摘要
自1966年K. S. Yee提出时域有限差分法(FDTD)以来,FDTD方法中的重要组成部分—吸收边界条件的研究就一直是研究热点。目前,效果最好的吸收边界条件是完全匹配层(PML)。完全匹配层是数学上在FDTD区域截断边界处虚拟设置一种特殊介质层,通过合理地选择PML的本构参数,能够使FDTD计算域的外行电磁波无反射地穿过分界面而进入PML层,并且无反射条件与外行电磁波的入射角、极化和频率都无关;由于PML层为有耗介质,进入PML层的透射波将迅速衰减。迄今为止,有三种完全匹配层方案:(1)Berenger的分裂场完全匹配层。(2)拉伸坐标完全匹配层(SC-PML)。(3)各向异性完全匹配层(APML或者UPML)。后来,为了吸收低频隐失波,提出了复频率偏移完全匹配层(CFS-PML)。本文的主要内容是研究各种PML公式的非分裂场形式实现方法。针对已有的PML算法中存在的缺点,提出了一些新颖并且有效的PML算法,并且对所提出的PML新算法进行了数值算例验证。
本文的主要研究内容及创新点如下:
1.基于SC-PML公式,利用在时域离散微分方程的方法和Z变换方法(包括双线性变换(Bilinear Transform)和零极点匹配Z变换(Matched Z-Transform)),提出了三种以非分裂场方式实现PML的新算法。它们的创新点是在三维PML的角和一些棱上每个元胞每个场分量的更新仅需要一个辅助变量。然而,在此之前所有的SC-PML实现方法在这些PML区域上都需要两个辅助变量。所以,三种SC-PML新算法具有节约内存的优点。特别地,三种SC-PML新算法在二维和一维情况下更简单和有效。这是因为在二维情况下两个横向场分量(例如,在TMz模式下的H_x和H_y两个场分量)的更新方程不需要辅助变量,而在一维情况下两个场分量的更新方程都不需要辅助变量,从而既节约内存又节约计算时间。
2.基于APML公式,利用Z变换方法(包括双线性变换和零极点匹配Z变换)提出了两种实现PML的新算法。与Ramadan利用Z变换方法实现APML的算法相比,本文提出的APML新算法需要的辅助变量和计算步骤少,从而节约了内存和计算时间。
3.基于带有CFS因子的SC-PML公式,分别利用辅助微分方程(ADE)方法和零极点匹配Z变换方法提出了两种实现CFS-PML的新算法。由于避免了卷积完全匹配层(CPML)所使用的卷积计算,所以两种新算法推导过程更简单,更容易理解。
4.基于带有CFS因子的APML公式,分别利用零极点匹配Z变换和双线性变换方法提出了4种实现CFS-PML的新算法。
5.利用由Sullivan所引入的Z变换方法提出了一种新的非线性FDTD公式,并结合SC-PML新算法给出了完整的更新方程组。
6.提出了一种针对高阶差分方程的最小化内存新算法。
Since K. S. Yee developed the finite difference time domain (FDTD) method in 1966, absorbing boundary conditions (ABCs), as an important component in the FDTD, has always been the focus of extensive and deep research. Nowadays, the perfectly matched layer (PML) ABC has proven to be the most efficient technique. The PML is an artificial lossy material in mathematics as an absorber for truncating numerical solution domains in the FDTD computations. By properly setting the constitutive parameters of the PML, outgoing waves in the FDTD computational domains, regardless of arbitrary incidence, polarization, and frequency, propagate without reflection across the interface between the FDTD computational domains and the PML and then are effectively absorbed due to the fact that the PML is lossy. So far, there are three different formulations that have been used for the PML ABC: i) Berenger’s split-field PML, ii) the stretched-coordinate PML (SC-PML), and iii) the anisotropic PML (APML, or uniaxial PML, UPML). In order to efficiently attenuate low frequency and evanescent waves and reduce late-time reflections, the complex frequency shifted PML (CFS-PML) was introduced. The emphasis of this thesis is to research algorithms for implementing various PMLs in unsplit-field formulations. As there are a few drawbacks in the published PMLs’implementations, some novel and efficient algorithms for implementing the PMLs are proposed and validated by numerical tests in this dissertation. The main achievements and originality are listed as follows:
1. Based on the SC-PML formulations, three efficient and unsplit-field algorithms for implementing the PML are proposed by using the method of discretizing differential equations in time-domain and the Z-transform methods (including the bilinear transform and the matched Z-transform). As compared with the published SC-PML-based algorithms which require two auxiliary variables per field component per cell in all corners and some edges of the three-dimensional PML regions, the novelty of the proposed algorithms is that only one auxiliary variable is required in these PML regions and then memory is saved. Especially in two-dimensional (2-D) and one-dimensional (1-D) cases, three new algorithms are much simpler so that they save both memory and computational time due to the fact that they require no auxiliary variable to update two transverse field components (e.g. H_x and H_y in the TM z mode) in all 2-D PML regions and to update two field components in 1-D PML regions.
2. Based on the APML formulations, two novel algorithms are proposed by using the Z-transform methods (including the bilinear transform and the matched Z-transform). As compared with the Z-transform-based APML algorithms by Ramadan, the main advantage is that the proposed algorithms require less auxiliary variables and computational steps so that their implementations save both memory and computational time.
3. Based on the SC-PML formulations with the CFS factor, two new algorithms for implementing the CFS-PML are proposed by using the auxiliary differential equation (ADE) method and the matched Z-transform method. The proposed algorithms are simpler and more suitable for understanding than the convolutional PML (CPML) as they need no computations of the convolutional terms.
4. Based on the APML formulations with the CFS factor, four new algorithms for implementing the CFS-PML are proposed by using the bilinear transform and the matched Z-transform methods.
5. A new nonlinear FDTD algorithm is proposed by using the Z-transform method introduced by Sullivan. A set of field-updated equations for truncating 1-D nonlinear FDTD lattices are given by combining the proposed nonlinear FDTD formulations and new SC-PML formulations.
6. A new memory-minimized algorithm for high-order difference equations is proposed.
引文
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