Second-order Vector Wave Equation Research Articles

A Convolution-Free Finite-Element Time-Domain Method for the Nonlinear Dispersive Vector Wave Equation

In this article, a finite-element time-domain method is presented for the solution of the second-order vector wave equation (VWE) subject to electrically complex materials, including general combinations of linear dispersion, instantaneous nonlinearity, and dispersive nonlinearity. The presented method is novel in that it offers greater geometric flexibility than existing finite-difference methods, incorporates both instantaneous and dispersive nonlinearity, scales to arbitrary dispersive and nonlinear orders, and is simpler, faster, and requires less computational complexity than existing mixed formulations due to the use of edge elements only.

Physical DC Modes in the Microwave Resonator With Complex Geometric Topology

The microwave resonator considered in this paper is not only filled with some piecewise homogeneous, lossless, and anisotropic media, but also has a complex geometric topology. This resonant cavity may have several physical dc eigenmodes if the geometry region occupied by the cavity is not homeomorphic to the sphere in 3-D Euclidean space. We point out that the nonzero physical eigenmodes simulated by the original source-free Maxwell’s equations of first order and a system consisting of a second-order vector wave equation, a scalar equation constrained by Gauss’s law and boundary conditions are equivalent, but the physical dc eigenmodes simulated by these systems are not equivalent. Furthermore, the governing equations with magnetic field as working variable is successfully solved by mixed finite-element method, which can eliminate all the nonphysical modes, including nonphysical dc modes. Finally, four numerical experiments are carried out to test the correctness of our theory.

An Unconditionally Stable Radial Point Interpolation Meshless Method Based on the Crank–Nicolson Scheme Solution of Wave Equation

A 2-D unconditionally stable radial point interpolation meshless method (RPIM) based on the Crank–Nicolson (CN) scheme is presented. The CN algorithm in the proposed method is applied to only one of the Maxwell equations. It leads to solving the second-order vector wave equation in the time domain. Therefore, only one electromagnetic field is implicitly updated at each iteration. Furthermore, the computational cost of the CN-RPIM scheme is lesser than the conventional RPIM. The Courant–Friedrichs–Lewy condition in the proposed vector wave equation method does not constrain the time step due to its implicit formulation. Moreover, the stability condition of the proposed method is investigated in this paper through its amplification matrix. A 2-D waveguide and a filter are used as examples to validate and to demonstrate the effectiveness and the accuracy of the proposed method. Numerical results are compared with those obtained by the explicit RPIM method and the conventional finite-difference time-domain method.

High-order accurate FDTD schemes for dispersive Maxwell’s equations in second-order form using recursive convolutions

Abstract We propose a novel finite-difference time-domain (FDTD) scheme for the solution of Maxwell’s equations in which linear dispersive effects are present. The method uses high-order accurate approximations in space and time for the dispersive Maxwell’s equations written as a second-order vector wave equation with a time-history convolution term. The modified equation approach is combined with the recursive convolution (RC) method to develop high-order approximations accurate to any desired order in space and time. High-order-accurate centered approximations of the physical Maxwell interface conditions are derived for the dispersive setting in order to fully restore accuracy at discontinuous material interfaces. Second- and fourth-order accurate versions of the scheme are presented and implemented in two spatial dimensions for the case of the Drude linear dispersion model. The stability of these schemes is analyzed. Finally, our approach is also amenable to curvilinear numerical grids if used with an appropriate generalized Laplace operator.

Finite-Element Time-Domain Solution of the Vector Wave Equation in Doubly Dispersive Media Using Möbius Transformation Technique

Several finite-element time-domain (FETD) formulations to model inhomogeneous and electrically/magnetically/doubly dispersive materials based on the second-order vector wave equation discretized by the Newmark-β scheme are developed. In contrast to the existing formulations, which employ recursive convolution (RC) approaches, we use a Möbius transformation method to derive our new formulations. Hence, the obtained equations are not only simpler in form and easier to derive and implement, but also do not suffer from the intrinsic limitations of the RC methods in modeling arbitrary high-order media. To obtain the formulations, we first demonstrate that the update equation for the electric field strength {e} in the mixed Crank-Nicolson (CN) FETD formulation, which is based on expanding the electric and magnetic field in terms of the edge and face elements in space and discretizing the resultant first-order differential equations using Crank-Nicolson scheme in time, is equivalent to the unconditionally stable (US) second-order vector wave equation for the same variable ( {e}) discretized by the Newmark- β method with β = 1/4. In addition, we show that the update equation for the magnetic flux density {b} in CN-FETD is the same as the second-order vector wave equation for {b} on the dual grid discretized again by a similar Newmark-β method. Subsequently, thanks to the mixed FETD formulation properties, we derive update equations for the constitutive relations using a Möbius transformation method separately. In addition, we use the shown equivalence to derive formulations based on the vector wave equation. Finally, several numerical examples are solved to validate the developed formulations.

Conformal Perfectly Matched Layer for the Mixed Finite Element Time-Domain Method

We introduce a conformal perfectly matched layer (PML) for the finite-element time-domain (FETD) solution of transient Maxwell equations in open domains. The conformal PML is implemented in a mixed FETD setting based on a direct discretization of the first-order coupled Maxwell curl equations (as opposed to the second-order vector wave equation) that employs edge elements (Whitney 1-form) to expand the electric field and face elements (Whitney 2-form) to expand the magnetic field. We show that the conformal PML can be easily incorporated into the mixed FETD algorithm by utilizing PML constitutive tensors whose discretization is naturally decoupled from that of Maxwell curl equations (spatial derivatives). Compared to the conventional (rectangular) PML, a conformal PML allows for a considerable reduction on the amount of buffer space in the computational domain around the scatterer(s).

Time-domain finite-element modeling of dispersive media

A general formulation is described for time-domain finite-element modeling of electromagnetic fields in a general dispersive medium. The formulation is based on the second-order vector wave equation and incorporates the dispersion effect of a medium via a recursively evaluated convolution integral. This evaluation is kept to second order in accuracy using linear interpolation within each time step. Numerical examples are given to validate the proposed formulation.