A hybrid finite-element / finite-difference method with implicit-explicit time stepping scheme for Maxwell's equations

Finite-difference time-domain (FDTD) methods have long been the most popular solution for transient electromagnetic (TEM) modeling. Compared with FDTD methods, finiteelement time-domain (FETD) methods have the potential to reduce the number of unknowns and the number of time steps. However, with a requirement to solve an irregular and large sparse matrix equation at every time step, it is generally difficult for FETD methods to take advantage of massively parallel computers. In our work, we propose a parallel FETD solution for TEM diffusion phenomena. We design and implement a parallel iterative solver, which includes a preconditioner that is customized for the FETD method to achieve both fast convergence and good scalability. Using a seabed model, our experiments show that our parallel FETD solution provides the same accuracy as the FDTD methods. On the performance side, using 8 CPU cores, the iterative solver part runs 5 times faster than the serial verison, while the preconditioner part runs 3 times faster. In total, our parallel FETD method runs 3.5 times faster than the serial verison when simulating a seabed model with around 100,000 unknowns.

The finite difference time domain method and finite element methods have proven to be highly effective techniques for the full wave analysis of microwave circuits and resonators. The finite difference time domain (FDTD) method is highly efficient, explicit and parallelizable. Finite element methods have been used in the frequency domain and in the time domain using implicit methods. Implicit methods require a solution of a linear system of equations at each time step. For the finite element time domain (FETD) algorithm to be competitive with the FDTD, the number of time iterations required for a simulation must be significantly less. An unconditionally stable FETD algorithm is used. The variational formulation is based on the vector wave equation. Unconditional stability is achieved by using a modified Newmark-Beta time integration technique. In addition, this algorithm is implemented on parallel message passing computers using a finite element tearing and interconnecting scheme. Some numerical examples based on cavity resonator problems are presented to demonstrate the efficiency and accuracy of the method and the parallel efficiency of the algorithm.

In this paper, a hybrid algorithm of explicit finite-element time-domain (FETD) and finite-difference time-domain (FDTD) methods with a large time step is proposed. The FETD method based on the unstable modes filtering (UMF-FETD) is adopted in the region involving fine structures while the FDTD method is employed in the other region. The proposed hybrid algorithm reduces the scale of the system matrix in UMF-FETD and guarantees that the whole calculation region adopts the uniform, large time step. The result proves the good ability of the proposed hybrid algorithm.

A new 3-D domain decomposition based hybrid finite-difference time-domain (FDTD)/finite-element time-domain (FETD) method is introduced to facilitate electromagnetic modeling by exploiting both the computational efficiency of FDTD and the meshing flexibility of FETD. The proposed hybrid method allows the FETD mesh and the FDTD grid to be nonconformal based on domain decomposition technique. It implements the hybridization with a buffer zone, which functions as a transition region between FDTD and FETD. The buffer zone helps the proposed hybrid method obviate the interpolation approach for field coupling of the nonconformal mesh and hence overcome the late-time instability issue. The discontinuous Galerkin method is utilized to couple different regions, thus improving the coupling accuracy compared with that using the Dirichlet boundary condition. Moreover, the hybrid method allows further division of the FETD region into multiple subdomains when the degrees of freedom in this region are large. For temporal discretization, a global leapfrog time integration scheme is implemented to sequentially update the fields in the FDTD, buffer, and FETD regions. The numerical results are shown to demonstrate the meshing flexibility and computational efficiency of the proposed hybrid method inherited from FETD and FDTD methods.

In order to analyze the switching transient electromagnetic fields generated by the aerial busbars in substation, the one-dimensional (1-D) time domain finite element (TDFE) method is presented to analyze the transient wave process in multi-conductor transmission lines (MTLs) with branches. Based on the Pade's approximation and on Prony's method, the iterative equations of the aerial MTLs are reported. Iterative formulae are also presented to determine the boundary conditions at the nodes of the branches in this paper. The proposed method avoids the Gibbs effect of the finite difference time domain (FDTD) method. An aerial busbars model in a 500 kV substation is analyzed when circuit breakers and disconnect switches are closed. The numerical results are in good agreement with those calculated by Bergeron's method and FDTD method.

Utilizing the knowledge of the singular behavior of the fields at a three‐dimensional corner, a simple algorithm is derived to account for the singularities in the numerical computations using the time domain finite difference or finite element methods. It is shown that the effects of the singularities have strong local effect near the corners or wedges for scattering problems.

The discrete origin of FETD-Newmark late time instability, and a correction scheme

Chapter 8 - Finite-Element Time Domain Method

The purpose of this work is to compute the electromagnetic fields generated by lightning return stroke using the TDFE (time domain finite element) method. Using the TL (transmission line) lightning return stroke model, the point-matched TDFE method is used to solve the telegraph equations that represent the line dynamic behavior, and the electromagnetic fields at ground level are computed using the dipole method. The results show the advantages of the TDFE method, where the channel boundary conditions and transmission line parameters are easy to compute using a very small code and processing time.

This paper presents an efficient hybrid method consisting of finite-difference time-domain (FDTD) method, transmission line (TL) equations, and a novel time-domain S -parameter cascade (TD-SC) technique, which can be directly applied in time domain for the EMI analysis of transmission line network (TLN) in shielded electronic device excited by ambient electromagnetic wave. The FDTD method combined with an auto mesh generator is utilized to model the shielded structure of an arbitrary electronic device rapidly, and obtain the excitation fields of the TLN exactly. Then, the TD-SC technique based on TL equations, FDTD method, and S parameter of circuits is proposed for the further EMI analysis of the TLN consisting of multistage TLs and circuits. It has the advantage of without modeling the fine structures of TLs and circuits directly. Numerous examinations exhibit that this hybrid method is an efficient method for the EMI analysis of TLN by comparing with a three-dimensional FDTD method. The computation time and memory cost by this hybrid method is much less than that of the FDTD method, because the hybrid method provides a fast way to compute the transient responses on TLN in which the TLs and circuits do not need to be meshed.

The numerical dispersion is one of the main factors to affect the accuracy of the finite-difference time-domain (FDTD) method. It can be easy to be taken for granted that a smaller time step leads to smaller simulation errors. This paper reveals that smaller time steps do not always make more accurate results in FDTD simulations. We analytically investigated how time steps affect the numerical dispersion of two FDTD methods: the leapfrog FDTD(2,2) method and the FDTD(2,4) method. The investigations in this paper show that in the FDTD(2,2) method, a larger time step limited by the Courant-Friedrichs-Lewy (CFL) condition is more helpful to reduce the numerical dispersion error. However, in the FDTD(2,4) method, as the time step grows, the numerical dispersion error decreases at the beginning and then increases. Several numerical examples are carried out to verify our analysis.

Use of a hybrid pseudospectral time-domain–finite-difference time-domain method for optical simulation of liquid crystals

New version of FDTD with GFM program in Schwarzschild space–time

The stable hybrid finite element time domain - finite difference time domain (FETD-FDTD) method is extended by incorporating higher order hierarchical basis functions in the finite element region. The use of unstructured tetrahedral elements in the modeling of antenna structure enables the application of the hybrid method to accurately model geometrically complex radiators. Pyramidal elements are used in the transition from unstructured tetrahedral elements to structured hexahedral elements of the FDTD grid. The finite element formulation incorporates the excitation of antennas using coaxial line or stripline feed with transverse electromagnetic mode (TEM). Traditional FDTD method with anisotropic perfectly matched layer (PML) is used to simulate unbounded media. The technique is extended for scattering problems, enabling the modeling and simulation of reception by antennas. Application of this method in the modeling of typical wideband antennas along with the numerical results is presented.

A finite-element time-domain (FETD) method for solving Maxwell's equations was developed by combining prism-based edge elements with central differencing in the time domain. This method solves for the electric field as a vector function of space and time in finite cylindrical cavity geometries of arbitrary geometry. Two stability analyses of this method are performed using the growth factor technique and the Z-transform. Furthermore, the computational complexity of the present approach is also investigated, This method has been implemented and its validity tested by studying the resonant frequencies of various microwave cavities.