单位圆到任意多边形区域的Schwarz Christoffel变换数值解法

发布时间：2018-03-11 16:32

本文选题：修正高斯雅克比型积分　切入点：单位圆　出处：《浙江大学学报(理学版)》2017年02期 　论文类型：期刊论文

【摘要】：Schwarz Christoffel变换技术在处理某些工程问题时具有重要作用.从黎曼存在定理出发,建立了单位圆到任意多边形区域的映射函数Schwarz Christoffel变换模型,采用Levenberg-Marquardt算法求解含约束条件的非线性映射函数Schwarz Christoffel变换模型参数系统.针对映射函数中出现的奇异积分问题,对映射函数进行2次参数变换,将其化为高斯雅克比型积分,以积分路径中的奇异点为界,缩短积分路径,对子路径采用修正高斯积分方法进行计算.通过指数变换、连乘变换和累加变换,使任意初值问题均可进行迭代计算并满足初值的约束条件.提出以边长绝对误差和顶点绝对误差为迭代计算的收敛条件,并保证了映射函数的精度.给出了11顶点多边形区域映射函数的求解算例,4种方案的计算结果表明,Schwarz Christoffel变换数值解法操作简单、精度高、收敛快.
[Abstract]:Schwarz Christoffel transform technology plays an important role in dealing with some engineering problems. From Riemann's existence theorem, establish the mapping function Schwarz Christoffel transform model unit circle to an arbitrary polygonal region, the constrained Levenberg-Marquardt algorithm was used to solve the nonlinear mapping function of Schwarz Christoffel transform model parameters of the system. To solve the problem of singular integral appearing in the mapping functions are. The 2 parameter transformation of mapping function, the Gauss Jacobi integral, to the singular point integral in the path for the community, to shorten the path of integration, on the path by using the method of integral calculation. Through the modified Gauss index transform, multiplicative transform and accumulated transformation, the initial value problem can be arbitrarily calculate and satisfy the constraint condition of initial value the length. The absolute error and the absolute error of vertex iteration convergence Conditions and guarantee the accuracy of mapping function. An example of solving the mapping function of 11 vertex polygonal region is given. The calculation results of the 4 schemes show that the Schwarz Christoffel transform numerical method is simple, accurate and fast convergent.

【作者单位】：陇东学院数学与统计学院;陇东学院能源工程学院;
【基金】：甘肃省科技计划资助项目(1606RJZM092,1606RJYM259,1506RJYM324)
【分类号】：O241.8

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