Semi-separable potential representation in the electrostatic calculation of cylindrical structures

Abstract

In High Voltage [HV] engineering understanding the electrostatic fields behavior is important for reliable and safe design. The goal in this thesis is to present a method to calculate these fields in cylindrical structures. The problem to solve is the 3D Laplace equation in multiply connected domains. Solving this directly in 3D is difficult and time consuming with Finite Element Method [FEM]. In order to solve such problems more efficiently the expected solution is expressed in semi-separable form. This will break down the 3D problem into two simpler parts which are a 2D Helmholtz equation and a 1D ordinary differential equation. The 2D problem is solved using a multipole expansion method which is an improvement on the classical Charge Simulation Method [CSM]. To build the full 3D solution Proper Generalized Decomposition [PGD] method is used. This method builds the final 3D solution step by step by summing up terms gradually until the solution reaches the desired accuracy. Each of the terms in the final 3D solution are the product of the 1D and 2D problem solutions. The MATLAB functions were written to solve the 1D and 2D problems. Their functionality tested and proved separately. These functions where used finally in the PGD MATLAB code to compute the full 3D solution. This method was tested on three different geometries which are cylindrical structures with one, two and three holes. The results are showing that the method works well. The computed 3D solutions are accurate and as expected. All the test cases simulated in COMSOL. This comparison proved the run time and memory efficiency of the PGD method. Even when the geometry becomes more complex and more parameters are added to the problem just by adding more terms it is possible to reach desired accuracies, while COMSOL as a FEM based method must resolve the entire system whenever any new parameter or complexity introduced.

Date of Award	2025
Original language	English
Supervisor	Rastko Zivanovic (Supervisor)