【作者】 张玉强；

【导师】 葛德彪；

【作者基本信息】 西安电子科技大学 ， 无线电物理， 2009， 博士

【摘要】 鉴于色散介质瞬态电磁问题在隐身技术、生物电磁学等许多领域中的重要性,结合数字信号处理(DSP)理论和技术,对色散介质瞬态电磁问题的FDTD算法及相关问题进行了深入研究,并应用所提出的新方法对一些实际问题进行了分析。结合DSP理论中的半解析递归卷积(SARC)算法提出了一种用于色散介质的半解析递归卷积时域有限差分(SARC-FDTD)方法。SARC算法是建立在数值逼近理论基础上的一种卷积递归计算方法,具有绝对稳定、高精度、低内存、高效率等优点。将SARC引入色散介质FDTD,在保留RC方法优点的基础上,提高了算法的通用性,使得在应用时只需给出色散介质模型的极点及其留数,即可进行计算,无需再针对色散模型计算更新公式的系数。解决了现有色散介质RC-FDTD方法对于各种色散介质模型需要单独处理,公式推导复杂、不具有通用性的问题。结合DSP技术中的IIR数字滤波器级联思想,研究了一种高阶有理分式模型色散介质的级联SARC FDTD算法。有理分式是描述介质色散特性的精确、高效的模型之一,常用于复杂介质的色散特性描述。在该方法中,通过先将高阶有理分式频域模型表示为一系列低阶有理分式之积,再将各低阶有理分式表示为时域复指数函数形式,使得高阶色散模型转化为一系列时域复指数函数的卷积,然后通过SARC-FDTD得到求解。解决了现有RC方法处理高阶有理分式模型和渐进稳定色散介质电磁问题时的困难,并将所提算法应用于宽带Lorentz介质、窄带Lorentz介质及其临界情况的分析,表明了算法的有效性。研究并实现了一种改进的Z变换方法。DSP理论中的脉冲响应不变法当脉冲响应h ( t )在t = 0时刻不连续时是不精确的。如果将传统Z变换直接应用于色散介质FDTD方法将导致高的数值误差。在对脉冲响应不变法改进的基础上,从极化矢量P和电场E的关系入手,重新推导了Z变换FDTD更新公式,解决了原Z变换FDTD方法内存占用大的问题。同时,对于极化率函数时域表达式χ( t)在t = 0时刻不连续的色散介质,还使FDTD计算的精度得到提高。提出一种基于DSP的色散介质改进SO-FDTD方法。借鉴IIR数字滤波器转置直接Ⅱ型结构和级联实现思想,并对其状态变量方程进行改进,将其引入色散介质SO-FDTD方法中,得到改进SO-FDTD更新公式。减少了高阶色散介质SO方法内存占用、简化了SO方法在处理高阶有理分式形式色散介质时更新公式系数计算。使内存减少量不低于33%,达到与RC方法相同,更新公式系数计算不超过二阶。提出一种适用于高阶Debye、Lorentz和Drude模型的SO-FDTD方法。借鉴IIR数字滤波器并联思想,从极化矢量P与电场E的关系入手,利用SO方法推导了适用于三种常用色散模型的SO-FDTD更新公式,避免原SO方法在对三种常用色散介质处理时需将低阶有理分式转化为高阶有理分式,简化了公式推导,且三种常用模型对应的更新公式具有统一的形式,适于处理高阶色散介质以及混合模型色散介质,并便于编制通用的计算程序。同时,极大地减少了SO-FDTD方法的内存占用和计算时间。研究并实现了基于改进SO-FDTD方法和SARC-FDTD方法的UPML吸收边界和CFS-PML吸收边界,可有效地实现色散介质FDTD计算时的计算区域截断。结合所提出的SO-FDTD方法或SARC-FDTD方法,新的PML吸收边界不仅独立于被截断介质,而且PML区域的FDTD更新公式具有不随所截断介质变化的统一形式。提出一种同时适用于SARC、ADE、改进Z变换及改进SO方法的色散介质FDTD更新公式的统一形式,解决了不同方法因FDTD更新公式不同而带来的使用不便问题。在该统一更新公式中,使用者可以根据所熟悉的FDTD方法和要处理的色散介质,通过给出色散介质模型参数,由独立的建模程序给出建模数据和更新公式所需系数即可进行计算。更多还原

【Abstract】 The transient electromagnetic problems with dispersive medium play an important role in stealth technology, bio-electromagnetics, and many other areas. Combination of digital signal processing (DSP) theory and technology, the Finite-Difference Time-Domain (FDTD) algorithms for dispersive medium and related problems are studied and a number of practical issues are analyzed using the proposed new methods.Combining with the Semi-Analytical Recursive Convolution (SARC) algorithm in DSP techniques, a novel FDTD update formulas for the analysis of electromagnetic characteristic of dispersive objects is proposed, namely as SARC-FDTD algorithm in here. In this scheme, the absolute stability, high accuracy, less storage and high effectiveness are retained, and a unified update formulation for general dispersive media, i.e., Debye, Drude, Lorentz and hybrid model is possessed. The SARC FDTD can therefore be applied to the analysis of general dispersive media provided that the poles and corresponding residues in dispersive medium model of interest are given. The complex derivation of coefficients in update formulas for the specified dispersive model is not requested and the separate treatment for a variety of dispersion media model in the existing RC-FDTD method is avoided.A novel SARC-FDTD algorithm for the dispersive medium described with high order rational fraction model based on the cascade implementation of Infinite Impulse Response (IIR) filter in DSP techniques is presented. The rational fraction is one of the dispersive models that can accurately and efficiently approximate to the complicated variation of the parameters of dispersive medium with the frequency. In this method, the high-order rational fraction model in frequency domain is firstly expressed as the product of a series low-order rational fraction and is further transformed into the convolution of a series complex exponential function responding to the each low-order rational fractions in time domain. Finally, the convolution is solved by the SARC-FDTD. The FDTD analysis of transient electromagnetic problems for dispersive medium with high-order rational fraction and asymptotic stability model is solved. It is applied to the analysis of dielectric described with wide-band Lorentz model, narrow-band Lorentz model and border-line case.An improved Z transformation is presented and improved FDTD method is implemented. The classical impulse invariance method in Z-transform theory is found to be incorrect and inaccurate when the impulse response is discontinuous at initial time t=0. Such inaccuracy results in higher numerical errors if it is used to develop the update equations for frequency-dependent FDTD. Thorough discussion of corrected impulse invariance method in the realm of Z-transform, a novel ZT-FDTD update formulas for dispersive media is discussed by the relation of polarization P with electric field E. Compared with the commonly used ZT-FDTD, the storage variables in the improved ZT-FDTD are reduced and high accuracy is achieved for dispersive media which exhibit discontinuity at t=0 in the time domain susceptibility function.An improved shifted operator (SO) FDTD method for general dispersive medium based on the implementation of IIR filter in DSP techniques is presented. Combining with the improved state-space equation of transposed direct-Ⅱstructure of IIR filter and cascade implementation, the update formulas of improved SO-FDTD are obtained. Compared with the commonly used SO-FDTD, the storage variables in the improved SO-FDTD are reduced by 33% and computational efficiency is also increased. The simple coefficients for high-order rational fraction model can be obtained in the novel scheme.A novel SO-FDTD scheme for high-order Debye, Lorentz and Drude dispersion model is presented as well. The transformation of low-order rational fraction into high-order form in commonly used SO-FDTD is avoided and derivation of the update formulas is simplified. A unified update formula is possessed for three typical kinds of dispersive model i.e. Debye, Lorentz and Drude medium and is suitable for dealing with high-order dispersive media as well as the hybrid model media. The general computational program can be conveniently developed.The novel Uniaxial Perfectly matched layers (UPML) absorbing boundary and Complex Frequency-Shifted PML (CFS-PML) absorbing boundary based on improved SO-FDTD and SARC-FDTD are presented and is suitable for truncating the FDTD computational domain for calculation of dispersive medium. Combined with the proposed SO-FDTD and SARC-FDTD, the implementation of PML is not only independent for the truncated medium, and the FDTD update formula in PML region is a unified form for the different dispersive medium.A unified FDTD update formulas simultaneously applied to SARC, Auxiliary Differential Equation (ADE), improved Z transform and improved SO algorithm is proposed. The inconvenience bringing with the different FDTD formulas for the above several algorithm is avoided. In this scheme, the coefficients in the unified update formulas are obtained providing the dispersion model parameters are specified. Then, FDTD calculation using the user-familiar FDTD algorithm can be conveniently carried out.更多还原