Explicit Finite-difference Time-domain Research Articles

A fast local embedded boundary method suitable for high power electromagnetic sources

High power sources of electromagnetic energy often require complicated structures to support electromagnetic modes and shape electromagnetic fields to maximize the coupling of the field energy to intense relativistic electron beams. Geometric fidelity is critical to the accurate simulation of these High Power Electromagnetic (HPEM) sources. Here, we present a fast and geometrically flexible approach to calculate the solution to Maxwell’s equations in vector potential form under the Lorenz gauge. The scheme is an implicit, linear-time, high-order, A-stable method that is based on the method of lines transpose (MOLT). As presented, the method is fourth order in time and second order in space, but the A-stable formulation could be extended to both high order in time and space. An O(n) fast convolution is employed for space-integration. The main focus of this work is to develop an approach to impose perfectly electrically conducting (PEC) boundary conditions in MOLT by extending our past work on embedded boundary methods. As the method is A-stable, it does not suffer from small time step limitations that are found in explicit finite difference time domain methods when using either embedded boundary or cut-cell methods to capture geometry. This is a major advance for the simulation of HPEM devices. While there is no conceptual limitation to develop this in 3D, our initial work has centered on 2D. The extension to 3D requires validation that the proposed fixed point iteration will converge and is the subject of our follow-up work. The eventual goal is to combine this method with particle methods for the simulations of plasma. In the current work, the scheme is evaluated for EM wave propagation within an object that is bounded by PEC. The consistency and performance of the scheme are confirmed using the ping test and frequency mode analysis for rotated square cavities—a standard test in the HPEM community. We then demonstrate the diffraction Q value test and the use of this method for simulating an A6 magnetron. The ability to handle both PEC and open boundaries in a standard device test problem, such as the A6, gives confidence on the robustness of this new method.

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 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.

Comparative study of grids based on the cubic crystal system for the FDTD solution of the wave equation

In this paper, a scalar wave is solved in the fully explicit finite difference time domain scheme for different stencils based on the cubic crystal system. In particular, we study four systems: the simple cubic, the body-centered cubic, the face-centered cubic and the compact packing cubic. In many papers that are focused on the artificial anisotropy induced by these grids in the propagated wave, one candidate is often better than all the others. In this manuscript, we study the stability, the physical phase velocity error and the anisotropy under two views. Firstly, we consider the same burdens or density of nodes per cubic wavelength and secondly we look at the asymptotic case. We also investigate the computational complexity based on several considerations: burdens, asymptotic time and implementation difficulties. Therefore, we pointed out how each problem or application, due to its different characteristics, has an appropriate grid in order to be treated properly.

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

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

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

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

One-Step Leapfrog HIE-FDTD for Drude Media

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

An explicit finite difference time domain method with high order accuracy in time for Maxwell's equations

An explicit finite difference time domain method with high order accuracy in time for Maxwell's equations

An Optimized 3-D HIE-FDTD Method With Reduced Numerical Dispersion

Fast and accurate modeling of scatters with fine structural feature in one or two directions has significant practical and scientific importance. To fill such research gaps, a 3-D artificial anisotropy hybrid implicit–explicit finite-difference time-domain (AA-HIE-FDTD) method is proposed. The new algorithm is based on the introduction of AA parameters to the traditional HIE-FDTD method. The iterative formulations and theoretical stability condition are given in detail. The proposed method is an easy-to-use extension to HIE-FDTD with the nearly unchanged Courant–Friedrich–Levy condition. Numerical results indicate that the proposed method has higher accuracy under the same cost of time and memory, when compared with the conventional HIE-FDTD method. In addition, the proposed method allows for larger time-step size than the conventional FDTD method, which implies reduced time cost.

Hybrid Newmark-conformal FDTD modeling of thin spoof plasmonic metamaterials

The aim of this paper is to present a new efficient and accurate semi-implicit finite-difference time-domain (FDTD) method for modeling thin plasmonic metamaterials supporting spoof surface plasmons. The presented method is formulated by combining Yu–Mittra's conformal FDTD technique for dielectrics, the simplified conformal finite integration technique for perfectly electric conductors (PECs) and the Newmark-Beta method. The resultant scheme is magnetically implicit, has a more relaxed stability condition than that of Yee's FDTD scheme, and allows for accurate modeling of both PEC and dielectric curved surfaces. Energy conservation and stability of the presented scheme is theoretically proved with the energy inequality method. Its formulation can be generalized to a class of magnetically implicit FDTD schemes with weakly conditional and unconditional stability in a unified manner. The presented scheme is used to simulate thin structures supporting spoof surface plasmon polaritons and spoof localized surface plasmons, and also applied to recent plasmonic devices such as planar frequency splitter, ultrathin rainbow trapping device and compact planar metadisk with split-ring resonators. The accuracy and efficiency of the presented scheme are assessed in comparison with the conventional explicit and implicit FDTD methods, and their conformal variants.

A Hybrid Algorithm Based on FDTD and HIE-FDTD Methods for Simulating Shielding Enclosure

A novel hybrid technique based on the classical finite-difference time-domain (FDTD) algorithm and hybrid implicit explicit finite-difference time-domain (HIE-FDTD) algorithm is developed to simulate the complex objects, which have both fine and coarse structures. The classical FDTD method is utilized to update the field components in coarse main grids, and the HIE-FDTD is applied in the fine local grids. Through controlling the classical FDTD and the HIE-FDTD calculation regions stringently, this method employs the uniform marching time-step size in the entire computational space. This time step size satisfies the Courant–Friedrich–Levy condition of the classical FDTD in coarse grids. Therefore, without an excessive increase of the computational burden, the computation efficiency of the proposed method is increased. Practical applications of the hybrid algorithm in the analysis of the shielding effectiveness of the enclosure are presented, and the simulation results certify the high accuracy and efficiency of the novel scheme.

Frequency-Dependent MNL-FDTD Scheme for Wideband Analysis of Printed Circuit Boards With Debye Dispersive Media

A semiimplicit and frequency-dependent 3-D finite-difference time-domain (FDTD) scheme with relaxed numerical stability was presented for wideband analysis of multilayer printed circuit boards (PCBs) with thin conductive and dielectric layers. This new scheme is formulated by directionally and magnetically introducing the Newmark-beta method into the conventional FDTD explicit scheme based on the recursive convolution ( RC ) approach for the so-called extended Debye model with an effective dc conductivity. No dependence of mesh step in the normal direction to the PCB layers is included in its stable condition. Interestingly, the presented semiimplicit scheme can be reduced to the RC -based explicit FDTD when the Newmark parameter is chosen as zero. The scheme was carefully verified in applications to rectangular power-bus structures, an irregular power-bus structure with decoupling capacitors and multilayer PCBs with planar electromagnetic bandgap structures. It was demonstrated in these test problems that the calculation time was reduced by a factor of 1/4–1/30 in comparison with the explicit scheme, while maintaining the same level of accuracy.

Revisiting the stability of the HIE-FDTD technique for modeling graphene dispersion

Recently, the stability of the dispersive hybrid implicit–explicit finite difference time domain (HIE-FDTD) scheme has been studied for the time domain simulations of infinite-thin graphene structures. It has been shown that the time step stability limit is a function of the graphene parameters and more restrictive than the classical HIE-FDTD constraint. In this communication, the stability of this numerical scheme is revisited, and based on the Routh–Hurwitz criterion, alternative time step stability condition is derived. To retain the classical HIE-FDTD stability constraint, stability improved implementation of the graphene dispersion is introduced. These findings are validated by using the root-locus method from the discrete-control theory. In addition, the stability is verified numerically and the accuracy of the presented implementation is demonstrated by studying the transmission coefficient of a thin two-dimensional (2D) graphene sheet placed in a 3D domain and excellent agreement with the analytical result is observed.

Three-dimensional compact explicit-finite difference time domain scheme with density variation

In this paper, the density variation is implemented in the three-dimensional compact-explicit finite-difference time-domain (CE-FDTD) method. The formulation is first developed based on the continuity equation and the equation of motion, which include the density. Some numerical demonstrations are performed for the three-dimensional sound wave propagation in a two density layered medium. The numerical results are compared with the theoretical results to verify the proposed formulation.

Explicit higher-order accurate isogeometric collocation methods for structural dynamics

The objective of the present work is to develop efficient, higher-order space- and time-accurate, methods for structural dynamics. To this end, we present a family of explicit isogeometric collocation methods for structural dynamics that are obtained from predictor–multicorrector schemes. These methods are very similar in structure to explicit finite-difference time-domain methods, and in particular, they exhibit similar levels of computational cost, ease of implementation, and ease of parallelization. However, unlike finite difference methods, they are easily extended to non-trivial geometries of engineering interest. To examine the spectral properties of the explicit isogeometric collocation methods, we first provide a semi-discrete interpretation of the classical predictor–multicorrector method. This allows us to characterize the spatial and modal accuracy of the isogeometric collocation predictor–multicorrector method, irrespective of the considered time-integration scheme, as well as the critical time step size for a particular explicit time-integration scheme. For pure Dirichlet problems, we demonstrate that it is possible to obtain a second-order-in-space scheme with one corrector pass, a fourth-order-in-space scheme with two corrector passes, and a fifth-order-in-space scheme with three corrector passes. For pure Neumann and mixed Dirichlet–Neumann problems, we demonstrate that it is possible to obtain a second-order-in-space scheme with one corrector pass and a third-order-in-space scheme with two corrector passes, and we observe that fourth-order-in-space accuracy may be obtained pre-asymptotically with three corrector passes. We then present second-order-in-time, fourth-order-in-time, and fifth-order-in-time fully discrete predictor–multicorrector algorithms that result from the application of explicit Runge–Kutta methods to the semi-discrete isogeometric collocation predictor–multicorrector method. We confirm the accuracy of the family of explicit isogeometric collocation methods using a suite of numerical examples.

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.

A review of hybrid implicit explicit finite difference time domain method

The finite-difference time-domain (FDTD) method has been extensively used to simulate varieties of electromagnetic interaction problems. However, because of its Courant–Friedrich–Levy (CFL) condition, the maximum time step size of this method is limited by the minimum size of cell used in the computational domain. So the FDTD method is inefficient to simulate the electromagnetic problems which have very fine structures. To deal with this problem, the Hybrid Implicit Explicit (HIE)-FDTD method is developed. The HIE-FDTD method uses the hybrid implicit explicit difference in the direction with fine structures to avoid the confinement of the fine spatial mesh on the time step size. So this method has much higher computational efficiency than the FDTD method, and is extremely useful for the problems which have fine structures in one direction. In this paper, the basic formulations, time stability condition and dispersion error of the HIE-FDTD method are presented. The implementations of several boundary conditions, including the connect boundary, absorbing boundary and periodic boundary are described, then some applications and important developments of this method are provided. The goal of this paper is to provide an historical overview and future prospects of the HIE-FDTD method.

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.

Hybrid sub-gridded ADE-FDTD method for modeling ground-penetrating radar on dispersive soils

Based on the sub-gridding scheme, a hybrid method with explicit finite-difference time-domain (FDTD) and implicit Crank–Nicolson (CN) FDTD is presented for ground-penetrating radar (GPR) simulations on dispersive and conductive soils in this paper. The time step of CN-FDTD can be set to a large value so that the whole region can be run with a common time step determined by the coarse grid in a stable fashion. The missing information in the sub-gridded border region is obtained with an interpolation in space. The dispersion of the soil is modeled by a multi-pole Debye model and is solved with auxiliary differential equations (ADEs) for both FDTD and CN-FDTD. Additionally, examples of the GPR scenario are calculated to verify the accuracy and efficiency of the hybrid sub-gridded ADE-FDTD method.

Hybrid Sub-Gridded Time-Domain Method for Ground Penetrating Radar Simulations Including Dispersive Materials

Based on the explicit finite-difference time-domain (FDTD) and implicit Crank–Nicolson (CN) FDTD methods, this paper presents a hybrid sub-gridded scheme whose time step size depends on the coarse grid size for numerically simulating the 3-D ground penetrating radar (GPR) scenarios in lossy, dispersive, and inhomogeneous soils. The time step size of CN-FDTD is independent of the grid size in the dense grid region due to its unconditional stability. Thus, the whole region can be run with a time step size determined by the coarse grid in an absolutely stable fashion. Moreover, a multi-pole Debye dispersion model, solved with auxiliary differential equations (ADEs) for both FDTD and CN-FDTD, is incorporated to simulate realistic GPR scenarios, including the detection of different objects buried in dispersive soils. In order to reduce the matrix size of the 3-D implicit CN-FDTD method, the domain decomposition technique is originally employed to achieve fast calculation. Several numerical examples of the GPR scenarios are provided to demonstrate the accuracy and efficiency of the hybrid sub-gridded ADE-FDTD method.