Fast calculation of electrostatic fields based on conformal mapping

Abstract

To address the challenges of electrostatic field calculation for dynamic sources in complex terrains, this study proposes a computational strategy combining multi-level conformal mapping (CM) and the method of image (MI). By coordinating fractional linear transformations with the Cauchy integral theorem, the actual terrain boundary was converted into a standard half-plane, establishing a mathematical model based on a single integral equation. The method demonstrates good adaptability for 2D smooth non-intersecting boundaries and 3D symmetric boundary scenarios, with the method of image simplifying repeated calculations in dynamic source scenarios. Numerical experiments reveal that in 2D planar fields, the maximum relative deviation between this method (CM-MI) and finite element method (FEM)/boundary element method (BEM) was <0.12 %. For 3D highly symmetric fields, deviations remained within 5 % for approximately 97 % of the spatial regions. Compared with FEM, the proposed method shows significant computational efficiency improvements in lightning leader development simulations - time consumption was reduced by two orders of magnitude during 40-step simulations. Preliminary experiments indicate that the method's simulation results for mountain lightning strike distribution align with actual statistical trends.