The Discontinuous Galerkin Time-Domain (DGTD) method has gain great popularity for its capability of obtaining highly accuracy, flexibility, efficiency and parallel-computing. Different basis functions have their own merit in their advantageous applications. In this work, we focus on reseaches of DGTD’s basis functions and their advantageous applications.

We firstly designed a nodal high-order based DGTD algorithm and studied its improvement on accuracy. As we known, an ultra-dense-mesh h-refinement will lead to the low-frequency breakdown which has been studied for decades. However, for highorder p-refinement, it is conventionally regarded that the accuracy is only limited by the basis orthogonality. While in fact, not only the basis orthogonality, but also the interpolation placement of these bases, will bring conditioning problems. Some types of grids are capable of ideal interpolation. But for triangular grids, we demonstrate that the accuracy improvement of p-refinement in a triangular grid is limited when the basis order becomes ultra-high, even though the basis orthogonality is guaranteed. This accuracy limit does not come from the lack of orthogonality, but rather from the nonideal interpolation placement of high-order bases.

Then we developed the prism-vector-based DGTD algorithm. It employs numerical flux which is defined based on the Rankine-Hugoniot jump relations. Thanks to the numerical flux, all operations of DGTD are localized and easily parallelizable. The dispersion effect is taken into account by auxiliary differential equations (ADE). We then developed the prism-based DGTD to analyze power-ground planes. Because of its planar structure, it is more optimal to discretize power-ground planes by triangular prisms, instead of general tetrahedrons.

Moreover, because the parallel-waveguide mode dominates in most parts of the power-ground planes. We can apply simplified two-dimensional (2D) algorithm for these domains, which helps to reduce unknowns and enhance efficiency. Meanwhile, for accuracy, we apply three-dimensional (3D) algorithm to treat high-order modes. With well-designed basis functions, 2D and 3D elements can be easily coupled together. We then developed a novel 2D/3D hybrid DGTD method to solve this kind of problem with both high efficiency and accuracy.

In reality, the high-order modes propagate along time. Therefore, the 2D/3D domain decomposition should also be time dependent. Since mode simplification error might accumulate along explicit time iteration, an adaptive criterion has to be adopted to ensure an accurate and stable result. We propose two adaptive criterion based on the causality principle and the semi-discrete DGTD formulas, respectively. Both adaptive criterions are shown with good control of the mode simplification error. Besides, modified nodal analysis (MNA) are integrated with DGTD for an efficient analysis with both lumped element and full-wave simulation.

We then use prism-based DGTD method to analyze periodic structures like metamaterials. As one type of metamaterials, chiral mirrors reflect circularly polarized light of a certain helicity in a handedness preserving manner, while absorbing circular polarization of the opposite handedness. However, most absorbing chiral mirrors operate only in a narrow frequency band, as limited by the causality principle. Our work provides an analytical bandwidth analysis of chiral mirrors, and then proposes a type of transparent chiral mirror that allows undesired waves to pass through. Most frequency selective surface (FSS) and metamaterial structures contain patterned planar conductive layers and supporting dielectric layers. These layers are very thin compared to the wavelength. Therefore, general tetrahedral discretization of space will unnecessarily increase the number of mesh elements, as well as the number of unknowns. Instead, we propose using prismatic elements, which are more optimal for planar structures, resulting in less unknowns, less memory usage and higher efficiency. The accuracy of the proposed prism-based DGTD method is verified by numerical examples compared with commercial software packages. Moreover, the CPU and memory costs are also compared, demonstrating clear advantages of the proposed prism-based DGTD method.

This paper mainly focuses on the research and application of different basis functions of DGTD method. We apply prism-based DGTD in analysis of circuits, FSS and metamaterials. The algorithm we proposed has high accuracy, high efficiency, and
high flexibility, which is advantageous in electronic engineering applications.

Ph. D. Disseration

Wending Mai