半无限体任意开挖断面外域到同心圆环域共形映射的计算方法

Calculating method for conformal mapping from exterior of cavern with arbitrary excavation cross-section in half-plane to the area between two concentric circles

保角映射函数的求取是采用复变函数法对埋隧道开挖问题进行分析的前提。根据边界对应原理,给出半无限体任意形状隧道断面外域到同心圆环域的保角映射函数形式,并提出计算此共形映射系数的方法。联合采用奇偶点插值法及遗传与序列二次规划算法,实现映射函数的高精度求解。基于映射函数的解析性,给出在求解复应力函数过程中,映射函数的化简处理方法。研究结果表明:(1)提出的映射函数求解方法可解决由地平线引起的求解困难,且可直接应用现有研究成果应用于函数系数的求解。(2)求得的隧道边界映射精度很高,相对误差可控制在0.5%以内,地表边界映射精度稍差,但都可控制在5%以内,且映射精度随系数个数的增多而提高,可满足工程及研究需要。(3)以一个浅埋矩形的工程为例,采用本映射函数的求解方法及映射函数的化简处理方法所得结果与有限元结果在基本完全符合,两者相差1%以内。

It is the essential prerequisite for mechanical analysis of the underground cavern with the complex variable theory that the conformal mapping function must be solved. According to the boundary correspondence principle,a conformal mapping function is established that can transform the exterior of the arbitrary excavation cross-section in half-plane to the area between two concentric circles. A method for calculating the conformal mapping coefficients is also proposed. By using the method of odd even point interpolation and genetic optimization and sequential quadratic programming algorithm,the high resolution is realized. Based on the analytic property of mapping function,a simplified method of mapping function in the process of solving complex stress function is given. The results show that：（1） The proposed mapping function method can solve the difficulty caused by horizon,and the existing research results can be applied to the solution of the function coefficients directly.（2） The tunnel boundary mapping accuracy is very high,the relative error can be controlled within 0.5%,the surface boundary mapping accuracy is slightly worse,but all can be controlled within 5%,and the mapping accuracy increases with the increase of the number of coefficients,which can meet the needs of engineering and research.（3） Taking a shallow buried rectangular project as an example,the proposed solution method and simplification method of mapping function are adopted. The results are in good agreement with the finite element results,with a difference of less than 1%.