A finite-element time-domain method using prism elements for microwave cavities

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.

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, an efficient algorithm for implementing Crank-Nicolson scheme in the finite-element time-domain (FETD) method is presented. Based on a direct discretization of the first-order coupled Maxwell curl equations, this algorithm employs edge elements (Whitney 1-form) to expand the electric field and face elements (Whitney 2-form) for the magnetic field. Since the curl of an edge-element is the linear combination of those face elements whose faces contain the given edge, only the Maxwell-Ampere equation composes a sparse linear matrix equation for the electric field update; the Maxwell-Faraday equation is explicit. The Crank-Nicolson scheme is implemented leading to an unconditionally stable vector FETD method and the matrix inverse is not required to be computed explicitly. Therefore, only one matrix equation is required to be solved at each time step. Numerical results demonstrate that the proposed method is efficient when compared with the conventional leap-frog mixed FETD method and the Crank-Nicolson FDTD method.

Summary form only given. Many materials in the nature or fabricated in laboratories exhibit nonlinear behavior where the permittivity and/or permeability is a function of the magnitude of the electric or magnetic field intensity. Such nonlinearity results in a nonlinear system of Maxwell's equations, which are more challenging to solve. Generally speaking, there are two kinds of methods commonly adopted to solve nonlinear problems. The first kind of methods solves the linearized versions of the original problems, while the second kind of methods deals with the nonlinear problems directly. Although more difficult to solve with a high efficiency, the second kind of methods is more desirable in order to capture the nonlinear phenomenon more accurately. Among various nonlinear solvers developed in the applied mathematics community, the Newton-Raphson method is widely used due to its second-order convergence, if started from an initial guess sufficiently close to the true solution. However, in a real application, there is no straightforward way to choose a good initial guess. As a result, the Newton-Raphson method could be unstable or divergent in solving complicated problems. Another difficulty of applying the Newton-Raphson method is that the Jacobian matrix needs to be updated and solved in every Newton iteration, which could be very time consuming for solving large problems. When the nonlinear electromagnetic problems are formulated and solved with the finite element method in time domain (J. M. Jin, The Finite Element Method in Electromagnetics, Second Edition. New York: Wiley, 2002), the numerical solution at a previous time step can be used as a natural choice for the initial guess of the Newton iteration at the successive time step. The convergent of the Newton-Raphson method can be achieved if the time step is small enough such that the solutions of two successive time steps are sufficiently close. In this presentation, we employ the time-domain finite element method (TDFEM) combined with the Newton-Raphson method and its variations to solve nonlinear electromagnetic problems. We first study the TDFEM formulation of nonlinear problems, and then show how it can be solved with Newton(-like) methods. We demonstrate and compare the performance of different nonlinear solvers including the basic Newton-Raphson method and its variations through a number of numerical examples.

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.

Chapter 8 - Finite-Element Time Domain 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.

Based on high order hierarchical vector function, the finite-element time-domain (FETD) method combined with tearing and interconnecting (TI) algorithm is presented. This method is efficient for time domain analysis of substrate integrated waveguide (SIW) structures, since the SIW structures have a lot of fine via holes. By means of the high order hierarchical vector function, the accuracy and ability of FETD simulation of complicated SIW problems are improved. With the aid of non overlapping domain decomposition tearing and interconnecting algorithm, the ill-conditioned spare linear system obtained by the high order hierarchical vector finite element can be solved with direct methods even if the SIW structures lead to very large linear systems. Simulation results have verified this hybrid algorithm.

In this paper we develop a leap-frog-type finite element method for modeling the electromagnetic wave propagation in metamaterials. The metamaterial model equations are represented by integrodifferential Maxwell's equations, which are quite challenging to analyze and solve in that we have to solve a coupled problem with different partial differential equations given in different material regions. Our method is based on a mixed finite element method using edge elements, which can easily handle the tangential continuity of the electric field. Stability analysis and optimal error estimate are carried out for the proposed scheme. The scheme is implemented and confirmed to obey the proved optimal convergence rate by using a smooth analytical solution. Then the scheme is extended to model wave propagation in heterogeneous media composed of metamaterials and free space, and extensive numerical results (using a rectangular edge element, a triangular edge element, and mixed edge elements) demonstrate the effectiveness of our algorithm for modeling the exotic backward wave propagation phenomenon in metamaterials. To the best of our knowledge, this is the first paper with an exhaustive simulation of backward wave propagation in metamaterials using time-domain finite element methods.

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

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.

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.

Integral method is employed in this paper to alleviate the error accumulation of differential equation discretization about time variant t in Time Domain Finite Element Method (TDFEM) for electromagnetic simulation. The error growth and the stability condition of the presented method and classical central difference scheme are analyzed. The electromagnetic responses of 2D lossless cavities are investigated with TDFEM; high accuracy is validated with numerical results presented.

Applications of magnetic-resonant wireless power transfer technology have been extensively researched because of its high efficiency, security and convenience. To recharge batteries, the induced ac voltage in the receiver coil of wireless systems must be rectified to dc, hence the power transfer systems are inevitably nonlinear. In this paper, a novel approach based on time-domain finite element method (FEM) is presented to analyze the nonlinear system in order to obtain frequency-sweeping solutions quickly. At the first step of the proposed method, the wireless power transfer system, excluding the rectifier circuit, will be equivalent to a voltage source and a general impedance at different frequencies. The equivalent circuit is then connected to the rectifier circuit. Numerical example being reported in this paper indicates that the results obtained using the proposed methodology are consistent with those obtained using traditional time-domain FEM method. If solutions at different operating frequencies are required, the proposed method can save a lot of computation time, because one just needs to solve the field problem using the time-domain FEM twice. From the numerical example on the performance analysis of the wireless power transfer system being studied, the computing time of the proposed time-domain method is only 2% of that required if traditional time-domain FEM method is used.