LOD-FDTD Method Research Articles

Multiple LOD-FDTD Method for Multiconductor Coupled Transmission Lines

This article presents the multiple locally one-dimensional (MLOD) finite-difference time-domain (FDTD) method for multiconductor coupled transmission lines. The method is aptly referred to as the MLOD coupled line (CL)-FDTD method, applicable for general asymmetrical multiconductor lines. The differential equations for multiconductor coupled transmission lines are expressed into general compact matrix form and the appropriate set of split matrices are proposed. Utilizing the proposed split matrices, the update procedures and equations of the MLOD CL-FDTD method are provided for general multiconductor lines. The update procedures comprise number of substeps depending on the number of conductor lines while the left-hand side of the implicit update equation constitutes a tridiagonal matrix that can be solved efficiently. The stability analysis is performed using the von Neumann method in the Fourier domain, as well as the two-media reduced-matrix method in the spatial domain for inhomogeneous media. It is found that our proposed method remains stable beyond the Courant time step. Numerical examples demonstrate that the MLOD CL-FDTD method is able to efficiently simulate time domain fields of multiconductor coupled transmission lines at time step larger than the Courant stability limit while maintaining good accuracy.

Multiple LOD-FDTD Method for Inhomogeneous Coupled Transmission Lines and Stability Analyses

A multiple locally 1-D (MLOD) finite-difference time-domain (FDTD) method for inhomogeneous coupled transmission lines and stability analyses are presented. The method is aptly called the MLOD coupled line (CL)-FDTD method. Various split matrices are proposed, and the corresponding update equations are formulated and discussed. All the proposed split matrices yield implicit electric field update equations with tridiagonal or block tridiagonal matrices on the left-hand sides. For more efficiency, the block tridiagonal matrices for implicit electric field may be reformulated and replaced with tridiagonal matrices for implicit magnetic field. The stability analysis is first performed using the von Neumann method in the Fourier domain. It is shown that the von Neumann method alone may not be sufficient to ascertain stability for inhomogeneous media. To include media inhomogeneity, the two-media reduced-matrix stability analysis is proposed. It allows us to efficiently analyze the key stability characteristics in inhomogeneous media and is useful for quick detection of any potential instability that is not apparent via the von Neumann method. The stability characteristics with variation of media parameters are also investigated and discussed. The numerical results are provided to validate the stability and accuracy of the proposed MLOD CL-FDTD method with various split matrices.

On the Behavior of the LOD-FDTD Method at Dielectric Interfaces

The electric field computed by the locally 1-D finite-difference time-domain (FDTD) method at dielectric interfaces is investigated. To this end, two comprehensive problems are considered, namely a dielectric step in a rectangular waveguide and a dielectric-loaded metallic cavity. We found that, in both problems, the electric field patterns exhibit an unexpected phase shift at the dielectric interface. By contrast, the alternating-direction implicit FDTD method does not present this unphysical behavior.

Efficient Three-Step LOD-FDTD Method in Lossy Saturated Ferrites With Arbitrary Magnetization

A three-step unconditionally stable locally one-dimensional finite-difference time-domain method is proposed to represent an arbitrarily magnetized lossy ferrite medium. The small loss in the saturated ferrite is introduced to the algorithm through a simple explicit relation. A step-by-step procedure is established to minimize the number of operations and variables. It is shown that the processing time and occupied memory variables in the proposed method are about 90% and 50% of those of the one-step leapfrog alternating direction implicit method, respectively. The accuracy and stability of the method are verified through the simulation of wave propagation in two ferrite-loaded devices, including a microstrip phase shifter, and a stripline circulator.

Non-uniform Time-Step FLOD-FDTD Method for Multiconductor Transmission Lines Including Lumped Elements

This paper presents a novel non-uniform time-step (NUTS) fundamental locally one-dimensional (FLOD) finite-difference time-domain (FDTD) method for multiconductor transmission lines (MTLs) including lumped elements. The NUTS scheme adopts different (non-uniform) time steps for different periods during simulation, in order to reduce the large errors caused by unconditionally stable FDTD methods with uniform time step larger than Courant-Friedrichs-Lewy limit. It is found that NUTS scheme is potentially unstable for alternating-direction-implicit (ADI) FDTD method, but it is stable for LOD-FDTD method or its implementation based on fundamental (FLOD) scheme. The NUTS FLOD-FDTD method is useful to simulate MTLs with lumped elements including resistor, capacitor, and inductor in series and parallel connections. Moreover, the method based on multiple 1-D approach is also extended to incorporate lumped elements in cross connection to model near-field coupling between the MTLs. While the electric current density is commonly used for field-circuit coupling of parallel connected lumped elements, the magnetic current density will be utilized for field-circuit coupling of series and cross-connected lumped elements. Numerical results for MTLs with lumped elements in series, parallel, and cross connections are provided to show the trade-off between efficiency and accuracy of the proposed NUTS FLOD-FDTD method. Compared with the FLOD-FDTD method with uniform time step, the NUTS FLOD-FDTD method may achieve higher accuracy by using smaller time step for certain periods of simulation.

Stochastic LOD-FDTD method for two-dimensional electromagnetic uncertainty problems

PurposeRandom media uncertainties exhibit a significant impact on the properties of electromagnetic fields that usually deterministic models tend to neglect. As a result, these models fail to quantify the variation in the calculated electromagnetic fields, leading to inaccurate outcomes. This paper aims to introduce an unconditionally stable finite-difference time-domain (FDTD) method for assessing two-dimensional random media uncertainties in one simulation.Design/methodology/approachThe proposed technique is an extension of the stochastic FDTD (S-FDTD) scheme, which approximates the variance of a given field component using the Delta method. Specifically in this paper, the Delta method is applied to the locally one-dimensional (LOD) FDTD scheme (hence named S-LOD-FDTD), to achieve unconditional stability. The validity of this algorithm is tested by solving two-dimensional random media problems and comparing the results with other methods, such as the Monte-Carlo (MC) and the S-FDTD techniques.FindingsThis paper provides numerical results that prove the unconditional stability of the S-LOD-FDTD technique. Also, the comparison with the MC and the S-FDTD methods shows that reliable outcomes can be extracted even with larger time steps, thus making this technique more efficient than the other two aforementioned schemes.Research limitations/implicationsThe S-LOD-FDTD method requires the proper quantification of various correlation coefficients between the calculated fields and the electrical parameters, to achieve reliable results. This cannot be known beforehand and the only known way to calculate them is to run a fraction of MC simulations.Originality/valueThis paper introduces a new unconditional stable technique for measuring material uncertainties in one realization.

Unconditionally Stable LOD-FDTD Method in Anisotropic Magnetized Plasma

A three-step unconditionally stable locally 1-D approach is proposed, representing a dispersive anisotropic magnetized plasma medium. The accuracy and stability of the method are verified through simulation of wave propagation in a microwave cavity partially filled with a plasma slab. Finally, Faraday rotation across a plasma slab is simulated by the proposed method.

Three-Dimensional Unconditionally Stable LOD-FDTD Methods With Low Numerical Dispersion in the Desired Directions

In order to get low numerical dispersion in the desired directions, four-step locally one-dimensional finite-difference time-domain (LOD4-FDTD), five-step LOD-FDTD (LOD5-FDTD), and six-step LOD-FDTD (LOD6-FDTD) methods are developed here. These methods are derived from the three-step LOD-FDTD method, and each of their substeps has variation in either the $x$ -, $y$ -, or $z$ -directions. The LOD4-FDTD, LOD5-FDTD, and LOD6-FDTD methods have two substeps in one, two, and three directions, respectively. These methods give low numerical dispersion in the directions in which they have two substeps. In addition, the LOD5-FDTD method gives low anisotropy error (AE) for the directions in which it has two substeps, and the LOD6-FDTD method gives low 3-D AE. Furthermore, to validate the analytical results, the simulation results are given here.

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.

3‐D non‐uniform time step locally one‐dimensional FDTD method

A 3-D non-uniform time step locally one-dimensional finite-difference time-domain (NUTS LOD-FDTD) method is presented. Unlike conventional and the other LOD-FDTD methods, NUTS LOD-FDTD method employs NUTS for different periods during the simulation. The update equation of NUTS LOD-FDTD method with arbitrary time step is given. Numerical results are provided to compare with conventional LOD-FDTD method and trade-off between efficiency and accuracy for different NUTSs.

Implementation of TF/SF boundary for unconditionally stable five-step LOD-FDTD method

The implementation of total-field/scattered-field (TF/SF) boundary for the unconditionally stable five-step locally one-dimensional finite-difference time-domain (LOD5-FDTD) method is proposed in this paper. There are 12 cases to be considered for each sub step, among them 8 cases for electric field grid points, based on the computational formulations and the particular location of grid points at the TF/SF face. Numerical results show that the plane wave can be successfully introduced into the TF region and maintains unconditional stabilization. Finally, the radar cross section (RCS) of a metal plate is computed, which further verifies the effectiveness of the proposed TF/SF connecting boundary.

Leapfrog Formulation of the 3-D LOD-FDTD Algorithm

This letter presents an efficient leapfrog formulation of the three-dimensional locally one dimensional finite-difference time-domain (3-D LOD-FDTD) method, denoted by the leapfrog LOD-FDTD method. By adopting the leapfrog scheme, the new algorithm, whose right-hand sides are much simpler and more concise than those in the original LOD-FDTD method, is implemented in the same manner as those in the conventional FDTD method. Numerical results demonstrate that the proposed algorithm requires the less amount of memory and the less CPU time while keeping the same numerical properties in comparisons with the original LOD-FDTD method.

Stability and dispersion analysis of higher order unconditionally stable three‐step locally one‐dimensional finite‐difference time‐domain method

Stability and dispersion analysis of higher-order three-step locally one-dimensional (LOD) finite-difference time-domain (FDTD) method is presented here. This method uses higher order cell-centred finite-difference approximation for the spatial differential operator and second-order finite difference approximation for the time differential operator. Unconditional stability of the higher-order three-step LOD-FDTD method is analytically proven and numerically verified. Numerical results show improvement in the overall performance of higher-order three-step LOD-FDTD method compared with that of second order. Also, the effects of the order of approximation, the time step and the mesh size on numerical dispersion are explained through analytical results and verified by simulation results.

Nonorthogonal LOD-FDTD Method for EM Scattering from Two-Dimensional Structures

In this paper, a nonorthogonal locally 1-D finite difference time-domain (LOD-NFDTD) method is presented for electromagnetic (EM) scattering involving curvilinear coordinate system. Formulations of scattered field and convolutional perfectly matched layer absorbing boundary condition in generalized nonorthogonal grids for LOD-NFDTD are also presented. The nonorthogonal grids are used to fully mesh the computational domain which leads to efficient computation. Moreover, the proposed technique requires fewer arithmetic operations than the alternating direction implicit NFDTD method leading to a reduction of CPU time. The numerical dispersion of the proposed method as a function of Courant-friedrich-lewy number (CFLN) is also presented. Computational results for EM scattering from 2-D conducting, dielectric, and coated cylinders as well as overfilled dielectric and bent cavity structures are presented.

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.

Efficient implementation of frequency-dependent 3D LOD-FDTD method using fundamental scheme

A fundamental scheme is utilised to efficiently implement the frequency-dependent 3D locally one-dimensional FDTD method. The formulation is performed in matrix-operator-free forms in the right-hand sides of resultant equations. A modified procedure, in which implicit and explicit equations are alternately calculated, needs to retain only 10 field arrays, while the traditional explicit FDTD requires 12. Analysis of a gap plasmonic waveguide validates the presented 3D LOD-FDTD method.

Frequency-Dependent 3-D LOD-FDTD Method for the Analysis of Plasmonic Devices

A simple trapezoidal recursive convolution technique is utilized to develop a frequency-dependent locally one-dimensional finite-difference time-domain (FDTD) method for the three-dimensional analysis of dispersive media. A gap plasmonic waveguide is analyzed and the numerical results are compared with those of the traditional explicit FDTD. A time step ten times as large as that determined from the stability criterion can be allowed to reduce computational time by 40%, offering acceptable numerical results. A plasmonic grating is analyzed as an application.

Unconditionally stable LOD-FDTD method based on the simplified sampling biorthogonal algorithm

This letter presents an unconditionally stable locally one-dimensional (LOD) finite-difference time-domain method based on the simplified sampling biorthogonal algorithm (SSB-LOD). The suggested approach requires fewer arithmetic operations than in the SSB-ADI scheme. This leads to a reduction of CPU time relative to the SSB-ADI method while equivalent numerical accuracy is retained. The numerical implementation of two parallel conducting plate shows that the CPU time of the SSB-LOD has been reduced to approximately 85% of that of the SSB-ADI. © 2010 Wiley Periodicals, Inc. Microwave Opt Technol Lett 52:2183–2185, 2010; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.25461

An Efficient Method to Reduce the Numerical Dispersion in the LOD-FDTD Method Based on the (2, 4) Stencil

A parameter-optimized (2, 4) stencil based locally-one-dimensional (LOD) finite-difference time-domain (FDTD) is presented with much reduced numerical dispersion errors. The method is first proved to be unconditionally stable. Then by using different optimization schemes, the method is optimized to satisfy different accuracy requirements, such as minimum dispersion errors in the axial directions, in the diagonal direction, and in the specified angles. Performances of the parameter-optimized LOD-FDTD with different time steps and frequencies are also studied. It is found that the parameter optimization can significantly reduce numerical dispersion errors, bringing them down to the level of the conventional FDTD but with the time step exceeding the CFL limit and without much additional computational cost. In addition, the optimized parameters are not sensitive to frequencies; in particular, the optimized parameters obtained at a higher frequency still present low numerical dispersion errors at a lower frequency.

Development of the CPML for Three-Dimensional Unconditionally Stable LOD-FDTD Method

A convolutional perfectly matched layer (CPML) is developed for three-dimensional unconditionally stable, locally one dimensional (LOD)-finite-difference time-domain (FDTD) method. The formulation of the LOD-FDTD CPML is derived and numerical results are demonstrated at different positions for different Courant Friedrich Levy numbers in the simulation domain. The method is validated numerically with FDTD-CPML.