Research On Algorithms For Implementing Perfectly Matched Layers In The Finite Difference Time Domain Method

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.