Explicit Finite-difference Time-domain Method Research Articles

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 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.

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 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.

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.

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.

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.

A Fundamental LOD-FDTD Method in Cylindrical Coordinates

An efficient implicit finite-difference time-domain (FDTD) algorithm in cylindrical coordinates is developed on the basis of a fundamental locally 1-D scheme. The Sherman–Morrison formula is introduced to treat a cyclic matrix and the image theory is utilized to impose the perfect electric conductor boundary condition. The effectiveness of the present method is investigated through the analysis of a metal disc-type terahertz surface wave splitter. The computation time is found to be reduced to less than half that of the explicit FDTD method, while maintaining a comparable accuracy.

An in-depth stability analysis of nonuniform FDTD combined with novel local implicitization techniques

This work focuses on efficient full-wave solutions of multiscale electromagnetic problems in the time domain. Three local implicitization techniques are proposed and carefully analyzed in order to relax the traditional time step limit of the Finite-Difference Time-Domain (FDTD) method on a nonuniform, staggered, tensor product grid: Newmark, Crank–Nicolson (CN) and Alternating-Direction-Implicit (ADI) implicitization. All of them are applied in preferable directions, alike Hybrid Implicit–Explicit (HIE) methods, as to limit the rank of the sparse linear systems. Both exponential and linear stability are rigorously investigated for arbitrary grid spacings and arbitrary inhomogeneous, possibly lossy, isotropic media. Numerical examples confirm the conservation of energy inside a cavity for a million iterations if the time step is chosen below the proposed, relaxed limit. Apart from the theoretical contributions, new accomplishments such as the development of the leapfrog Alternating-Direction-Hybrid-Implicit-Explicit (ADHIE) FDTD method and a less stringent Courant-like time step limit for the conventional, fully explicit FDTD method on a nonuniform grid, have immediate practical applications.

Three-dimensional numerical simulation of parametric speaker by compact explicit finite-difference time-domain method

In this study, the three-dimensional numerical simulation of a parametric speaker is carried out. To simulate the secondary sound field of difference frequency generated by the high intense ultrasonic wave, the Westervelt equation is numerically solved by the compact explicit finite-difference time-domain (CE-FDTD) method which is a high-accuracy version of the standard FDTD method. In the simulation, the interpolated wideband (IWB) scheme that is one of the most accurate scheme of the CE-FDTD method and whose cut-off frequency is in agreement with the Nyquist frequency, is chosen to reduce computer resources. The IWB scheme is then implemented on a graphics processing cluster system. Three-dimensional numerical models whose size is 1.5 m x 1.5 m x 8 m are developed with cell sizes of 1 mm. Some demonstrations are made for the nonlinear sound wave propagation. The propagation characteristic and the directivity of the parametric speaker are obtained with reasonable accuracy.

A New Hybrid Implicit–Explicit FDTD Method for Local Subgridding in Multiscale 2-D TE Scattering Problems

The conventional finite-difference time-domain (FDTD) method with staggered Yee scheme does not easily allow including thin material layers, especially so if these layers are highly conductive. This paper proposes a novel subgridding technique for 2-D problems, based on a hybrid implicit–explicit scheme, which efficiently copes with this problem. In the subgrid, the new method collocates field components such that the thin layer boundaries are defined unambiguously. Moreover, aspect ratios of more than a million do not impair the stability of the method and allow for very accurate predictions of the skin effect. The new method retains the Courant limit of the coarse Yee grid and is easily incorporated into existing FDTD codes. A number of illustrative examples, including scattering by a metal grating, demonstrate the accuracy and stability of the new method.

Three-dimensional dispersive hybrid implicit–explicit finite-difference time-domain method for simulations of graphene

A dispersive hybrid implicit–explicit finite-difference time-domain (HIE-FDTD) method is presented in this paper. Surface conductivity of the graphene is incorporated into the HIE-FDTD method directly through an auxiliary difference equation. The time step size in proposed method has no relation with the fine spatial discretization, so it is very useful for the simulation of the graphene when it needs to be discretized across its thickness. The stability condition of this method is not only determined by the spatial cell sizes Δx and Δz, but also related with the surface conductivity of the graphene. The computational accuracy and efficiency of this method are demonstrated through numerical examples. The results show that with reasonable accuracy, the memory requirement and computation time of the dispersive HIE-FDTD method are both considerably reduced as compared with those of the conventional FDTD method and LOD-FDTD method.

A Hybrid Implicit–Explicit and Conformal (HIE/C) FDTD Method for Efficient Electromagnetic Simulation of Nonorthogonally Aligned Thin Structures

In a conventional finite-difference time-domain (FDTD) method, the computational cost increases as a grid spacing decreases because the number of unknowns increases and a time step size decreases to fulfill a numerical stability condition. Especially, it is inherently time-consuming to analyze thin structures such as a printed circuit board (PCB) and nonorthogonally aligned objects such as interconnection patterns on the PCB. In this paper, we propose an efficient electromagnetic simulation technique which is constructed by combining a hybrid implicit–explicit (HIE)-FDTD method and a conformal FDTD (CFDTD) method. The proposed technique can avoid the aforementioned problems by exploiting advantages of the HIE-FDTD and CFDTD methods. An example model of the interconnection pattern on the PCB is analyzed by the proposed and conventional methods, and we compare their accuracy and efficiency.

Simulations and measurements of 3-D ultrasonic fields radiated by phased-array transducers using the westervelt equation.

The purpose of this work is to validate, by comparing numerical and experimental results, the ability of the Westervelt equation to predict the behavior of ultrasound beams generated by phased-array transducers. To this end, the full Westervelt equation is solved numerically and the results obtained are compared with experimental measurements. The numerical implementation of the Westervelt equation is performed using the explicit finite-difference time-domain method on a three-dimensional Cartesian grid. The validation of the developed numerical code is first carried out by using experimental data obtained for two different focused circular transducers in the regimes of small-amplitude and finite-amplitude excitations. Then, the comparison of simulated and measured ultrasonic fields is extended to the case of a modified 32-element array transducer. It is shown that the developed code is capable of correctly predicting the behavior of the main lobe and the grating lobes in the cases of zero and nonzero steering angles for both the fundamental and the second-harmonic components.

3P2-2 Reflective boundary condition with arbitrary boundary shape for compact explicit-finite difference time domain method(Poster Session)

3P2-2 Reflective boundary condition with arbitrary boundary shape for compact explicit-finite difference time domain method(Poster Session)

Unconditionally Stable Fundamental LOD-FDTD Method With Second-Order Temporal Accuracy and Complying Divergence

An unconditionally stable fundamental locally one-dimensional (LOD) finite-difference time-domain (FDTD) method with second-order temporal accuracy and complying divergence (CD) (denoted as LOD2-CD-FDTD) is presented for three-dimensional (3-D) Maxwell's equations. While the conventional LOD-FDTD method does not have complying divergence, the LOD2-CD-FDTD method has complying divergence in a manner analogous to the conventional explicit FDTD method. The update procedures for a family of LOD-FDTD methods that employ similar splitting matrix operators are presented. By extending the previous concept of achieving second-order temporal accuracy for the LOD2-FDTD method via implicit output processing, we hereby propose novel, explicit output processing that not only retains second-order temporal accuracy, but also complying divergence for the LOD2-CD-FDTD method. The current source implementation for the LOD2-CD-FDTD method involves source-incorporation in only the first procedure. To further enhance efficiency, the LOD2-CD-FDTD method is formulated into the fundamental LOD2-CD-FDTD method with efficient matrix-operator-free right-hand sides. Subsequently, detailed implementation for the fundamental LOD2-CD-FDTD method is presented. Analytical proof is provided to ascertain the second-order temporal accuracy of the LOD2-CD-FDTD method. Numerical results and examples are also presented to validate the divergence-complying property of the LOD2-CD-FDTD method.