Approximate Crank-Nicolson schemes for the 2-D finite-difference time-domain method for TE/sub z/ waves

Two implicit finite-difference time-domain (FDTD) methods are presented in this paper for a two-dimensional TE/sub z/ wave, which are based on the unconditionally-stable Crank-Nicolson scheme. To treat PEC boundaries efficiently, the methods deal with the electric field components rather than the magnetic field. The "approximate-decoupling method" solves two tridiagonal matrices and computes only one explicit equation for a full update cycle. It has the same numerical dispersion relation as the ADI-FDTD method. The "cycle-sweep method" solves two tridiagonal matrices, and computes two equations explicitly for a full update cycle. It has the same numerical dispersion relation as the previously-reported Crank-Nicolson-Douglas-Gunn algorithm, which solves for the magnetic field. The cycle-sweep method has much smaller numerical anisotropy than the approximate-decoupling method, though the dispersion error is the same along the axes as, and larger along the 45/spl deg/ diagonal than ADI-FDTD. With different formulations, two algorithms for the approximate-decoupling method and four algorithms for the cycle-sweep method are presented. All the six algorithms are strictly nondissipative, unconditionally stable, and are tested by numerical computation in this paper. The numerical dispersion relations are validated by numerical experiments, and very good agreement between the experiments and the theoretical predication is obtained.     

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.     

A three-dimensional unconditionally stable ADI-FDTD method in the cylindrical coordinate system

An unconditionally stable finite-difference time-domain (FDTD) method in a cylindrical coordinate system is presented in this paper. The alternating-direction-implicit (ADI) method is applied, leading to a cylindrical ADI-FDTD scheme where the time step is no longer restricted by the stability condition, but by the modeling accuracy. In contrast to the conventional ADI method, in which the alternation is applied in each coordinate direction, the ADI scheme here performs alternations in mixed coordinates so that only two alternations in solution matching are required at each time step in the three-dimensional formulation. Different from its counterpart in the Cartesian coordinate system, the cylindrical ADI-FDTD includes an additional special treatment along the vertical axis of the cylindrical coordinates to overcome singularity. A theoretical proof of the unconditional stability is shown and numerical results are presented to demonstrate the effectiveness of the cylindrical algorithm in solving electromagnetic-field problems.     

ADI-FDTD algorithm in curvilinear co-ordinates

The alternating direction implicit (ADI) scheme has been successfully applied to the finite-difference time-domain (FDTD) method to achieve an unconditionally stable algorithm. The ADI-FDTD method is extended to the curvilinear co-ordinate system to form an alternating direction implicit nonorthogonal FDTD (ADI-NFDTD) method. The numerical results show that the proposed ADI-NFDTD algorithm demonstrates better late time stability compared to the conventional NFDTD scheme.     

一种非条件稳定的隐式时域有限差分法

介绍一种基于交替方向隐式(ADI)技术的时域有限差分法(FDTD).该方法是非条件稳定的,时间步长不再受到Courant稳定条件的限制,而是由数值色散误差来确定.与传统的FDTD相比,ADI-FDTD增大了时间步长,从而缩短了总的计算时间,特别是当空间网格远小于波长时,优点更加突出.首次把完全匹配层(PML)边界条件应用到ADI-FDTD计算中,采用幂指数形式的时间步进算法,推导了相应的迭代公式.进行了实例计算,并与传统FDTD的结果对比,验证了ADI-FDTD的有效性与优越性.     

3-D ADI-FDTD method-unconditionally stable time-domain algorithmfor solving full vector Maxwell's equations

We previously introduced the alternating direction implicit finite-difference time domain (ADI-FDTD) method for a two-dimensional TE wave. We analytically and numerically verified that the algorithm of the method is unconditionally stable and free from the Courant-Friedrich-Levy condition restraint. In this paper, we extend this approach to a full three-dimensional (3-D) wave. Numerical formulations of the 3-D ADI-FDTD method are presented and simulation results are compared to those using the conventional 3-D finite-difference time-domain (FDTD) method. We numerically verify that the 3-D ADI-FDTD method is also unconditionally stable and it is more efficient than the conventional 3-D FDTD method in terms of the central processing unit time if the size of the local minimum cell in the computational domain is much smaller than the other cells and the wavelength     

Unconditionally stable ADI-FDTD formulations of the APML for frequency-dependent media

Unconditional stable formulations of the anisotropic perfectly matched layer (APML) are presented for truncating frequency-dependent media. The formulations are based on the auxiliary differential equation and the alternating direction implicit finite-difference time-domain (ADI-FDTD) methods. Numerical examples carried out in one and two dimensions show that the proposed formulations remain unconditionally stable with inclusion of material dispersion into ADI-FDTD implementation of APML.     

Theoretical Proof of Unconditional Stability of the 3-D ADI-FDTD Method

In order to eliminate Courant-Friedrich-Levy(CFL) condition restraint and improvecomputational efficiency,a new finite-difference time-domain(FDTD)method based on the alternating-direction implicit(ADI) technique is introduced recently.In this paper,a theoretical proof of the stabilityof the three-dimensional(3-D)ADI-FDTD method is presented.It is shown that the 3-D ADI-FDTDmethod is unconditionally stable and free from the CFL condition restraint.     

On the modeling of conducting media with the unconditionally stable ADI-FDTD method

The Courant-Friedrich-Levy stability condition has prevented the conventional finite-difference time-domain (FDTD) method from being effectively applied to conductive materials because of the fine mesh required for the conducting regions. In this paper, the recently developed unconditionally stable alternating-direction-implicit (ADI) FDTD is employed because of its capability in handling a fine mesh with a relatively large time step. The results show that the unconditionally alternating-direction-implicit-finite-difference time-domain (ADI-FDTD) method can be used as an effective universal tool in modeling a medium regardless of its conductivity. In addition, the unsplit perfectly matched layer combined with the ADI-FDTD method is implemented in the cylindrical coordinates and is proven to be very effective even with the cylindrical structures that contain open conducting media.     

ADI-FDTD方法在乳腺癌检测正向计算中的应用

传统的时域有限差分(Finite-Difference Time-Domain, FDTD)算法受到稳定性条件的制约, 时间步长受限于空间网格的尺寸.医学应用讲究即时性, 为提高成像的速度, 文中采用无条件稳定的交替隐式时域有限差分(Alternating-Direction Implicit Finite-Difference Time-Domain, ADI-FDTD)算法替代传统的FDTD算法进行正向计算, 通过实验得出采用ADI-FDTD算法在保证精度的前提下, 计算时间可缩短为FDTD算法的四分之一, 为乳腺癌微波即时成像提供了可能.     

ADI-FDTD Method With Fourth Order Accuracy in Time

This letter presents an unconditionally stable alternating direction implicit finite-difference time-domain (ADI-FDTD) method with fourth order accuracy in time. Analytical proof of unconditional stability and detailed analysis of numerical dispersion are presented. Compared to second order ADI-FDTD and six-steps SS-FDTD, the fourth order ADI-FDTD generally achieves lower phase velocity error for sufficiently fine mesh. Using finer mesh gridding also reduces the phase velocity error floor, which dictates the accuracy limit due to spatial discretization errors when the time step size is reduced further.     

An Unconditionally Stable Split-Step FDTD Method for Low Anisotropy

Split-step unconditionally stable finite-difference time-domain (FDTD) methods have higher dispersion and anisotropic errors for large stability factors. A new split-step method with four sub-steps is introduced and shown to have much lower anisotropy compared with the well known alternating direction implicit finite-difference time-domain (ADI-FDTD) and other known split step methods. Another important aspect of the new method is that for each space step value there is a stability factor value that the numerical propagation phase velocity is isotropic.     

An Arbitrary-Order LOD-FDTD Method and its Stability and Numerical Dispersion

An arbitrary-order unconditionally stable three-dimensional (3-D) locally-one- dimensional finite-difference time-method (FDTD) (LOD-FDTD) method is proposed. Theoretical proof and numerical verification of the unconditional stability are shown and numerical dispersion is derived analytically. Effects of discretization parameters on the numerical dispersion errors are studied comprehensively. It is found that the second-order LOD-FDTD has the same level of numerical dispersion error as that of the unconditionally stable alternating direction implicit finite-difference time-domain (ADI-FDTD) method and other LOD-FDTD methods but with higher computational efficiency. To reduce the dispersion errors, either a higher-order LOD-FDTD method or a denser grid can be applied, but the choice has to be carefully made in order to achieve best trade-off between the accuracy and computational efficiency. The work presented in this paper lays the foundations and guidelines for practical uses of the LOD method including the potential mixed-order LOD-FDTD methods.      

Investigation on the characteristics of the envelope FDTD based on the alternating direction implicit scheme

This letter presents numerical characteristics of recently developed the envelope FDTD based on the alternating direction implicit scheme (envelope ADI-FDTD). Through numerical simulations, it is shown that the envelope ADI-FDTD is unconditionally stable and we can get better dispersion accuracy than the traditional ADI-FDTD by analyzing the envelope of the signal. This fact gives the opportunity to extend the temporal step size to the Nyquist limit in certain cases. Numerical results show that the envelope ADI-FDTD can be used as an efficient electromagnetic analysis tool especially in the single frequency or band limited systems.     

Development of split-step FDTD method with higher-order spatial accuracy

A split-step FDTD method with higher-order spatial accuracy is presented, which is proved to be unconditionally stable. From the dispersion analysis, it is justified that the method achieves improved accuracy compared with lower-order cases and its dispersion error is comparable with the higher-order ADI-FDTD method.     

A hybrid 2-D ADI-FDTD subgridding scheme for modeling on-chipinterconnects

This paper presents a novel technique for extracting the propagation characteristics of on-chip interconnects. A hybrid two-dimensional subgridding scheme, based on a combination of the finite-difference time-domain (FDTD) method and the alternating-direction implicit (ADI-)FDTD technique, is utilized. The ADI-FDTD scheme is used for fine grid in the vicinity of the metallic etch, while the coarse FDTD grid is used outside this region. The advantage of the ADI-FDTD scheme is that it can be synchronized with the time marching step employed in the coarse FDTD scheme, obviating the need for the temporal interpolation of the fields in the process. This helps to render the hybrid ADI-FDTD subgridding scheme to be more efficient than the conventional FDTD subgridding algorithm in terms of the run time. The phase and attenuation constants of the dominant mode of a lossy stripline are computed by the proposed scheme to validate the technique     

Three New Unconditionally-Stable FDTD Methods With High-Order Accuracy

Three novel finite-difference time-domain (FDTD) methods based on the split-step (SS) scheme with high-order accuracy are presented, which are proven to be unconditionally stable. In the first novel method, symmetric operator and uniform splitting are adopted simultaneously to split the matrix derived from the classical Maxwell's equations into six sub-matrices. Accordingly, the time step is divided into six sub-steps. The second and third proposed methods are obtained by adjusting the sequence of the sub-matrices deduced in the first method, so all the novel methods presented in the paper have similar formulations, of which the numerical dispersion errors and the anisotropic errors are lower than the alternating direction implicit finite-difference time-domain (ADI-FDTD) method, the initial SS-FDTD method and the modified SS-FDTD method based on the Strang-splitting scheme. Specifically, for the second method, corresponding to a certain cell per wavelength (CPW), there is a Courant number value making the numerical anisotropic error to be zero, while in the third novel method, corresponding to a certain Courant number value, there exists a CPW making the numerical anisotropic error to be zero. In order to demonstrate the high-order accuracy and efficiency of the proposed methods, numerical results are presented.     

Multispecies ADI-FDTD Algorithm for Nanoscale Three-Dimensional Photonic Metallic Structures

The unconditionally stable alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method is extended to model multispecies dispersive media for simulation of nanoscale three-dimensional metallic structures based on optical plasmon resonances. Examples involving modeling of Au nanoparticles show that the proposed ADI-FDTD yields improved computational performance versus standard FDTD in highly refined grids and for moderate Courant numbers     

Transient Chip-Package Cosimulation of Multiscale Structures Using the Laguerre-FDTD Scheme

Transient simulation using Laguerre polynomials is unconditionally stable and is ideally suited for modeling structures containing both small and large feature sizes. The focus of this paper is on the automation of this technique and its application to chip-package cosimulation. Laguerre finite-difference time-domain (FDTD) requires using the right number of basis coefficients to generate accurate time-domain waveforms. A method for generating the optimal number of basis functions is presented in this paper. Equivalent circuit models of the FDTD grid have been developed. In addition, a method for simulation over a long time period is also presented that enables the extraction of the frequency response both at low and high frequencies. A node numbering scheme in the circuit model of the FDTD grid that is suitable for implementation has been discussed. Results from a chip-package example that shows the scalability of this technique to solve multiscale problems have been presented.      

色散媒质中ADI-FDTD

基于交替方向隐式(ADI)技术的时域有限差分(FDTD)法是一种非条件稳定的计算方法，该方法的时间步长不受Courant稳定条件限制，而由数值色散误差决定。与传统的FDTD相比，ADI-FDTD增大了时间步长，从而缩短了总的计算时间。本文采用递归卷积方法将ADI-FDTD推广应用于色散媒质，推导了二维情况下色散媒质中的ADI-FDTD迭代公式。应用推导公式计算了色散土壤中目标的散射，并与色散媒质FDTD结果对比，在大量减少计算时间的情况下，两者结果符合很好。