有耗介质中时域精细积分方法的数值色散分析

分析了时间步长、空间步长、电导率和电磁波传播方向对时域精细积分(PITD)方法的数值损耗和数值色散的影响。结果表明:PITD的数值损耗大于电磁波的真实损耗,其数值波速可以大于电磁波的真实波速。PITD的数值损耗和数值色散都基本上不受时间步长的影响。随着空间步长的减小,PITD的数值损耗和数值色散的误差都逐步减小。当电导率较小时,PITD的数值损耗和数值色散的误差比时域有限差分(FDTD)方法的大。但当电导率较大时,PITD的数值波速却比FDTD的数值波速更加接近于电磁波的真实波速。PITD的数值损耗和数值色散的各向异性在三维情况下的值要大于其在二维情况下的值。数值算例表明:对良导体而言,PITD比FDTD拥有更高的计算精度和更快的计算速度。     

An efficient method to reduce the numerical dispersion in the ADI-FDTD

A new approach to reduce the numerical dispersion in the finite-difference time-domain (FDTD) method with alternating-direction implicit (ADI) is studied. By adding anisotropic parameters into the ADI-FDTD formulas, the error of the numerical phase velocity can be controlled, causing the numerical dispersion to decrease significantly. The numerical stability and dispersion relation are discussed in this paper. Numerical experiments are given to substantiate the proposed method.     

ADI-MRTD算法的数值色散性分析

针对传统的时域多分辨分析(MRTD)方法的稳定性不足问题,讨论了一种将交替方向隐式技术(ADI)与MRTD算法相结合的交替方向隐式时域多分辨分析算法(ADI-MRTD)。导出了基于Daubechies小波尺度函数的ADI-MRTD算法的差分公式和色散性方程,同时证明了其仍然满足无条件稳定方程。并讨论了空间步长、时间步长和电磁波传播方向等因素对ADI-MRTD算法的数值色散影响。结果表明:ADI-MRTD算法的数值色散特性优于传统的时域有限差分(FDTD)算法。     

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。     

UPML媒质中无条件稳定的二维ADI-FDTD方法

对单轴各向异性PML（UPML）媒质中二维TM波的交变方向隐式时域有限差分方向（ADI-FDTD），通过计算实例表明，ADI-FDTD方法在UMPL媒质中是无条件稳定的，其时间步长不受CFL稳定性条件的限制，并且当计算区域内具有精细差分网格时，其计算效率明显优于传统的时域有限差分方向（FDTD）。     

Rigorous analysis of the influence of the aspect ratio of Yee's unit cell on the numerical dispersion property of the 2-D and 3-D FDTD methods

In this paper, the influence of the aspect ratio of Yee's unit cell on the numerical dispersion errors [in terms of the physical phase-velocity error (PVE) and the velocity-anisotropy error (VAE)] of two-dimensional (2-D) finite-difference time-domain (FDTD) and three-dimensional (3-D) FDTD methods is comprehensively investigated. Numerical results reveal that, for a fixed mesh resolution, the physical PVE and the VAE of both the 2-D and 3-D FDTD methods converge to certain limits for higher aspect ratio. Most importantly, it is found for the first time that for the 2-D and 3-D cases the converged dispersion errors (i.e., the limits) are, respectively, about 2.0 and 1.5 times of the corresponding square and cubic unit cells; and the validity of the above theoretical prediction is verified through numerical tests. The investigation carried out in this paper certainly confirms, from the numerical dispersion point of view, that very accurate numerical results can still be obtained even when the aspect ratio of the cells is higher. Consequently, it gives design engineers more freedom and confidence to use the FDTD methods, especially when the aspect ratio of the cells has to be greatly adjusted due to the special requirement of structures under study.     

Toward the development of a three-dimensional unconditionallystable finite-difference time-domain method

In this paper, an unconditionally stable three-dimensional (3-D) finite-difference time-method (FDTD) is presented where the time step used is no longer restricted by stability but by accuracy. The principle of the alternating direction implicit (ADI) technique that has been used in formulating an unconditionally stable two-dimensional FDTD is applied. Unlike the conventional ADI algorithms, however, the alternation is performed in respect to mixed coordinates rather than to each respective coordinate direction, Consequently, only two alternations in solution marching are required in the 3-D formulations. Theoretical proof of the unconditional stability is shown and numerical results are presented to demonstrate the effectiveness and efficiency of the method. It is found that the number of iterations with the proposed FDTD can be at least four times less than that with the conventional FDTD at the same level of accuracy     

二维无条件稳定时域有限差分方法

基于半隐式的Crank-Nicolson差分格式给出了一种无条件稳定时城有限差分方法。和传统FDTD法中采用的显式差分格式不同，对Maxwell方程组采用半隐式差分格式，在时间和空间上仍然是二阶精确的。但时间步长不再受稳定性条件的限制，只需考虑数值色散误差对其取值的制约。利用分裂场完全匹配层吸收边界截断计算空间，为保证PML空间的无条件稳定性，其方程也采用半隐式差分格式。数值结果表明相同条件下US-FDTD方法与传统FDTD方法的计算精度是相同的，而且在增大时间步长时US-FDTD方法是稳定的和收敛的。可以预见US-FDTD方法在模拟具有电小结构问题时具有实际意义。     

平面波在时域有限差分法中的引入方法研究

基于角点延迟的思想,研究了时域有限差分(FDTD)法中,两种在连接边界上引入平面波的方法--一维平面波推进法和解析法.仿真结果表明:一维平面波推进法与解析法同样有效;一维平面波推进法很大程度上抵消了FDTD法的数值色散误差,溢出波较少,而解析法出现了较多的溢出波;使用一维平面波推进法时,减小网格尺寸能够进一步减小溢出波...     

三维最优时域有限差分方法

数值色散是时域有限差分方法(FDTD)中最主要的误差来源,导致数值相速成为频率和方向的函数。文中讨论了一种基于最优有限冲激滤波器设计方法的最优差分格式,从频率空间或者波数空间中实现对理想偏微分算子的逼近,构造一种新的具有低数值色散关系的最优时域有限差分方法。文中导出了其数值色散关系和进行了稳定性分析,并通过与常用的基于泰勒级数展开定理的高阶(2,4)时域有限差分法相比较,发现最优时域有限差分法的数值色散得到了极大的改善。最后通过一个数值例子来验证其有效性。     

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.      

An analysis of new and existing FDTD methods for isotropic coldplasma and a method for improving their accuracy

Over the past few years, a number of different finite-difference time-domain (FDTD) methods for modeling electromagnetic propagation in an isotropic cold plasma have been published. We have analyzed the accuracy and stability of these methods to determine which method provides the greatest accuracy for a given computation time. For completeness, two new FDTD methods for cold plasma, one of which is based on the concept of exponential fitting, are introduced and evaluated along with the existing methods. We also introduce the concept of cutoff modification which can be easily applied to most of the FDTD methods, and which we show can improve the accuracy of these methods with no additional computational cost. Von Neumann's stability analysis is used to evaluate the stability of the various methods, and their accuracy is determined from a straightforward time-and-space harmonic analysis of the dispersion and dissipation errors. Results of numerical experiments to verify the accuracy analysis are presented. It is found that for low-loss plasma, the piecewise linear recursive convolution method (PLRC) method is the most accurate, but the method of Young (see Radio Sci., vol.29, p.1513-22, 1994) can use less memory and is nearly as accurate. In this low-loss plasma regime, cutoff modification can significantly reduce the error near cutoff at the expense of slightly greater error at lower frequencies. For strongly collisional plasmas, the PLRC method also provides the most accurate solution     

Properties of the FDTD method relevant to the analysis of microwave power problems

The objective of the paper is to provide a systematic consideration and generalization of properties and features of the FDTD method in the context of its use in solving microwave power problems. This is aimed at filling the gap between the general theory of the FDTD method and the specific practice of its applications by microwave power engineers. The paper starts with a comparison of FDTD to other methods like FEM, from the perspective of microwave power simulations. It then discusses FDTD-specific models of lossy and dispersive media, conformal boundaries, field singularities, and modal excitation as well as error bounds due to numerical dispersion. Theoretical overview is illustrated with examples. References are provided to the literature where more details and application notes can be found.     

A Study of Infrared Photonic Crystal Devices Using New ADI-FDTD Algorithm

To investigate the infrared photonic crystal devices numerically, the traditional finite-difference time-domain (FDTD) method has been modified by combining with a new alternating direction implicit (ADI) algorithm. An improvement of two-five in speed over previous FDTD methods can be obtained by calculating the envelope rather than the fast-varying field, and the numerical errors are minimized. Consider the isolated localized coupled-cavity modes, the phenomenon of eigenmode splitting has been observed when the coupled-cavity structures in two dimension triangular dielectric photonic crystals are simulated. The results are in good agreement with experiments.     

Viable Three-Dimensional Medical Microwave Tomography: Theory and Numerical Experiments

Three-dimensional microwave tomography represents a potentially very important advance over 2D techniques because it eliminates associated approximations which may lead to more accurate images. However, with the significant increase in problem size, computational efficiency is critical to making 3D microwave imaging viable in practice. In this paper, we present two 3D image reconstruction methods utilizing 3D scalar and vector field modeling strategies, respectively. Finite element (FE) and finite-difference time-domain (FDTD) algorithms are used to model the electromagnetic field interactions in human tissue in 3D. Image reconstruction techniques previously developed for the 2D problem, such as the dual-mesh scheme, iterative block solver, and adjoint Jacobian method are extended directly to 3D reconstructions. Speed improvements achieved by setting an initial field distribution and utilizing an alternating-direction implicit (ADI) FDTD are explored for 3D vector field modeling. The proposed algorithms are tested with simulated data and correctly recovered the position, size and electrical properties of the target. The adjoint formulation and the FDTD method utilizing initial field estimates are found to be significantly more effective in reducing the computation time. Finally, these results also demonstrate that cross-plane measurements are critical for reconstructing 3D profiles of the target.      

Dispersion-relation-preserving FDTD algorithms for large-scale three-dimensional problems

We introduce dispersion-relation-preserving (DRP) algorithms to minimize the numerical dispersion error in large-scale three-dimensional (3D) finite-difference time-domain (FDTD) simulations. The dispersion error is first expanded in spherical harmonics in terms of the propagation angle and the leading order terms of the series are made equal to zero. Frequency-dependent FDTD coefficients are then obtained and subsequently expanded in a polynomial (Taylor) series in the frequency variable. An inverse Fourier transformation is used to allow for the incorporation of the new coefficients into the FDTD updates. Butterworth or Chebyshev filters are subsequently employed to fine-tune the FDTD coefficients for a given narrowband or broadband range of frequencies of interest. Numerical results are used to compare the proposed 3D DRP-FDTD schemes against traditional high-order FDTD schemes.     

Efficient compact 2-D time-domain method with weighted Laguerre polynomials

An efficient time-domain method based on a compact two-dimensional (2-D) finite-difference time-domain (FDTD) method combined with weighted Laguerre polynomials has been proposed to analyze the propagation properties of uniform transmission lines. Starting from Maxwell's differential equations corresponding to the compact 2-D FDTD method, we use the orthonormality of weighted Laguerre polynomials and Galerkin's testing procedure to eliminate the time variable. Thus, an implicit relation, which results in a marching-on-in-degree scheme, can be obtained. To verify the accuracy and efficiency of the hybrid method, we compare the results with those from the conventional compact 2-D FDTD and compact 2-D alternating-direction-implicit (ADI) FDTD methods. The hybrid method improves the computational efficiency notably, especially for complex problems with fine structure details that are restricted by stability constrains in the FDTD method.     

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。     

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.     

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.