Discontinuous Galerkin Finite-element Time-domain Method Research Articles

A partially staggered discontinuous Galerkin method for transient electromagnetics

Conservation of energy, spurious-free, and optimal convergence are among the most favorite ingredients of a discontinuous Galerkin finite-element time-domain method (DG-FETD) for the analysis of transient electromagnetic problems. Unfortunately, due to the reliance on dissipative upwind flux to suppress spurious modes, most existing DG-FETDs possess only one or two of them. In this paper, we present a novel two-dimensional DG-FETD method, which combines all of the three advantages. The proposed method is based on the electric field E and magnetic field H, whose approximation spaces are composed of Whitney's edge functions. To inhibit the spurious modes while keeping the conservation of numerical energy, the numerical flux with dissipation is not adopted. Instead, a carefully designed interface condition along with the non-dissipative central flux is employed for this purpose. That is, across each face of an element except for those coinciding with boundaries of the computational domain, if E is strongly enforced to be tangential continuous then the continuity of H is weakly imposed, and vice versa. For the convenience of complying with this requirement and without loss of the inherent parallelism of the DG-FETD method, two partially staggered sets of subdomains are built for E and H, respectively. These subdomains are made up of sub-elements generated through a refinement of each cell on the initial mesh into four similar elements. The final discrete system is advanced with a second order leap-frog time-stepping scheme. A series of numerical examples demonstrate that the present method is robust and much more superior to the conventional DG-FETD methods.

Numerical Dispersion in DG-FETD Method Using Brick and Tetrahedral Edge Elements

A numerical dispersion analysis (NDA) of the discontinuous Galerkin finite-element time-domain (DG-FETD) method is presented. Vector basis functions under the NDA are with brick and tetrahedral elements. This NDA shows that there exist both normal and spurious modes in the DG-FETD method. A DG-FETD modeling of transverse electromagnetic (TEM) wave propagation in a parallel waveguide is applied to verify the NDA for zeroth-order vector bases. The effect of wave propagation direction and electrical size of elements on numerical dispersion is investigated. It is shown from the NDA for higher-order interpolatory vector bases that the phase error of normal modes can be significantly reduced by using higher-order bases.

Simulation of Electromagnetic Waves in the Magnetized Cold Plasma by a DGFETD Method

The properties of electromagnetic waves propagating in magnetized plasma slabs and scattering from plasma coated spheres are investigated using the discontinuous Galerkin finite-element time-domain (DGFETD) method. The magnetized plasma is a kind of gyroelectric anisotropic dispersive medium. This anisotropy must be described by an electric permittivity tensor with both nonzero diagonal and off-diagonal elements. Instead of doing the convolution directly in the time domain, an auxiliary differential equations (ADE) method is employed to represent the constitutive relation. The algorithm is validated by studying the transmission characteristics via electron-composed plasma slabs and radar cross section (RCS) of plasma-coated spheres. All numerical results demonstrate very good agreements with exact solutions and comply with the plasma physics.

Higher-Order DG-FETD Modeling of Wideband Antennas With Resistive Loading

A discontinuous Galerkin finite-element time-domain (DG-FETD) method with higher-order tetrahedral elements is developed to model wideband antennas with resistive loading. To fully characterize the wideband fundamental parameters of the antennas with multiscaled structures, several schemes are adopted to effectively model resistive loading and accurately calculate return loss, directivity, and radiation pattern. Successful simulation of interesting antennas using the proposed DG-FETD method demonstrates its capability of modeling wideband antennas without or with resistive loading.

SPICE Lumped Circuit Subcell Model for the Discontinuous Galerkin Finite-Element Time-Domain Method

A SPICE lumped circuit subcell model is formulated within the discontinuous Galerkin finite-element time-domain (DGFETD) discretization of Maxwell's equations. A fourth-order exponential time difference (ETD) algorithm is used for circuits that lead to stiff systems. The ETD method reduces to a standard fourth-order Runge-Kutta (RK4) time-integration for nonstiff regions. A number of test cases, including a microstrip transmission line terminated with general RLC networks, load arrays, and a diode detector are presented for the validation of the proposed hybrid DGFETD/SPICE solution method.

An Auxiliary Differential Equation Formulation for the Complex-Frequency Shifted PML

An efficient auxiliary-differential equation (ADE) form of the complex frequency shifted perfectly matched layer (CPML) absorbing media derived from a stretched coordinate PML formulation is presented. It is shown that a unit step response of the ADE-CPML equations leads to a discrete form that is identical to Roden's convolutional PML method for FDTD implementations. The derivation of discrete difference operators for the ADE-CPML equations for FDTD is also presented. The ADE-CPML method is also extended in a compact form to a multiple-pole PML formulation. The advantage of the ADE-CPML method is that it provides a more flexible representation that can be extended to higher-order methods. In this paper, it is applied to the discontinuous Galerkin finite element time-domain (DGFETD) method. It is demonstrated that the ADE-CPML maintains the exponential convergence of the DGFETD method.