Yee's FDTD Research Articles

Time-dependent Schrödinger Equation based on HO-FDTD Schemes

A high order finite-different time-domain methods using Taylor series expansion for solving time-dependent Schrodinger equation has been systematically discussed in this paper. Numerical characteristics have been investigated of the schemes for the Schrodinger equation. Compared with the standard Yee FDTD scheme, the numerical dispersion has been decreased and the convergence has been improved. The general update equations of the methods have been presented for wave function. Numerical results of potential well in one-dimension show that the application of the schemes is more effective than the Yee’s FDTD method and the higher order has the better numerical dispersion characteristics.

EM modelling of arbitrary shaped dispersive chiral dielectric objects using a 3D leapfrog scheme on unstructured meshes

The standard Yee FDTD algorithm is widely used in computational electromagnetics because of its simplicity and divergence free nature. A generalization of this classical scheme to 3D unstructured co-volume meshes is adopted, based on the use of a Delaunay primal mesh and its high quality Voronoi dual. This circumvents the problem of accuracy losses, which are normally associated with the use of a staircased representation of curved material interfaces in the standard Yee scheme. The procedure has been successfully employed for modelling problems involving both isotropic and anisotropic lossy materials. Here, we consider the novel extension of this approach to allow for the challenging modelling of chiral materials, where the material parameters are frequency dependent. To adequately model the dispersive behaviour, the Z-transform is employed, using second order Padé approximations to maintain the accuracy of the basic scheme. To validate the implementation, the numerical results produced are compared with available analytical solutions. The stability of the chiral algorithm is also studied.

Implementation and optimization of GPU‐based parallel one‐step leapfrog ADI‐FDTD for far‐field scattering problems

The one-step leapfrog alternative-direction-implicit finite-difference time-domain (ADI-FDTD), free from the Courant-Friedrichs-Lewy (CFL) stability condition and sub-step computations, is efficient when dealing with fine grid problems. However, solution of the numerous tridiagonal systems still imposes a great computational burden and makes the method hard to execute in parallel. In this paper, we proposed an efficient graphic processing unit (GPU)-based parallel implementation of the one-step leapfrog ADI-FDTD for the far-field EM scattering simulation of objects, in which we present and analyze the manners of calculation area division and thread allocation and a data layout transformation of z components is proposed to achieve better memory access mode, which is a key factor affecting GPU execution efficiency. The simulation experiment is carried out to verify the accuracy and efficiency of the GPU-based implementation. The simulation results show that there is a good agreement between the proposed one-step leapfrog ADI-FDTD method and Yee's FDTD in solving the far-field scattering problem and huge benefits in performance were encountered when the method was accelerated using GPU technology.

A Multi-GPU Accelerated Parallel Domain Decomposition One-Step Leapfrog ADI-FDTD

In this letter, a multi-GPU accelerated one-step leapfrog alternative-direction-implicit finite-difference time-domain (ADI-FDTD) based on parallel SPIKE tridiagonal systems solver is presented. Through our algorithm simplification and execution optimization, online computation and data exchange between GPUs of the SPIKE solver is dramatically reduced. In the simulation experiment, the bistatic RCS of the benchmark model calculated by the proposed method is consistent with the solutions given by one-step leapfrog ADI-FDTD based on Thomas tridiagonal systems solver and Yee's FDTD. Compared with the single-GPU-based implementation, the multi-GPU accelerated one-step leapfrog ADI-FDTD solver can achieve 3.94× speedups at most when executed at a workstation with 4 GPUs.

Efficient one‐step leapfrog ADI‐FDTD for far‐field scattering calculation of lossy media

AbstractFree from the Courant‐Friedrichs‐Lewy (CFL) stability condition and sub‐step computations, the one‐step leapfrog alternative‐direction‐implicit finite‐difference time‐domain (ADI‐FDTD) method is very efficient in dealing with fine grid problems. In this letter, a one‐step leapfrog ADI‐FDTD method for three‐dimensional far‐field scattering simulation method for lossy media is proposed. A novel total field/scattering field (TF/SF) implementation is derived for the one‐step leapfrog ADI‐FDTD, providing the computational flexibility in far‐field electromagnetic (EM) scattering calculation. The auxiliary matrices are introduced to further improve its efficiency for lossy media. The simulation results show that there is a good agreement between the proposed one‐step leapfrog ADI‐FDTD method and Yee's FDTD in solving the far‐field scattering problem and benefits in performance were encountered when using the proposed method.

Complex-Frequency Shifted PML Formulation for the 3-D MNL-FDTD Method

This letter presents a complex-frequency shifted perfectly matched layer (CFS-PML) formulation for the partially implicit Magnetically-mixed Newmark-Leapfrog finite-difference time-domain (MNL-FDTD) method. Its formulation is based on time-dependent Maxwell's equations in CFS-PML media and the auxiliary differential equation (ADE) of PML auxiliary variables. In the PML region, the efficiency of auxiliary variable terms of the formulated scheme is same as that of Roden's convolutional PML for Yee's FDTD. It is demonstrated that reflection error of MNL-FDTD with CFS-PML is comparable to those of the conventional explicit and implicit FDTDs with CFS-PML.

MNL-FDTD/SPICE Method for Fast Analysis of Short-Gap ESD in Complex Systems

This paper presents a magnetically mixed Newmark-Leapfrog finite-difference time-domain (MNL-FDTD) method for efficient three-dimensional electromagnetic simulations of transient interactions between a spark channel of air-discharge electrostatic discharge (ESD) occurred at a short gap and its surrounding environment. The formulation is based on introducing the implicit Newmark-Beta method into the explicit leapfrog scheme of Yee's FDTD method directionally and magnetically. The stability condition of the algorithm does not include the mesh step in the channel direction, and therefore, is more relaxed than the Courant–Friedrichs–Lewy condition of the Yee scheme. For combined full-wave/circuit systems involving air-discharge ESD, both the sequential and simultaneous solutions are discussed in the context of MNL-FDTD. A stable direct linking of MNL-FDTD with SPICE is presented to include a discharge current characterized by arbitrary spark resistance. The relaxed stability is maintained in the combined systems. The presented method is verified with three spark resistance formulae. The accuracy, stability, and computational efficiency of the method are demonstrated in comparison with the conventional approaches in several numerical examples.

On the numerical dispersion of electromagnetic particle-in-cell code: Finite grid instability

The Particle-In-Cell (PIC) method is widely used in relativistic particle beam and laser plasma modeling. However, the PIC method exhibits numerical instabilities that can render unphysical simulation results or even destroy the simulation. For electromagnetic relativistic beam and plasma modeling, the most relevant numerical instabilities are the finite grid instability and the numerical Cherenkov instability. We review the numerical dispersion relation of the Electromagnetic PIC model. We rigorously derive the faithful 3-D numerical dispersion relation of the PIC model, for a simple, direct current deposition scheme, which does not conserve electric charge exactly. We then specialize to the Yee FDTD scheme. In particular, we clarify the presence of alias modes in an eigenmode analysis of the PIC model, which combines both discrete and continuous variables. The manner in which the PIC model updates and samples the fields and distribution function, together with the temporal and spatial phase factors from solving Maxwell's equations on the Yee grid with the leapfrog scheme, is explicitly accounted for. Numerical solutions to the electrostatic-like modes in the 1-D dispersion relation for a cold drifting plasma are obtained for parameters of interest. In the succeeding analysis, we investigate how the finite grid instability arises from the interaction of the numerical modes admitted in the system and their aliases. The most significant interaction is due critically to the correct representation of the operators in the dispersion relation. We obtain a simple analytic expression for the peak growth rate due to this interaction, which is then verified by simulation. We demonstrate that our analysis is readily extendable to charge conserving models.

Computation of Optical Force on Nanoparticles Using Locally Non-Orthogonal Overlapping Yee FDTD Method

In this paper, a locally non-orthogonal overlapping Yee (OY) FDTD method is proposed in order to accurately calculates the optical force on dielectric and dispersive nanoparticles. It extends our previous work to geometries with sharp corners and dispersive materials. In addition to consistently achieving the smallest errors in comparison to the standard FDTD method, the OY approach is a stable non-orthogonal FDTD method that attains second-order convergence when sharp corners are present.

3차원 ID-FDTD 알고리즘의 Stability Condition과 광대역 특성 분석

Abstract The stability condition and wideband characteristics of 3D ID-FDTD algorithm which has low dispersion error with isotropic dispersion are presented in this paper. 3D ID-FDTD method was proposed to improve the defect of the Yee FDTD such as the anisotropy and large dispersion error. The published paper calculated the stability condition of 3D ID-FDTD algorithm by using numerical method, however, it is thought that the examples were not sufficient to verify the stability condition. Thus, in this paper, various simulations are included in order to hold reliability under the con-ditions that the plane wave propagation is assumed with a single frequency and a wideband frequency. Also, the 3D ID-FDTD algorithm is compared to those that have the similar FDTD algorithm with ID-FDTD such as Forgy's method and non-standard FDTD method in a wideband. Finally, the radar cross section(RCS) for the large sphere with high dielectric constant is calculated. Key words : FDTD, Isotropic Dispersion, Rcs, Stability Condition

Development of a Self-Consistent Truly Multiphysics Algorithm Based Upon the Courant-Insensitive Space–Time Conservation-Element Solution-Element Method

This paper reports on the theoretical aspects and current development status of a self-consistent truly multiphysics algorithm. The algorithm is based upon the Courant-insensitive space-time conservation-element solution-element methodology. Previous attempts for electromagnetic solutions have applicability only in constant material domains with PEC boundary conditions. This paper reports on the extension of this algorithm for the solution of the generalized Maxwell equations, including linear-dispersive materials. The numerical solution is shown to be extremely accurate on highly nonuniform meshes and reduces to the classical Yee FDTD error properties in the uniform Cartesian grid limit. Validation problems and comparison with the ubiquitous baseline FDTD algorithm will be presented in 1-D (2-D space-time). Results show that the second-order CESE method has an accuracy equivalent to fourth-sixth order FDTD for equal grids with highly discontinuous coefficients (e.g., permittivity).

A diagonal split-cell model for the overlapping Yee FDTD method

In this paper, we present a nonorthogonal overlapping Yee method for solving Maxwell's equations using the diagonal split-cell model. When material interface is presented, the diagonal split-cell model does not require permittivity averaging so that better accuracy can be achieved. Our numerical results on optical force computation show that the standard FDTD method converges linearly, while the proposed method achieves quadratic convergence and better accuracy.

A New 3-D Conformal PEC FDTD Scheme With User-Defined Geometric Precision and Derived Stability Criterion

A new conformal finite-difference time-domain (CFDTD) updating scheme for metallic surfaces nonaligned in the grid is presented in this paper. In contrast to existing conformal models, the new model can be formulated with the original Yee FDTD update equation. Therefore, the proposed scheme can be easily added in standard FDTD codes even if the codes are already parallelized or hardware-accelerated. In addition, based on the commonly used conventional stability criterion, a derivation of the stability is presented and based on the conformal geometric information, a time step reduction formula is presented. The time step reduction is used as a user-defined parameter to tradeoff speed versus accuracy. The achievable geometric precision is optimized to a given time step. Therefore, even with the conventional time step (no reduction) the presented scheme profits from the conformal discretization. To show the performance and the robustness of the proposed scheme canonical validations and two real world applications were investigated. A broadband low profile (circular) antenna was successfully simulated showing the benefit of the conformal FDTD method compared to the conventional scheme. Furthermore, a CAD based mobile phone was conformally discretized and successfully simulated showing that the proposed scheme is highly suited for the simulation of advanced engineering problems.

On the dispersion relation of ADI-FDTD

In this letter, we analyze the alternating direction implicit finite-difference time-domain (ADI-FDTD) dispersion relation and find the numerical plane-wave relationship between the magnetic and electric fields, showing that the scheme is not divergence-free. We also show that negative-group-velocity modes with positive phase velocities may appear for high Courant numbers. A parallel comparison is made with the behavior of the classical Yee FDTD and the Crank-Nicolson schemes.

A novel separable backward–central FDTD method

AbstractIn this paper, we present a novel separable backward–central FDTD (SBC‐FDTD) method in which the backward and central differences are used to discretize both time and spatial directive of the Maxwell's equations. For the type of problem in which the FDTD cell size in one direction is much smaller than those in the other directions, the time step in the proposed separable backward‐central FDTD technique can be chosen based on the largest cell size instead of the smallest one used in the conventional Yee's FDTD. Compared to the alternating‐direction implicit FDTD (ADI‐FDTD), although the proposed approach is not unconditionally stable, it allows us to explicitly solve electric and magnetic fields in one direction. Both the numerical experiments and the stability property demonstrate that the proposed scheme is stable and accurate. A TE mode wave is used to validate the proposed technique. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 48: 212–215, 2006; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21308

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

Stability and dispersion analysis of ADI-MRTD and ADI high-order schemes

The maximum time-step size of the alternating-direction implicit finite-difference time-domain (ADI-FDTD) method is not limited by the Courant–Friedrich–Levy (CFL) stability condition. However, the numerical-dispersion error of the ADI-FDTD method is much greater than that of Yee's FDTD method. In this paper, the numerical dispersion is improved by approximating the spatial derivatives using cubic spline Battle–Lemarie scaling functions and the high-order centered differences. The stability condition and the numerical-dispersion relations are derived using the Fourier series method and validated by a numerical simulation. The new scheme is unconditionally stable and the numerical dispersion error can be reduced to the limit of the conventional ADI-FDTD method with the 6th-order centered difference. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 45: 43–46, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20717

Numerical validation of dispersion relations using a cylindrical wave for 2D FDTD methods

AbstractThe numerical dispersion of a proposed new FDTD scheme is often evaluated and compared with that of well‐established FDTD methods. Because there may be a theoretical deficiency in the dispersion analysis, numerical experimentation is often used to validate the dispersion relation. This paper describes the “matching method” for validation using a cylindrical wave in the two‐dimensional (2D) case. The matching method is demonstrated by evaluating the numerical dispersion of Yee's FDTD and ADI‐FDTD. The result is compared with the analytical numerical dispersion relation, and excellent agreement is obtained. The difference between the theoretical value for the numerical dispersion and that from numerical experiments using the matching method is 10−5 for Yee's FDTD and 10−4–10−5 for ADI‐FDTD. The numerical velocity obtained from the matching method satisfies the theoretical dispersion relation with residual errors to the order of 10−6. Such accuracy is sufficient to conclude that a numerical dispersion relation is correct. Although the matching method uses a cylindrical wave to test the numerical dispersion relation, the cylindrical wave is not a solution of the difference equations, thus it cannot be used to derive a theoretical dispersion relation. © 2004 Wiley Periodicals, Inc. Microwave Opt Technol Lett 43: 138–142, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20400

Three-dimensional FDTD modeling of a ground-penetrating experiment

This paper deals with a numerical FDTD approach to solve a particular ground-penetrating problem. Classical Yee FDTD formulation is used to simulate layered media on a perfectly electrically conductive plate (PEC). It is shown that an alternative of the step 3-D PML formulated by Sullivan is well adaptated to surround the computational domain. In particular, the PEC can be easily extended to the PML medium without any extra reflection error.

A simple method to determine the time‐step size to achieve a desired dispersion accuracy in ADI‐FDTD

AbstractThis paper presents a simple approach to determine the time‐step size required in the alternate‐direction‐implicit finite‐difference time‐domain (ADI‐FDTD) method in order to obtain a desired numerical dispersion accuracy. The Courant number, the desired dispersion accuracy, and the maximum mesh size Δmax = max(Δx, Δy, Δz) are governed by the numerical dispersion relation, which can be solved by a simple root‐finding algorithm to evaluate the Courant number and hence the time‐step size for a given mesh size and accuracy. The time‐step size is independent of the aspect ratio. To determine if ADI‐FDTD is more efficient than the Yee's FDTD, this paper provides a simple relation to evaluate the relative Courant–Friedrich–Levy number (CFLN) from the Courant number and the aspect ratio. The ADI‐FDTD method is more efficient than Yee's FDTD when the aspect ratio is high or the mesh density is very large. © 2004 Wiley Periodicals, Inc. Microwave Opt Technol Lett 40: 487–490, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20012