保角变换的相关概念和定理进行介绍.(2)给出了以心脏线和星形线为外边界的多连通区域的模拟电荷点和约束点的取法.(3)通过对于双连通和有界多连通区域的约束方程进行预处理,构造了一个对称正定线形方程组.进而,分别利用Runge-Kutta法和(n,m)-Pade迭代法求解对称正定线性方程组得到电量和变换半径,构造了近似高精度的保角变换函数.(4)对基于模拟电荷法的无界多连通区域的数值保角变换计算法进行了详细阐述,将本文的方法运用到了流体力学,并模拟出了圆柱体周围的流线.最后,对文中提出的数值保角变换改善法其进行了数值实验,数值结果表明了本文算法改善数值保角变换精度的有效性.
[Abstract]:Conformal transformation is a basic problem of complex function, which is widely used in physics and engineering. The main methods of conformal transformation are analytical method and numerical method. The analytical method is based on the existence and uniqueness theorem of Riemann, and points out the existence of conformal transformation function, but it can not give the concrete form of the function. Therefore, the transformation function must be solved by numerical method for many complex regions. At present, conformal transformation is divided into two categories: one is conformal transformation of simply connected region and the other is conformal transformation of multi-connected region. In this paper, the numerical conformal transformation method based on the simulated charge method for multi-connected regions is studied. The main work of this paper is to introduce the concepts and theorems of conformal transformation. (2) the method of selecting the simulated charge points and constraint points of multi-connected regions with the outer boundary of heart line and star line is given. The constraint equation of the connected region is pretreated, A system of symmetric positive definite linear equations is constructed. Furthermore, the electric quantity and the transformation radius are obtained by using the Runge-Kutta method and the Pade iteration method, respectively, for solving the symmetric positive definite linear equations. The conformal transformation function of approximate high precision is constructed. The numerical conformal transformation calculation method of unbounded multi-connected region based on simulated charge method is described in detail. The method in this paper is applied to hydrodynamics. The streamline around the cylinder is simulated. Finally, numerical experiments are carried out to improve the accuracy of the numerical conformal transform proposed in this paper. The numerical results show that the proposed algorithm is effective in improving the accuracy of the numerical conformal transformation.
【学位授予单位】：昆明理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241

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