Finite-element time-domain methods for nonlinear dispersive media

Today’s telecommunications infrastructure is increasingly reliant upon complex material interactions with the electromagnetic field. For instance, effects such as dispersion, in which a material’s response to an applied field depends on its frequency, and nonlinearity, where the response is a complex function of field strength, form the cornerstones of fields such as nonlinear fibre optics. In consequence, efficient, accurate, and reliable numerical simulation tools capable of modeling these complex interactions are increasingly in demand, as cost effective alternatives to physical experimentation and prototyping.In this thesis, a family of Finite-Element Time-Domain (FETD) based numerical methods for the simulation of electromagnetic problems containing electrically complex material interactions is presented. Making use of both the mixed and vector wave equation formulations, the derived methods are capable of modeling very general combinations of linear dispersion,instantaneous nonlinearity, and dispersive nonlinearity within the nonlinear Maxwell’s Equations, free from any simplifying assumptions about the nature of the field solutions. In contrast to existing methods, these techniques permit increased geometric freedom, improved stability, and are capable of handling arbitrarily high nonlinear and dispersive orders.This thesis also presents several additional tools and methods to increase the effectiveness and versatility of these techniques. For instance, a Perfectly Matched Layer (PML) is derived which is compatible with nonlinear dispersive media, permitting the emulation of infinite domains as well as the truncation of finite systems within the nonlinear FETD method. More specifically, by utilizing the stretched coordinate formulation of the PML, the resulting technique not only saves computational resources, but does so without significantly altering the original underlying algorithms.Furthermore, while the derived techniques permit a much more accurate and general solution to Maxwell’s Equations for complex media, they unfortunately do so with a significant added computational burden. To mitigate this fact, this thesis also presents an analysis and breakdown of the computational overhead and bottlenecks associated with these methods. Building upon this analysis, a scheme is then presented by which these algorithms may be accelerated via parallelism and implemented on Graphics Processing Units (GPUs) to help alleviate some of the burden they pose.Lastly, in each case the FETD, PML, and GPU algorithms proposed in this dissertation are tested via numerical studies to verify their proper functioning, convergence, and accuracy. These include convergence studies as well as the demonstration of several well-known and physically significant nonlinear phenomena, such as spatial solitons, temporal solitons, and supercontinuum generation. Moreover, a parallel GPU implementation of the nonlinear algorithm is benchmarked against an equivalent traditional serial Central Processing Unit (CPU) version, and is shown to perform up to 150 times faster, significantly increasing the applicability and usefulness of these algorithms

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