Accuracy Investigation of FDM, FEM and MoM for a Numerical Solution of the 2D Laplace’s Differential Equation for Electrostatic Problems

Laplace’s differential equation is one of the most important equations which describe the continuity of a system in various fields of engineering. As a system gets more complex, the need for solving this equation numerically rises. In this paper we present an accuracy investigation of three of the most significant numerical methods used in computational electromagnetics by applying them to solve Laplace’s differential equation in a two-dimensional domain with Dirichlet boundary conditions. We investigate the influence of discretization on the relative error obtained by applying each method. We point out advantages and disadvantages of the investigated computational methods with emphasis on the hardware requirements for achieving certain accuracy.

Improving the Efficiency of Hybrid Boundary Element Method for Electrostatic Problems Solving

This paper describes a modification of the Hybrid Boundary Element Method (HBEM) for electrostatic problems solving. Such improved method is applied for transmission lines analyses. By taking a quasi-static TEM approach, the Hybrid Boundary Element Method is applied to determine the effective relative permittivity and the characteristic impedance of different stripline structures. Comparisons with already published numerical results and software simulation have been also performed with an aim to test the validity of the proposed approach. A close results match can be noticed. The main novelties of the proposed HBEM modification are better accuracy and ability to determine the polarization charges distribution on the separating surface between a strip and a dielectric layer. This was not possible before, using the previous version of the method.

Application of the Schwarz-Christoffel Transformation to the Solution of Harmonic Dirichlet Problems in Electrostatics

In this paper, a purely conformal mapping method for efficiently solving harmonic Dirichlet problems of electrostatic in domains free of charge and with charge whose boundaries have inconvenient geometries consisting of straight-line segments is presented. The method which uses the inverse of an appropriately determined Schwarz-Christoffel transformation as the mapping function, was applied to harmonic Dirichlet problems in an infinite strip and infinite sector and the solution or electrostatic potential for the problem obtained for each case. Furthermore, the equipotential lines of the electric field were also obtained in order to show the features of the solution and the field analysed accordingly. The electric field intensity was also analysed to show its variation in the field. This method could therefore be a suitable alternative method for solving Laplace's equation in two dimensional electrostatic problems.

An Artificial Neural Network Based Solution Scheme for Periodic Computational Homogenization of Electrostatic Problems

The present work addresses a solution algorithm for homogenization problems based on an artificial neural network (ANN) discretization. The core idea is the construction of trial functions through ANNs that fulfill a priori the periodic boundary conditions of the microscopic problem. A global potential serves as an objective function, which by construction of the trial function can be optimized without constraints. The aim of the new approach is to reduce the number of unknowns as ANNs are able to fit complicated functions with a relatively small number of internal parameters. We investigate the viability of the scheme on the basis of one-, two- and three-dimensional microstructure problems. Further, global and piecewise-defined approaches for constructing the trial function are discussed and compared to finite element (FE) and fast Fourier transform (FFT) based simulations.