Mixed Finite-element Time-domain Research Articles

Total and scattered field decomposition technique in mixed FETD methods and its applications for electromagnetic cloaks

In this paper, we apply the total and scattered field plane wave excitation method to the mixed finite element time-domain (FETD) method for solving Maxwell’s equations to compute the scattered field of electromagnetic cloaks. This method can quantify the performance and operating frequency range of the cloak. We also present a mixed FETD scheme, which is coupling with perfectly matched layers technique. It can deal with various physical quantities more flexibly when solving the scattered field of complex dispersive media. Numerical results confirm the effectiveness of our algorithm, and quantify the performances of our electromagnetic cloaks at different frequency. We can conclude that the farther the incident wave’s frequency is from the cloak’s optimal operating frequency, the worse its performance.

The Dilemma of Resolving the Low-Frequency Breakdown Problem in Microwave Components via Traditional and Improved Finite-Element Time-Domain Techniques

This paper developed an improved numerical methods for calculating electromagnetic field at low frequencies, and compared the performance with the traditional method. Traditional finite-element time-domain (FETD) is known to suffer from low frequency breakdown (LFB) problem, where system matrix becomes ill-conditioned, and leads to instability of small size electrical structures. In this paper, we proposed a mixed FETD (mFETD) method in resolving the LFB anomaly in the traditional FEM, with a particular reference to transmission line, inductor, and coaxial cable. We considered wave equation of E-field with incorporation of divergence constraint equation (DCE) as a function of Lagrange multiplier using current continuity equation (CCE) and Gauss’s law. For spatial discretization, both nodal basis functions and Curl conforming vector basis functions were selected, and we employed implicit Newmark beta algorithm (NBA) for integration of time. We describe how the components constructed via DCE in system matrix mitigate the singularity impact of the stiffness matrix, which consequently led to significant improvement of the system matrix. Numerical experiment and results demonstrate how the mFETD obtains stable numerical solution and attains faster rate of convergence while using an iterative solver, as such, improves the computational efficiency. Therefore, the mFETD method handles the transient problems in transmission line, which causes LFB anomaly in the traditional FETD, but not efficient in terms of computational time and iteration at solving the coaxial cable problem.

A Combination of Mixed FETD Method and SPICE to Simulate Nonlinear Multiport Circuits

A combination of the mixed finite-element time-domain method (MFETD) and SPICE solver is proposed in this letter to analyze the circuit response and crosstalk effects in complex microwave circuits with linear/nonlinear, passive/active components. The MFETD method in the EM part is adopted as an unconditionally stable time integration scheme to account for the low-frequency breakdown problem. SPICE is introduced to solve the nonlinear circuit equations in the circuit part by taking into account the coupling between multiport circuits. A microwave circuit based on an AD633 chip and an MCP3221 board is simulated to verify that this hybrid MFETD-SPICE method not only is superior in mesh flexibility to the finite-difference time-domain (FDTD)-based method but also exploits the well-developed open-source software SPICE to solve the nonlinear equations.

New Mixed SETD and FETD Methods to Overcome the Low-Frequency Breakdown Problems by Tree-Cotree Splitting

Traditional time-domain finite-element and spectral-element methods are known to suffer from the so-called low-frequency breakdown problem, where the system matrix can become ill-conditioned, and results can be unstable for electrically fine structures. In this article, a new mixed spectral-element time-domain (SETD) and finite-element time-domain (FETD) methods are proposed to overcome this low-frequency breakdown problem by constraint equations and tree-cotree splitting. Two forms of gradient matrices or the null space of the stiffness matrix of high-order basis functions are calculated to construct the constraint equations. Then, the constraint equations are directly applied in the time-stepping matrix equation by tree-cotree splitting to improve the properties of system matrices when the implicit Newmark-beta algorithm is adopted in two different forms. In the calculation of the first form of gradient matrix, the electric field is expanded by high-order edge basis functions; in the second form of gradient matrix, high-order edge basis functions associated with cotree edges and nodal basis functions defined on the free nodes are both employed to expand the electric field. Several numerical examples demonstrate that the new mixed SETD and mixed FETD methods can effectively overcome the low-frequency breakdown with high accuracy and without increasing the number of degrees of freedom over the conventional SETD and FETD methods.

Wideband Low-Frequency Design of Inductors and Wireless Power Transfer Coils Using the Mixed Finite-Element Time-Domain Method

Wideband low-frequency designs of inductors and wireless power transfer (WPT) coils are challenging because of the low-frequency breakdown in traditional numerical methods. In this letter, we apply a recently developed mixed finite-element time-domain (mixed FETD) method to overcome such low-frequency breakdown difficulties in designing inductors and WPT coils and verify the feasibility of the mixed FETD method for such models at low frequencies. This letter provides a demonstration for the design of the low-frequency electrically small electromagnetic devices.

An Efficient Mixed Finite-Element Time-Domain Method for Complex Electrically Small Problems

A new mixed finite-element time-domain (FETD) method is developed to eliminate the low-frequency breakdown phenomenon in the conventional finite-element method when solving electrically small transient problems. Constructed by Gauss’s law and the current continuity equation, the divergence constraint equation is incorporated into the wave equation for the electric field in the form of a Lagrange multiplier. Curl-conforming vector basis functions and nodal basis functions are chosen for spatial discretization and the implicit Newmark-beta algorithm is adopted for time integration. It is shown that the terms constituted by the divergence constraint equation in the system matrix help to suppress the effect of the singularity of the stiffness matrix, and the property of the system matrix is significantly improved. Numerical results show that for electrically small problems that cannot be solved by the conventional FETD method, the proposed method not only obtains a stable numerical solution but also achieves a faster convergence rate when an iterative solver is used, thus improving the computational efficiency.

A Convolution-Free Mixed Finite-Element Time-Domain Method for General Nonlinear Dispersive Media

In this paper, a mixed finite-element time-domain (FETD) method is presented for the simulation of electrically complex materials, including general combinations of linear dispersion, instantaneous nonlinearity, and dispersive nonlinearity. Using both edge and face elements, the presented method offers greater geometric flexibility than existing finite-difference time-domain (FDTD) implementations, and in contrast to existing nonlinear FETD methods, also incorporates both linear and nonlinear material dispersions. Dielectric nonlinearity is incorporated into the Crank–Nicolson mixed FETD formulation via a straightforward Newton–Raphson approach, for which the associated Jacobian is derived. Moreover, the dispersion is modeled via the Mobius $z$ transform method, yielding a simpler more general algorithm. The method’s accuracy and convergence are verified, and its capability demonstrated via the simulation of several nonlinear phenomena, including temporal and spatial solitons in two spatial dimensions.

Stabilized Mixed Finite-Element Time-Domain Method for Fast Transient Analysis of Multiscale Electromagnetic Problems

This paper describes the mixed E-B finite-element time-domain (FETD) method that is stabilized for an arbitrary time step size. Since the existing mixed FETD (M-FETD) method is based on a fully explicit leapfrog time marching procedure, it has a numerical stability condition: if not satisfied, the instability arises. The condition gives the upper limit of the time step size, and it becomes much strict if there exist small meshes in a computational domain. As a result, the smaller the time step size is, the higher the computational cost becomes. In this paper, we propose the stabilized M-FETD method through developing the stabilization process after specifying the root cause of the instability of the existing method. By virtue of the stabilization process, we can derive the fully explicit updating process which is efficient and numerically stable for the arbitrary time step size without degrading the accuracy. Numerical results of example problems demonstrate that our approach is superior to the existing one, especially in a multiscale electromagnetic problem in two and three dimensions.

Data-Sparse Matrix Decomposition Algorithm for Mixed Finite-Element Time-Domain Solution of Maxwell Equations

In this article, an efficient algorithm is presented for the mixed finite-element time-domain solution of Maxwell equations. The implicit finite-element time-domain method based on the Crank–Nicolson time integration is unconditionally stable but involves solving a linear system at each time step. A data-sparse matrix decomposition algorithm combined with the nested dissection technique is developed to solve the large linear system. Based on the hierarchical matrix (H-matrix) formatted arithmetic, this algorithm can reduce the computational costs to be almost linear. Moreover, an iterative improvement method is introduced to further improve the solution performance. Numerical results demonstrate the effectiveness of the proposed method for the analysis of microwave circuits.

Performance of the Alternating Direction Implicit Scheme With Recursive Sparsification for the Finite Element Time Domain Method

In this paper, we solve the $ \mathcal {E}$ and $ \mathcal {B}$ mixed finite element time domain formulation using the alternating-direction implicit (ADI) scheme in time. To save computational cost, the algorithm is improved using a recursive sparsification procedure that approximates the inverse matrix needed to calculate the electric field. Although the method is unconditionally stable, we show that the numerical performance of the algorithm is strongly related to the time step. Numerical results demonstrated that for the same accuracy our approach is much faster than the ADI with Cholesky factorization.

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.

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.

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

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.

Conformal Perfectly Matched Layer for the Mixed Finite Element Time-Domain Method

We introduce a conformal perfectly matched layer (PML) for the finite-element time-domain (FETD) solution of transient Maxwell equations in open domains. The conformal PML is implemented in a mixed FETD setting based on a direct discretization of the first-order coupled Maxwell curl equations (as opposed to the second-order vector wave equation) that employs edge elements (Whitney 1-form) to expand the electric field and face elements (Whitney 2-form) to expand the magnetic field. We show that the conformal PML can be easily incorporated into the mixed FETD algorithm by utilizing PML constitutive tensors whose discretization is naturally decoupled from that of Maxwell curl equations (spatial derivatives). Compared to the conventional (rectangular) PML, a conformal PML allows for a considerable reduction on the amount of buffer space in the computational domain around the scatterer(s).

Mixed Finite-Element Time-Domain Method for Transient Maxwell Equations in Doubly Dispersive Media

We describe a mixed finite-element time-domain algorithm to solve transient Maxwell equations in inhomogeneous and doubly dispersive linear media where both the permittivity and permeability are functions of frequency. The mixed finite-element time-domain algorithm is based on the simultaneous use of both electric and magnetic field as state variables with a mix of edge (Whitney 1-form) and face (Whitney 2-form) elements for discretization of the coupled first-order Maxwell curl equations. The constitutive relations are decoupled from the curl equations and cast in terms of (auxiliary) ordinary differential equations involving time derivatives. Permittivity and permeability dispersion models considered here are quite general and recover Lorentz, Debye, and Drude models as special cases. The present finite-element time-domain algorithm also incorporates the perfectly matched layer absorbing boundary conditions in a natural way.