Investigating algebraic topological method for solving 3D Laplace equation in electrostatic boundary value problems

In this paper we investigate the adaptability of a non-conventional numerical technique called Algebraic Topological Method (ATM) for solving the Laplace equation in Electrostatic Boundary Value Problems (ESBVP). The main advantage of the ATM is that we can solve ESBVP directly in 3D domain without using complex differential or integral vector calculus tools. As a case study, we solved the problem of an ideal Parallel Disc Capacitor (PDC) using ATM by applying the Dirichlet boundary conditions at the domain boundaries. The ATM solution in the 3D domain between the disc's electrodes of PDC is presented as plots of electric potential distribution across different planes. The accuracy of scattered numerical solutions of ATM was compared with the analytical solutions for the ideal PDC. Further, the mesh convergence of ATM scattered solutions was studied by reducing the mesh size with a minimum and maximum edge length ratio as the control parameter. The ATM solution obtained from the optimum mesh size was interpolated, and the continuous electric potential distributions on various planes across the solution domain were plotted. In addition to that, the fringe field effect in a real capacitor was modeled using the ATM, and the electric potential distribution was compared with the solutions from commercial FEM software (COMSOL Multiphysics). Finally, the accuracy of ATM in case of asymmetric electrode geometries was compared with the COMSOL Multiphysics by solving the 3D Laplace equation for Semi Circular Disc Capacitor. The accuracy and solution time of the ATM for solving 3D Laplace equation in the capacitor models show that it is an efficient numerical method for solving the ESBVP.