An Efficient Algorithm for Implementing the Crank–Nicolson Scheme in the Mixed Finite-Element Time-Domain Method

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 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.

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.

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.

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.

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.

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.

In this paper, we present a new time-domain method that has a naturally diagonal mass matrix and thereby a strict linear computational complexity per time step, regardless of whether the discretization is a structured grid or an unstructured mesh. This property is obtained independent of the element shape used for discretization. No interpolations, projections, and mass lumping are required. The accuracy and stability of the proposed method are both theoretically guaranteed. In addition, no dual mesh is needed and the tangential continuity of the fields is satisfied across the element interface. The flexible framework of the proposed method also allows for a straightforward extension to higher order accuracy in both electric and magnetic fields. Numerical experiments have been conducted on a variety of unstructured triangular-element meshes. Correlations with analytical solutions and the time-domain finite-element method have validated the accuracy and generality of the proposed new time-domain method.

A new time-domain finite element method (TDFEM) based on B-spline temporal basis function is presented for the first time in this paper. This approach is motivated by aims of achieving higher computational efficiency. Compared with the reported TDFEM scheme, which is based on Galerldnpsilas method with a piecewise linear temporal expansion of electric field[t], in our scheme, the electric field is parameterised in terms of piecewise polynomial B-spline basis function. B-spline is chosen as temporal basis function both for a smoother interpolation and a more efficient approximation[2]. Moreover, this method allows for a consistent time discretization of the electric field vector wave equation and its augmented PML regions. The novel scheme is first verified on cavity problem, in which computed results agree with the analytical solution successfully. Then the scheme is applied to a bow-tie antenna and the results have good agreement with those in frequent-domain 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.

A discontinuous Galerkin time-domain finite-element method (DG-TDFEM) is developed for modeling wideband EM response of dielectric loaded waveguide structures. It can be considered as a kind of domain decomposition method (DDM). The hierarchical vector basis functions are used to expand the electric and magnetic fields to maintain high order accuracy and Crank-Nicolson (CN) difference scheme is utilized for the time-partial equation for each subdomain. The fields in adjacent subdomains are directly related to each other by the fields on their interfaces. Thus, there is no need to solve a global matrix equation related to all boundary values. The complex frequency-shifted perfectly matched layers (CFS-PML) are used for the truncation of solution domain. 3-dimensional cavity and waveguide structure with dielectric loading are simulated to demonstrate the accuracy and efficiency of the proposed method.

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.

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.

This paper proposes an extension of Finite Element Time Domain (FETD) method for the global electromagnetic analysis of complex active microwave circuits. The equivalent current sources and capacitances of the distributed part of the circuit, derived from FETD analysis, are incorporated into state equations to solve the lumped element part of the circuit. Benchmark tests on a microwave amplifier and self-oscillating mixer indicate that this extended FETD is not only superior in mesh flexibility but also gives much better results than a simulated FDTD-based algorithm.

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