关键词: 任意开挖断面, 共形映射, 三角插值, 法线逼近法, 收敛条件

Abstract: It is the essential prerequisite for mechanical analysis of the surrounding rock around underground cavern with the complex variable theory that the mechanical boundary must be transformed from the actual excavation cross-section to the unit circle. According to the Riemann’s existence theorem, the mapping function is established with the finite Laurent series, by which the exterior of unit circle can be conformally mapped to the exterior of carven with the arbitrary excavation cross-section. Based on the boundary correspondence principle, the conformal mapping of source and target domains is transformed into that of the boundary lines of unit circle and cavern. Research on solving the mapping function is carried out with the triangle interpolation theory, by which even and odd interpolation points are repeatedly iterated each other. In order to accelerate the iterative calculation, the calculating points are adjusted by the normal approximation method. Precision of the conformal mapping is ensured when the absolute error between the mapping boundary and the actrual excavation boundary is set as the convergence condition. Moreover, examples of the calculating mapping functions are provided for caverns with three excavation cross-sections. Compared to other methods in the literatures, the conformal mapping of the exteriors of unit circle and cavern with the arbitrary excavation cross-section is achieved more simply, accurately and rapidly by the method in this study.

Key words: arbitrary excavation cross-section, conformal mapping, triangle interpolation, normal approximation method, convergence condition