Bidirectional conformal mapping for over-break and under-break tunnelling and its application in complex variable method

Over-break and under-break excavation is very common in practical tunnel engineering with asymmetrical cavity contour, while existing conformal mapping schemes of complex variable method generally focus on tunnelling with theoretical and symmetrical cavity contour. Besides, the solution strategies of existing conformal mapping schemes for noncircular tunnel generally apply optimization theory, and are thereby mathematically complicated. This paper proposes a new bidirectional conformal mapping for over-break and under-break tunnels of asymmetrical contours by incorporating Charge Simulation Method, which only involves a pair of forward and backward linear systems, and is therefore logically straight-forward, computationally efficient, and practically easy in coding. New numerical strategies are developed to deal with possible sharp corners of cavity by small arc simulation and densified collocation points. Several numerical examples are presented to illustrate the geometrical usage of the new bidirectional conformal mapping. Furthermore, the new bidirectional conformal mapping is embedded into two complex variable solutions of under-break shallow tunnelling in gravitational geomaterial with reasonable far-field displacement. The respective result comparisons with finite element solution and existing analytical solution show good agreements, indicating that the new bidirectional conformal mapping would extend the mechanical application range of the complex variable method in practical over-break and under-break tunnelling.