Theoretical Developments and Simulation Tools for Discrete Geometric Computational Electromagnetics in the Time Domain

The original Finite Difference Time Domain (FDTD) method, devised by Yee in 1966, inspired a conspicuous amount of research in the field of numerical schemes for solving Maxwell's equations in the time domain, thanks to its simplicity and computational efficiency. The original algorithm, which computes the values of electric and magnetic fields on the points of two interlocked Cartesian orthogonal grids, has also been rewritten as a Finite Integration Technique (FIT) algorithm, where the computed quantities are the integrals of the field over geometric elements of the grids. Both formulations suffer from the so-called staircase approximation problem: when an interface between regions with discontinuous material properties is not flat, the expected convergence properties of the numerical solution are not guaranteed if an exaggeratedly fine grid is not used. In this regard, even recent improved techniques based on combined arithmetic and harmonic averaging techniques cannot achieve second order accuracy in time in the neighborhood of the interface. This problem is inherent to the Cartesian orthogonal discretization of the domain, as unstructured grids (tetrahedral or polyhedral) mesh generators avoid it with grids conformal to the discontinuities in material properties. Approaches that have had some degree of success in adapting the FDTD algorithm to unstructured grids include schemes based on the Finite Element method (FEM), on the Cell Method and, more recently, formulations based on the Discontinuous Galerkin (DG) approach. Yet, consistency issues of discontinuous methods question their accuracy, since these methods do not explicitly force tangential continuity of the fields across mesh element interfaces, weakening the local fulfillment of physical conservation laws (charge conservation in particular). On the other hand, classical FEM formulations, which do not share this drawback, trade their geometric flexibility with an implicit time-stepping scheme, i.e. the computation includes solving a linear system of algebraic equations at each time-step. This severely limits the scalability of the algorithm. Recently, a technique has been introduced by Codecasa et al., based on a Discrete Geometric Approach (DGA) which instead yields an explicit, consistent and conditionally stable algorithm on tetrahedral grids. Due to the promising features of this approach, a thorough analysis of its performance and accuracy is in order, since neither have been widely tested yet. This work addresses the issue and shows that the latter approach compares favorably with equal order FEM approaches on unstructured grids. An important drawback of the DGA approach is that it was originally formulated for strictly dielectric materials. The way to overcome this limitation is unfortunately not obvious. The present work addresses this issue and solves it without sacrificing any property of the original algorithm. Furthermore, although the properties of the material operators in the original formulation show that the resulting scheme is conditionally stable, a Courant--Friedrich--Lewy (CFL) condition equivalent to the one of the original FDTD algorithm is not given. This is also dealt with in the bulk of this thesis and a sufficient condition for the stability of this algorithm is given with proof. Finally a practical toolbox for time domain electromagnetic simulations, tentatively named TetFIT and resulting from the coding efforts of the author is presented, with preliminary results on its performance when running on Graphical Processing Units (GPUs).