Explicit Finite-difference Time-domain Research Articles

Relaxation of the Courant Condition in the Explicit Finite-Difference Time-Domain(2,6) Method with Third- and Fifth-degree Differential Terms

Relaxation of the Courant Condition in the Explicit Finite-Difference Time-Domain(2,6) Method with Third- and Fifth-degree Differential Terms

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.

Relaxation of the Courant Condition in the Explicit Finite-Difference Time-Domain Method With Higher-Degree Differential Terms

A new explicit and nondissipative finite-difference time-domain (FDTD) method in two and three dimensions is proposed for the relaxation of the Courant condition. The third-degree spatial difference terms with second- and fourth-order accuracies are added with coefficients to the time-development equations of FDTD(2,4). Optimal coefficients are obtained by a brute-force search of the dispersion relations, which reduces phase velocity errors but satisfies the numerical stabilities as well. The new method is stable with large Courant numbers, whereas the conventional FDTD methods are unstable. The new method also has smaller numerical errors in the phase velocity than conventional FDTD methods with small Courant numbers.

A benchmark study on room acoustic simulations with various material input complexities

This work investigates and evaluates the capabilities of an explicit finite-difference time-domain (FDTD), a fast multipole indirect boundary element method (FMBEM), and a ray-tracing (RT) solver in the context of room acoustic modeling and simulations. In room acoustic simulations, the wave-based FDTD and FMEBM methods are known for generating accurate results at low frequencies, while the RT technique is in principle more valid at higher frequencies. The numerical aspects of setting up the solutions for each solver are discussed. Special attention is given to the influence of material input data of various degrees of detail. Single/multiple frequency-independent/dependent materials are considered in the model setup. The modeling capabilities of the three solvers in handling material input with various complexities are analyzed. Numerical results are evaluated in both frequency and time domains. Room acoustic parameters, including the reverberation time (T_20), early decay time (EDT), clarity (C_80), and definition (D_50), are compared. These results are also compared with available measurement data. Last but not least, the computational efficiencies of the three solvers are briefly discussed.

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.

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.

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.

Three-dimensional finite difference-time domain simulation of moving sound source and receiver

In this paper, moving sound source and receiver with an arbitrary trajectory are implemented in the three-dimensional compact explicit finite-difference time-domain method. To implement a moving sound source, a driving method in which the grid points around the source position are driven by the source distribution function is proposed. It is confirmed that the Gaussian distribution driving is suitable for the analysis of the moving sound sources. For a moving receiver, the sound pressure at the receiver is interpolated from the sound pressures of the adjacent eight grid points. The formulations and the numerical experiments are made for the three-dimensional sound field, and the accuracy of the proposed method is discussed. It is confirmed that the proposed method can be applied accurately to the moving sound source and receiver including the Doppler effect.

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.

A Hybrid MPI/OpenMP Parallelization Scheme Based on Nested FDTD for Parametric Decay Instability

Parametric decay instability (PDI) generated in milliseconds is an important physical phenomenon in ionospheric heating. Usually, numerical simulations are used to study PDI mechanisms. They can intuitively investigate the generation and development process of PDI, which is necessary in experimental studies. When simulating the PDI phenomenon through the explicit finite-difference time-domain (FDTD), the spatial scale spans from kilometers to centimeters, and the time scale needs to meet the Courant–Friedrichs–Lewy condition. Simulating the PDI phenomenon is time-consuming and difficult due to the high spatial resolution and strict restriction on the discrete time step. Although a nested mesh technique can boost the computational efficiency, the application of a parallel strategy is imperative to further improve it. In this study, we present a hybrid Message Passing Interface (MPI)/OpenMP parallelization scheme to solve the above-mentioned problems. This scheme can achieve an adaptive calculation and automatic allocation of MPI tasks and OpenMP threads, proving its flexibility and portability. Under the EISCAT background parameters, the PDI phenomenon was simulated. The results of the wave mode conversion and intense localized turbulence were identical to those of the serial program. Furthermore, a new simulation example and the effect of the cavity depth on electrostatic waves and negative ion density cavity were investigated. By utilizing the proposed parallelization scheme, the simulation time can be reduced from 70 h for the serial program to 3.6 h.

Explicit predictor–corrector method for nonlinear acoustic waves excited by a moving wave emitting boundary

We present an explicit finite difference time domain method to solve the lossless Westervelt equation for a moving wave emitting boundary in one dimension and in spherical symmetry. The approach is based on a coordinate transformation between a moving physical domain and a fixed computational domain. This allows to simulate the combined effects of wave profile distortion due to the constitutive nonlinearity of the medium and the nonlinear Doppler modulation of a pressure wave due to the acceleration of the wave emitting boundary. A predictor–corrector method is employed to enhance the numerical stability of the method in the presence of shocks and grid motion. It is demonstrated that the method can accurately predict the Doppler shift of nonlinear wave distortion and the amplitude modulation caused by an oscillating motion of the wave emitting boundary. The novelty of the presented methodology lies in its capability to reflect the Doppler shift of the rate of nonlinear wave profile distortion and shock attenuation for finite amplitude acoustic waves emitted from an accelerating boundary.

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.

Bandpass Simulation of Anisotropic Magnetized Ferrite Material With Alternating Direction Implicit Scheme in Open Region

Stability of explicit finite-difference time-domain (FDTD) algorithm is limited by the established relationship which results in computation duration unacceptable in certain circumstances. To enhance the development of FDTD algorithm, unconditionally stable scheme is implemented to overcome such problem, especially for the open region simulation. Here, for bandpass simulation of anisotropic magnetized ferrite material, complex envelope (CE) method and its modified scheme are employed into the implementation. Generally speaking, the proposed scheme incorporates the alternating direction implicit (ADI) with the CE method during the simulation obtaining the bandpass ADI form in perfectly matched layer (PML) regions. Meanwhile, the proposed scheme establishes the bandpass calculation method of constitutive relationship in ferrite material regions. Such method can also be employed into the unmagnetized isotropic materials and other media. Through ferrite slab model and ferrite waveguide structure, accuracy, absorption, and efficiency are demonstrated in the bandpass open region simulation numerically. The proposed scheme can not only benefit an impressive performance but also maintain both stability and accuracy with decrement of iteration steps which can be proved to be an unconditionally stable implementation. Meanwhile, it also alleviates time increment problem within implicit schemes of small time steps. Thus, benefits of the proposed scheme can be summarized from the following categories: 1) enhanced absorption and efficiency for the ADI procedure; 2) decrement of duration with a small time step; 3) bandpass form inside the magnetized ferrite material; and 4) extensively employment in different kinds of materials.

Modeling of dispersive media in ADI-FDTD method with complex–conjugate pole residue pairs

This work presents an alternating-direction implicit (ADI) finite-difference time-domain (FDTD) scheme for the study of structures that involve materials with arbitrary frequency dispersion. The material dispersion is fitted to the complex–conjugate pole-residue (CCPR) terms model, and a novel, to the best of our knowledge, numerical formulation is presented based on auxiliary differential equations and two-step ADI methodology. Additionally, the proposed technique is combined with the concept of the perfectly matched layer, and a new implicit scheme is introduced for the termination of media with CCPR dispersion in the ADI-FDTD framework. The ADI-FDTD formulation is compared with the explicit FDTD scheme for several benchmark two-dimensional problems in terms of accuracy and efficiency. The suggested algorithm is proven to be robust and capable of simulating applications in different frequency regions, spanning from microwaves to optical frequencies. It can provide a powerful tool for the analysis of nanostructures involving both strongly dispersive and nanosized materials, such as plasmonic metasurfaces, antennas, core–shell nanoparticle systems, light-trapping plasmonic solar cells, surface-enhanced Raman spectroscopy substrates, or nanodevices based on epsilon-near-zero materials.

Explicit FDTD method based on iterated Crank–Nicolson scheme

An implicit Crank–Nicolson procedure can be replaced with an explicit iteration process. An explicit finite-difference time-domain method based on the iterated Crank–Nicolson scheme that has been widely used for solving Einstein's equations is newly developed. The formulation is presented with two iterations and its stability condition is also derived. Numerical results are found to agree well with those obtained from the traditional explicit finite-difference time-domain method, showing the validity of the present iterated Crank–Nicolson–finite-difference time-domain method.

Full Explicit Numerical Modeling in Time-Domain for Nonlinear Electromagnetics Simulations in Ultrafast Laser Nanostructuring

The purpose of this paper is to present a new and accurate, fully explicit finite-difference time-domain method for modeling nonlinear electromagnetics. The approach relies on a stable algorithm based on a general vector auxiliary differential equation in order to solve the curl Maxwell’s equation in a frequency-dependent and nonlinear medium. The energy conservation and stability of the presented scheme are theoretically proved. The algorithms presented here can accurately describe laser pulse interaction with metals and nonlinear dielectric media interfaces where Kerr and Raman effects, as well as multiphoton ionization and metal dispersion, occur simultaneously. The approach is finally illustrated by simulating the nonlinear propagation of an ultrafast laser pulse through a dielectric medium transiently turning to inhomogeneous metal-like states by local free-electron plasma formation. This free carrier generation can also be localized in the dielectric region surrounding nanovoids and embedded metallic nanoparticles, and may trigger collective effects depending on the distance between them. The proposed numerical approach can also be applied to deal with full-wave electromagnetic simulations of optical guided systems where nonlinear effects play an important role and cannot be neglected.

A Leapfrog Scheme for Complying-Divergence Implicit Finite-Difference Time-Domain Method

A leapfrog scheme for the unconditionally stable complying-divergence implicit (CDI) finite-difference time-domain (FDTD) method is presented. The leapfrog CDI-FDTD method is formulated to comprise distinctive implicit and explicit update procedures. The implicit update procedures are in the fundamental form with operator-free right-hand sides (RHS), whereas the explicit ones are compatible and correspond to the familiar explicit FDTD method. Using the Fourier (von Neumann) approach, the stability of the leapfrog CDI-FDTD method is analyzed. The eigenvalues of amplification matrix are obtained to prove the unconditional stability. Furthermore, the leapfrog alternating direction implicit (ADI) FDTD and CDI-FDTD methods are discussed and compared, including the RHS floating-point operations (flops) count, for-loops and memory. The leapfrog CDI-FDTD requires merely half the flops of leapfrog ADI-FDTD at RHS while retaining the same left-hand sides (LHS) of implicit update equations and the same number of for-loops without much extra memory. Discussions and numerical results are presented to demonstrate the advantages of leapfrog CDI-FDTD method including unconditional stability, complying divergence, and efficient leapfrog scheme with reduced RHS flops.

EMI Analysis of Multiscale Transmission Line Network Using a Hybrid FDTD Method

A hybrid time-domain numerical method based on the full wave finite-difference time-domain (FDTD) method combined with a modified 1-D FDTD method has been proposed to analyze the electromagnetic interference (EMI) of multiscale transmission line networks in a metallic cabin in this article. This method is implemented through two numerical solvers for full wave and transmission line simulations, respectively, by adopting a set of reasonable physical boundary conditions between external fields and transmission line network. By using Ampere's law, the external fields can be transferred to a voltage source that can be further used as the forcing terms in the telegraph equation of the transmission line network. To overcome the challenge introduced by the multiscale size of the transmission line network, we combine 1-D explicit and implicit FDTD methods together to solve the coaxial cables and interconnects in PCB, respectively. Thus, among different subdomains, the hybrid method can be carried with the same time step without the restriction of the stability condition. Numerical examples show the accuracy as well as efficiency of the proposed method compared with the HSPICE and traditional FDTD method. In addition, the method is used successfully in analyzing the EMI effects of a complex transmission line network inside a cabin illuminated by external electromagnetic wave.

An Explicit Newmark-FDTD Algorithm With Subgridding Technique

We address the electromagnetic problem involving small structures, and put forward an improved, explicit finite-difference time-domain (FDTD) algorithm based on the Newmark discretization with subgridding technique. This algorithm achieves a stable, low-reflection, and straightforward subgridding with an arbitrary grid refinement ratio by adopting hanging variables. The Newmark method is employed to discretize the numerical system generated by the subgridding method. Besides, the Neumann series is used to obtain the inverse of the coefficient matrix. It makes the time-marching explicit and stable with a uniform large time step only determined by the coarse grid size in the whole computational domain. The series expansion avoids the matrix inversion operation, which largely enhances the calculation efficiency compared to the traditional Newmark-FDTD method. Numerical results prove the reliability and robustness of the proposed method.