Chapter 8 - Finite-Element Time Domain Method

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.

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.

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.

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.

Time domain modeling is a fascinating world which brings together several complex phenomena and methods of essential interest to engineers. This book is a reference guide which discusses the most advanced time-domain modeling methods and applications in electromagnetics and electrical engineering. The book starts by clearly explaining why time-domain modeling may be worth doing; then, it provides guidelines about why some choices must be made among the principal modeling approaches and next guides the reader through the state of the art in time domain modeling, concerning either numerical and analytical methods, and applications. Finally, it highlights areas for future time-domain modeling research. The book is a collection of chapters written by leading research groups in the fields, following a logical development set out by the editor. Topics covered include finite element methods in time domain with applications to low-frequency problems; transient analysis of scattering from composite objects using late-time stable TDIEs; the transmission-line modeling method, partial element equivalent circuit method in time-domain; unconditionally stable time-domain methods; time-domain linear macromodeling, analytical techniques for transient analysis; the application of the finite-difference time-domain (FDTD) technique to lightning studies; modeling of lightning and its interaction with overhead conductors; transient behaviour of grounding systems; and statistics of electromagnetic reverberation chambers and their simulation through time domain modeling.

A hybrid scheme for time-domain electromagnetic simulations, combining the explicit finite-difference time-domain (FDTD) method and the implicit finite-element time-domain (FETD) method, is developed. In the hybrid method, the FETD part uses the unconditionally stable Crank-Nicholson method; while the standard FDTD part employs staggered Cartesian grid for spatial discretization and the leap-frog scheme for time stepping. Non-conforming meshes are allowed between the FETD and the FDTD sub-domains. The hybrid method takes advantages of the modeling flexibility of the FETD method for complex structures and the efficiency of the FDTD method for simple structures. The hybrid implicit-explicit time stepping scheme allows time steps as long as the stability limit for the FDTD method, which can be much larger than the stability criterion of the explicit FETD scheme with small elements. Numerical examples demonstrate the efficiency of the proposed method.

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

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.

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.

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.

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.

Simulation of ground penetrating radar on dispersive media by a finite element time domain algorithm

This paper deals with recent progress in diagnosis of electric power apparatus using non-conventional partial discharge (PD) measurements; i.e. acoustic signal detection method, electromagnetic wave detection method, and transient earth voltage (TEV) detection method. The author describes simulation technology of electromagnetic (EM) wave propagation and induced surface current, and acoustic wave propagation by PD using finite difference time domain (FDTD) method and finite element method (FEM) for understanding and visualization of the phenomena. Diagnosis and reliability evaluation are also described based on the understanding of the PD induced electromagnetic and acoustic waves propagation, and surface current generated in high voltage equipment such as gas insulated switchgear (GIS), and oil-filled transformers.

The perfectly matched layer (PML) is very popular and efficient for grid truncation of open-region problems. The concept of PML was introduced by J. P. Berenger (see J. Comput. Phys., vol.114, p.185-200, 1994) together with a numerical implementation based on the finite-difference time-domain (FDTD) scheme. However, the basic FDTD scheme is formulated on Cartesian grids and in many cases an unnecessarily large free-space region must be discretized between the PML and the object under investigation. F.L. Teixeira et al. (see IEEE Trans. Antennas Propag., vol.49, p.902-7, 2001) developed a conformal PML for the FDTD scheme. We continue this effort by presenting a new conformal PML formulation for the finite element method (FEM) in the time domain. It is based on the anisotropic material which has been used for the time domain FEM by D. Jiao et al. (see IEEE Antennas and Propag. Soc. Int. Symp., vol.2, p.158-61, 2002). We present the 2D-version of the conformal PML, which is a special case of our 3D-formulation. Furthermore, we consider only radar cross section (RCS) computations based on the scattered field formulation and emphasize that our PML is applicable to the total field formulation as well as radiating structures.

The site-specific radio wave propagation prediction in outdoor is increasingly important for electromagnetic interference analysis. Since the outdoor environment is usually very complex and electrically large, the shooting and bouncing ray (SBR) method (one technique of ray tracing) rather than other numerical methods such as finite difference time domain method and finite element method, is often applied in the computation. However, a large amount of rays have to be launched to guarantee the accuracy in conventional ray tracing methods. This, in turn, increases the simulation time significantly. Recently, many acceleration algorithms have been reported including space partitioning and wavefront decomposition. This paper presents a novel adaptive ray launching (ARL) method based on the pattern of transmitting antenna, which reduces the launched ray number significantly while maintaining the computation accuracy