Ανάπτυξη υψηλής ακρίβειας αριθμητικών σχημάτων για την επίλυση της κυματικής εξίσωσης και των εξισώσεων Μaxwell στο πεδίο του χρόνου με εφαρμογή σε θέματα ηλεκτρομαγνητικής συμβατότητας

Abstract

The scope of this doctoral thesis is the development of high precision explicit time domain schemes, suitable for the solution of wideband electromagnetic problems featuring complex geometries. The traditional finite differencing framework and associated theory traditionally utilized for the development of time domain numerical schemes is abandoned in favor of a more intuitive and mathematically consistent approach. In this thesis the direct temporal discretization technique of the field partial differential equations is introduced and developed from first principles. The discretization is performed on the transfer function representation of the partial differential equation directly, leading to the immediate discretization of both space and time upon inversion of the ensuing discrete transfer function. The final time domain scheme is determined in terms of the realization of the discrete transfer function. In higher than one dimensions, the determination of a realizable transfer function requires the use of Fourier series truncation and approximation techniques, in conjunction with a new set of stability and precision criteria also developed in this thesis. The stability and performance criteria are based on the study of the behavior of the discrete transfer function poles, which provide immediate insight into the intrinsic properties of the time domain scheme, without the computational difficulties and complexities of the traditional von Neumann analysis methods. The use of the “Characteristic diagram” i.e. the graphical representation of the behavior of the temporal frequency as a function of the spatial frequency introduced here further aid the assessment of the behavior of the schemes developed here. The time domain schemes resulting from the use of the above methods, feature a distinctively dense nodal interconnection pattern, whereby all nodes within a given radius from the central computational node are retained. This guarantees the maximization of the nodal interconnection density, while retaining the smallest cell footprint leading to highly isotropic propagation characteristics while at the same time minimizing the dispersion error of the scheme. The numerical results confirm the predicted superior accuracy capabilities of the numerical schemes developed in this thesis through a variety of different problems of propagation and shielding applications.