Solving Conformal Mapping Issues in Tunnel Engineering

Abstract

The calculation of conformal mapping for irregular domains is a crucial step in deriving analytical and semi-analytical solutions for irregularly shaped tunnels in rock masses using complex theory. The optimization methods, iteration methods, and the extended Melentiev’s method have been developed and adopted to calculate the conformal mapping function in tunnel engineering. According to the strict definition and theorems of conformal mapping, it is proven that these three methods only map boundaries and do not guarantee the mapping’s conformal properties due to inherent limitations. Notably, there are other challenges in applying conformal mapping to tunnel engineering. To tackle these issues, a practical procedure is proposed for the conformal mapping of common tunnels in rock masses. The procedure is based on the extended SC transformation formulas and corresponding numerical methods. The discretization codes for polygonal, multi-arc, smooth curve, and mixed boundaries are programmed and embedded into the procedure, catering to both simply and multiply connected domains. Six cases of conformal mapping for typical tunnel cross sections, including rectangular tunnels, multi-arc tunnels, horseshoe-shaped tunnels, and symmetric and asymmetric multiple tunnels at depth, are performed and illustrated. Furthermore, this article also illustrates the use of the conformal mapping method for shallow tunnels, which aligns with the symmetry principle of conformal mapping. Finally, the discussion highlights the use of an explicit power function as an approximation method for symmetric tunnels, outlining its key points.

Keywords

• conformal mapping
• tunnel engineering
• complex analysis
• Schwarz–Christoffel mapping
• existence and uniqueness theorem
• boundary correspondence theorem