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将复杂开挖断面为边界的力学问题转化成单位圆为边界的力学问题,是利用复变函数理论开展地下隧洞围岩力学分析的前提。从黎曼存在定理出发,建立了以洛朗级数有限项表示的单位圆外域到任意开挖断面隧洞外域的共形映射函数。根据边界对应原理,将域到域的共形映射问题转化成单位圆周线到任意开挖断面隧洞边界线的共形映射问题。从三角插值理论出发,采用奇偶插值点反复相互迭代对映射函数的求解开展了研究。为快速地进行迭代计算,采用法线逼近法进行计算点(映射点)调整。提出了以映射边界线与实际开挖洞形边界线之间的绝对误差作为迭代计算的收敛条件,保证了共形映射精度。给出了3个开挖断面隧洞映射函数的求解算例,并将研究结果与文献中的方法进行了比较。结果表明,提出的单位圆外域到任意开挖断面隧洞外域共形映射的计算方法具有操作简单、精度高和收敛快等特点。
Transforming mechanical problems with complex excavation section boundaries into mechanics problems with unit circle as the boundary is a prerequisite for mechanics analysis of surrounding rock mass of underground tunnels based on complex variable theory. Based on the existence theorem of Riemann, the conformal mapping function of the outer circle of the unit circle to the outer zone of the tunnel with arbitrary excavation section is established, which is represented by the Laurent series finite element. According to the principle of boundary correspondence, the problem of conformal mapping from domain to domain is transformed into the conformal mapping problem from the unit circumference to the tunnel boundary of any excavation section. Starting from the theory of trigonometric interpolation, the iterative iteration of parity points is used to solve the mapping function. For fast iterative calculation, the normal point approximation method to calculate the point (mapping point) adjustment. The absolute error between the boundary line between the mapping boundary and the actual excavated tunnel boundary is proposed as the convergence condition for the iterative calculation, which ensures the conformal mapping accuracy. The calculation examples of the tunnel mapping function of three excavation sections are given, and the results are compared with those in the literature. The results show that the proposed method for calculating the conformal mapping of the outer circle of the unit circle to the outer tunnel of arbitrary excavation section has the characteristics of simple operation, high precision and fast convergence.