Ali Akbarzadeh-Sharbaf

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.

A Stable and Efficient Direct Time Integration of the Vector Wave Equation in the Finite-Element Time-Domain Method for Dispersive Media

The finite-element time-domain (FETD) solution of the vector wave equation (VWE) is directly extended to doubly dispersive media using the bilinear transform approach. The proposed formulation is quite general and flexible in the sense of material dispersion model in contrast to limitations of existing convolutional-based approaches. In addition, it can be implemented in a very straightforward and efficient manner. A stability analysis is performed that demonstrates the unconditional stability of the original formulation is preserved in the dispersive case. However, in order to avoid the well-known late-time instabilities that arise in this formulation, an alternative formulation is considered and extended to the dispersive case with the same approach. Several numerical examples are simulated to verify the validity and accuracy of the proposed formulations. The late-time stability of the alternative formulation is tested through a numerical example with 20 million time steps. The solution is completely free from late-time growth.

A Provably Stable and Simple FDTD Formulation for Electromagnetic Modeling of Graphene Sheets

A new finite-difference time-domain (FDTD) formulation for modeling graphene is proposed, in which the graphene is modeled as a resistive sheet with a frequency-dependent conductivity. The formulation is first developed in the context of the vector wave finite-element time-domain and then reduced to the FDTD based on the equivalence between these two techniques. The obtained formulation is easy to implement and does not alter the original FDTD update equations. It can be applied to an existing FDTD code by simply adding a correction term to appropriate variables. One of the main contributions of this paper is analyzing the stability of the proposed formulation, which has not been done previously.

On the Development of Nonoverlapping and Stable Hybrid FETD-FDTD Formulations

A general framework to combine the finite-difference time-domain (FDTD) and the finite-element time-domain (FETD) formulations, both based on the vector wave equation, is proposed. In contrast to the existing stable hybrid FETD-FDTD, there is no transition layer between two subdomains. In addition, the stability of the proposed approach is analytically proved. This framework allows combining different FDTD and FETD formulations together. Particularly, a fully unconditionally stable hybrid method is proposed, which is proved to be energy conservative too. The key ingredient is a finite-element tearing and interconnecting method for electromagnetic problems with a new interface condition that preserves the stability of the numerical method in each region. Several numerical examples are considered in order to validate the proposed methods. The numerical results match with the reference solutions very well in all cases.

Implementation of a First-Order ABC in Mixed Finite-Element Time-Domain Formulations Using Equivalent Currents

In this letter, we describe an easy approach to implement the first-order Bayliss-Turkel-like absorbing boundary condition (ABC) in two mixed finite-element time-domain (FETD) formulations, namely the Crank-Nicolson FETD (CN-FETD) and the leap-frog FETD (LF-FETD). The idea is to introduce a current source distribution on the outer boundary of the domain such that it cancels outgoing waves. The current distribution is obtained based on the ABC relation. In addition, we show that the CN-FETD and the LF-FETD are equivalent to the FETD based on the vector wave equation discretized by the Newmark-β method in time with β = 1/4 and 0, respectively. Having utilized these equivalences, we demonstrate that our approach to implement the ABC in the mixed formulations lead to the same result as the vector wave FETD truncated with the same ABC. A numerical example is provided to validate our formulations.

WIDEBAND FINITE-DIFFERENCE TIME-DOMAIN MODELING OF GRAPHENE VIA RECURSIVE FAST FOURIER TRANSFORM

An efficient method based on the recursive fast Fourier transform (FFT) to incorporate both the intraband and interband conductivity terms of Graphene into the finite-difference time-domain (FDTD) method is proposed. As it only requires numerical values of the conductivity, it not only does not force any restriction of the conductivity models, but also can directly take into account material properties obtained from measurement. It reduces the total computational cost from O(N2) to O(Nlog2N).

Solving Finite-Element Time-Domain Problems With GaBP

In this paper, a new finite element Gaussian belief propagation (FGaBP) method is presented for time-domain applications. The unconditionally stable Newmark time-stepping scheme is combined with FGaBP for this purpose. As shown empirically, the method converges for increasing time step sizes without losing stability. The combined FGaBP-time stepping is able to retain the parallel scalability from FGaBP as in previous work. In addition, this paper also shows that lossy material properties can be easily supported by the method with minimal changes to its formulation.

Solving finite-element time-domain problems with GaBP

In this paper, we extend the recently introduced Finite Element Gaussian Belief Propagation (FGaBP) method beyond the scope of electrostatic problems, to address time-domain applications. The FGaBP inference algorithm was adapted to perform time-stepping computations based on the vector wave equation (VWE) discretized by the Newmark method. We validated the new algorithm using a reference finite element method (FEM) solution of a high-frequency waveguide application, where FGaBP reported consistent results.

Wideband finite-difference time-domain modeling of graphene via recursive fast fourier transform

An efficient method based on the recursive fast Fourier transform (FFT) to incorporate both the intraband and interband conductivity terms of Graphene into the finite-difference time-domain (FDTD) method is proposed. As it only requires numerical values of the conductivity, it not only does not force any restriction of the conductivity models, but also can directly take into account material properties obtained from measurement. It reduces the total computational cost from O(N2) to O(Nlog2N).

Efficient transient full-wave analysis of high-speed interconnects in multilayer PCBs

A new approach to efficiently simulate 3-D high-speed multilayer printed circuit boards (PCBs) using the finite-element time-domain (FETD) method is proposed. The method is based on the discretization of these structures using multiple layers of prism finite elements. In order to eliminate coupling between different layers in the mass matrix, a combined exact/approximate numerical integration technique is applied to the finite element matrices. This makes the final mass matrix fully block-diagonal and greatly reduces the inversion/factorization cost and time. The formulation is readily extended to lossy materials without any additional adverse effect on the stability condition.