Analytical Green's Functions for Two-Dimensional Electrostatics and Boundary-Element Based Solver

We derive analytical Green's functions for two-dimensional (2D) electrostatic problem, based on which a boundary-element based 2D electrostatic solver is developed. The system is composed of perfect electric conductors and linear dielectric materials; the boundary conditions are specified by an applied field and the total charge of each conductor. The solver is benchmarked by computing the macroscopic parameter of the system and comparing them to the exact values. Two applications are considered. First we determine the characteristic impedance of a microstrip transmission line; the results agree well with established values in literature. Second we compute the sensitivity of the field-induced torque with respect to the shape deformation. The consistent results using the finite difference and the adjoint method indicate that our solver is sufficiently precise to tell the fine difference from two very similar shapes.