Nonstandard Finite-difference Time-domain Research Articles

Calculation of Diffraction Characteristics of Sub wavelength Conducting Gratings Using a High Accuracy Nonstandard Finite-Difference Time-Domain Method

Gratings with subwavelength groove depth and period are frequently used in optics for various purposes. The polarization dependent diffraction characteristics of subwavelength (high frequency) gratings can only be calculated by solving Maxwell’s equations of electromagnetism. In this paper, we calculate the classical diffraction characteristics of subwavelength conducting gratings numerically, using a new high accuracy version of nonstandard finite-difference time-domain (NS-FDTD) algorithm. For the purpose of analysis, we employ a gold sinusoidal grating with light incident at a large angle. We have compared high accuracy NS-FDTD simulation results with those obtained from standard finite-difference time-domain (S-FDTD), and the finite element method (FEM) simulations.

New optimization parameters for the nonstandard FDTD method

In this paper, new simpler optimization parameters for the nonstandard FDTD (NS-FDTD) method are proposed by redefining the spatial-difference operator. Consequently, the number of optimization parameters required in the method is reduced to one from two in 2D space, and three from nine in 3D space, respectively. In addition, it is shown that the NS-FDTD method using new optimization parameters has more accurate characteristics than the NS-FDTD method using conventional parameters. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 47: 161–163, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21112

A subgridding technique for the complex nonstandard FDTD method

AbstractIn this paper, a subgridding method is proposed for the two‐dimensional Complex Nonstandard FDTD (CNS‐FDTD) method. Taking into account the characteristics of the CNS‐FDTD method, new spatial interpolation, difference operator interpolation, and time interpolation are introduced. In order to demonstrate the usefulness of the proposed subgridding method, it is compared with the cases in which linear interpolation and quadratic interpolation of the conventional method are used for the main grids and local grids. It is found that the present subgridding method is superior to the conventional method in terms of stability and accuracy. Further, in the scattering analysis of a large rectangular cavity loaded with a lossy medium, the present subgridding method demonstrated substantial improvement in computation time and number of grids over CNS‐FDTD calculations with the local grid width over the entire space. As a subgridding method, a connection method is explained in a general case where the spatial division ratio of the main grids and local grids is 1:3. © 2004 Wiley Periodicals, Inc. Electron Comm Jpn Pt 2, 87(8): 1–9, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjb.20078

The Nonstandard FDTD Method Using a Complex Formulation

In this paper, a complex nonstandard finite-difference time-domain (CNS-FDTD) method is proposed in order to simulate wave propagation in lossy media. To clarify the characteristics of the method, expressions for the numerical propagation constant and the stability condition are derived. It is found that the CNS-FDTD method is much more accurate and stable than the conventional finite-difference time-domain (FDTD) method. The method is applied to the analysis of a fin ferrite electromagnetic wave absorber with a periodic structure.

Higher Order Tangential Vector Finite Elements for 3-D Antenna Array Structures

The radiation properties of several contemporary array structures in three-dimensional problems are numerically investigated by implementing an advanced second-order tangential vector finite element (TVFE) methodology. The choice and the development of the proposed strategy is justified, since it provides a more precise field representation compared to the first-order edge elements and enables its utilization within coarse meshes without reducing the accuracy of the calculated solution. The latter observation becomes more important when electrically large domains are modeled, as their study with low-order elements usually calls for a fine discretization and, consequently, results in a large number of unknowns. In this paper, we mainly focus on arrays of patch antennas with linear and planar configurations, while cases where beam steering is investigated are also taken into account. For comparison purposes, our study incorporates results obtained with a higher order nonstandard finite-difference time-domain (FDTD) method, which efficiently overcomes some of the deficiencies concerning the conventional Yee algorithm.

On the duality of electric and magnetic fields using the nonstandard FDTD method

AbstractVerifying the duality of electric and magnetic fields using the nonstandard FDTD method is important for far‐field calculations using the equivalence principle and for the reduction of reflection from boundaries of different size lattices. In this paper, the duality of the fields using this method is confirmed in both physically and numerically. © 2004 Wiley Periodicals, Inc. Microwave Opt Technol Lett 40: 148–151, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.11310

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

Phase velocity errors of the nonstandard FDTD method and comparison with other high-accuracy FDTD methods

Several high accuracy finite-difference time-domain (FDTD) methods have been developed to overcome the phase velocity errors present in the FDTD method. The nonstandard FDTD method has been developed as one of those. The phase velocity errors of the method are investigated and the characteristics are compared with other high-accuracy FDTD methods. As a result, the numerical dispersion characteristics of the method are clearly shown and the extremely high-accuracy characteristic at the desired frequency is confirmed.

Numerical dispersion and stability condition of the three‐dimensional nonstandard FDTD method

AbstractIn order to reduce the phase error in FDTD analysis, the NS‐FDTD method has been conceived by Cole. However, numerical dispersion characteristics and stability condition were not presented by Cole; only the definition for cubic meshes was given. In the present paper, the three‐dimensional NS‐FDTD method is extended to rectangular meshes and the numerical dispersion characteristics and stability condition are derived. From the analysis of the stability condition, it is found that the stability condition of the present method is larger than that in the FDTD method. From the analysis of the numerical dispersion characteristics, it is shown that the method has better accuracy than the FDTD method for both cubic and rectangular meshes. © 2003 Wiley Periodicals, Inc. Electron Comm Jpn Pt 2, 86(5): 66–76, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjb.1122

The nonstandard FDTD method for three‐dimensional acoustic analysis and its numerical dispersion and stability condition

AbstractRecently, the present authors have applied the FDTD (Finite‐Difference Time‐Domain) method to acoustic problems. FDTD is one of the finite‐difference methods in the time domain used in electromagnetic problems. However, in this method, phase errors cannot be neglected in a large‐scale propagation analysis such as indoor acoustic propagation and they cause problems in the analysis. In order to reduce the phase error, Cole has invented the NS‐FDTD (Non‐Standard FDTD) method for electromagnetic problems. However, the numerical dispersion and stability condition are not described in Cole's paper. Also, the definition is given only for a cubic lattice. In this paper, an application of the three‐dimensional NS‐FDTD method to acoustic problems is attempted. Further, the method is extended to a rectangular lattice. The numerical dispersion and stability condition of the present method are derived. As a result, it is shown that the present method has much higher accuracy than the FDTD method. © 2002 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 85(9): 15–24, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.1115

The phase velocity error and stability condition of the three-dimensional nonstandard FDTD method

The nonstandard finite-difference time-domain (NS-FDTD) method, using a rectangular parallelepipeds structured grid, has been proposed to overcome the dispersion and anisotropic errors of the FDTD method. However, the numerical dispersion and the stability condition have not been examined. Furthermore, the method has been defined only in the isotropic grids. This paper investigates the numerical dispersion and the stability condition of the three-dimensional NS-FDTD method for isotropic and nonisotropic grids. The method is compared with the FDTD method. As a result, this method demonstrates highly accurate characteristics and high Courant stability condition.

A nonstandard higher order FDTD algorithm for 3-D arbitrarily and fractal-shaped antenna structures on general curvilinear lattices

A new higher order finite-difference time-domain (FDTD) methodology for the consistent modeling of arbitrarily shaped antennas in three-dimensional (3D) curvilinear coordinates is presented in this paper. The generalized algorithm, which introduces novel conventional and nonstandard regimes, develops advanced PMLs and compact differences to handle the widened spatial increments. Also, a systematic leapfrog integrator with mesh expansion concepts is established. Beyond diverse 3D structures, analysis studies fractal arrays whose self-similarity renders them ideal for small-sized designs. Results indicate that the proposed method achieves a critical elimination of lattice errors and provides very precise radiation patterns.

Numerical Dispersion and Stability Condition of the Nonstandard FDTD Method

AbstractIn order to reduce the phase error in the FDTD analysis, Cole proposed the NS‐FDTD method. However, in the paper by Cole, numerical dispersion characteristics and accurate stability condition were not presented. Also, definition was given only for cubic lattices. In this paper, the NS‐FDTD method is extended to the rectangular lattices and the numerical dispersion characteristic and the stability condition are derived. From the analysis for the stability condition, it is found that the stability condition in the present method is larger than those for the FDTD method and the FDTD(2, 4) method. From the analysis of the dispersion characteristic, it is shown that the present method has much higher accuracy than the FDTD method whether the lattices are cubic or rectangular. An analysis of a waveguide model is carried out in which the NS‐FDTD method and the FDTD method are mixed. © 2001 Scripta Technica, Electron Comm Jpn Pt 2, 85(1): 22–30, 2002

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

A generalized methodology based on higher-order conventional and non-standard FDTD concepts for the systematic development of enhanced dispersionless wide-angle absorbing perfectly matched layers

A generalized theory of higher-order finite-difference time-domain (FDTD) schemes for the construction of new dispersionless Berenger and Maxwellian unsplit-field perfectly matched layers (PMLs), is presented in this paper. The technique incorporates both conventional and non-standard approximating concepts. Superior accuracy and modelling attributes are further attained by biasing the FDTD increments on generalizations of Padé formulae and derivative definitions. For the inevitably widened spatial stencils, we adopt the compact operators procedure, whereas temporal integration is alternatively performed via the four-stage Runge–Kutta integrator. In order to terminate the PML outer boundaries and decrease the absorber's necessary thickness, various higher-order lossy absorbing boundary conditions (ABCs) are implemented. Based on the previous theory, we finally introduce an enhanced reflection-annihilating PML for wide-angle absorption. The novel unsplit-field PML has a non-diagonal symmetric complex tensor anisotropy and by an appropriate choice of its parameters together with new conductivity profiles, it can successfully absorb waves of grazing incidence, thus allowing its imposition much closer to electrically large structures. Numerical results reveal that the proposed 2- and 3-D PMLs suppress dispersion and anisotropy errors, alleviate the near-grazing incidence effect and achieve significant savings in the overall computational resources. Copyright © 2000 John Wiley & Sons, Ltd.

Poynting's theorem for the finite‐difference–time‐domain method

AbstractA discretized version of Poynting's theorem is rigorously derived from the standard FDTD equations. By using the correct averaging expressions for the Poynting vector itself, the energy density, and the current power density, the well‐known form of the theorem in continuous space is obtained. The theorem is checked by calculating the radiated power of an antenna in two ways. It is also shown that the use of nonstandard FDTD equations for thin wires violates the theorem. © 1995 John Wiley & Sons. Inc.

The influence of boundary conditions on resonant frequencies of cavities in 3-D FDTD algorithm using non-orthogonal co-ordinates

The 3-dimensional finite-difference time-domain method in non-orthogonal co-ordinates (non-standard FDTD) is used to calculate the frequencies of resonators. The numerical boundary conditions of the method are presented. The influences of boundary conditions and discrete meshes on the numerical accuracy are investigated. We present the nonstandard FDTD method using the boundary-orthogonal mesh and equivalent dielectric constant so that the error is reduced from 8.6% to 3.0% for the cylindrical cavity loaded by a dielectric button.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>