Conventional Finite-difference Time-domain Research Articles

Iteration-Based Temporal Subgridding Method for the Finite-Difference Time-Domain Algorithm

A novel temporal subgridding technique is proposed for the finite-difference time-domain (FDTD) method to solve two-dimensional Maxwell’s equations of electrodynamics in the TEz mode. Based on the subgridding FDTD algorithm with a separated spatial and temporal interface, our method focuses on the temporal subgridding region, as it is the main source of late-time instability. Different from other subgridding algorithms that work on the interpolation between coarse and fine meshes, our method stabilizes the solution by using iterative updating equations on the temporal coarse–fine mesh interface. This new method presents an alternative approach aimed at improving the stability of the subgridding technique without modifying the interpolation formulas. We numerically study the stability of the proposed algorithm via eigenvalue tests and by performing long-term simulations. We employ a refinement ratio of 2:1 in our study. Our findings indicate the stability of the conventional temporal subgridding FDTD algorithm with a magnetic field (Hz) interpolation. However, when electric fields (Ex and Ey) are utilized in interpolation, late-time instability occurs. In contrast, the proposed iteration-based method with an electric field interpolation appears to be stable. We further employ our method as the forward problem solver in the Through-the-Wall Radar (TWR) imaging application.

Unified GSTC-FDTD Algorithm for the Efficient Electromagnetic Analysis of 2D Dispersive Materials

The finite-difference time-domain (FDTD) method has been widely used for the electromagnetic wave analysis of complex media. Conventional FDTD analyses of very thin two-dimensional (2D) dispersive materials require overwhelming computing resources because they should use very refined FDTD spatial grids. In this work, we propose a computationally efficient and unified FDTD formulation for 2D dispersive materials based on a combination of the generalized sheet transition condition (GSTC) and the modified Lorentz dispersion model. The proposed FDTD formulation can lead to a significant improvement in computational efficiency compared to the conventional FDTD method, while maintaining high accuracy. Numerical examples validate the improved computational efficiency of the proposed FDTD formulation.

Extension of the contour‐path effective‐permittivity finite‐difference time‐domain method to three‐dimensional problems for the analysis of arbitrarily shaped dielectric media

AbstractThe conventional finite‐difference time‐domain (FDTD) method has been formulated in the Cartesian coordinate system. In this case, the staircase approximation should be used even for the analysis of arbitrarily shaped dielectric media. As a result, small meshes are required to accurately model these media. In this letter, we extend the contour‐path effective‐permittivity (CP‐EP) technique to three‐dimensional (3D) problems in which a reasonably accurate solution can be obtained with larger meshes for arbitrarily shaped dielectric media. The formulation is performed with the position of the dielectric medium being considered in each cell, leading to its effective permittivity. We compare the result from the present method with the Mie solution and that from the conventional staircase FDTD method to confirm the effectiveness of the 3D CP‐EP technique. The relative error of the scattering cross‐section is reduced to about half that of the staircase FDTD method.

Effective Numerical Simulations of Multipactor by WBPAT Theory

We propose a fast electromagnetic particle-in-cell (EM-PIC) method to simulate multipactor in waveguide devices. First, complex amplitudes (amplitudes and phases) at multiple frequencies are quantitatively extracted by using the wideband power-to-amplitude transformation (WBPAT) algorithm in the finite-difference time-domain (FDTD) method. Then the charged particles are advanced one way from the time-domain (TD) electromagnetic (EM) fields transformed from the complex amplitudes without considering the space-charge effects. Therefore, the conventional FDTD iterations and interpolations to cell nodes are avoided at the iterative stage. As a result, the simulation efficiency can be greatly improved. The efficiency can be further promoted by using the time step magnification and local region simulation. The proposed method can also be utilized for analyzing the multicarrier multipactor. For the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textit{C}$</tex-math> </inline-formula> -band impedance transformer under the excitations by different multicarrier waveforms, the simulation speed of the proposed method can be improved by several orders of magnitude and the accuracy of the power threshold is comparable with the self-consistent simulation.

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.

Fast Computation of Resonant Metasurfaces in FDTD Scheme Using Dispersive Surface Susceptibility Model

A new framework of scattering analysis of metasurfaces composed of an assemble of resonant scatterers, with a multistage analysis procedure, is presented. The metasurface analysis using this framework utilizes a dispersive surface susceptibility model (SSM) and a finite-difference time-domain (FDTD) method, replacing the traditional brute force simulation with the physics-based SSM. The SSM extraction for characterizing scatterers is provided, which then is used to replace detailed practical scatterers in numerical simulation. The SSM with the frequency-dependent features’ behavior is described by multiple-Lorentz pole pairs, and the coefficients of the Lorentz model are obtained by using a rational fitting method. An easy-to-implement FDTD algorithm combined with the Lorentz-described SSM is proposed and presented. The efficiency of the proposed framework is proven by comparing it with conventional FDTD. The validity of the proposed framework is verified using commercial EM simulation software. The use of this analysis procedure is demonstrated by two example scatterers. Through the examples, we discuss the requirement of the meshing lattice size, sensitivity of the algorithm to the incident angle, and influence of the edge effects on the bistatic radar cross section (RCS) computation.

Modeling of catenary electromagnetic emission with electrified train passing through neutral section

Rail transit entering the airport facilitates people’s travel, but it also brings potential electromagnetic interference. In this paper, based on the transmission line model of the arc harmonic conduction current generated by the pantograph and the catenary with the highly insulated ballastless railway, the electromagnetic emission model of the traction network is proposed in the working frequency band of the nondirectional beacon when the train passes through the neutral section of the electric phase separation. And the simplification of the transmission line model is realized by deforming the transmission line equation. Then, the finite difference time domain algorithm is used to solve the distributed current on the transmission line. However, the terminal of the conventional finite difference time domain algorithm can only be a pure resistive load. Therefore, the backward difference and the Implicit Wendroff difference format are used to extend the finite difference time domain algorithm to a more general case. Combined with a real ballastless railway in China, the effectiveness and accuracy of the proposed method are verified. From the perspective of magnetic field attenuation, ballastless railway is easier to pass the electromagnetic environment assessment of the airport than ballasted railway.

Efficacious GPR Implementations of Z-Transform-Based Hybrid LOD-FDTD with Subgridding Scheme: Theoretical Formalism and Numerical Study

Ground penetrating radar (GPR) forward modeling is one of the core geophysical research topics and also the primary task of simulating ground penetrating radar system. It is a process of simulating the propagation laws and characteristics of electromagnetic waves in simulated space when the distribution of internal parameters in the exploration region is known. And the finite-difference-time-domain (FDTD) method has the characteristics of simulating the space-time transient evolution of electromagnetic wave, whose numerical method is simple and easy to program, so it has become one of the most extensively utilized methods in GPR forward modeling. It is generally known that the conventional FDTD approach requires finer uniform Yee cell all the time to produce satisfactory accuracies from numerical simulations of the GPR. However, the smaller temporal incremental has to be adopted due to the lower spatial incremental, which would dramatically weaken the advantage of the FDTD method. To solve this issue, the subgridding-technique-based hybrid local-one-dimensional FDTD (LOD-FDTD) is applied in this work to modeling the classical GPR scenarios. In this method, the unconditional-stable LOD-FDTD is employed in the fine-grid domain, while the traditional FDTD is used in the coarse-grid domain, which could avoid the oversampling problem in the local domain if the uniform fine-grid scheme is adopted. Meanwhile due to the unconditional stability of the LOD-FDTD, the larger time step, derived from the coarse grid which satisfies the Courant-Friedrichs-Lewy (CFL) stability condition, could be utilized in the whole domain so that the long-time interpolation process could be circumvented. Additionally, the proposed approach could be arbitrarily adjusted by means of different ratio of both coarse- and fine-grid, and hence it holds much higher generality. As compared with the auxiliary differential equation (ADE) technique, the Z-transform method is integrated into FDTD methods for modeling multi-pole Debye-based dispersive media in this method, resulting in more direct numerical implementations and fewer computing steps. Finally, three different classical GPR problems are carried out to validate accuracies and efficiencies of the proposed method.

Analysis on Uncertainty of Field-to-Wire Coupling Model in Time Domain

This paper presents an approach for the time domain statistical simulation of field-to-wire coupling model with uncertain wire position in traverse direction. The parameters of governing equations of field-to-wire coupling model about the wire position are expanded to the augmented governing equations by using the generalized polynomial chaos technique with normalized orthogonal basis functions. And the coefficients of the polynomial chaos expansions are solved by combining with the stochastic Galerkin method. However, the solution of augmented governing equations is a complex problem. Therefore, a new finite-difference time-domain (FDTD) algorithm based on the implicit Wendroff difference format is proposed to solve the governing equations, and the general iterative equations are given in this paper. The proposed methods are verified by a field-to-wire coupling model. The effectiveness and advantages of the new FDTD algorithm and the uncertainty analysis method based on generalized polynomial chaos technique are verified, which are achieved by comparing with the conventional leapfrog fashion FDTD algorithm and the Monte Carlo technique, respectively.

Fast 3D transient electromagnetic forward modeling using BEDS-FDTD algorithm and GPU parallelization

The time-step sizes of the conventional finite-difference time-domain (FDTD) method are severely restricted by the Courant-Friedrichs-Lewy stability condition. This may result in excessive iterations, high cost and large runtime, and increasing late-stage cumulative errors in 3D transient electromagnetic (TEM) forward modeling. At present, forward modeling speed is a major obstacle in the development of 3D TEM inversion. To address this issue, a new 3D TEM forward modeling algorithm is presented. In the new algorithm, the backward Euler (BE) scheme is used with a view toward relaxing the stability constraint. However, after using the BE approach, a huge sparse matrix needs to be inverted in each iteration. Directly solving such matrices by Gaussian elimination or an iterative method is still time-consuming and requires large amounts of computer memory. To improve computational efficiency, the direct-splitting (DS) strategy is adopted to transform the large sparse matrices into a series of small diagonally dominant tridiagonal matrices, which are easier to solve. Then, the complex-frequency-shifted perfectly matched layer boundary condition is implemented using the bilinear transform method to reduce the size of the model. To further speed up the calculation process, the codes are then programmed to run on graphics processing unit (GPU) devices. Three models are used for calculations, and the results are compared with other numerical methods to verify the accuracy of our algorithm. After that, to demonstrate the BEDS-FDTD algorithm’s ability to model complex earth structures, a model with a very complex shape is used for the calculation via the conformal mesh technique. In addition, the modeling speed is tested using different GPU devices. We find that an NVIDIA Tesla A100 is able to calculate the 50 × 50 × 50 cell model in only 8 s.

Efficient Finite-Difference Time-Domain Modeling of Time-Varying Dusty Plasma

The finite-difference time-domain (FDTD) method has been widely used for the electromagnetic analysis of dusty plasma sheath in reentering hypersonic vehicles. The time-varying characteristics of dusty plasma should be considered to accurately analyze THz wave propagation in dusty plasma. In this work, we propose an efficient FDTD modeling of time-varying dusty plasma based on the combination of the bilinear transform and the state-space approach. The proposed FDTD formulation for time-varying dusty plasma can lead to a significant improvement in computational efficiency against the conventional shift operator FDTD counterpart while maintaining numerical accuracy. Numerical examples are performed to validate the proposed FDTD modeling of time-varying dusty plasma.

Compact System-Combined-Based FDTD Implementations With Approximate Crank–Nicolson Scheme and Electromagnetic Scattering

In this article, we develop the efficacious system-combined-based approximate Crank–Nicolson finite-difference time-domain (SC-ACN-FDTD) method which can be applied to electromagnetic scattering. After being processed by the matrix form with the discrete time, the electric and magnetic fields from Maxwell’s equations are both unified as the unique vector at the same time step. By means of the vector transformation, iterative procedures in the time domain of both electric and magnetic fields can be obtained in the Crank–Nicolson format. To model an open-domain problem, we apply convolutional perfectly matched layer (CPML) to truncate the finite computational region. To guarantee the numerical accuracy of the SC-ACN-FDTD method, we employ the empirical formula to forecast the maximum Courant–Friedrichs–Lewy (CFL) factor based on the spatial sampling density. Compared with the conventional finite-difference time-domain (FDTD) methods, the proposed SC-ACN-FDTD method can be well demonstrated and further validated by adopting electromagnetic scattering in the near and far fields. To further validate our results, the commercial software COMSOL is also adopted for the far field’s pattern in the single frequency.

Improved FDTD Method for Coupling Analysis of a Dielectric-Coated Wire Above Rough Soil Surface Under HPEM Pulse

An improved finite difference time domain (FDTD) method is proposed to calculate the transient coupling of a dielectric-coated wire above the rough soil surface under the high power electromagnetic (HPEM) pulse. The improved method is based on the domain decomposition technique, in which the dielectric-coated wire and the rough surface are allocated into the target subdomain and the rough surface subdomain, respectively. The reflected waveform from the rough surface is computed by applying the FDTD method combined with the time-domain Huygen's technique. The incident field on the target subdomain is calculated through the superposition of the excitation fields generated with the directly incident wave and the reflected wave from the rough surface. The near-field distribution and the energy-flux density (EFD) inside the dielectric-coated wire are simulated by running the FDTD in the target subdomain. The accuracy and high efficiency of the improved method are validated and demonstrated by comparing with the conventional FDTD method. Finally, the influences of some parameters related to the rough soil surface and dielectric-coated wire on the EFD are analyzed through the typical examples.

From Time-Collocated to Leapfrog Fundamental Schemes for ADI and CDI FDTD Methods

The leapfrog schemes have been developed for unconditionally stable alternating-direction implicit (ADI) finite-difference time-domain (FDTD) method, and recently the complying-divergence implicit (CDI) FDTD method. In this paper, the formulations from time-collocated to leapfrog fundamental schemes are presented for ADI and CDI FDTD methods. For the ADI FDTD method, the time-collocated fundamental schemes are implemented using implicit E-E and E-H update procedures, which comprise simple and concise right-hand sides (RHS) in their update equations. From the fundamental implicit E-H scheme, the leapfrog ADI FDTD method is formulated in conventional form, whose RHS are simplified into the leapfrog fundamental scheme with reduced operations and improved efficiency. For the CDI FDTD method, the time-collocated fundamental scheme is presented based on locally one-dimensional (LOD) FDTD method with complying divergence. The formulations from time-collocated to leapfrog schemes are provided, which result in the leapfrog fundamental scheme for CDI FDTD method. Based on their fundamental forms, further insights are given into the relations of leapfrog fundamental schemes for ADI and CDI FDTD methods. The time-collocated fundamental schemes require considerably fewer operations than all conventional ADI, LOD and leapfrog ADI FDTD methods, while the leapfrog fundamental schemes for ADI and CDI FDTD methods constitute the most efficient implicit FDTD schemes to date.

A Theory-Guided Deep Neural Network for Time Domain Electromagnetic Simulation and Inversion Using a Differentiable Programming Platform

In this communication, a trainable theory-guided recurrent neural network (RNN) equivalent to the finite-difference-time-domain (FDTD) method is exploited to formulate electromagnetic propagation, solve Maxwell’s equations, and the inverse problem on differentiable programming platform Pytorch. For forward modeling, the computation efficiency is substantially improved compared to conventional FDTD implemented on MATLAB. Gradient computation becomes more precise and faster than the traditional finite difference method benefiting from the accurate and efficient automatic differentiation on the differentiable programming platform. Moreover, by setting the trainable weights of RNN as the material-related parameters, an inverse problem can be solved by training the network. Numerical results demonstrate the effectiveness and efficiency of the method for forward and inverse electromagnetic modeling.

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.

Optimized 3D Finite-Difference-Time-Domain Algorithm to Model the Plasmonic Properties of Metal Nanoparticles with Near-Unity Accuracy

The finite difference time domain (FDTD) method is a grid-based, robust, and straightforward method to model the optical properties of metal nanoparticles (MNPs). Modelling accuracy and optical properties can be enhanced by increasing FDTD grid resolution; however, the resolution of the grid size is limited by the memory and computational requirements. In this paper, a 3D optimized FDTD (OFDTD) was designed and developed, which introduced new FDTD approximation terms based on the physical events occurring during the plasmonic oscillations in MNP. The proposed method not only required ~52% less memory than conventional FDTD, but also reduced the calculation requirements by ~9%. The 3D OFDTD method was used to model and obtain the extinction spectrum, localized surface plasmon resonance (LSPR) frequency, and the electric field enhancement factor (EF) for spherical silver nanoparticles (Ag NPs). The model’s predicted results were compared with traditional FDTD as well as experimental results to validate the model. The OFDTD results were found to be in excellent agreement with the experimental results. The EF accuracy was improved by 74% with respect to FDTD simulation, which helped reaching a near-unity OFDTD accuracy of ~99%. The λLSPR discrepancy reduced from 20 nm to 3 nm. The EF peak position discrepancy improved from ±5.5 nm to only ±0.5 nm.

Charge transfer plasmons in the arrays of nanoparticles connected by conductive linkers.

Charge transfer plasmons (CTPs) that occur in different topology and dimensionality arrays of metallic nanoparticles (NPs) linked by narrow molecular bridges are studied. The occurrence of CTPs in such arrays is related to the ballistic motion of electrons in thin linkers with the conductivity that is purely imaginary, in contrast to the case of conventional CTPs, where metallic NPs are linked by thick bridges with the real optical conductivity caused by carrier scattering. An original hybrid model for describing the CTPs with such linkers has been further developed. For different NP arrays, either a general analytical expression or a numerical solution has been obtained for the CTP frequencies. It has been shown that the CTP frequencies lie in the IR spectral range and depend on both the linker conductivity and the system geometry. It is found that the electron currents of plasmon oscillations correspond to minor charge displacements of only few electrons. It has been established that the interaction of the CTPs with an external electromagnetic field strongly depends on the symmetry of the electron currents in the linkers, which, in turn, are fully governed by the symmetry of the investigated system. The extended model and the analytical expressions for the CTPs frequencies have been compared with the conventional finite difference time domain simulations. It is argued that applications of this novel type of plasmon may have wide ramifications in the area of chemical sensing.

3-D FDTD Method for Fast Calculation of Geomagnetic Storm Electromagnetic Fields

In this article, a 3-D finite-difference time-domain (FDTD) method is proposed for rigorous modeling and fast calculation of ultralow frequency geomagnetic storm electromagnetic fields (GS-EMFs). The proposed FDTD approach enables modeling of large-scale and complex 3-D earth structure (e.g., multilayer ground and earth curvature). Unlike conventional FDTD methods, the proposed FDTD method does not require extensive computational resources when dealing with ultralow frequency problems. In this article, the ground surface GS-EMFs produced by an electrojet (line current and sheet current sources) are compared with those obtained using the classical 1-D approaches for case of uniform and horizontally stratified ground. The effect of horizontally vertically multilayer ground and earth curvature on the GS-EMFs is evaluated by the proposed FDTD method. The presented results fully support the adequacy and efficiency of the proposed FDTD method for modeling of large-scale and complex 3-D earth structures for calculation of GS-EMFs.

An FDTD-Based Method for Difference Scattering From a Target Above a Randomly Rough Surface

The difference scattering field has been defined for studying the scattering from a target above a randomly rough surface, which is independent of the selection of rough surface's length theoretically. It is feasible to extend the conventional finite-difference time-domain (FDTD)-based method for calculating composite scattering field from the target/surface model to the calculation of the difference scattering field. However, the dependence of its solution on the choice of rough surface's length is significant, which is inconsistent with the intention of the proposal of difference scattering field. In this communication, a novel FDTD-based method is proposed for difference scattering calculation, which adopts different output boundary and incident wave introduction methods compared with the conventional method. The simulation results are given to verify that the proposed method can effectively reduce the dependence on the choice of the rough surface's length in the difference scattering calculation.