Stretched coordinate PML in TLM

As with all differential equation based numerical methods, open boundary problems in TLM require special boundary treatments to be applied at the edges of the computational domain in order to accurately simulate the conditions of an infinite propagating medium. Particular consideration must be given to the choice of the domain truncation technique employed since this can result in the computation of inaccurate field solutions. Various techniques have been employed over the years to address this problem, where each method has shown varying degree of success depending on the nature of the problem under study. To date, the most popular methods employed are the matched boundary, analytical absorbing boundary conditions (ABCs) and the Perfectly Matched Layer (PML). Due to the low absorption capability of the matched boundary and analytical ABCs a significant distance must exist between the boundary and the features of the problem in order to ensure that an accurate solution is obtained. This substantially increases the overall computational burden. On the other hand, as extensively demonstrated in the Finite Difference Time Domain (FDTD) method, minimal reflections can be achieved with the PML over a wider frequency range and for wider angles of incidence. However, to date, only a handful of PML formulations have been demonstrated within the framework of the TLM method and, due to the instabilities observed, their application is not widely reported. The advancement of the PML theory has enabled the study of more complex geometries and media, especially within the FDTD and Finite Element (FE) methods. It can be argued that the advent of the PML within these numerical methods has contributed significantly to their overall usability since a higher accuracy can be achieved without compromising on the computational costs. It is imperative that such benefits are also realized in the TLM method. This thesis therefore aims to develop a PML formulation in TLM which demonstrates high effectiveness in a broad class of electromagnetic applications. Motivated by its suitability to general media the stretched coordinate PML theory will be basis of the PML formulation developed. The PML method developed in this thesis is referred to as the mapped TLM-PML due to the implementation approach taken which avoids the direct discretization of the PML equations but follows more closely to the classical TLM mapping of wave equations to equivalent transmission line quantities. In this manner the highly desired unconditionally stability of the TLM algorithm is maintained. Based on the mapping approach a direct stretching from real to complex space is thus applied to the transmission line parameters. This is shown to result in a complex propagation delay and complex frequency dependent line admittances/impedances. Consequently, this modifies the connect and scatter equations. A comprehensive derivation of the mapped TLM-PML theory is provided for the 2D and 3D TLM method. The 2D mapped TLM-PML formulation is demonstrated through a mapping of the shunt node. For the 3D case a process of mapping the Symmetrical Condensed Node (SCN) is formulated. The reflection performance of both the 2D and 3D formulations is characterised using the canonical rectangular waveguide application. Further investigation of the capability of the developed method in 3D TLM simulations is demonstrated by applying the mapped TLM-PML in: (i) the simulation of planar-periodic structures, (ii) radiation and scattering applications, and (iii) in terminating materially inhomogeneous domains. A performance comparison with previously proposed TLM-PML schemes demonstrates the superior temporal stability of the mapped TLM-PML

Efficient discrete modelling of axisymmetric radiating structures

This thesis describes research on Efficient Discrete Modelling of Axisymmetric Radiating Structures . Investigating the possibilities of surmounting the inherent limitation in the Cartesian rectangular Transmission Line Modelling (TLM) method due to staircase approximation by efficiently implementing the 3D cylindrical TLM mesh led to the development of a numerical model for simulating axisymmetric radiating structures such as cylindrical and conical monopole antennas. Following a brief introduction to the TLM method, potential applications of the method are presented. Cubic and cylindrical TLM models have been implemented in MATLAB and the code has been validated against microwave cavity benchmark problems. The results are compared to analytical results and the results obtained from the use of commercial cubic model (CST) in order to highlight the benefit of using a cylindrical model over its cubic counterpart. A cylindrical TLM mesh has not previously been used in the modelling of axisymmetric 3D radiating structures. In this thesis, it has been applied to the modelling of both cylindrical monopole and the conical monopole. The technique can also be applied to any radiating structure with axisymmetric cylindrical shape. The application of the method also led to the development of a novel conical antenna with periodic slot loading. Prototype antennas have been fabricated and measured to validate the simulated results for the antennas

Application of TLM for optical microresonators

Optical microresonators can form the basis of all-optical switching and control devices. The presented study is an exploration of the Transmission Line Modelling (TLM) method as a suitable candidate for designing optical microresonators. Chalcogenide glasses were identified as promising materials, with which to fabricate optical microresonators. The study presents the formulation of TLM in two dimensions to model nonmagnetic dielectric materials and a suitable computationally efficient yet flexible software design. Some methods for extracting spectral properties of resonators are compared and the modified difference Prony method was identified as a suitable tool to extract resonant frequencies and Q factors from a limited time signal. When applying TLM to microresonators of sub-wavelength dimensions it was understood that the method of discretisation plays an important role in accurately modelling microresonators. Two novel methods of discretisations -the same area method and the anti-aliasing method- were used to improve the accuracy significantly compared to existing mesh refinement techniques. Perfect matched layers (PMLs) were implemented to improve reflections from domain truncation using several methods. A Convolutional PML(CPML) was identified as the best, but it does not reach the efficiency of PMLs in the Finite Difference Time Domain (FDTD) method. Several frequency dependent refractive index models were proposed and implemented in TLM. A Tauc-Lorentz model was identified as the best fit to the experimental refractive index of three chalcogenide glasses, but a Sellmeier model with one term and a coefficient was efficient for TLM implementation. The main concern in the use of these models within TLM was shown to be the error arising due to mesh dispersion. Kerr nonlinear models were formulated and implemented in TLM and the models applied to the study of a waveguide junction. Compared to an equivalent implementation in a time domain beam propagation method, TLM models better represent the waveguide junction reflections

Modelling multi-scale problems in the transmission line modelling method

Modern electromagnetic problems are becoming increasingly complex and their simulation must take into account geometrical features that are both large and small compared to the wavelength of interest. These multi-scale problems lead to a heavy computational burden in a discretised computational simulation approach since the small features require fine mesh to be used in the simulation, resulting in large run time and memory storage. To overcome such problems, this thesis presents an efficient and versatile method for embedding small features into an otherwise coarse mesh. The embedded model eliminates the need for discretising the small features and allows for a relative large mesh size to be used, thus saving the computational costs. The subject of the thesis is embedding a thin film as a small feature into the numerical Transmission Line Modelling (TLM) method, although any small feature with known analytical response can be implemented in practice. In the embedded model, the thin film is treated as a section of transmission line, whose admittance matrix is used to describe the frequency response of the thin film. The admittance matrix is manipulated by expanding the constituent cotangent and cosecant functions analytically, and then transforming them from the frequency domain to the time domain using the inverse Z transform and general digital filter theory. In this way the frequency responses of the thin film are successfully embedded into the TLM algorithm. The embedded thin film model can be applied to both single and multiple thin film layers. The embedded thin film model has been implemented in the one-dimensional (1D) and two-dimensional (2D) TLM method in the thesis. In the 1D TLM method, the embedded thin film model is used to investigate the reflection and transmission properties of lossy,anisotropic and lossless thin films, e.g. carbon fibre composite (CFC) panels, titanium panels, antireflection (AR) coatings and fibre Bragg gratings (FBG). The shielding performance of CFC panels is also discussed. In the 2D TLM method, the embedded thin film model is extended to model arbitrary excitations and curved thin films. The electromagnetic behaviour of infinitely long CFC panels with oblique incidence and a CFC panel of finite length with a point source excitation are studied using the embedded thin film model. The resonant effects of CFC circular and elliptical resonators and the shielding performance of a CFC airfoil with the profile of NACA2415 are investigated using the embedded curved thin film model. In addition, the effects of small gaps in the airfoil structure on the shielding performance are also reported. All the examples discussed in the thesis have validated the accuracy, stability, convergence and efficiency of the embedded thin film model developed. At the same time, the embedded thin film model has been proven to have the advantage of significantly saving computational overheads

Application of TLM for optical microresonators

Optical microresonators can form the basis of all-optical switching and control devices. The presented study is an exploration of the Transmission Line Modelling (TLM) method as a suitable candidate for designing optical microresonators. Chalcogenide glasses were identified as promising materials, with which to fabricate optical microresonators. The study presents the formulation of TLM in two dimensions to model nonmagnetic dielectric materials and a suitable computationally efficient yet flexible software design. Some methods for extracting spectral properties of resonators are compared and the modified difference Prony method was identified as a suitable tool to extract resonant frequencies and Q factors from a limited time signal. When applying TLM to microresonators of sub-wavelength dimensions it was understood that the method of discretisation plays an important role in accurately modelling microresonators. Two novel methods of discretisations -the same area method and the anti-aliasing method- were used to improve the accuracy significantly compared to existing mesh refinement techniques. Perfect matched layers (PMLs) were implemented to improve reflections from domain truncation using several methods. A Convolutional PML(CPML) was identified as the best, but it does not reach the efficiency of PMLs in the Finite Difference Time Domain (FDTD) method. Several frequency dependent refractive index models were proposed and implemented in TLM. A Tauc-Lorentz model was identified as the best fit to the experimental refractive index of three chalcogenide glasses, but a Sellmeier model with one term and a coefficient was efficient for TLM implementation. The main concern in the use of these models within TLM was shown to be the error arising due to mesh dispersion. Kerr nonlinear models were formulated and implemented in TLM and the models applied to the study of a waveguide junction. Compared to an equivalent implementation in a time domain beam propagation method, TLM models better represent the waveguide junction reflections

Modelling multi-scale problems in the transmission line modelling method

Modern electromagnetic problems are becoming increasingly complex and their simulation must take into account geometrical features that are both large and small compared to the wavelength of interest. These multi-scale problems lead to a heavy computational burden in a discretised computational simulation approach since the small features require fine mesh to be used in the simulation, resulting in large run time and memory storage. To overcome such problems, this thesis presents an efficient and versatile method for embedding small features into an otherwise coarse mesh. The embedded model eliminates the need for discretising the small features and allows for a relative large mesh size to be used, thus saving the computational costs. The subject of the thesis is embedding a thin film as a small feature into the numerical Transmission Line Modelling (TLM) method, although any small feature with known analytical response can be implemented in practice. In the embedded model, the thin film is treated as a section of transmission line, whose admittance matrix is used to describe the frequency response of the thin film. The admittance matrix is manipulated by expanding the constituent cotangent and cosecant functions analytically, and then transforming them from the frequency domain to the time domain using the inverse Z transform and general digital filter theory. In this way the frequency responses of the thin film are successfully embedded into the TLM algorithm. The embedded thin film model can be applied to both single and multiple thin film layers. The embedded thin film model has been implemented in the one-dimensional (1D) and two-dimensional (2D) TLM method in the thesis. In the 1D TLM method, the embedded thin film model is used to investigate the reflection and transmission properties of lossy,anisotropic and lossless thin films, e.g. carbon fibre composite (CFC) panels, titanium panels, antireflection (AR) coatings and fibre Bragg gratings (FBG). The shielding performance of CFC panels is also discussed. In the 2D TLM method, the embedded thin film model is extended to model arbitrary excitations and curved thin films. The electromagnetic behaviour of infinitely long CFC panels with oblique incidence and a CFC panel of finite length with a point source excitation are studied using the embedded thin film model. The resonant effects of CFC circular and elliptical resonators and the shielding performance of a CFC airfoil with the profile of NACA2415 are investigated using the embedded curved thin film model. In addition, the effects of small gaps in the airfoil structure on the shielding performance are also reported. All the examples discussed in the thesis have validated the accuracy, stability, convergence and efficiency of the embedded thin film model developed. At the same time, the embedded thin film model has been proven to have the advantage of significantly saving computational overheads

13th Annual Review of Progress in Applied Computational Electromagnetics at the Naval Postgraduate School, Monterey, CA, March 17-21, 1997, Conference Proceedings Volumes I & II

Includes Volumes 1 &