Implicit-explicit Finite-difference Time-domain Research Articles

A Parallel 3-D HIE-FDTD Method using the MPI Library

This paper presents the implementation of the parallel hybrid implicit-explicit finite-difference time-domain (HIE-FDTD) method using the Message Passing Interface (MPI) library. The method proves to be very effective in simulating large-scale three-dimensional electromagnetic problems with fine structures in one direction. For the decomposition of the computational volume in the HIE-FDTD method, an MPI Cartesian 2D topology is implemented, allowing arbitrary division of the volume in two directions. Derived data types provided in the MPI library are employed to optimize inter-process communication. High accuracy and efficiency are subsequently demonstrated through a numerical example of a frequency-selected surface (FSS). It shows that the proposed method is very suitable for parallel computing, and the parallel efficiency maintains above 80% for different numbers of processes.

The HIE-FDTD Method for Simulating Dispersion Media Represented by Drude, Debye, and Lorentz Models

The hybrid implicit-explicit finite-difference time-domain (HIE-FDTD) method is a weakly conditionally stable finite-difference time-domain (FDTD) method that has attracted much attention in recent years. However due to the dispersion media such as water, soil, plasma, biological tissue, optical materials, etc., the application of the HIE-FDTD method is still relatively limited. Therefore, in this paper, the HIE-FDTD method was extended to solve typical dispersion media by combining the Drude, Debye, and Lorentz models with hybrid implicit-explicit difference techniques. The advantage of the presented method is that it only needs to solve a set of equations, and then different dispersion media including water, soil, plasma, biological tissue, and optical materials can be analyzed. The convolutional perfectly matched layer (CPML) boundary condition was introduced to truncate the computational domain. Numerical examples were used to validate the absorbing performance of the CPML boundary and prove the accuracy and computational efficiency of the dispersion HIE-FDTD method proposed in this paper. The simulated results showed that the dispersion HIE-FDTD method could not only obtain accurate calculation results, but also had a much higher computational efficiency than the finite-difference time-domain (FDTD) method.

An HIE‐FDTD algorithm for dispersive media with modified Lorentz model

AbstractIn this letter, a hybrid implicit–explicit finite difference time domain (HIE‐FDTD) method is developed to simulate general dispersive media with modified Lorentz model. The dispersive media is described through modified Lorentz model and is incorporated into the HIE‐FDTD method by two different auxiliary differential equations (ADEs). One ADE establishes the relationship between the electric displacement and electric field intensity and the other one establishes the connection between the electric current density and electric field intensity. Numerical examples are conducted and it can be seen from the numerical results that the proposed algorithm can simulate dispersive media correctly and also improve computational efficiency compared with the FDTD method and one unconditionally stable FDTD method.

Electromagnetic–Thermal Co-Simulation of Planar Monopole Antenna Based on HIE-FDTD Method

This paper presents an efficient approach to implement electromagnetic-thermal (EM-T) co-simulation of a planar monopole antenna based on hybrid implicit-explicit finite-difference time-domain method (HIE-FDTD-M). First, the EM simulation is carried out by solving Maxwell’s curl equation. Once the EM field reaches steady state, the EM power loss is computed according to the electric conductivity of the material. Finally, the thermal field is simulated by taking the EM power loss as the heat source in the heat transfer equation (HTE). For comparison, HIE-FDTD-M and FDTD-M are adopted respectively in the computation of the EM field. The simulated EM parameters of the planar monopole antenna, including S11 and radiation pattern, are consistent with those obtained by using CST software. The thermal field distribution on the surface of the antenna computed by the proposed method in this paper is approximately similar to that obtained using COMSOL software. However, the EM-T co-simulation of the antenna using HIE-FDTD-M takes only 1/11 of the time required using FDTD-M.

Current Source Implementations for the HIE-FDTD Methods

In this letter, the current source implementations are presented for the hybrid implicit–explicit finite-difference time-domain (HIE-FDTD), the improved HIE-FDTD, and the leapfrog HIE-FDTD methods. The asymmetry and the field errors of these HIE-FDTD methods are investigated for all possible values of the time index parameters. The proposed implementations give very low asymmetry and field errors for all the three HIE–FDTD methods. The field errors given by these HIE-FDTD methods are lower than those given by the alternating direction implicit finite-difference time-domain (ADI-FDTD) method.

Runge–Kutta HIE-FDTD formulations for analyzing optical transmission through periodic array of magnetically-biased graphene micro-patch structures

In this paper, simple and efficient hybrid implicit–explicit finite difference time domain (HIE-FDTD) formulations are presented for analyzing optical transmission through periodic array of magnetically-biased graphene structures. In this respect, the Runge–Kutta (RK) time differencing scheme is extended for incorporating the gaphene’s anisotropic conductivity into the HIE-FDTD scheme. The temporal discretization of the presented formulations, which are named as RK-HIE-FDTD, is not confined by the graphene’s nano thickness and this will allow using larger time-step than that of the classical explicit-FDTD counterpart. Compared with the previous magnetized HIE-FDTD approaches, the presented formulations not only involve less memory indexing overhead and fewer arithmetic operations, but also found to be numerically stable at the conventional HIE-FDTD stability limit. The accuracy and stability of the presented RK-HIE-FDTD formulations is verified by first investigating the transmission coefficient for a plane-wave normally incident on an infinite magnetized graphene layer and very good agreement with the analytical solution is observed. Furthermore, a numerical test conducted on a periodic array of magnetically-biased graphene micro-patches deposited on top of SiO2 glass substrate shows that the number and the intensity of the generated resonance frequencies are greatly affected by the graphene’s chemical potential as well as by the magnetic field bias intensity.

Stability study of the modified HIE-FDTD method for modeling graphene as a surface boundary condition

An impressive approach to model graphene in the finite-difference time-domain (FDTD) method is using a surface boundary condition (SBC). In our previous work, a method for modeling graphene as an SBC in the hybrid implicit–explicit finite-difference time-domain (HIE-FDTD) was extracted. In this paper, the impact of applying an SBC on the stability of the HIE-FDTD method is numerically analyzed using an amplification matrix method. We show that the application of the SBC does not degrade the stability of the original HIE-FDTD algorithm.

Universal HIE-FDTD method and progarm implementation for one-dimensional fine structure electromagnetic target simulation

The hybrid implicit-explicit finite-difference time-domain (HIE-FDTD) method is a weakly conditionally stable finite-difference time-domain (FDTD) method. The time step size of the HIE-FDTD method is only confined by the two coarse spatial cell sizes, so it is widely used in the simulation of electromagnetic targets with fine structures along one direction. In this work, the basic iterative formulations of the HIE-FDTD method are proposed by approximating the formulations of the FDTD method. In these formulations, the iterative coefficients are marked with spatial grid numbers. Therefore, the coefficients can be calculated automatically according to the medium parameters of the spatial cells by indexing the grid numbers. Since the triangular matrix which is used to calculate the electric field is based on the iterative coefficients, the triangular matrix can also be updated automatically. In addition, a method to reduce the number of tridiagonal matrices is proposed in this work, which can effectively reduce the calculation memory and improve the calculation efficiency. In the proposed HIE-FDTD method, equivalent parameters are employed at the interfaces of different media and the convolution perfectly matched layer boundary condition is used to truncate the computational region. Based on the proposed HIE-FDTD method, a series of programs are implemented, which can simulate arbitrary electromagnetic targets with fine structure in one dimension in linear and non-dispersive space. A dielectric plate irradiated by planar wave and a dual-frequency microstrip inverted F antenna are simulated by using these programs. The numerical results are in good agreement with those from the traditional FDTD method and CST software, and the computational efficiency of the proposed HIE-FDTD method is greatly improved in comparison with that of the traditional FDTD method. This study provides a reliable simulation tool for the wide application of the HIE-FDTD method.

Efficient HIE-FDTD method for designing a dual-band anisotropic terahertz absorption structure

The finite-difference time-domain (FDTD) method is considered to be one of the most accurate and common methods for the simulation of optical devices. However, the conventional FDTD method is subject to the Courant-Friedrich-Levy condition, resulting in extremely low efficiency for calculating two-dimensional materials (2DMs). Recent researches on the hybrid implicit-explicit FDTD (HIE-FDTD) method show that the method can efficiently simulate homogeneous and isotropic 2DMs such as graphene sheet; however, it is inapplicable to the anisotropic medium. In this paper, we propose an in-plane anisotropic HIE-FDTD method to simulate optical devices containing graphene and black phosphorus (BP) sheets. Numerical analysis shows that the proposed method is accurate and efficient. With this method, we present a novel multi-layer graphene-BP-based dual-band anisotropic terahertz absorption structure (GBP-DATAS) and analyze its optical characteristics. Combining the advantages of graphene and BP localized surface plasmons, the GBP-DATAS demonstrates strong anisotropic plasmonic resonance and high absorption rate in the terahertz band.

Low Numerical Dispersion 3-D HIE-FDTD Methods With Weaker Stability Condition

In order to analyze electromagnetic targets with fine structure in one direction more effectively and accurately, optimized 3-D hybrid implicit-explicit finite-difference time-domain (HIE-FDTD) algorithms are formulated based on isotropic dispersion (ID) finite difference (FD) schemes, which are weighted summation of new FD schemes and conventional second-order central FD scheme. Weighting factors for the proposed optimized HIE-FDTD methods can be analytically determined to minimize the anisotropy of the dispersion. The methods also adopt scaling factors for material properties, such as the permittivity and permeability, which can generate nearly the exact phase velocity for a single frequency. Compared with previous several HIE-FDTD methods, the numerical dispersion error of the new methods is significantly reduced, and the Courant-Friedrichs-Lewy stability conditions of the proposed HIE-FDTD methods are weaker, which means that the proposed methods are more flexible in choosing time step size and satisfy the requirement of accuracy and efficiency simultaneously. Numerical simulation demonstrates the effectiveness of the methods.

An Efficient 3-D Stochastic HIE-FDTD Algorithm for Investigation of Statistical Variation in Electromagnetic Field

We propose a stochastic hybrid implicit–explicit finite-difference time-domain method (S-HIE-FDTD) to compute the mean and variance of the electromagnetic (EM) fields using a single simulation, given those of the conductivity and permittivity in the computation domain. The mean and variance field update equations underlying the proposed method are derived from the field update equations of the “traditional” deterministic HIE-FDTD. The Courant–Friedrichs–Lewy condition of the S-HIE-FDTD depends on the spatial discretization sizes only in two dimensions; therefore, for computation domains with fine geometric features only in the remaining dimension, it uses a time step size that is larger than that of fully explicit schemes. Indeed, numerical results demonstrate that the proposed method is faster than the previously developed stochastic FDTD in computing the mean and variance of the EM fields in two different problems: wave propagation through a multilayer human tissue and transmission through a frequency selective surface.

Applications of anisotropic one‐step leapfrog HIE‐FDTD method in microwave circuit and antenna

To verify the effect of artificial anisotropy parameters in one-step leapfrog hybrid implicit-explicit finite-difference time-domain (FDTD) method, we calculated several microwave components with different characteristics. Introduced auxiliary field variable can reduce the program difficulty and improve the computational efficiency without additional computational time and memory cost. Analyses of the numerical results are proved that the calculation time is reduced to about one-sixth compared to the traditional FDTD method for the same example simulated. The memory cost and relative error are remained at a good level. The numerical experiments for microwave circuit and antenna have been well demonstrated the method available.

An Artificial Anisotropy Four-Step HIE-FDTD Method With Lower Numerical Dispersion Error

In this article, by introducing artificial anisotropic (AA) parameters in the four-step hybrid implicit-explicit finite-difference time domain (HIE-FDTD) method, an AA four-step HIE-FDTD method is proposed, which can reduce the numerical dispersion error and improve the computational accuracy. Firstly, the Maxwell's matrix equation is decomposed into four sub-matrix equations by using the split-step scheme, and artificial anisotropy parameters are introduced, then the proposed AA four-step HIE-FDTD is obtained. Furthermore, the stability analysis of the AA four-step HIE-FDTD method shows that the stability condition of the proposed method is closed to the original four-step HIE-FDTD method. Next, the numerical dispersion characteristics of the proposed method are analyzed and compared with other HIE-FDTD methods. The results show that the numerical dispersion error of the proposed method is significantly reduced compared with the four-step HIE-FDTD method. Finally, the performance of the proposed method is further verified by the numerical simulation. Numerical results show that the proposed method has lower numerical dispersion error and higher computational accuracy than that of the four-step HIE-FDTD method.

Computer simulation of polarization rotation characteristics of graphene

The magnetically biased graphene has the polarization rotation characteristics which is useful to design the polarizer. But this characteristic is difficult to simulate by using the finite-difference time-domain (FDTD) method, not only due to the graphene's thin layer which confines the time step size, but also because of graphene's isotropic surface conductivity when it is biased by static magnetic field. To solve this problem, this study presents an anisotropic hybrid implicit-explicit FDTD method. This method uses auxiliary difference equations to represent graphene's conductivity, and removes the confinement of graphene's thickness on time step size by using hybrid implicit-explicit technique. So, compared with FDTD method, the presented method can save a large number of computational time, which are validated by numerical examples.

An Optimized One-Step Leapfrog HIE-FDTD Method With the Artificial Anisotropy Parameters

By introducing the artificial anisotropy (AA) parameters, a 3-D one-step leapfrog hybrid implicit–explicit finite-difference time domain (HIE-FDTD) method is proposed, which can reduce the numerical dispersion error without increasing the computational cost. The formulation of the AA one-step leapfrog HIE-FDTD method is obtained by introducing the relative permittivity and permeability tensors in the original one-step leapfrog HIE-FDTD method. The Courant–Friedrichs–Lewy (CFL) stability condition of the AA one-step leapfrog HIE-FDTD method is close to that of the original one-step leapfrog HIE-FDTD method. In addition, the proposed HIE-FDTD method has lower numerical dispersion error and higher calculation accuracy than that of the one-step leapfrog HIE-FDTD method. Finally, to testify the characteristics of the proposed HIE-FDTD method, numerical simulation experiments are given.

A 3-D Hybrid Implicit Explicit Single-Field Finite-Difference Time-Domain Method

This paper presents a new 3-D hybrid implicit explicit (HIE) finite-difference time-domain (FDTD) method with a stability criterion relaxed from the discretization space steps in one direction. The proposed method is particularly useful and efficient for electromagnetic (EM) simulation of structures having fine details in one Cartesian direction such as thin shields and thin slots on a metallic enclosure whose EM analysis are popular and important in electromagnetic compatibility. In this algorithm, unlike common FDTD methods in which both electric and magnetic fields are updated, only the electric field needs to be updated during the time-marching process of the algorithm; hence, it is called as a “single-field” FDTD method. This could be achieved by applying the Crank-Nicolson (CN) scheme only to the electric field, while in earlier HIE-FDTD (and also unconditionally stable FDTD) methods, the CN scheme is applied to both electric and magnetic fields. The proposed algorithm consists of two implicit and three explicit updating equations that can be solved simultaneously (in parallel) by applying multithreading. Thanks to few and simple updating equations of the proposed algorithm, it is computationally more efficient than other HIE-FDTD methods in terms of runtime. The high computational efficiency and accuracy of the proposed method are demonstrated by producing several numerical examples and providing comparison to the results obtained using methods available in the literature.

Implementation of convolutional perfectly matched layer for three‐dimensional hybrid implicit‐explicit finite‐difference time‐domain method

The hybrid implicit-explicit (HIE) finite-difference time-domain (FDTD) method with the convolutional perfectly matched layer (CPML) is extended to a full three-dimensional scheme in this article. To demonstrate the application of the CPML better, the entire derivation process is presented, in which the fine scale structure is changed from y-direction to z-direction of the propagation innovatively. The numerical examples are adopted to verify the efficiency and accuracy of the proposed method. Numerical results show that the HIE-FDTD with CPML truncation has the similar relative reflection error with the FDTD with CPML method, but it is much better than the methods with Mur absorbing boundary. Although Courant-Friedrich-Levy number climbs to 8, the maximum relative error of the proposed HIE-CPML remains more below than −71 dB, and CPU time is nearly 72.1% less than the FDTD-CPML. As an example, a low-pass filter is simulated by using the FDTD-CPML and HIE-CPML methods. The curves obtained are highly fitted between two methods; the maximum errors are lower than −79 dB. Furthermore, the CPU time saved much more, accounting for only 26.8% of the FDTD-CPML method while the same example simulated.

Improved Cylindrical HIE-FDTD Method and Its Application to a Metal Disk-Type Surface Wave Splitter

The cylindrical hybrid implicit–explicit finite-difference time-domain method, in which the implicit scheme is adopted only in the circumferential direction, is improved in order to treat general electromagnetic problems. The connection of the field in a specific plane gives the problem with a cyclic matrix, the solution of which is performed with the Sherman–Morrison formula. The convolutional perfectly matched layer is also introduced in cylindrical coordinates. The present method is used to analyze a metal disk-type surface wave splitter. The computational time is reduced to less than 20% of the explicit method, yielding almost the same numerical results. Calculated radiation patterns are compared with experimental results.

One-Step Leapfrog HIE-FDTD for Drude Media

A one-step leapfrog hybrid implicit–explicit finite-difference time-domain (HIE-FDTD) method is developed for Drude dispersive media. The Drude media is modeled through a current source characterized by a semi-implicit auxiliary differential equation. The stability condition is the same as that of conventional leapfrog HIE-FDTD and is determined by the grid cell size along only one direction. The proposed method is used to efficiently investigate the properties of surface plasmon polaritons propagating along a one-atom-thick graphene sheet in low terahertz band with a very fine grid.

One-Step Leapfrog HIE-FDTD Modeling of Plasmonic Structures Using Partial-Fraction Dispersion Models

An improved one-step Leapfrog hybrid implicit-explicit finite-Difference Time-Domain (HIE-FDTD) method is developed for simulating complex plasmonic structures that comprise dispersive materials over an ultra-wide optical frequency range. The dispersive media is described by a partial fraction (PF) dispersion model by using the vector fitting technique and further implemented into the Leapfrog HIE-FDTD scheme via an auxiliary difference equation (ADE) formulation. The performance of the proposed method is evaluated in optical response of a multi-functional plasmonic structure which can act as a dynamically tunable band-stop filter and refractive index sensor. Simulation results demonstrate good numerical accuracy and high computational efficiency and may find some potential applications in sensing, detecting and optical communications.