Stable Finite-difference Time-domain Method Research Articles

Explicit and unconditionally stable finite-difference time-domain subgridding algorithm based on hanging variables

Due to the limitation of the stability condition, the finite-difference time-domain (FDTD) method cannot efficiently deal with electromagnetic problems containing fine structures. The explicit and unconditionally stable (EUS) FDTD method can eliminate the constraint of the stability condition and improve the simulation efficiency of fine structures by filtering out the unstable modes for the system matrix. However, the EUS-FDTD method needs to solve the eigenvalues of the numerical system matrix, and the symmetry of the numerical system matrix needs to be ensured when the subgridding scheme is used to discretize targets containing fine structures. The existing EUS-FDTD subgridding method encounters some problems such as complex implementation and insufficient accuracy. In order to solve the above problems, in this work, the hanging variables subgridding (HVS) algorithm is applied to the EUS-FDTD algorithm. Starting from the symmetry of the system matrix, the stability of the hanging variables subgridding algorithm is proven, and a high-precision, stable, and easy-to-implement HVS-EUS-FDTD scheme is proposed. Numerical examples of the radiation of linear magnetic currents in free space, electromagnetic scattering of multiple dielectric objects, and a three-dimensional cavity containing a medium demonstrate the stability, high accuracy and efficiency of the proposed method. Numerical experiments show that the computational efficiency of the HVS-EUS-FDTD algorithm can be improved hundreds of times compared with that of the uniform fine grid FDTD algorithm, and the highest computational efficiency can be improved up to ratio (the size ratio of coarse grid to fine grid) times compared with that of the HVS-FDTD algorithm.

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.

The Unconditionally Stable Associated Hermite FDTD Method for 3-D Periodic Structure Analysis

In this letter, the unconditionally stable associated hermite (AH) finite-difference time-domain (FDTD) method is extended to analyze three-dimensional (3-D) periodic structure. By combining the 3-D AH FDTD method and split-field technique, a 13-points coefficient matrix equation to eliminate phase shift is obtained with the corrected nonzero elements. To deal with the overflow at the periodic boundaries, the coefficient matrix is updated by the Floquet theory as long as the treatment of the angular points at the periodic boundary. The updating equation of the source term is derived by the total-field/scattered-field formula. The proposed method is verified by calculating the reflection coefficients of the infinite dielectric slab and the Jerusalem Cross FSS structure, respectively. The former shows a higher accuracy compared with the analytical solution with the average relative difference below <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$-$</tex-math></inline-formula> 60 dB, and reflects less sensitive to the angle of incidence. And the later represents 7 times efficiency than that of the split-field FDTD method while keeping a sound accuracy.

Improved convolutional perfectly matched layer with auxiliary differential equation for the weakly conditionally stable finite‐difference time‐domain method

On basis of the auxiliary differential equation, an improved convolutional perfectly matched layer (CPML) scheme has been proposed and applied to the three-dimensional weakly conditionally stable (WCS) finite-difference time-domain (FDTD) method. With this method, the maximum error is reduced effectively, and the absorption effect is more obvious. The simulation results show the proposed improvement scheme with more calculation accuracy than initial CPML under various Courant-Friedrich-Levy number (CFLN) conditions while ensuring similar calculation time and memory cost. Moreover, when CFLN is increased to eight, the calculation time is <30% of the traditional method, which proved the proposed method with more calculation efficiency.

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.

Incident Plane-Wave Source Formulations for Leapfrog Complying-Divergence Implicit FDTD Method

The commonly used unconditionally stable finite-difference time-domain (FDTD) methods such as alternating direction implicit (ADI)-FDTD, and its one-step formulation, leapfrog ADI-FDTD, have been found to violate the divergence condition of Gauss's law. The recently proposed leapfrog complying-divergence implicit (CDI)-FDTD not only addresses this problem, but also features many advantages, including unconditional stability, minimal floating-point operations and one-step leapfrog update. To further expand its application, this paper presents the incident plane-wave source formulations for leapfrog CDI-FDTD. Two stable and efficient formulations with different advantages are presented for introducing the far-zone plane-wave source into the FDTD problem space, namely, the scattered-field (SF) formulation and total-field / scattered field (TF/SF) formulation. To deal with the discontinuity and inconsistency across TF/SF boundaries, the fields on the boundaries need special treatments with careful modifications to ensure stability and proper plane-wave injection. Numerical results show that the incident fields can be effectively injected into the problem space with the stability of leapfrog CDI-FDTD maintained in both formulations. In addition, comparisons of radar cross sections computed using leapfrog CDI-FDTD, leapfrog ADI-FDTD and explicit FDTD with both SF and TF/SF formulations are presented. These demonstrate the advantages of leapfrog CDI-FDTD method in solving far-zone plane-wave source problems, including high efficiency, unconditional stability and complying divergence.

Optimised low dispersive ID‐WLP‐FDTD method based on scaling factor

An optimised low numerical dispersion unconditionally stable finite-difference time-domain (FDTD) method based on the isotropic dispersion (ID) scheme and weighted Laguerre polynomials (WLP) is presented. The proposed method adopts a scaling factor for permittivity and permeability in the ID-WLP-FDTD method, which can generate almost the exact phase velocity for a single frequency. In contrast with the traditional WLP-FDTD method and ID-WLP-FDTD method, the numerical dispersion of the new method is significantly reduced, and the proposed method can guarantee high accuracy and high efficiency simultaneously.

Efficient Modeling of Dispersive Media in Leapfrog WCS-FDTD Method

A leapfrog weakly conditionally stable finite-difference time-domain method is extended to simulate general dispersive media modeled through a partial fraction model via a vector fitting tool and auxiliary difference equation. The computational accuracy and efficiency of this algorithm are demonstrated by numerical examples of metal and graphene.

Analysis of FSS Using NWCS-FDTD Method

In this paper, the frequency selective surfaces (FSS) are analyzed by using a novel weakly conditional Stable(NWCS) finite-difference time-domain (FDTD) method, which is extremely useful for periodic problems with thin slots in two directions. Compared with the ADI-FDTD method, the presented method has higher accuracy and efficiency. Two examples are provided

Unconditionally stable FDTD algorithm for 3-D electromagnetic simulation of nonlinear media.

For the first time, an unconditionally stable finite-difference time-domain (FDTD) method for 3-D simulation of dispersive nonlinear media is presented. By applying a new adopted alternating-direction implicit (ADI) time-splitting scheme and the auxiliary differential equation (ADE) technique, the time-step in the FDTD simulations can be increased much beyond the Courant-Friedrichs-Lewy (CFL) stability limit. Thus, in comparison to the classical nonlinear FDTD method, the computational time for the proposed approach is decreased significantly while maintaining a reasonable level of accuracy. Numerical examples are presented to demonstrate the validity, stability, accuracy and computational efficiency of the proposed method.

One‐step leapfrog WCS‐FDTD for drude dispersive models

A one-step leapfrog weakly conditionally stable finite-difference time-domain method for modelling drude dispersive models is developed. Numerical results show that its stability condition remains invariant with respect to that in lossless media. The new algorithm is applied to efficiently investigate the transmission properties of a graphene sheet and a graphene-based frequency selective surface.

Using WCS-FDTD Method to Study the Plasma Frequency Selective Surface

The plasma is very useful to design the dynamically tunable Frequency Selective Surface (FSS) because of its tunable property. But it is difficult to simulate the plasma FSS by using the conventional finite-difference time-domain (FDTD) method, because the FSS often has fine structure which confines the time step size. To solve this problem, the weakly conditionally stable finite-difference time-domain (WCS-FDTD) method is developed in this paper. The WCS-FDTD method uses auxiliary difference equation to represent plasma's dispersive characteristic, and removes the confinement of FSS's fine structure on time step size by using hybrid implicit-explicit (HIE) technique. So, compared with the FDTD method, the presented method can save a large number of computational time, which are validated by numerical examples. Also, by using the proposed method, the tunability of plasma FSS are validated and analyzed numerically.

A Novel Unconditionally Stable SS-FDTD Method With Isotropic Dispersion

A novel unconditionally stable split-step finite-difference time domain (SS-FDTD) method based on an isotropic dispersion finite-difference scheme is introduced. The proposed method reformed the weighting factor and scaling factor of the isotropic-dispersion FDTD for the SS-FDTD method with four substeps, which can generate nearly exact phase velocity for a single frequency. Compared with the conventional four-stage split-step FDTD method, the numerical dispersion error of the new method is significantly reduced even for a large stability factor. The new method is validated by numerical simulations.

Tunable resonators based on hybrid metal–graphene structures at mid-infrared frequencies using an improved Leapfrog WCS-FDTD method

In this paper, tunable resonators based on hybrid metal–graphene structures at mid-infrared frequencies are investigated using an improved Leapfrog weakly conditional stable finite-difference time-domain (WCS-FDTD) method. Simulation results reveal that the transmission responses of the proposed devices can be predicted efficiently and controlled conveniently by means of changing chemical potentials and operating temperatures. Moreover, the structures with symmetry and asymmetry metal ribbons can be used as high-sensitivity sensors and switches, respectively. This work may provide a novel method for designing ultrafast active optoelectronic devices in optical communication.

High Spatial Order Energy Stable FDTD Methods for Maxwell’s Equations in Nonlinear Optical Media in One Dimension

In this paper, we consider electromagnetic (EM) wave propagation in nonlinear optical media in one spatial dimension. We model the EM wave propagation by the time-dependent Maxwell’s equations coupled with a system of nonlinear ordinary differential equations (ODEs) for the response of the medium to the EM waves. The nonlinearity in the ODEs describes the instantaneous electronic Kerr response and the residual Raman molecular vibrational response. The ODEs also include the single resonance linear Lorentz dispersion. For such model, we will design and analyze fully discrete finite difference time domain (FDTD) methods that have arbitrary (even) order in space and second order in time. It is challenging to achieve provable stability for fully discrete methods, and this depends on the choices of temporal discretizations of the nonlinear terms. In Bokil et al. (J Comput Phys 350:420–452, 2017), we proposed novel modifications of second-order leap-frog and trapezoidal temporal schemes in the context of discontinuous Galerkin methods to discretize the nonlinear terms in this Maxwell model. Here, we continue this work by developing similar time discretizations within the framework of FDTD methods. More specifically, we design fully discrete modified leap-frog FDTD methods which are proved to be stable under appropriate CFL conditions. These method can be viewed as an extension of the Yee-FDTD scheme to this nonlinear Maxwell model. We also design fully discrete trapezoidal FDTD methods which are proved to be unconditionally stable. The performance of the fully discrete FDTD methods are demonstrated through numerical experiments involving kink, antikink waves and third harmonic generation in soliton propagation.

A Stable FDTD Method With Embedded Reduced-Order Models

The computational efficiency of the Finite-Difference Time-Domain (FDTD) method can be significantly reduced by the presence of complex objects with fine features. Small geometrical details impose a fine mesh and a reduced time step, significantly increasing computational cost. Model order reduction has been proposed as a systematic way to generate compact models for complex objects, that one can then instantiate into a main FDTD mesh. However, the stability of FDTD with embedded reduced models remains an open problem. We propose a systematic method to generate reduced models for FDTD domains, and embed them into a main FDTD mesh with guaranteed stability up to the Courant-Friedrichs-Lewy (CFL) limit of the fine mesh. With a simple perturbation technique, the CFL of the whole scheme can be further extended beyond the fine grid's CFL limit. Reduced models can be created for arbitrary domains containing inhomogeneous and lossy materials. Numerical tests confirm the stability of the proposed method, and its potential to accelerate multiscale FDTD simulations.

Optimization of the Artificially Anisotropic Parameters in WCS-FDTD Method for Reducing Numerical Dispersion

A 3-D artificial anisotropy weakly conditionally stable finite-difference time-domain (AA-WCS-FDTD) method is presented to reduce the dispersion error without introducing additional computational cost and complexity. The motivation is to equal the speed of light in the numerical simulation to the physical speed through manipulating the permittivity and permeability. The proposed AA-WCS-FDTD algorithm achieves the reduced dispersion error for the arbitrary incident angles, different time-step sizes, and wideband frequency. The analyses of the stability and dispersion indicate that the proposed algorithm is WCS, and the Courant–Friedrich–Levy condition is slightly changed. The numerical example demonstrates that the proposed method has a reduced dispersion error and high accuracy as compared to the WCS-FDTD method.

Non-uniform Time-Step FLOD-FDTD Method for Multiconductor Transmission Lines Including Lumped Elements

This paper presents a novel non-uniform time-step (NUTS) fundamental locally one-dimensional (FLOD) finite-difference time-domain (FDTD) method for multiconductor transmission lines (MTLs) including lumped elements. The NUTS scheme adopts different (non-uniform) time steps for different periods during simulation, in order to reduce the large errors caused by unconditionally stable FDTD methods with uniform time step larger than Courant-Friedrichs-Lewy limit. It is found that NUTS scheme is potentially unstable for alternating-direction-implicit (ADI) FDTD method, but it is stable for LOD-FDTD method or its implementation based on fundamental (FLOD) scheme. The NUTS FLOD-FDTD method is useful to simulate MTLs with lumped elements including resistor, capacitor, and inductor in series and parallel connections. Moreover, the method based on multiple 1-D approach is also extended to incorporate lumped elements in cross connection to model near-field coupling between the MTLs. While the electric current density is commonly used for field-circuit coupling of parallel connected lumped elements, the magnetic current density will be utilized for field-circuit coupling of series and cross-connected lumped elements. Numerical results for MTLs with lumped elements in series, parallel, and cross connections are provided to show the trade-off between efficiency and accuracy of the proposed NUTS FLOD-FDTD method. Compared with the FLOD-FDTD method with uniform time step, the NUTS FLOD-FDTD method may achieve higher accuracy by using smaller time step for certain periods of simulation.

Analyzing Graphene-Based Absorber by Using the WCS-FDTD Method

A weakly conditionally stable finite-difference time-domain (WCS-FDTD) method is used to simulate a graphene-based absorber. By using the auxiliary differential equation and Pade fitting method, both the interband and intraband conductivities of the graphene are incorporated into the WCS-FDTD method. The time step increment in the proposed method is not determined by the fine meshes in the graphene layer, so the computational efficiency of this method is greatly improved from that of the conventional FDTD method. By using the proposed WCS-FDTD method, a graphene-based absorber is simulated and analyzed. The numerical result shows that the graphene can achieve tunable absorption through controlling its chemical potential, and the interband conductivity of the graphene has important effects on the performance of the absorber.

Stochastic LOD-FDTD method for two-dimensional electromagnetic uncertainty problems

PurposeRandom media uncertainties exhibit a significant impact on the properties of electromagnetic fields that usually deterministic models tend to neglect. As a result, these models fail to quantify the variation in the calculated electromagnetic fields, leading to inaccurate outcomes. This paper aims to introduce an unconditionally stable finite-difference time-domain (FDTD) method for assessing two-dimensional random media uncertainties in one simulation.Design/methodology/approachThe proposed technique is an extension of the stochastic FDTD (S-FDTD) scheme, which approximates the variance of a given field component using the Delta method. Specifically in this paper, the Delta method is applied to the locally one-dimensional (LOD) FDTD scheme (hence named S-LOD-FDTD), to achieve unconditional stability. The validity of this algorithm is tested by solving two-dimensional random media problems and comparing the results with other methods, such as the Monte-Carlo (MC) and the S-FDTD techniques.FindingsThis paper provides numerical results that prove the unconditional stability of the S-LOD-FDTD technique. Also, the comparison with the MC and the S-FDTD methods shows that reliable outcomes can be extracted even with larger time steps, thus making this technique more efficient than the other two aforementioned schemes.Research limitations/implicationsThe S-LOD-FDTD method requires the proper quantification of various correlation coefficients between the calculated fields and the electrical parameters, to achieve reliable results. This cannot be known beforehand and the only known way to calculate them is to run a fraction of MC simulations.Originality/valueThis paper introduces a new unconditional stable technique for measuring material uncertainties in one realization.