Complex variable solution on over-/under-break shallow tunnelling in gravitational geomaterial with reasonable far-field displacement

TL;DR

The paper addresses the challenge of modeling over-/under-break shallow tunnelling in gravitational geomaterial by introducing a mixed boundary problem with far-field displacement constraint and a bidirectional composite conformal mapping that combines Charge Simulation Method and Verruijt's mapping. This transforms the problem into a homogeneous Riemann–Hilbert formulation and yields an infinite complex potential series, which is truncated and stabilized with Lanczos filtering for numerical reliability. Verification against finite-element results and Lin's analytical solutions demonstrates accurate stress and displacement predictions for asymmetrical, noncircular cavities, with clear convergence with respect to mapping parameters. The approach offers a computationally efficient, objective tool for engineering analysis of shallow tunnels with far-field constraints and nonaxisymmetric cavities, with open-source code available for practitioners.

Abstract

Over-/under-break excavation is a common phenomenon in shallow tunnelling, which is nonetheless not generally considered in existing complex variable solutions. In this paper, a new equilibrium mechanical model on over-/under-break shallow tunnelling in gravitational geomaterial is established by fixing far-field ground surface to form a corresponding mixed boundary problem. With integration of a newly proposed bidirectional composite conformal mapping using Charge Simulation Method, a complex variable solution of infinite complex potential series is subsequently derived using analytic continuation to tranform the mixed boundaries into a homogenerous Riemann-Hilbert problem, which is iteratively solved to obtain the stress and displacement in geomaterial. The infinite complex potential series of the complex variable solution are truncated to obtain numerical results, which is rectified by Lanczos filtering to reduce the oscillation of Gibbs phenomena. The bidirectional conformal mapping is discussed and validated via several numerical cases, and the subsequent complex variable solution is verified by examining the Lanczos filtering and solution convergence, and comparing with corresponding finite element solution and existing analytical solution. Further discussions are made to disclose possible defects of the proposed solution for objectivity.

Figures (16)

• Figure 1: Shallow asymmetrical cavity excavation in gravitational geomaterial
• Figure 2: New mechanical model construction with constrained far-field ground surface
• Figure 3: Schematic diagram of bidirectional composite conformal mapping and geomaterial simulation
• Figure 4: Bidirectional conformal mapping of Case 1
• Figure 5: Bidirectional conformal mapping of Case 2