Alternating-direction Implicit Finite-difference Time-domain Method Research Articles

Concise Current Source Implementation for Efficient 3-D ADI-FDTD Method

A concise current source implementation is presented for the unconditionally stable 3-D alternating direction implicit finite-difference time-domain (ADI-FDTD) method. Unlike the conventional implicit symmetric source scheme applied in both updating procedures, the new implementation involves implicit current formulation in the first procedure only. Note however that the resultant accuracy does not deteriorate but remains to be identical as before. This is achieved using our recent efficient algorithm for the 3-D ADI-FDTD method. Moreover, the current formulations applied in both procedures therein are simplified, along with further reduction of the floating point operations count for the main iterations. Analytical validation of the equivalence among conventional implicit symmetric source scheme and our efficient algorithms with alternative current implementations are discussed. Numerical results provide closer scrutiny of the asymmetry errors, which may still exist even with symmetric source.

Application of the Complementary Derivatives Method in the ADI-FDTD Solution of Maxwell's Equations

The alternating-direction implicit finite-difference time-domain (ADI-FDTD) method is well suited for simulating structures with large aspect ratios or problems with large gradient fields where different grid sizes can be used to yield greater computational efficiency. However, using different grid sizes increases the truncation error at the interface between domains having different grid sizes. The truncation error is manifested as a spurious reflection from the grid boundary, thus decreasing the simulation accuracy. In this paper, we apply the complementary derivatives method (CDM) to reduce the spurious reflections arising from the use of different grid size domains when using the ADI-FDTD method. It is shown that, the CDM guarantees uniform second-order accuracy throughout the computational domain. When the CDM is implemented in the ADI-FDTD method, the implicit updating equations cannot be written in a tri-diagonal matrix and the computational efficiency of the ADI-FDTD method is not preserved. By employing the Sherman-Morrison formula, we retain the numerical efficiency of the conventional ADI-FDTD. A representative numerical example is presented to demonstrate the accuracy of CDM in the ADI-FDTD simulations.

Envelope ADI–FDTD Method and Its Application in Three-Dimensional Nonuniform Meshes

The envelope alternating-direction-implicit finite difference time domain (ADI-FDTD) method in 3-D nonuniform meshes was proposed and studied. The phase velocity error for the envelope ADI-FDTD and ADI-FDTD methods in uniform and nonuniform meshes and different temporal increments were studied. A cavity problem was studied using the envelope ADI-FDTD and ADI-FDTD methods in graded meshes and the conventional FDTD method in a uniform mesh. The simulation results show that the envelope ADI-FDTD performs better than the ADI-FDTD in numerical accuracy

NUMERICAL MODELING OF OPTICALLY CONTROLLED DIELECTRIC REASONATORS

To analyze the optically controlled dielectric resonators accurately, an efficient numerical modelling technique is proposed in this paper. By using alternating-direction implicit finite-difference time-domain method and conformal technique, the resonant frequency of a dielectric resonator is calculated. In addition, an optical generation of plasma is used as a possible means of controlling the resonate frequency, and the effect that solid state plasmas have on the resonator’s frequency is described. The numerical results agree very well with measurements in estimation of the optically induced resonator’s frequency-shift.

Stability analysis of the Mur's absorbing boundary condition in the alternating direction implicit finite-difference method

The stability analysis of the Mur's first order absorbing boundary condition (ABC) in the alternating direction implicit finite-difference time-domain (ADI-FDTD) method is presented. The theoretical stability analysis of this scheme is studied by deriving the amplification matrix. The effect of wave propagation direction on the stability of this scheme is investigated. From the stability analysis, it is found that the Mur's first order ABC in the ADI-FDTD method will be unstable. The instability of the scheme is validated by means of actual numerical simulations.

Unconditionally Stable ADI-FDTD Method Including Passive Lumped Elements

From the stability analysis for the conventional explicit finite-difference time-domain (FDTD) method including the passive lumped element, one can find that its stability is related either to the mesh size or to both the mesh size and the values of the elements. This paper presents a three-dimensional (3-D) alternating-direction-implicit FDTD (ADI-FDTD) method, including various lumped elements, such as resistors, capacitors, and inductors. The stability of these schemes is proved to be neither related to the mesh size, nor to the values of the elements. This unconditional stability is different from that of the previous explicit schemes including lumped elements and is therefore a distinct advantage. These 3-D schemes can be justified to achieve improved computation efficiency. Some numerical experiments are performed for the demonstration of the higher efficiency and validation of these schemes

Efficient numerical methods for simulation of high-frequency active devices

We present two new numerical approaches for physical modeling of high-frequency semiconductor devices using filter-bank transforms and the alternating-direction implicit finite-difference time-domain method. In the first proposed approach, a preconditioner based on the filter-bank and wavelet transforms is used to facilitate the iterative solution of Poisson's equation and the other semiconductor equations discretized using implicit schemes. The second approach solves Maxwell's equations which, in conjunction with the semiconductor equations, describe the complete behavior of high-frequency active devices, with larger time-step size. These approaches lead to the significant reduction of the full-wave simulation time. For the first time, we can reach over 95% reduction in the simulation time by using these two techniques while maintaining the same degree of accuracy achieved using the conventional approach

Conformal Method to Eliminate the ADI-FDTD Staircasing Errors

The alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method is an unconditionally stable method and allows the time step to be increased beyond the Courant-Friedrich-Levy (CFL) stability condition. This method is potentially very useful for modeling electrically small but complex features often encountered in applications. As the regular FDTD method, however, the spatial discretization in the ADI-FDTD method is only first-order accurate for discontinuous media; several researchers have shown that the errors can be very high when the regular ADI-FDTD method is applied to such discontinuous media. On the other hand, the conformal FDTD method has recently emerged as an efficient FDTD method with higher order accuracy. In this work, a second-order accurate ADI-FDTD method using the conformal approximation of spatial derivatives is proposed. This new scheme, called the ADI-CFDTD method, retains the second-order accuracy in both temporal and spatial discretizations even for discontinuous media with metallic structures, and is unconditionally stable. 2D and 3D examples demonstrate the efficacy of this method and its application in EMC problems.

Efficient implementations of the Crank-Nicolson scheme for the finite-difference time-domain method

When a finite-difference time-domain (FDTD) method is constructed by applying the Crank-Nicolson (CN) scheme to discretize Maxwell's equations, a huge sparse irreducible matrix results, which cannot be solved efficiently. This paper proposes a factorization-splitting scheme using two substeps to decompose the generalized CN matrix into two simple matrices with the terms not factored confined to one sub-step. Two unconditionally stable methods are developed: one has the same numerical dispersion relation as the alternating-direction implicit FDTD method, and the other has a much more isotropic numerical velocity. The limit on the time-step size to avoid numerical attenuation is investigated, and is shown to be below the Nyquist sampling rate. The intrinsic temporal numerical dispersion is discussed, which is the fundamental accuracy limit of the methods.

Low numerical dispersion two-dimensional (2,4) ADI-FDTD method

The numerical dispersion of the alternating direction implicit finite-difference time-domain (ADI-FDTD) method can be improved by approximating the spatial derivatives using the four-point centered finite-difference formula. However, the improvement is not significant when the time step size increases. In this paper, we develop a low numerical dispersion 2-D (2,4) ADI-FDTD method, by which the finite-difference operators are determined by minimizing the error terms in the numerical dispersion relation. The numerical dispersion error is shown to be significantly reduced for any time-step size. In addition, there is an alternative method that results in zero numerical dispersion errors at a specified propagation angle. The numerical dispersion relation of the proposed method is investigated theoretically and compared with the standard ADI-FDTD method as well as with the conventional FDTD method

Numerical performance and applications of the envelope ADI-FDTD method

The numerical performance of the envelope alternating-direction-implicit-finite-difference time-domain (ADI-FDTD) method and its applications are studied in this paper. The ADI-FDTD method is independent of the Courant-Friedrich-Levy stability condition, but its numerical dispersion grows with the increase of the time-step size. By introducing the envelope technique in the ADI-FDTD method, the numerical accuracy can be improved efficiently. In this paper, the phase velocity error of a propagating Gaussian pulse was studied for the envelope ADI-FDTD and ADI-FDTD and conventional FDTD methods with different cell size and time-step increment, then two waveguide problems and a scattering problem were simulated with the envelope ADI-FDTD and ADI-FDTD methods in graded meshes and the conventional FDTD method in a uniform mesh. The simulation results show the superior performance of the envelope ADI-FDTD over the ADI-FDTD in numerical accuracy.

A performance study of the iterative ADI-FDTD method

The performance of the iterative alternating-direction implicit finite-difference time-domain (ADI-FDTD) method is studied. By interpreting the ADI-FDTD method as an iterative solver for the Crank-Nicolson scheme, we present analytical results of its convergence rate. Numerical examples further to demonstrate the performance of different variants of the iterative ADI-FDTD method.

An ADI-FDTD method for periodic structures

In this communication, the alternating-direction implicit finite-difference time-domain (ADI-FDTD) method is extended to analyze periodic structures. In the implicit updates of the ADI-FDTD method, the periodic boundary condition leads to a cyclic matrix. Instead of inverting the cyclic matrix directly, the problem is converted into two auxiliary linear systems that can be solved using the tridiagonal matrix solver. Consequently, only 7n arithmetic operations are required for each implicit update and the efficiency of the ADI-FDTD method is retained. Numerical examples further demonstrate the effectiveness of this periodic ADI-FDTD method.

Pre-iterative ADI-FDTD method for conductive medium

An efficient accuracy-improvement scheme is proposed to analyze electromagnetic problems with conductive medium. This scheme is based on interpreting the alternating-direction implicit finite-difference time-domain (ADI-FDTD) method as a special iterative solver for the Crank-Nicholson scheme. By applying an additional number of iterations to locations with relatively large field variation, the overall accuracy can be improved with little computational overhead. Special treatment of lossy medium in the ADI-FDTD method is also addressed. Finally, numerical examples demonstrate the effectiveness of the proposed method.

Symmetric source implementation for the ADI-FDTD method

The alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method has been introduced to overcome the Courant limit present in the standard FDTD scheme. However, the ADI-FDTD scheme may produce small asymmetries in the field distributions according to the source implementation. In this communication, we compare different source implementation strategies and show that ADI-FDTD scheme has no asymmetry errors up to the numerical noise level, and can be implemented most accurately if the excitation is directly incorporated within the tridiagonal matrix and if the time discretization of the source is done appropriately within each full time step.

PLJERC-ADI-FDTD method for isotropic plasma

The unconditionally stable alternating-direction-implicit-finite-difference-time-domain (ADI-FDTD) method is extended to dispersive media-isotropic plasma based on the piecewise linear JE recursive convolution (PLJERC) method. Two-dimensional ADI-FDTD formulations for isotropic plasma are derived. Two examples are calculated, the numerical results of the ADI-FDTD method for isotropic plasma are compared to those obtained by the FDTD method to show the efficiency of the proposed method.

An iterative ADI-FDTD with reduced splitting error

We present a new iterative alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method. By recognizing the ADI-FDTD method as a special case of a more general iterative approach to solve the Crank-Nicolson (CN) FDTD scheme, the splitting error in ADI-FDTD can be reduced systematically. Numerical examples are used to illustrate the improved accuracy of this method.

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

Efficient ADI-FDTD/modified matrix pencil method for microwave integrated circuit analysis

This paper presents a new combination of the 3D alternating-direction implicit finite-difference time-domain (ADI-FDTD) method and the modified matrix pencil (MMP) method for the efficient calculation of microwave integrated circuits. It is demonstrated that the time step in the ADI-FDTD algorithm can be four times greater than that of the conventional FDTD without a large loss of accuracy. The MMP method can be used to further reduce the computational time by more than 60% with a reasonable accuracy comparable to that obtained using the ADI-FDTD method in the frequency domain. The iteration number of the ADI-FDTD algorithm combined with the MMP method can be 12 times less than that of the conventional FDTD method for the analysis of microstrip circuits. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 46: 391–395, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20997

Analysis of planar circuits using an unconditionally stable 3D ADI-FDTD method

In this paper, a 3D alternating-direction implicit finite-difference time-domain (ADI-FDTD) algorithm is used to analyze planar circuits. The source excitation in ADI-FDTD is described and two kinds of absorbing boundary conditions (ABCs) are jointly employed for ADI-FDTD algorithm: the Gedney's uniaxial PML scheme is applied in the propagation direction, and Mur's 1st-order absorbing boundary condition (ABC) is set on the other outer surfaces. Both time-domain waveforms and S-parameters are presented. The numerical simulations of several examples show that the number of iterations with this ADI-FDTD algorithm can be six times less than that with the conventional FDTD without a large loss of accuracy. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 46: 175–179, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20936