基于Schwarz Christoffel变换的单位圆到任意开挖巷道断面共形映射的计算方法

Calculating Method for Conformal Mapping from Boundary of Unit Circle to Boundary Ofthe Arbitrary cross Section Roadway with Schwarz Christoffel Transform

将复杂开挖巷道断面的边界力学问题转化成单位圆的边界力学问题,是利用复变函数理论开展地下巷道围岩力学分析的前提。从黎曼存在定理出发,建立了以Schwarz Christoffel变换表示的单位圆边界到任意开挖巷道断面边界的共形映射模型,采用高斯雅克比积分方法和Levenberg-Marquardt算法对模型进行求解,提出了以相邻两点映射边界长度与实际开挖巷道边界线之间的绝对误差作为迭代计算的收敛条件。以矩形巷道、椭圆巷道和复杂断面巷道为例,计算结果表明,提出的基于Schwarz Christoffel变换的单位圆边界到任意开挖巷道边界共形映射的计算方法操作简单、精度高、收敛快。

It was the essential prerequisite for mechanical analysis of the surrounding rock around underground roadway with the complex variable theory that the mechanical boundary must be transformed from the actual complex cross section of roadway to the unit circle. According to the Riemann＇s existence theorem,the model of conformal mapping was established with the Christoffel Schwarz transfor,by which the boundary of unit circle can be conformally mapped to the boundary of roadway with the arbitrary cross section of roadway. The method of Gauss Jacobi integral and the algorithm of Levenberg-Marquardt were used to solve the model. The absolute error between the boundary lengths of adjacent two point mapping and the boundary lengths of actual excavation roadway was set as the convergence condition. As an example of rectangular,elliptical and complex section of roadway,the calculation results showed that the proposed computing method of conformal mapping was simple,high precision and fast convergence with Schwarz Christoffel transform,by which the boundary of unit circle can be conformally mapped to the boundary of roadway with the arbitrary cross section of roadway.