Ανάπτυξη τοπολογικά συνεπούς σύμμορφης μεθόδου των πεπερασμένων διαφορών στο πεδίο του χρόνου για τη μοντελοποίηση μη ορθογωνικών διατάξεων στις 3 διαστάσεις
Authors: Κοσμάνης Θεόδωρος 
Published: August 2021

The scope of this doctoral thesis is the development of a topologically consistent conformal finite-difference time-domain method (FDTD), which is appropriate for the accurate simulation of non-orthogonal dielectric interfaces in three dimensions. An outline of the thesis content is given below. A thorough bibliographical survey of the attempts performed for the adjustment of the original FDTD to non-orthogonal geometries is presented in the second chapter. The reasons that led to the inquiry of FDTD algorithms for generalized non-Cartesian grids, the most important techniques that gave greater prospects of implementation to the Yee structure, together with their advantages and disadvantages, are mentioned. The second chapter is devoted to the detailed description of the most popular variations of the FDTD that have been proposed for the accurate modeling of generally non-orthogonal structures. The analysis focuses on the theory of locally conformal grids (CPFDTD and CFDTD), which appears to be much more attractive due to its simplicity, its broad implementation and its accuracy. The study of their fundamental elements brought to light their main disadvantages that are responsible for the appearance of late time instabilities or for difficulty in defining a proper stability condition. Furthermore, the inability to simulate dielectric interfaces in three dimensions that characterizes the two techniques, is noticed. An interpretation of the consistency of numerical simulations in the time domain is attempted in the third chapter. Utilizing the theories of differential forms and of algebraic topology, the mathematical model is represented in the discrete space. The algebraic properties which the discrete Maxwell’s system must satisfy so that the laws of the continuous space (zero divergence, duality of Maxwell’s equations, properties of the material tensors) are preserved in the discrete one, are defined. These restraining conditions are connected to the grid structure and the form of the discrete material tensors, and are the base for the evaluation of the two conformal FDTD methods that have been described in the second chapter. A new methodology of consistent conformal modeling of non-orthogonal dielectric materials in three dimensions is introduced in the fourth chapter. The main element of the proposed algorithm is the detection of the problematic topological parameters that are responsible for the instabilities of the conformal methods. Aiming at their systematic obliteration, an improved grid is created, in which new types of cells appear. The locally non-orthogonal discretization structure enforces the connection of the algorithm with special projection coefficients, connecting the discrete form of the field intensities to the corresponding discrete form of the field fluxes. The appropriate computation of these coefficients is an important element of the proposed methodology, mainly because of their critical part for the preservation of the algorithm’s consistency. Simultaneously, a technique calculating the effective dielectric constants along the dielectric boundary between the two materials is introduced, for the optimal description of the dielectric discontinuity. Having enforced the algorithm’s topological consistency, the stability of the time discretization scheme is studied thoroughly, so that the convergence of the numerical method is assured. The concept of consistency is extended to more general materials (dielectrics with losses, anisotropic materials). The chapter ends with the study of the combination of various absorbing boundary conditions, and particularly of the Perfectly Matched Layer (PML), with the improved grid structure. The evaluation of the proposed algorithm through the solution of appropriate three dimensional problems of resonance and electromagnetic radiation is the object of the fifth chapter. Thanks to the perfect conformation with the geometry of the materials and the choice of additional parameters framing it, the proposed methodology achieves remarkable accuracy with significant low computational cost and without instabilities even for long simulation times. It is noted that the comparison with the existing conformal FDTD methods is not possible due to their complete inability to model dielectric interfaces in three dimensions. Moreover, the procedure of computing the antenna radiation pattern, demanding special modifications, due to the non-orthogonality of several cells of the new grid structure, is described. Finally, in the sixth chapter, the conclusions of the thesis are extracted and some aspects of future research, as extensions of the thesis, are proposed.