Conventional FDTD Research Articles

Stability and numerical dispersion analysis of CE-FDTD method

The complex-envelope finite-difference time-domain (CE-FDTD) method has been developed in order to efficiently simulate electromagnetic structures with band-limited signals. This method has most often been formulated in terms of wave equations, leaving its numerical properties significantly understudied. In this paper, full-wave CE-FDTD formulations, and a comprehensive analysis of the numerical stability and dispersion of this method are presented. It is found that that the maximum time step allowed for stability is strongly dependent upon the ratio of the carrier wavelength to spatial step sizes. When this ratio is larger than a threshold value, the time step is bounded by a Courant-Frederick-Levy (CFL) condition. When this ratio is lower than the threshold value, the time step is no longer bounded by this stability condition. However, in such instances, the associated numerical dispersion errors become unacceptably large (more than 200%). Therefore, in comparison with the conventional FDTD method, the CE-FDTD method is proven to be not advantageous in terms of computation efficiency for an accuracy of a similar level.

3D low-dispersion IFD-FDTD based on 3D isotropic finite difference

Usually, 1D finite difference based on the Taylor series expansion theorem is used to approximate the spatial ideal partial-differential operator (IPDO) using conventional FDTD methods. Such a treatment is simple, but its severe numerical dispersion is untenable. In this paper, a kind of 3D isotropic finite difference (IFD) is introduced and a new FDTD method, called IFD-FDTD, is presented. A numerical-dispersion analysis shows that it is superior to Yee's conventional FDTD method. In addition, through the analysis of stability, it is found that its stability condition is the same as that of Yee's FDTD method. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 46: 381–384, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20993

An Upwind Leapfrog Scheme for Computational Electromagnetics: CL-FDTD

This paper extends upwind-leapfrog scheme, initially developed from computational aeroacoustics and elastodynamics, into computational electromagnetics, and develops a novel Characteristic Line FDTD (CL-FDTD) method. In illustrating 2-D problems, this paper gives the implementation of the CL-FDTD method, and points out its merits over the FDTD method chosen for comparison. The results demonstrate that the CLFDTD method is less numerical dispersion, no requirement to deal with outer boundary conditions and it can precisely simulate accident signals that are difficult to treat with the conventional FDTD method.

Enlarged cells for the conformal FDTD method to avoid the time step reduction

Recently, the conformal finite-difference time-domain (CFDTD) method has emerged as an efficient FDTD method with a higher order accuracy than the conventional FDTD methods that are degraded by staircasing errors. The only obvious point to further improve on the CFDTD method is its requirement for a smaller time step increment due to the existence of small irregular cells near the boundary. In this letter, an enlarged cell technique is introduced to ensure the stability of the CFDTD method without the time step reduction. The introduction of the enlarged cells therefore makes the CFDTD method much more efficient and suffers from a smaller dispersion error, as shown in several two-dimensional examples.

A hybrid ADI‐FDTD subgridding scheme for efficient electromagnetic computation

AbstractSubgridding has been a challenge in FDTD modelling. While it can significantly decrease memory requirements by using coarse and dense grids or meshes wherever they are needed, a small time step must normally be applied to the dense or fine mesh due to the Courant–Friedrich–Levy (CFL) stability condition. In this paper, a technique that combines FDTD and ADI‐FDTD methods for subgridding is proposed to circumvent the problem. The solution domain is divided into coarse grid regions and fine subgridded regions whenever necessary. The conventional FDTD is then applied to the coarse grid regions, while the ADI‐FDTD is used in the finely subgridded regions. In comparison with subgridding schemes using solely the conventional FDTD, the hybrid method allows the use of a much larger time step and therefore reduces the CPU time. In comparison between the subgridding scheme and pure ADI‐FDTD schemes, the hybrid method minimizes the use of the memory because the conventional FDTD algorithm is applied to the coarse grid region. Numerical examples are given to validate these advantages. Copyright © 2004 John Wiley & Sons, Ltd.

General Formulation of Unconditionally Stable ADI–FDTD Method in Linear Dispersive Media

The unconditionally stable alternating-direction-implicit-finite-difference time-domain (ADI-FDTD) method is used to model wave propagation in dispersive media. A formulation is presented by introducing the Z-transform method into the ADI-FDTD scheme to handle the frequency-dependent features of the media. This formulation is applicable to arbitrary dispersive media, and can be easily coded. Numerical results are compared to those based on the conventional FDTD method to show the efficiency of the proposed method.

A Novel Miniaturization Technique of a Microstrip Patch Antenna using Patch Resonators

Microstrip patch antennas have been widely used in mobile and satellite communication systems due to their great advantages of low cost, low profile, lightweight and easy fabrication. However, the dimensions of a classical patch antenna are on the order of half a wavelength. This paper proposes a new approach to reduce the size of the antenna by embedding several patch resonators in an antenna substrate. Periodically installed resonators are expected to exhibit slow-wave effects. First of all, a microstrip delay line having a train of patch resonators in its substrate is demonstrated theoretically by the conventional FDTD method, and the slow-wave effect is discussed. Next, a 2-dimentional patch resonator array is applied to a microstrip patch antenna, and the effectiveness of the proposed structure is evaluated in the respect of antenna dimensions. Also, several experiments have been carried out to confirm the theoretical predictions. Using a prototype model fabricated on an LTCC substrate, the size reduction of more the 50% is attained.

FDTD analysis of ELF wave propagation and Schumann resonances for a subionospheric waveguide model

The space formed by the ground and ionosphere is known to act as a resonator for extremely low frequency (ELF) waves. Lightning discharges trigger this global resonance, which is known as Schumann resonance. Even though the inhomogeneity (like day‐night asymmetry, local perturbation, etc.) is important for such a subionospheric ELF propagation, the previous analyses have been always made by some approximations because the problem is too complicated to be analyzed by any exact full‐wave analysis. This paper presents the first application of the conventional numerical FDTD method to such a subionospheric ELF wave propagation, in which any kind of inhomogeneity can be included in this analysis. However, the present paper is intended to demonstrate the workability of this method only for a uniform waveguide (without day‐night asymmetry), by comparing the results from this method with those by the corresponding analytical method.

A hybrid implicit‐explicit FDTD scheme with weakly conditional stability

AbstractA hybrid implicit‐explicit FDTD (HIE‐FDTD) algorithm is proposed in this paper. In the HIE‐FDTD algorithm, semi‐implicit‐ and explicit‐difference schemes are imposed on two different field components, respectively, and the field component with implicit difference is updated by solving simple tridiagonal matrix equations. The HIE‐FDTD method, although conditionally stable, allows for larger time‐step size than the conventional FDTD method because the stability condition of the HIE‐FDTD method is weaker than that of the conventional Yee's FDTD algorithm. The demonstrated numerical dispersion relation of the HIE‐FDTD method is found to be identical to that of the ADI‐FDTD method. The accuracy and efficiency of the HIE‐FDTD are verified by numerical simulation results. © 2003 Wiley Periodicals, Inc. Microwave Opt Technol Lett 39: 97–101, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.11138

Periodic‐structure‐type electromagnetic wave absorber analysis by the Non‐Standard FDTD method using complex formulation

AbstractIn this paper, in order to analyze a periodic‐structure electromagnetic wave absorber by the Non‐Standard FDTD (NS‐FDTD) method, an NS‐FDTD method using complex formulation is proposed. In order to clarify the characteristics of this method, the numerical dispersion equation and the stability condition are newly derived. It is found that the proposed NS‐FDTD method using complex formulation is much more accurate and stable than the conventional FDTD method. The present method is also applied to the analysis of a fin ferrite electromagnetic wave absorber with a periodic structure. It is demonstrated that the proposed NS‐FDTD method using complex formulation is an effective method of analysis with higher accuracy than the conventional FDTD method. © 2003 Wiley Periodicals, Inc. Electron Comm Jpn Pt 2, 86(8): 20–27, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjb.10145

A high definition, finite difference time domain method

A high definition, finite difference time domain (HD-FDTD) method is presented in this paper. This new method allows the FDTD method to be efficiently applied over a very large frequency range including low frequencies, which are problematic for conventional FDTD methods. In the method, no alterations to the properties of either the source or the transmission media are required. The method is essentially frequency independent and has been verified against analytical solutions within the frequency range 50 Hz–1 GHz. As an example of the lower frequency range, the method has been applied to the problem of induced eddy currents in the human body resulting from the pulsed magnetic field gradients of an MRI system. The new method only requires approximately 0.3% of the source period to obtain an accurate solution.

Variational expression for the analysis of photonic crystal devices

Variational expressions of the eigenfrequencies for photonic crystals are presented. We demonstrate that, by combining the expression with the conventional FDTD algorithm, the features of photonic band structures can be obtained efficiently. The proposed method can also be applied to the dispersion curves of photonic crystal waveguides. By employing the Rayleigh-Ritz method, anti-crossing features of the dispersion curves caused by the coupling of eigenmodes can be also successfully treated.

An improved ADI-FDTD method and its application to photonic simulations

When the alternating direction implicit-finite difference time domain method (ADI-FDTD) is applied to simulating photonic devices, full efficiency can not be achieved if reasonable accuracy is to be kept, due to numerical errors such as numerical dispersion. A simple modification to ADI-FDTD is proposed by calculating the envelope rather than the fast-varying field, so that errors are minimized. A factor of two-five in speed can usually be gained while retaining the same level of accuracy compared with conventional FDTD. The efficiency and the accuracy of this improved approach is demonstrated on several problems, from simple waveguide structures to complex photonic crystal structures.

Scattering analysis of large‐scale cavities using the nonstandard FDTD method

AbstractThe nonstandard FDTD method has been proposed for highly accurate numerical analysis of large structures. However, the conventional nonstandard FDTD has been defined only with isotropic grids, and hence the treatment of arbitrary cavity shapes has been severely restricted. In order to remove this restriction in this paper, the nonstandard FDTD method is extended to more general noniso‐tropic grids. First, the FD Laplacian for the nonisotropic grids is newly defined. From the expression for decomposition by the first‐order difference operator, the nonstandard FDTD equations are derived for nonisotropic grids. The physical meaning of the nonstandard FD method used in the nonstandard FDTD method is clarified. Next, by means of the nonstandard FDTD method extended to nonisotropic grids, a scattering analysis of large‐scale cavities is performed and its effectiveness is demonstrated. The results obtained by the nonstandard FDTD method agree well with the measured values, confirming that the nonstandard FDTD method is effective as a highly accurate scattering analysis method for large‐scale cavities. © 2001 Scripta Technica, Electron Comm Jpn Pt 2, 84(12): 8–16, 2001

Hybrid Ray-Fdtd Moving Coordinate Frame Approach for Long Range Tracking of Collimated Wavepackets

Modeling of long range propagation of collimated wavepackets poses some major difficulties with the conventional FDTD scheme. The difficulties arise from the vast computer resources needed to discretize the entire region of interest and the accumulation of numerical dispersion error. As a means for circumventing these difficulties, the moving frame FDTD approach is in this work. In this approach, the computational grid size is limited to the order of the pulse length, and it and moves along with the pulse. The issues discussed in conjunction with this method are those of numerical dispersion, which is shown to be reduced substantially compared with the stationary formulation, numerical stability, and absorbing boundary conditions at the leading, trailing and side boundaries, Numerical results of pulsed beam propagation in both homogeneous and plane stratified media are shown, and the capability of the method is demonstrated with propagation distances exceeding the order of 10 4 pulse lengths.

A combined MRTD/FDTD approach to analyze local characteristics of lumped elements

This letter presents a modeling method of lumped elements by using the Haar-wavelet MRTD (multiresolution time-domain) technique. With the MRTD technique, the conventional FDTD method is partly used to describe the localized characteristics of the lumped elements, which improves the analysis accuracy of the circuits with lumped elements. © 2000 John Wiley & Sons, Inc. Microwave Opt Technol Lett 27: 190–192, 2000.

Hybrid Ray-Fdtd Moving Coordinate Frame Approach for Long Range Tracking of Collimated Wavepackets - Abstract

Modeling of long range propagation of collimated wavepackets poses some major difficulties with the conventional FDTD scheme. The difficulties arise from the vast computer resources needed to discretize the entire region of interest and the accumulation of numerical dispersion error. As a means for circumventing these difficulties, the moving frame FDTD approach is in this work. In this approach, the computational grid size is limited to the order of the pulse length, and it and moves along with the pulse. The issues discussed in conjunction with this method are those of numerical dispersion, which is shown to be reduced substantially compared with the stationary formulation, numerical stability, and absorbing boundary conditions at the leading, trailing and side boundaries, Numerical results of pulsed beam propagation in both homogeneous and plane stratified media are shown, and the capability of the method is demonstrated with propagation distances exceeding the order of 104 pulse lengths.

Moving Coordinate Frame FDTD Analysis of Long Range Tracking of Pulsed Fields in Graded Index Waveguides

Modeling of long range propagation of pulsed fields along guiding structures poses some major difficulties with the conventional FDTD scheme, arising from the vast computer resources needed to discretize the entire domain and the accumulation of numerical dispersion error. The moving frame FDTD approach, presented in this work, overcomes these difficulties by limiting the computational grid to a window centered about the propagating pulse and moving along with it at the local wavespeed of the medium. In this work, we first present an FDTD solution of the field equations as formulated in the moving coordinate frame, and then explore such issues as the numerical dispersion, which is shown to be reduced substantially compared with the stationary formulation, the numerical stability, and the absorbing boundary conditions at the leading, trailing and side boundaries of the moving window. Numerical results of pulsed field propagation along graded index waveguides are shown, and the capability of the method is d...

Moving Coordinate Frame FDTD Analysis of Long Range Tracking of Pulsed Fields in Graded Index Waveguides- Abstract

Modeling of long range propagation of pulsed fields along guiding structures poses some major difficulties with the conventional FDTD scheme, arising from the vast computer resources needed to discretize the entire domain and the accumulation of numerical dispersion error. The moving frame FDTD approach, presented in this work, overcomes these difficulties by limiting the computational grid to a window centered about the propagating pulse and moving along with it at the local wavespeed of the medium. In this work, we first present an FDTD solution of the field equations as formulated in the moving coordinate frame, and then explore such issues as the numerical dispersion, which is shown to be reduced substantially compared with the stationary formulation, the numerical stability, and the absorbing boundary conditions at the leading, trailing and side boundaries of the moving window. Numerical results of pulsed field propagation along graded index waveguides are shown, and the capability of the method is demonstrated with propagation distances exceeding the order of 104 pulse lengths. We choose initial field distributions that give rise to pulsed beam fields. The propagation characteristics of these fields along nonuniform guides are explored analytically in the Appendix. The numerical results are also checked against frequency domain adiabatic mode solutions developed in the Appendix.

Investigation of numerical errors of the two-dimensional ADI-FDTD method [for Maxwell's equations solution

We previously proposed the ADI-FDTD method as a means of solving two-dimensional Maxwell's equations. The algorithm used in this method is unconditionally stable, which means the time-step size ran be set arbitrarily when this method is used. The limitation of the time-step size is not dependent on the Courant-Friedrich-Levy (CFL) condition, but on numerical errors such as numerical dispersion. In this paper, we investigate the numerical errors of the method quantitatively and discuss the calculation accuracy and efficiency of the method. We found that a large time-step size results in high numerical dispersion. However, the limit of the time-step size due to numerical dispersion is greater than the CPL limit if the size of the local minimum cell in the computational domain is much smaller than the other cells and the wavelength. In that case, because the large time-step size reduces the number of time-loop iterations, the ADI-FDTD method is more efficient than the conventional FDTD method in terms of computer resources such as central processing unit time.