Study of the discontinuous galerkin time-domain method applied to electromagnetic problems involving dispersive materials

Abstract

The discontinuous Galerkin time-domain (DGTD) method is a promising technique for solving electromagnetic problems, particularly electromagnetic wave scattering. This approach discretizes the domain into non-overlapping elements. Within each element, the solution uses high-order polynomial basis functions, continuous internally but discontinuous across element boundaries. The resulting system of semi-discrete ordinary differential equations is advanced in time using an explicit step-by-step scheme. A key feature of this process is that each time step involves solving only local problems within each element, contributing significantly to the method's efficiency. This methodology offers several distinct advantages, including its inherent flexibility for handling complex geometries and its straightforward adaptability to non-uniform grids with local refinement. High accuracy is achieved through the use of high-order basis functions. Furthermore, the discontinuous Galerkin method is computationally efficient, often requiring less memory compared to other discretization techniques like traditional finite element methods. The efficacy of the DGTD method has been validated across a wide range of applications, encompassing scattering analysis for both simple and complex objects, wave propagation studies, and antenna modeling. The framework has also proven adaptable, successfully incorporating various essential boundary conditions, such as perfectly matched layers (PMLs) and other absorbing boundary treatments for domain truncation. This research specifically focuses on advancing the application of the DGTD method to the analysis of complex media, with a primary emphasis on frequency-dependent (dispersive) materials and PMLs. Both of these are incorporated into the DGTD formulation through the use of auxiliary differential equations. The main goal is to enhance the DGTD method's utility for electromagnetic simulations involving these challenging material types. The central contribution of this thesis lies in the extension of conventional uniaxial PMLs to novel dispersive uniaxial PMLs. In this advanced formulation, both the material dispersion and the frequency dependence within the PML itself are represented using the complex conjugate pole-residue model. This unified and modular modeling approach, potentially leveraging efficient parameter extraction techniques like vector fitting, facilitates easier integration and enables more accurate and robust DGTD of wave interactions in complex, anisotropic, dispersive environments.