Controlling Tip-Enhanced Electric Fields with Parametric Singular Basis Functions in Boundary Element Simulations

Sharp conductive tips are known to induce strong electric field enhancements due to geometric singularities, which pose significant challenges to conventional numerical methods. This paper proposes a boundary element method (BEM) framework enhanced with parametric singular basis functions for controlling tip-induced electric fields. Firstly, a tip extraction algorithm based on proper orthogonal decomposition (POD) identifies regions with small curvature radii. Parametric singular basis functions are then introduced at these locations to capture the asymptotic behavior of the electric field near sharp tips. The singularity strength is controlled via a tunable parameter, allowing the basis functions to adapt to different geometric configurations. Finally, through a series of numerical experiments, the relationship between the singular parameter and key geometric factors—cone angle, tip curvature radius, and mesh resolution—is analyzed. A compact empirical fitting expression is established using curve fitting techniques, enabling fast estimation of the singular parameter for arbitrary tip models. This work provides a robust and efficient modeling framework for electrostatic problems involving field singularities and is particularly relevant for applications such as lightning protection and discharge initiation studies.