Ανάπτυξη βελτιστοποιημένων σχημάτων πεπερασμένων διαφορών ανώτερης τάξης για την ακριβή επίλυση ηλεκτρομαγνητικών προβλημάτων στο πεδίο του χρόνου

Abstract

The scope of this doctoral thesis is the development and implementation of novel, higher order finite difference time domain (FDTD) algorithms for the reliable treatment of computationally demanding electromagnetic problems. A thorough bibliographical research on the discretization errors of Yee’s method and the existence of possible alternatives, based on higher order and/or optimized approaches, is given in the first chapter. The second chapter focuses on important issues related to the discretized form of Maxwell’s equations, such as dispersion and anisotropy artifacts, cutoff behaviour of the computational grid and performance degradation of analytical absorbing boundary conditions. A brief review of implicit and explicit spatial operators and time integration schemes, along with a modified FDTD method based on operator superposition, is also presented. The deficiency of the conventional (2, 4) technique when operated at the stability limit is dealt with in the third chapter, by introducing slightly modified materials in the simulations. The demand for accurate numerical wavenumbers at specific frequencies and propagation angles leads to corrected values for the constitutive parameters, which produce additional, but of correctional nature, dispersion and anisotropy effects and ensure good narrowband, as well as satisfactory wideband behaviour. The fourth chapter is devoted to the design of two novel FDTD algorithms, based on the (2, 4) discretization strategy. The first one attempts the cancellation of equally-ordered terms in an error expression based on the numerical dispersion relation, and produces second-order spatial operators that overwhelm the conventional ones throughout the entire frequency spectrum. The second scheme is based on treating spatial and temporal operators separately, by quantifying their effects on plane-wave solutions of Maxwell’s equations, leading to single-frequency optimization with good wideband characteristics. A novel family of optimized FDTD techniques is presented in the fifth chapter, which exploit multi-dimensional parametric definitions of the spatial operators and utilize innovative error estimators. The latter are expanded in terms of cylindrical or spherical harmonic functions, thus enabling error reduction for all propagation angles, as well as gradual algorithmic improvement through the vanishing of an increasing number of terms. Error minimization is carried out at either one or two frequency points, which ensures narrowband or wideband optimization, respectively. The final chapter summarizes the main conclusions of the presented research and suggests some potential extensions of this work.