Numerical conformal mapping for underground unlined tunnels with arbitrary shapes based on Symm’s method

Abstract

Conformal mapping technique is important in theoretical analysis and numerical computation for the fields of stress and displacement. In general, a unlined tunnel with arbitrary shape has no analytical solution for conformal mapping. Therefore, the study of numerical method for conformal mapping has great significance. The basic functions of numerical conformal mapping are given based on Symm’s method in this paper. Furthermore, the inverse mapping functions were deduced according to the relationships between the boundary nodes in physical and mapped plane. Compared to the other numerical methods, the presented method has some advantages such that, it is simple in concept to be understood, and can give the mapping function without iteration process. The method can be used to the forward and inverse numerical conformal mappings for multiple underground unlined tunnels with arbitrary shapes in finite and infinite domains. With the help of method of fundamental solutions (MFS), the interpolation equations were proposed for multiple underground unlined tunnels with arbitrary shapes. Finally, several numerical examples for the groups of U-shaped and rectangle tunnels have been given to verify the effectiveness of this method. The numerical results can convergent to real cases, which show that the proposed method has the properties of good accuracy and strong adaptability.