From 2D conformal mapping to 3D electrostatic modeling: a regularized integral equation approach

Abstract Electric field computation holds a significant position in electromagnetic field theory and its engineering applications. Particularly when addressing problems with complex boundary conditions, traditional numerical methods often encounter a trade-off between computational efficiency and accuracy. This study proposes a novel conformal mapping framework designed to efficiently compute electrostatic fields in domains featuring complex boundaries. By constructing a mapping from an arbitrary smooth domain to a standard half-plane, the original problem with intricate boundaries is transformed into a more regular and solvable form. The mapping function is developed using the Cauchy integral equation, and, in conjunction with the theory of Fredholm integral equations of the second kind, a comprehensive analysis is conducted on the singularity and continuity of the kernel function, as well as the compactness of the integral operator. To solve the integral equation, the Nyström discretization method is employed alongside high-precision numerical integration techniques. Additionally, spectral analysis and regularization techniques are utilized to effectively address the ill-conditioning issues of the discrete system. The proposed method is further validated through comparisons with the finite element method (FEM) using two-dimensional and three-dimensional electrostatic field case studies. Results demonstrate that, in two-dimensional scenarios, the relative deviation between the two methods is less than 0.5%, while in three-dimensional cases, the deviation remains within 5%, with a substantial enhancement in computational efficiency. The approach introduced in this paper offers a new theoretical and numerical tool for the high-precision, rapid solution of electrostatic fields with complex boundaries, establishing a foundation for future extensions to higher-dimensional problems and the development of adaptive algorithms.