单位圆到任意多边形区域的Schwarz Christoffel变换数值解法

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.%Schwarz Christoffel transformation technique has an important role in dealing with engineering problem.Based on the Riemann's existence theorem, a transform model of Schwarz Christoffel from unit circle to arbitrary polygon area was established.Levenberg-Marquardt algorithm was used to solve parameters of the nonlinear mapping function of Schwarz Christoffel with constraint conditions.As to the mapping function in the singular integral problem, the mapping function was twice transformed to Gauss Jacobi integral, then the singular point in the integral path was taken as the boundary;The length of integral path was narrowed, and the modified Gaussian integral was used to calculate the sub path.By the exponential transform, multiplicative transform and accumulation transform process, the arbitrary initial value could be iteratively calculated and the results satisfied the constraints of the initial value.The convergence conditions of the iterative copulation based on the absolute errors of length and absolute error of vertex were put forward to ensure the precision of the mapping function.The solution of the mapping function with 11 vertex polygon region was calculated by four calculation schemes.Results of the four schemes showed that the numerical solution of Schwarz Christoffel transform is simple with high precision and fast convergence.