Dispersion analysis of the three-dimensional complex envelope ADI-FDTD method

In this paper, numerical dispersion properties of the three-dimensional complex envelope (CE) alternate-direction implicit finite-difference time-domain (ADI-FDTD) method are studied. The variations of dispersion errors with propagation direction, ratio of carrier to envelope frequencies, and spatial and temporal steps are presented. It is found that the CE ADI-FDTD scheme have much better accuracy and efficiency over the ADI-FDTD, especially with a higher ratio of carrier to envelope frequencies. Therefore, the CE ADI-FDTD is recommended for use in efficient narrow bandwidth electromagnetic modeling.     

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     

On Numerical Artifacts of the Complex Envelope ADI-FDTD Method

We examine two spurious numerical artifacts of the complex envelope (CE) alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method, viz. spurious charges and anomalous wave propagation (modes with positive phase velocity and negative group velocity). These artifacts are also present in the conventional ADI-FDTD; however, the spurious charges in CE-ADI-FDTD have a fundamental distinction from those of ADI-FDTD: they are static in ADI-FDTD and implicitly time-harmonic in CE-ADI-FDTD. Spurious charges are particularly detrimental to CE-ADI-FDTD simulations because they produce secondary radiation. We also show that spurious charges can be reduced by a fixed-point iterative correction in CE-ADI-FDTD.      

An improved ADI-FDTD method and its application to photonicsimulations

When the alternating direction implicit-finite difference time domain method (ADI-FDTD) is applied to simulating photonic devices, full efficiency can not be achieved if reasonable accuracy is to be kept, due to numerical errors such as numerical dispersion. A simple modification to ADI-FDTD is proposed by calculating the envelope rather than the fast-varying field, so that errors are minimized. A factor of two-five in speed can usually be gained while retaining the same level of accuracy compared with conventional FDTD. The efficiency and the accuracy of this improved approach is demonstrated on several problems, from simple waveguide structures to complex photonic crystal structures     

Improvement on the numerical dispersion of 2-D ADI-FDTD with artificial anisotropy

In this letter, by introducing artificial anisotropy into computational space, a simple and efficient approach to reduce numerical dispersion of the two-dimensional alternating direction implicit finite-difference time-domain (ADI-FDTD) method is proposed. It is shown that performance of the ADI-FDTD method can be improved significantly for both single frequency simulations and relatively wideband problems. Consequently, the usefulness and effectiveness of the ADI-FDTD method can be notably enhanced.     

A study of the numerical dispersion relation for the 2-D ADI-FDTD method

This letter presents a numerical dispersion relation for the two-dimensional (2-D) finite-difference time-domain method based on the alternating-direction implicit time-marching scheme (2-D ADI-FDTD). The proposed analytical relation for 2-D ADI-FDTD is compared with those relations in the previous works. Through numerical tests, the dispersion equation of this work was shown as correct one for 2-D ADI-FDTD.     

一种有效减少ADI-FDTD数值色散的方法

ADI—FDTD算法的数值色散效应较为明显，本文的研究表明一种通过添加各向异性媒质来修正相速误差，从而减少FDTD数值色散的方法，同样适用于ADI-FDTD，且收效更为显著。数值运算结果证明该方法能够简单有效地去除较宽频带范围内的色散。     

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.     

UPML媒质中的包络ADI-FDTD

由于交替方向隐式时域有限差分法(Alternating-Direction Implicit Finite-Difference Time Domain,ADI-FDTD)的数值色散会随着时间步长的增加而增加,文中讨论了单轴各向异性完全匹配层(uniaxial perfectly matched layer,UPML)媒质中包络交替方向隐式时域有限差分法(Envelope ADI-FDTD),推导了二维Envelope ADI-FDTD UPML的迭代公式,并提出一种新的离散方法。与ADI-FDTD UPML相比,改进后的Envelope ADI-FDTD UPML的时间步长可以取得更大,且能有效地修正相速误差,从而减少数值色散,提高计算精度。     

高阶ADI-FDTD算法的数值色散分析

该文给出高阶交替方向隐时域优先差分(ADI-FDTD)算法,即在ADI-FDTD迭代公式的基础上对时间的差分仍然采用二阶中心差分格式,而对空间的差分则采用四阶中心差分格式,并解析地证明了所给出的高阶ADI-FDTD算法仍然满足无条件稳定方程,同时对增长因子相位的分析,得到数值色散关系,最后对其数值色散误差进行了分析,研究表明与普通ADI-FDTD相比,其色散误差较小。     

各向异性介质在三维ADI-FDTD中的应用

该文研究一种减小三维交替方向隐式时域有限差分法(ADI-FDTD)数值色散的新方法。通过在三维空间中合理添加各向异性介质,达到调整相速的目的,从而减小数值色散,使计算结果更加精确。首先对添加各向异性介质后的三维ADI-FDTD迭代公式进行变形,并得到新的数值色散关系,从而求解得到各向异性介质的相对介电常数。以空心波导和具有介质不连续性的波导作为数值算例,分析不同的各向异性介质和添加方法对计算精度的影响,并与传统ADI-FDTD得到的结果和计算资源占用情况进行比较。结果表明通过正确选择各向异性介质和添加方法,可以有效地减小三维ADI-FDTD数值色散。     

ADI-FDTD+GRT在波导电路分析中的应用

本文研究时域有限差分法(FDTD)的一种新的时空压缩技术,并应用于波导电路的分析.首先分析了软激励条件下的改进的几何重置技术(GRT),研究了合理选择源面与参考面的放置位置,使GRT不仅减小了吸收边界对计算结果的影响,而且节省了计算空间,还可以精确得到全部散射参量.另外阐述了与交替方向隐式时域有限差分法(ADI-FDTD)相结合,使计算空间和时间同时被压缩,达到节省计算资源的目的.为了衡量ADI-FDTD+GRT算法的计算精度和效率,分析了包含不连续结构的波导作为算例,将其数值计算结果分别与传统FDTD和HFSS作比较,并将端面和参考面不同间距的ADI-FDTD+GRT与传统ADI-FDTD在仿真结果和资源占用方面进行对比,结果表明本文算法是精确和高效的.     

色散媒质中ADI-FDTD的PML

基于交替方向隐式(ADI)技术的时域有限差分法(FDTD)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,而是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。该文采用递归卷积(RC)方法导出了二维情况下色散媒质中ADI-FDTD的完全匹配层(PML)公式。应用推导公式计算了色散土壤中目标的散射,并与色散媒质中FDTD结果对比,在大量减少计算时间的情况下,两者结果符合较好。     

Alternating-direction implicit (ADI) formulation of the finite-difference time-domain (FDTD) method: algorithm and material dispersion implementation

The alternating-direction implicit finite-difference time-domain (ADI-FDTD) technique is an unconditionally stable time-domain numerical scheme, allowing the /spl Delta/t time step to be increased beyond the Courant-Friedrichs-Lewy limit. Execution time of a simulation is inversely proportional to /spl Delta/t, and as such, increasing /spl Delta/t results in a decrease of execution time. The ADI-FDTD technique greatly increases the utility of the FDTD technique for electromagnetic compatibility problems. Once the basics of the ADI-FDTD technique are presented and the differences of the relative accuracy of ADI-FDTD and standard FDTD are discussed, the problems that benefit greatly from ADI-FDTD are described. A discussion is given on the true time savings of applying the ADI-FDTD technique. The feasibility of using higher order spatial and temporal techniques with ADI-FDTD is presented. The incorporation of frequency dependent material properties (material dispersion) into ADI-FDTD is also presented. The material dispersion scheme is implemented into a one-dimensional and three-dimensional problem space. The scheme is shown to be both accurate and unconditionally stable.     

GENETIC ALGORITHM IN REDUCTION OF NUMERICAL DISPERSION OF 3-D ADI-FDTD METHOD

A new method to reduce the numerical dispersion of the three-dimensional Alternating Direction Implicit Finite-Difference Time-Domain （3-D ADI-FDTD） method is proposed. Firstly, the numerical formulations of the 3-D ADI-FDTD method are modified with the artificial anisotropy, and the new numerical dispersion relation is derived. Secondly, the relative permittivity tensor of the artificial anisotropy can be obtained by the Adaptive Genetic Algorithm （AGA）. In order to demonstrate the accuracy and efficiency of this new method, a monopole antenna is simulated as an example. And the numerical results and the computational requirements of the proposed method are compared with those of the conventional ADI-FDTD method and the measured data. In addition the reduction of the numerical dispersion is investigated as the objective function of the AGA. It is found that this new method is accurate and efficient by choosing proper objective function.     

Comments on "An efficient PML implementation for the ADI-FDTD method"

For original paper see Wang and Teixeira (IEEE Microwave Wireless Comp. Lett., vol.13, p.72-4, 2003 February). In this paper, a more precise way to evaluate the actual performance of the perfectly matched layer (PML) used for the alternating direction implicit finite-difference time-domain (ADI-FDTD) method is presented. It is shown that the intrinsic numerical dispersion error of the ADI-FDTD method must be taken into account when the actual performance of the ADI-PML (as well as the ADI-FDTD method) is evaluated. Most importantly, it is demonstrated that the ADI-PMLs implemented with either the traditional manner or the way proposed in have almost the same level of accuracy when the performance of the ADI-PML is correctly evaluated.     

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.     

Stability and Dispersion Analysis for ADI-FDTD Method in Lossy Media

The stability and dispersion analysis for the alternating-direction-implicit finite-difference time-domain (ADI- FDTD) method in lossy media is presented. Although the stability and numerical dispersion have been analyzed for the ADI-FDTD method, most of the analysis is dedicated to the cases of lossless media. Here, the stability and dispersion analysis is performed for the method in lossy media. The stability analysis theoretically proves the unconditional stability of the ADI-FDTD method in lossy media. Meanwhile, the dispersion analysis reveals the numerical loss and dispersion characteristics of this method. This will be meaningful for the evaluation and further development of the ADI-FDTD method in lossy media     

ADI-FDTD方法在一维PBG结构中的应用

提出一种新的无时间约束时域有限差分(FDTD)法。与传统的方法不同的是在该方法中引进了交替隐式(ADI)技术，即将原来一个时间步分成两个子时间步，在两个子时间步中，显式和隐式差分误差相互弥补使精度仍保留在二阶小量。理论证明，这种混合技术不再要求时间上满足原有约束条件，对于长时间才能稳定的问题具有较高的实用价值。而光电子带隙结构(PBG)是一种周期性结构，多次反射与透射导致电磁波在该结构中形成较长时间振荡，采用ADI技术，时间步长的增加将明显减少FDTD的计算时间，提高FDTD的计算效率。     

色散媒质中采用Z变换的ADI-FDTD方法

基于Z变换方法将ADI-FDTD推广应用于色散媒质,得到了二维情况下色散媒质中的迭代差分公式,同时给出了一种采用ⅡR滤波器结构来减少存储量的方法.本文方法适合于任意M阶色散媒质,最后,给出了一个算例,数据仿真结果表明,本文的算法与传统色散媒质中的FDTD相比,在计算结果吻合的情况下,存储量相当,计算效率却更高,时间步长仅仅由计算精度来决定.