Hybrid Implicit-explicit Finite-difference Time-domain Method 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.

Analysis of electromagnetic/thermal coupling of Debye media using HIE-FDTD method

In this paper, the multi-physical working mechanism of the Debye dispersive media is studied in detail, and the coupling analysis of Maxwell's and heat conduction equations is carried out. The hybrid-implicit-explicit finite difference time domain (HIE-FDTD) method is applied to accelerate calculation of the electromagnetic(EM). The Debye model, introduced to HIE-FDTD through the auxiliary difference equation (ADE) technique by utilizing polarization current, describes the dispersion characteristics of a water-based absorber. In addition, a new expression of transient power loss density is proposed with the polarization current, which overcomes the negative power loss in the traditional method and makes the computation more accurate. Finally, the analysis proves that the ADE-HIE-FDTD method has good performance in the EM-thermal coupling calculation, and the absorption rate of the absorber shows the interaction effect between temperature distribution and dielectric material.

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.

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.

A Memory-Efficient Hybrid Implicit–Explicit FDTD Method for Electromagnetic Simulation

As the explicit finite-difference time-domain (FDTD) method is restricted by the Courant−-Friedrich−-Levy (CFL) stability condition and inefficient for simulation in some situations, implicit methods are developed. The hybrid implicit−-explicit (HIE) FDTD method is one popular method among them. In this paper, a memory-efficient HIE FDTD method is designed for electromagnetic simulation. The proposed HIE-FDTD method is based upon the divergence relationship of electric fields, nearly reduces one field component, and realizes a memory reduction rate of 33% approximately. Two numerical experiments are carried out to validate the proposed method and the results indicate that the proposed memory-efficient HIE-FDTD method can work well.

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.

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.

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.

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.

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.

Analyzing the Shielding Effectiveness of a Graphene-Coated Shielding Sheet by Using the HIE-FDTD Method

A hybrid implicit explicit-finite difference time-domain (HIE-FDTD) method is proposed to simulate a graphene-coated shielding sheet. In this method, both the interband and intraband conductivity of the graphene are handled directly through an auxiliary difference equation. This method has much higher computational efficiency than the conventional FDTD method, because the time step size in this method is not related with the fine spatial cells in the graphene layer. A graphene-coated shielding sheet is simulated and analyzed by using proposed HIE-FDTD method. The numerical result shows that the shielding effectiveness of the sheet can be improved through controlling graphene's chemical potential, and the interband conductivity of the graphene has important effect on the shielding performance of the sheet, especially at the frequency band larger than 10 THz.

3D optimised hybrid implicit–explicit FDTD method with suppressed numerical dispersion

A 3D optimised hybrid implicit–explicit finite-difference time-domain (HIE-FDTD) method aimed at suppressing the numerical dispersion is proposed. Compared with the original HIE-FDTD method, the optimised parameters are introduced without costing any additional computational resource. In addition, the proposed algorithm sacrifices Courant–Friedrichs–Lewy condition and just requires a modest amount of reprogramming. The update equations of the proposed HIE-FDTD method are given and numerical results demonstrate that the proposed method performs better in the numerical dispersion.

A Collocated 3-D HIE-FDTD Scheme With PML

This letter proposes a novel hybrid implicit-explicit finite-difference time-domain method with collocation in the direction of implicitization. In this direction, the method features increased accuracy, the reason being twofold: 1) the applied combination of interpolations and differentiations preserves second-order accuracy upon nonuniform discretization and 2) material boundaries are more accurately approximated by the grid. The method’s stability and accuracy are verified by a microstrip line example. An appropriate PML is included.

Designing Graphene-Based Absorber by Using HIE-FDTD Method

A hybrid implicit explicit-finite-difference time-domain (HIE-FDTD) method is used to simulate a graphene-based absorber. Both the interband conductivity and intraband conductivity of the graphene are incorporated into the HIE-FDTD method directly through an auxiliary difference equation. Because the time step size in the proposed method is not confined by the fine spatial cells in the graphene layer, the simulation time of the proposed method is greatly reduced compared with that of the conventional FDTD method. By using the proposed HIE-FDTD method, a graphene-based absorber is simulated and analyzed. The numerical result shows that the operating frequency of the absorber is tunable through controlling graphene’s chemical potential. It also shows that the interband conductivity has an important effect on the performance of the graphene device, especially at the frequency band larger than 10 THz.

Using HIE-FDTD method to simulate graphene’s interband conductivity

A hybrid implicit–explicit-finite-difference time-domain (HIE-FDTD) method is modified to simulate the graphene. In this method, both the intraband and interband conductivity of grahene are considered. A simple Pade fit method is used to represent the interband conductivity which has complex relation with the angular frequency, and is incorporated into the HIE-FDTD method directly through an auxiliary difference equation. An infinite graphene sheet is simulated using presented technique and the result is compared with that of the conventional FDTD scheme and theoretical value. The results show that the HIE-FDTD method has excellent computational accuracy and considerably higher computational efficiency than FDTD scheme. Besides, the numerical example presents that the interband conductivity of the graphene has important effect on the numerical result at high frequency range, especially when the chemical potential is small.

Dispersion HIE‐FDTD method for simulating graphene‐based absorber

A dispersion hybrid implicit-explicit finite-difference time-domain (HIE-FDTD) method is proposed for simulating the graphene-based absorber at terahertz frequencies. The surface conductivity of the graphene derived from the Kubo formula is incorporated into the conventional HIE-FDTD algorithm by using the auxiliary differential equation technique. The time step size of this method is not related with the fine cells inside the graphene layer. The simulation result of the infinite graphene sheet shows that the dispersion HIE-FDTD method has high computational accuracy and the computational time is greatly reduced compared with the dispersion FDTD method. By using the dispersion HIE-FDTD method, it is validated that the graphene sheet can be used as a broadband absorber.

Modified HIE-FDTD Algorithm for 2-D Periodic Structures With Fine Mesh in Either Direction

This letter presents a modified hybrid implicit–explicit finite-difference time-domain (HIE-FDTD) algorithm with weakly conditionally stability for periodic structures at the oblique incident, and the complex frequency-shifted perfectly matched layer is derived to truncate the computational space. This method is suitable for the refined structures in either direction ( x - or y -direction), which covers the shortage of the previous HIE-FDTD method for periodic structures. By splitting only one field component and applying the 1-D scheme, the proposed method can be implemented self-consistently, which does not need to introduce additional equations, and the Courant–Friedrichs–Lewy condition is only limited by the bigger one of grid cell sizes. The accuracy and efficiency of this method are verified by numerical results.

A one-step leapfrog HIE-FDTD method for rotationally symmetric structures

In this work, they propose a one-step leapfrog hybrid implicit-explicit finite-difference time-domain (HIE-FDTD) method for body-of-revolution (BOR). Meanwhile, its Convolutional Perfect Matched Layer (CPML) absorbing boundary condition is implemented. In this method, the implicit difference is applied in the angular direction. All the resultant updating equations are still explicit. However, the stability condition of the proposed method is relaxed. The analytical analysis shows that its time step is only determined by the smaller one of spatial increments Δρ and Δz. A scattering example is provided to demonstrate the new algorithm. At the same time, the relative of reflection error of the CPML is given with comparisons of Mur.