Stephen D. Gedney

An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices

A perfectly matched layer (PML) absorbing material composed of a uniaxial anisotropic material is presented for the truncation of finite-difference time-domain (FDTD) lattices. It is shown that the uniaxial PML material formulation is mathematically equivalent to the perfectly matched layer method published by Berenger (see J. Computat. Phys., Oct. 1994). However, unlike Berengers technique, the uniaxial PML absorbing medium presented in this paper is based on a Maxwellian formulation. Numerical examples demonstrate that the FDTD implementation of the uniaxial PML medium is stable, equal in effectiveness as compared to Berengers PML medium, while being more computationally efficient.

An Anisotropic PML Absorbing Media for the FDTD Simulation of Fields in Lossy and Dispersive Media

ABSTRACT A uniaxial anisotropic perfectly matched layer (PML) absorbing material is presented for the truncation of finite-difference time-domain (FDTD) lattices for the simulation of electromagnetic fields in lossy and dispersive material media. It is shown that by properly choosing the constitutive parameters of the uniaxial media both propagating and evanescent waves can be highly attenuated within the PML medium. This resolves the concern that the original Berengers formulation for a PML medium does not attenuate evanescent waves. FDTD formulations for the uniaxial PML method are presented for lossy and dispersive medium. Based on this formulation an equivalent modified representation of Berengers split equations is also derived. Through numerical examples, it is demonstrated that the uniaxial PML method provides a nearly reflectionless absorbing boundary for the FDTD simulation of evanescent and propagating waves encountered in highly dispersive and lossy medium.

An unconditionally stable finite element time-domain solution of the vector wave equation

This paper presents an implicit finite element time-domain (FETD) solution of the time-dependent vector wave equation. The time-dependent formulation employs a time-integration method based on the Newmark-Beta method. A stability analysis is presented demonstrating that this leads to an unconditionally stable solution of the time-dependent vector wave equation. The advantage of this formulation is that the time step is no longer governed by the spatial discretization of the mesh, but rather by the spectral content of the time-dependent signal. A numerical example of a three-dimensional cavity resonator is presented studying the effects of the Newmark-beta parameters on the solution error. Optimal choices of parameters are derived based on this example. >

Perfectly matched layer media with CFS for an unconditionally stable ADI-FDTD method

A perfectly matched layer (PML) medium with complex frequency shifted (CFS) constitutive parameters is introduced for the three-dimensional alternating direction implicit (ADI) formulation of the finite-difference time-domain (FDTD) method. The absorbing boundary is implemented using the convolutional PML (CPML) approach. It is demonstrated that the resulting ADI-CPML scheme is unconditionally stable. The effectiveness of the absorbing medium as a function of the time step is also demonstrated. The proposed method has the advantage that it allows the application of the ADI method to low-frequency analysis.

On deriving a locally corrected Nystrom scheme from a quadrature sampled moment method

A novel high-order method of moment procedure with quadrature point-based discretization is presented. The scheme is equivalent to a moment method employing smooth basis and testing functions applying a fixed-point numerical quadrature approximation for the outer integral. Mapping the current to the quadrature points then leads to a formulation that is equivalent to that derived via the locally corrected Nystrom method. The convergence properties of the current density and RCS for smooth and singular geometries in two and three dimensions are also studied.

Finite-difference, time-domain analysis of lossy transmission lines

An active and efficient method of including frequency-dependent conductor losses into the time-domain solution of the multiconductor transmission line equations is presented. It is shown that the usual A+B/spl radic/s representation of these frequency-dependent losses is not valid for some practical geometries. The reason for this the representation of the internal inductance the at lower frequencies. A computationally efficient method for improving this representation in the finite-difference time-domain (FDTD) solution method is given and is verified using the conventional time-domain to frequency-domain (TDFD) solution technique.

On the long-time behavior of unsplit perfectly matched layers

This paper shows how to eliminate an undesirable long-time linear growth of the electromagnetic field in a class of unsplit perfectly matched layers (PML) typically used as absorbing boundary conditions in computational electromagnetics codes. For the new PML equations, we give energy arguments that show the fields in the layer are bounded by a time-independent constant, hence they are long-time stable. Numerical experiments confirm the elimination of the linear growth, and the long-time boundedness of the fields.

A parallel finite-element tearing and interconnecting algorithm for solution of the vector wave equation with PML absorbing medium

A domain decomposition method based on the finite-element tearing and interconnecting (FETI) algorithm is presented for the solution of the large sparse matrices associated with the finite-element method (FEM) solution of the vector wave equation. The FETI algorithm is based on the method of Lagrange multipliers and leads to a reduced-order system, which is solved using the biconjugate gradient method (BiCGM). It is shown that this method is highly scalable and is more efficient on parallel platforms when solving large matrices than traditional iterative methods such as a preconditioned conjugate gradient algorithm. This is especially true when a perfectly matched layer (PML) absorbing medium is used to terminate the problem domain.

A combined FEM/MoM approach to analyze the plane wave diffraction by arbitrary gratings

The diffraction of TE- and TM-polarized plane waves by planar gratings is numerically analyzed using a combined finite-element-method/method-of-moments (FEM/MoM) algorithm based on the generalized network formulation. The interior region, treated using the FEM, is truncated to a single unit cell with the introduction of an exact periodic boundary condition, which is enforced as a natural boundary condition. Using the FEM to compute the fields within the periodic structure allows gratings of arbitrary cross section and material composition to be efficiently modeled. >

Numerical stability of nonorthogonal FDTD methods

In this paper, a sufficient test for the numerical stability of generalized grid finite-difference time-domain (FDTD) schemes is presented. It is shown that the projection operators of such schemes must be symmetric positive definite. Without this property, such schemes can exhibit late-time instabilities. The origin and the characteristics of these late-time instabilities are also uncovered. Based on this study, nonorthogonal grid FDTD schemes (NFDTD) and the generalized Yee (GY) methods are proposed that are numerically stable in the late time for quadrilateral prism elements, allowing these methods to be extended to problems requiring very long-time simulations. The study of numerical stability that is presented is very general and can be applied to most solutions of Maxwells equations based on explicit time-domain schemes.