Yee 's Finite-difference Time-domain Research Articles

A novel 1D-FDTD scheme to solve the nonlinear second-order thermoviscous hydrodynamic model

In this paper, we present a novel and simple Yee Finite-Difference Time-Domain (FDTD) scheme to solve numerically the nonlinear second-order thermoviscous Navier–Stokes and the Continuity equations. In their original form, these equations cannot be discretized by using the Yee’s mesh, at least, easily. As it is known, the use of the Yee’s mesh is recommended because it is optimized in order to obtain higher computational performance and remains at the core of many current acoustic FDTD softwares. In order to use the Yee’s mesh, we propose to rewrite the aforementioned equations in a novel form. To achieve this, we will use the substitution corollary. This procedure is novel in the literature. Although the scheme can be extended to more than one dimension, in this paper, we will focus only on the one-dimensional solution because it can be validated with two analytical solutions to the Burgers equation: the Mendousse mono-frequency solution and the Lardner bi-frequency solution. Numerical solutions are excellently consistent with the analytical solution, which demonstrates the effectiveness of our formulation.

A 3D Unstructured Mesh FDTD Scheme for EM Modelling

The Yee finite difference time domain (FDTD) algorithm is widely used in computational electromagnetics because of its simplicity, low computational costs and divergence free nature. The standard method uses a pair of staggered orthogonal cartesian meshes. However, accuracy losses result when it is used for modelling electromagnetic interactions with objects of arbitrary shape, because of the staircased representation of curved interfaces. For the solution of such problems, we generalise the approach and adopt an unstructured mesh FDTD method. This co-volume method is based upon the use of a Delaunay primal mesh and its high quality Voronoi dual. Computational efficiency is improved by employing a hybrid primal mesh, consisting of tetrahedral elements in the vicinity of curved interfaces and hexahedral elements elsewhere. Difficulties associated with ensuring the necessary quality of the generated meshes will be discussed. The power of the proposed solution approach is demonstrated by considering a range of scattering and/or transmission problems involving perfect electric conductors and isotropic lossy, anisotropic lossy and isotropic frequency dependent chiral materials.

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.

Finite-difference time-domain method for three-dimensional grid of hexagonal prisms

The finite-difference time-domain (FDTD) method was applied in a grid of hexagonal prisms, having as objective to yield less numerical anisotropy of phase velocity than the Yee FDTD method (with hexahedral cells). Comparisons of wave propagation are made between the FDTD method with grid of hexagonal prisms and the Yee FDTD method. The theoretical analyses of the numerical anisotropy, dispersion and stability condition are obtained using the Fourier analysis in the FDTD method with grid of hexagonal prisms. Measurements of numerical anisotropy are also accomplished in this FDTD method, and then ones are compared with the results of the Fourier analysis. As a result, the grid of hexagonal prisms yielded somewhat less numerical anisotropy and dispersion than the Yee grid. Additionally, a simplification in compensation of numerical dispersion in the grid of hexagonal prisms may improve on the accuracy and density of mesh for indoor buildings that are large, mainly in the xy-plane.

An overlapping Yee finite-difference time-domain method for material interfaces between anisotropic dielectrics and general dispersive or perfect electric conductor media

SUMMARY A novel stable anisotropic finite-difference time-domain (FDTD) algorithm based on the overlapping cells is developed for solving Maxwell's equations of electrodynamics in anisotropic media with interfaces between different types of materials, such as the interface between anisotropic dielectrics and dispersive medium or perfect electric conductor (PEC). The previous proposed conventional anisotropic FDTD methods suffer from the late-time instability due to the extrapolation of the field components near the material interface. The proposed anisotropic overlapping Yee FDTD method is stable, as it relies on the overlapping cells to provide the collocated field values without any interpolation or extrapolation. Our method has been applied to simulate electromagnetic invisibility cloaking devices with both anisotropic dielectrics and PEC included in the computational domain. Numerical results and eigenvalue analysis confirm that the conventional anisotropic FDTD method is weakly unstable, whereas our method is stable. Copyright © 2013 John Wiley & Sons, Ltd.

A New Trend in Propagation Prediction

Owing to its direct applicability in solving problems of the telecommunications industry, propagation prediction for a long time has been an important area of research and development. Because of the increasing complexity of wireless networks, growing number of smaller cells, and higher intercell interference, software tools that aid in network optimization are necessary. There fore, in the study of particular environments, where wireless networks are deployed, deterministic propagation models play an important role. Deterministic models can predict the path loss in a given scenario through the simulation of the main propagation phenomena such as reflections and diffractions. The most commonly used deterministic solutions for radio coverage prediction are ray models, which model electromagnetic (EM) waves through optical laws. However, since their appearance nearly 40 years ago, there has been an increasing interest in using other techniques such as finite-difference EM methods. These methods [the most well known is finite-difference time-domain (FDTD) method] solve Maxwell's equations on a discrete space time grid. They can provide highly accurate results, provided there are accurate boundary conditions, since the reflections, diffractions, and scattering effects are implicit. This contrasts with ray-based methods that must account explicitly for a limited number of rays. This also explains why Yee's FDTD methods, have been and are still widely used today, especially in confined environments such as for antenna design that involves reasonable grid sizes.

Characterization of the Numerical Group Velocity in Yee's FDTD Grid

A method is introduced for the optimization of the numerical group velocity in standard finite-difference time-domain (FDTD) electromagnetic simulations. Through this method analytical expressions for the extrema of are presented for the first time, thus also characterizing its anisotropy. The knowledge of these expressions is hence essential for the evaluation of the anisotropy error in FDTD-based electrodynamics simulations of the propagation of wavepackets in 2D and 3D. This can be of assistance, for example, in the design of error-bounded FDTD simulations with pulsed sources at low computational cost.

Overlapping Yee FDTD Method on Nonorthogonal Grids

We propose a new overlapping Yee (OY) method for solving time-domain Maxwell's equations on nonorthogonal grids. The proposed method is a direct extension of the Finite-Difference Time-Domain (FDTD) method to irregular grids. The OY algorithm is stable and maintains second-order accuracy of the original FDTD method, and it overcomes the late-time instability of the previous FDTD algorithms on nonorthogonal grids. Numerical examples are presented to illustrate the accuracy, stability, convergence and efficiency of the OY method.

Universally applicable uniaxial perfect matched layer formulation for explicit and implicit finite difference time domain algorithms

An important advance in the finite difference time domain (FDTD) algorithms is Berenger's perfect matched layer (PML), along with its derivatives. In recent years several new explicit and implicit FDTD algorithms have appeared on the scene, of which each requires a specific PML. To simplify programming, the authors derive a universal uniaxial PML formulation, which is universally applicable to the Yee FDTD, FDTD with higher-order stencils and implicit FDTD algorithms. Numerical results show the efficiency and versatility of the new method.

An Updated Review of General Dispersion Relation for Conditionally and Unconditionally Stable FDTD Algorithms

We present a general numerical dispersion equation which is applicable to all known conditionally and unconditionally stable finite-difference time-domain (FDTD) algorithms on staggered rectangular grid, including Yee's FDTD, wavelet-based FDTD, extended curl FDTD, alternating direction implicit (ADI)-FDTD, Crank-Nicolson (CN)-FDTD, Crank-Nicolson split-step (CNSS)-FDTD, and their modifications of higher order spatial stencils. The real part of the complex eigenvalue of the total amplification matrix defines and distinguishes the dispersion relation for each individual scheme. Easy-to-check conditions are provided, under which the numerical dispersion of a particular time-domain scheme is governed by the proposed dispersion equation. These conditions are on the amplification matrix eigenvalues. The proposed dispersion equation includes each considered dispersion relation as a special case, and presents itself a general governing equation to estimate 3-D numerical dispersion of the aforementioned schemes in the frame of plane waves.

The Lobatto Cell: Robust, Explicit, Higher Order FDTD That Handles Inhomogeneous Media

To improve numerical dispersion and second-order convergence behavior of the traditional Yee finite-difference time-domain (FDTD) algorithm, various higher order finite difference implementations have been proposed. Many have difficulty handling interfaces between distinct dielectric media or metallic sheets, because the high-order Taylor expansions on which these codes are based are invalidated by field or derivative discontinuities. One-sided extensions/extrapolations are a common approach to remedy interface discontinuities. Here, another approach based upon Whitney form, mass lumped finite elements is pursued to develop the ldquoLobatto Cell,rdquo a high-order replacement for the Yee cell with appropriate continuity behavior at contrasting media interfaces. The Lobatto Cell retains many robust features of the Yee method (discrete conservation laws, a guarantee of conditional stability and straightforward explicit updating), but provides higher order interpolation and dispersion error convergence. A notable drawback of the method is larger modeling granularity (staircasing), which is managed here by combining the Lobatto Cell method with preexisting, robust subgridding techniques.

Rigorous Analysis of Numerical Phase and Group Velocity Bounds in Yee's FDTD Grid

We present a systematic derivation of the extreme values of phase and group velocity in Yee's finite difference time domain (FDTD) lattice with unequal aspect ratios. Using a Lagrange multiplier based approach; we derive necessary conditions that propagation vector components need to satisfy to attain extreme values in phase and group velocity. Knowledge of these extreme values is useful in designing low numerical dispersion FDTD schemes and also seeding numerical inversion routines of FDTD dispersion relations.

A Finite Volumes-Based 3-D Low Dispersion FDTD Algorithm

An algorithm extension to three dimensions is developed and presented for the highly phase-coherent modified second-order in time, fourth-order in space (or M24) finite-difference time-domain (FDTD) algorithm. A finite-volumes approach in conjunction with Yee's standard FDTD lattice is used for algorithm development. The corresponding dispersion relation is also developed, analyzed and compared to both the standard second-order and fourth-order FDTD algorithms as well as to two closely related high-order phase-coherent algorithms. Wideband algorithm attributes are also presented as well as sets of ready to use optimized algorithm coefficients.

The unconditionally stable Crank Nicolson FDTD method for three-dimensional Maxwell's equations

In this paper, an accurate and computationally implicit 3D finite-difference time-domain (FDTD) method based on the unconditionally stable Crank–Nicolson scheme (3D CN-FDTD) is presented. The source excitation in 3D CN-FDTD is described and the numerical simulation of the 3D CN-FDTD method is demonstrated through numerical examples. The results of this method, the ADI-FDTD method, and traditional FDTD schemes are compared. A good agreement is obtained for the 3D CN-FDTD method with time steps greatly more than the Courant–Friedrich–Levy (CFL) limit and the traditional Yee FDTD method. © 2006 Wiley Periodicals, Inc. Microwave Opt Technol Lett 48: 1619–1622, 2006; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21684

Suppression of numerical anisotropy and dispersion with optimized finite-difference time-domain methods

To reduce numerical dispersion in finite-difference time-domain (FDTD) methods, large computational stencils are often used. This paper proposes an optimized two-dimensional method by weighting the (2,4) stencil and the neighborhood stencil. After obtaining the amplification factor and the numerical dispersion relation, the optimal value of the weight parameter is obtained to minimize the numerical dispersion at a designated frequency. The anisotropy, dispersion error and the accumulated phase errors are greatly reduced over a broad bandwidth. Both the maximum anisotropy and the maximum dispersion error are 8.9/spl times/10/sup -5/, and the accumulated phase error is 0.002367/spl deg/ per cell, respectively, for a broad band of frequencies if optimized at 10 cells per wavelength. Numerical experiments are performed which show very good agreement with theoretical analysis. The time step size bound is the same as for Yee's FDTD.

Numerical dispersion and numerical loss in explicit finite-difference time-domain methods in lossy media

The numerical dispersion relations of finite-difference time-domain (FDTD) methods have been analyzed extensively in lossless media. This paper investigates numerical dispersion and loss for Yee's FDTD in lossy media. It is shown that: the numerical velocity can be smaller or larger than the physical velocity; there is no magic time step size in lossy media; and the numerical loss is smallest at the Courant limit. It is shown that the numerical loss is always larger than its physical value, and so Yee's FDTD overestimates the absorption of electromagnetic energy in lossy media. The numerical velocity anisotropy can be positive or negative, but the numerical loss anisotropy is always positive. The anisotropies in the three-dimensional (3-D) case are usually larger than those in the 2-D case. Numerical experiments in 1-D are shown to agree with the theoretical prediction.

Off-grid perfect boundary conditions for the FDTD method

We implement off-grid boundary conditions within Yee's finite-difference time-domain (FDTD) method without disturbing the existing uniform mesh or changing the standard FDTD code. Both perfect electric conductor and perfect magnetic conductor walls are considered. Examples of straight, slanted, and curved walls are shown, the latter two being represented by an enhanced staircase approximation. We show that: 1) offsets comparable to the spatial step size lead to instabilities and 2) this issue is easily resolved by stepping into the neighboring cell. The method enhances the flexibility of the FDTD method with respect to complex geometrical domains without reducing the spatial step.

AN ACOUSTIC FINITE-DIFFERENCE TIME-DOMAIN ALGORITHM WITH ISOTROPIC DISPERSION

The classic Yee Finite-Difference Time-Domain (FDTD) algorithm employs central differences to achieve second-order accuracy, i.e., if the spatial and temporal step sizes are reduced by a factor of n, the phase error associated with propagation through the grid will be reduced by a factor of n2. The Yee algorithm is also second-order isotropic meaning the error as a function of the direction of propagation has a leading term which depends on the square of the discretization step sizes. An FDTD algorithm is presented here that has second-order accuracy but fourth-order isotropy. This algorithm permits a temporal step size 50% larger than that of the three-dimensional Yee algorithm. Pressure-release resonators are used to demonstrate the behavior of the algorithm and to compare it with the Yee algorithm. It is demonstrated how the increased isotropy enables post-processing of the simulation spectra to correct much of the dispersion error. The algorithm can also be optimized at a specified frequency, substantially reducing numerical errors at that design frequency. Also considered are simulations of scattering from penetrable spheres ensonified by a pulsed plane wave. Each simulation yields results at multiple frequencies which are compared to the exact solution. In general excellent agreement is obtained.

TE/TM field solver for particle beam simulations without numerical Cherenkov radiation

The Yee finite-difference time domain method (FDTD) is commonly used in wake field and particle-in-cell simulations. However, in accelerator modeling the high energy particles can travel in vacuum faster than their own radiation. This effect is commonly referred to as numerical Cherenkov radiation and is a consequence of numerical grid dispersion. Several numerical approaches are proposed to reduce the dispersion for all angles and for a given frequency range, that justifies itself for domains big in all three directions. On the contrary, in accelerator modeling the transverse dimensions and transverse beam velocity are small, but it is extremely important to eliminate the dispersion error in the well-defined direction of the beam motion for all frequencies. In this paper we propose a new two-level economical conservative scheme for electromagnetic field calculations in three dimensions. The scheme does not have dispersion in the longitudinal direction and is staircase-free (second order convergent). Unlike the FDTD method, it is based on a ``transversal-electric/transversal-magnetic'' (TE/TM)-like splitting of the field components in time. The scheme assures energy and charge conservation. Additionally, the usage of damping terms allows suppressing high frequency noise generated due to the transverse dispersion and the current fluctuations. The dispersion relation of the damping scheme is analyzed. As numerical examples show, the new scheme is much more accurate on the long-time scale than the conventional FDTD approach.

A fourth-order accurate FDTD scheme with long-time stability

This letter presents a new finite-difference time-domain (FDTD) scheme with fourth-order accuracy in time and space. A combination of staggered backward differentiation time integrator and high-order spatial discretization is used to derive fourth-order accurate FDTD equations. The fourth-order accuracy of the proposed FDTD method is verified by monitoring L/sub 2/ norm errors in a two-dimensional partially-filled cavity. Furthermore, the computational cost study shows that the proposed fourth-order scheme is much more efficient than Yee's second-order FDTD scheme in solving the inhomogeneous cavity problem.