Mesh-free Monte Carlo method for electrostatic problems with floating potentials

Numerical simulation plays a crucial role in the analysis and design of power equipment, such as lightning protection devices, which may become inefficient using traditional grid-based methods when handling complex geometries of large problems. The authors propose a grid-free Monte Carlo method to handle electrostatic problems of complex geometry for both the interior and exterior domains, which is governed by the Poisson equation with a floating potential boundary condition that is neither a pure Dirichlet nor a Neumann condition. The potential and gradient at any given point can be expressed in terms of integral equations, which can be estimated recursively within the walk-on-sphere algorithm. Numerical examples have been demonstrated, including the evaluation of the mutual capacitance matrix of multi-conductor structures and lighting striking near real fractal trees. The proposed method shows advantages in terms of geometric flexibility and robustness, output sensitivity, and parallelism, which may become a candidate for game-changing numerical methods and exhibit great potential applications in high-voltage engineering.