保角变换FDTD算法的数值稳定性与数值色散

本文提出了新的保角变换ＦＤＴＤ算法，推导了保角变换ＦＤＴＤ算法的时间稳定性和数值色散方程。此外，本文以圆波导为例了不同网格下ＴＥ模数值波长的相对误差。分析了不同传播常数和对极点不同半径的半圆电壁近似下数值波长的相对误差，通过适当地选择网格数，可得到高精度。     

电磁场与微波技术

()盆4 1 01040720保角变换FDTD算法的数值稳定性与数值色散/周晓军,喻志远,林为千(电子科技大学)“电子科学学刊.一2 000,22(一全)一618一625文中提出了新的保角变换FDTD算法,推导了保角变换F DTD算法的时间稳定性和数值色散方程.此外,文中以圆波导为例计算了不同网格下TE模数值波长的相对误差,分析了不同传播常数和对极点不同半径的半圆电壁近似下数值波长的相对误差通过适当地选择网格数,可得到高精度.图3参6(李)值作计算(通常对每一采徉时间选用12个S值)因而是一种计算速度较决的算法.但是,要对大量采徉时间作计算,其计算量仍太大.…     

用Cassinian变换FDTD方法分析几种特殊截面波导传输特性

本文把保角变换应用于紧凑格式2D/FDTD算法,给出了保角变换FDTD算法差分公式,提出了焦点的处理方法.用Cassinian变换分别计算了椭圆波导、茧形波导的截止波长与色散曲线,以及屏蔽平行双线高阶模的截止波长.     

二维非正交坐标系下FDTD方法的数值色散特性

本文详细讨论了二维非正交坐标系下FDTD方法的色散特性，导出了其数值色散方程。理论计算结果表明，非正交FDTD方法的空间色散与网络尺寸、网络内角、波传播方向有密切关系。同时指出，在应用非正交FDTD方法解决有关时域电磁问题时，网格的部分应分量选取网格边长相接近，夹角接近90的情况以减少此方法的数值色散。     

一种改进的FDTD网格剖分算法

分析了时域有限差分(FDTD)网格的生成原理,提出了一种新型非均匀FDTD网格生成算法.该算法通过读取模型获得轴线上的不连续分界点,将整个空间沿x轴、y轴和z轴方向各自分成多个区间;然后通过各区间的长度及其在模型中所处的位置,来确定该不同区间剖分时所采用的具体网格生成算法,由此得到一种新型的分区递变非均匀网格.此算法能够完全由程序实现,避免了普通网格剖分算法所带来的人为误差.在最大限度降低网格数量的同时,克服了剖分算法中由于最大网格过大而导致的高频数值色散问题,使得最后的FDTD程序具有计算时间短、收敛速度快的优点.     

基于Hybrid迭代法的多连通域数值保角变换计算法

针对模拟电荷法无界多连通区域的数值保角变换问题,文中提出了一种高精度的数值方法,即基于Hybrid迭代法的无界多连通区域的数值保角变换计算法.该方法通过模拟电荷法构造约束方程,对约束方程应用预处理构造对称正定的方程,获得新的模拟电荷和辐角,并构造近似保角变换函数.文中以多连通圆变换为径向狭缝域为例进行数值实验.使用解析...     

基于高阶时域有限差分算法的电磁波传播计算

利用FDTD(2,4)高阶时域有限差分(Finite-Difference Time-Domain,FDTD)算法并结合滑动窗口的思想,对电磁波传播特性进行了仿真计算.采用的高阶FDTD算法在空间上达到四阶精度,与二阶精度的传统FDTD算法相比,在相同每波长采样数的条件下,数值色散误差能得到进一步的减少.在源脉冲传播较长距离时,数值色散的减少使得时域下脉冲扩展现象得到改善,滑动子窗口仍然能包含着激励源脉冲的全部信息,从而可更加准确地计算长距离电波传播特性.另外,在相同的数值色散误差容限下,每波长采样数比传统二阶FDTD方法有所减少,从而节省存储空间,加快计算速度.     

辛算法的稳定性及数值色散性分析

引入一种新的数值计算方法 —辛算法求解Maxwell方程,即在时间上用不同阶数的辛差分格式离散,空间分别采用二阶及四阶精度的差分格式离散,建立了求解二维Maxwell方程的各阶辛算法,探讨了各阶辛算法的稳定性及数值色散性.通过理论上的分析及数值计算表明,在空间采用相同的二阶精度的中心差分离散格式时,一阶、二阶辛算法(T1S2、T2S2) 的稳定性及数值色散性与时域有限差分(FDTD)法一致,高阶辛算法的稳定性与FDTD法相当;四阶辛算法结合四阶精度的空间差分格式(T4S4) 较FDTD法具有更为优越的数值色散性.对二维TMz波的数值计算结果表明,高阶辛算法较FDTD法有着更大的计算优势.     

基于Taylor级数展开定理的高阶FDTD的色散分析

时域有限差分法(FDTD)是计算电磁领域中的一类非常重要的研究工具.而Taylor级数展开定理是构造差分格式的一种重要方法,例如Yee格式采用二阶Taylor格式,Fang格式采用四阶Taylor格式.本文借助于采样定理,详细分析了不同阶Taylor中心差分格式的谱特性以及计算误差,并将任意阶Taylor中心差分格式用于数值求解麦克斯韦方程中,严格导出了稳定性条件和数值色散关系的表达式,引入了新的误差定义来衡量算法的好坏.详细地研究了Courant数、网格分辨率CPW和网格长度比率等因素对于数值色散误差的影响,为基于Taylor差分格式的FDTD算法的研究提供了有用的参考.     

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。     

Numerical Stability and Numerical Dispersion of Compact Conformal Mapping 2D-FDTD Method Used for the Dispersion Analysis of Waveguides

In dispersion analysis of waveguides with particular cross-section by compact 2D-FDTD method, using conformal-boundary coordinates can obtain high computational accuracy. The transformation from conformal-boundary coordinates to rectangular coordinates can be done by conformal mapping technique in order to match Yee algorithm. In this paper, numerical stability and numerical dispersion equation of compact conformal mapping 2D-FDTD (CCM-2D-FDTD) method are derived. It is shown that the upper limit of Courant number for CCM-2D-FDTD is always smaller than 1/√2. As an example, the dispersion equation is used to examine the impact of number of cell for circular waveguide.     

Three-dimensional simulation of eccentric LWD tool response in boreholes through dipping formations

We simulate the response of logging-while-drilling (LWD) tools in complex thee-dimensional (3-D) borehole environments using a finite-difference time-domain (FDTD) scheme in cylindrical coordinates. Several techniques are applied to the FDTD algorithm to improve the computational efficiency and the modeling accuracy of more arbitrary geometries/media in well-logging problems: (1) a 3-D FDTD cylindrical grid to avoid staircasing discretization errors in the transmitter, receiver, and mandrel geometries; (2) an anisotropic-medium (unsplit) perfectly matched layer (PML) absorbing boundary condition in cylindrical coordinates is applied to the FDTD algorithm, leading to more compact grids and reduced memory requirements; (3) a simple and efficient algorithm is employed to extract frequency-domain data (phase and amplitude) from early-time FDTD data; (4) permittivity scaling is applied to overcome the Courant limit of FDTD and allow faster simulations of lower frequency tool; and (5) two locally conformal FDTD (LC-FDTD) techniques are applied to better simulate the response of logging tools in eccentric boreholes. We validate the FDTD results against the numerical mode matching method for problems where the latter is applicable, and against pseudoanalytical results for eccentric borehole problems. The comparisons show very good agreement. Results from 3-D borehole problems involving eccentric tools and dipping beds simultaneously are also included to demonstrate the robustness of the method.     

Mastering Conformal Meshing for Complex CAD-Based C-FDTD Simulations

A robust and automatic discretization algorithm for complex conformal finite-difference time-domain (C-FDTD) simulation is presented in this publication. The targeted application range is to enable C-FDTD simulations for real-word engineering problems. Based on computer-graphics methods, complex CAD models with thousands of distinct parts can be efficiently and robustly discretized. A versatile concept is introduced to avoid numerical inaccuracies while calculating intersections, and to lead to a symmetric discretization without the overhead of "virtual lines." In addition, a necessary three-dimensional consistency check/correction, as well as merging of conformal cells of different CAD parts, are explained. The conformal geometric information is incorporated into the conventional FDTD algorithm using the conventional updating coefficients, which are conformally enhanced. Due to the derived stability criterion, the conformal updating scheme is always stable. The robustness and performance of the discretization algorithm presented is demonstrated with CAD models of increasing complexity towards real-world benchmarks. A conformal FDTD simulation with 80 million computational cells and 229 distinguished parts, representing a complete mobile phone and a head with hand, demonstrates the capabilities of the versatile technique.     

Analysis of the numerical dispersion of the 2Dalternating-direction implicit FDTD method

The numerical dispersion property of the two-dimensional alternating-direction implicit finite-difference time-domain (2D ADI FDTD) method is studied. First, we notice that the original 2D ADI FDTD method can be divided into two sub-ADI FDTD methods: either the x-directional 2D ADI FDTD method or the y-directional 2D ADI FDTD method; and secondly, the numerical dispersion relations are derived for both the ADI FDTD methods. Finally, the numerical dispersion errors caused by the two ADI FDTD methods are investigated. Numerical results indicate that the numerical dispersion error of the ADI FDTD methods depends highly on the selected time step and the shape and mesh resolution of the unit cell. It is also found that, to ensure the numerical dispersion error within certain accuracy, the maximum time steps allowed to be used in the two ADI FDTD methods are different and they can be numerically determined     

A systematic and topologically stable conformal finite-difference time-domain algorithm for modeling curved dielectric interfaces in three dimensions

A systematic conformal finite-difference time-domain (FDTD) algorithm for the direct modeling of dielectric interfaces in three dimensions is presented in this paper. The straightforward procedure is based on the proper reformation of the grid cells in the vicinity of the dielectric surface, leading thus to the creation of five-faced prisms on the primary grid, apart from the standard hexagonal ones. The new scheme overcomes any topological deficiency that forbids the contour path FDTD and conformal FDTD technique to directly simulate dielectric boundaries, since it maintains the lattice duality. Therefore, no instabilities, even late-time ones, are observed. On the other hand, the accuracy obtained, even with very coarse meshes, is very good as is proved by the numerical analysis of various resonance problems.     

网格长度对FDTD计算的影响

基于平面波传播的物理性质，提出一种FDTD改进方法——在研究稳态问题时，为保证数字波前与入射方向垂直，网格划分不能是固定的，其长度之比应随人射方向变化。研究了改进方法对数值色散的影响，并与传统方法作了比较。在三维Yee网格中对平面波的模拟结果表明，改进方法的收敛性能大为提高，对远场散射的误差影响小。     

Development of a graphical user interface for the conformal finite-difference/time-domain (CFDTD) software package

We present the development of the graphical user interface of a conformal FDTD software package utilizing the Visual Basic programming language. We describe a complete interface, including the drawing of three-dimensional objects, the generation of non-uniform and conformal mesh, interfacing with AutoCad and GID, and data post-processing. The engine for the software package is based on a conformal FDTD algorithm, designed to handle general electromagnetic problems, including curved PEC and dielectric surfaces, without the use of "staircasing". To illustrate the versatility of the package, we present several representative simulation examples, viz., the modeling of a microwave stripline, a patch antenna, a triangular antenna, and the evaluation of the surface impedance of Earth. We also provide a comparison between the numerical results derived by using this package and other commercial software for a patch antenna.     

Enlarged cells for the conformal FDTD method to avoid the time step reduction

Recently, the conformal finite-difference time-domain (CFDTD) method has emerged as an efficient FDTD method with a higher order accuracy than the conventional FDTD methods that are degraded by staircasing errors. The only obvious point to further improve on the CFDTD method is its requirement for a smaller time step increment due to the existence of small irregular cells near the boundary. In this letter, an enlarged cell technique is introduced to ensure the stability of the CFDTD method without the time step reduction. The introduction of the enlarged cells therefore makes the CFDTD method much more efficient and suffers from a smaller dispersion error, as shown in several two-dimensional examples.     

Higher Stability Limits for the Symplectic FDTD Method by Making Use of Chebyshev Polynomials

A new scheme is introduced for obtaining higher stability performance for the symplectic finite-difference time-domain (FDTD) method. Both the stability limit and the numerical dispersion of the symplectic FDTD are determined by a function zeta. It is shown that when the zeta function is a Chebyshev polynomial the stability limit is linearly proportional to the number of the exponential operators. Thus, the stability limit can be increased as much as possible at the cost of increased number of operators. For example, the stability limit of the four-exponential operator scheme is 0.989 and of the eight-exponential operator scheme it is 1.979 for fourth-order space discretization in three dimensions, which is almost three times the stability limit of previously published symplectic FDTD schemes with a similar number of operators. This study also shows that the numerical dispersion errors for this new scheme are less than those of the previously reported symplectic FDTD schemes