The objective of all numerical methods in Electromagnetics is to obtain approximate solutions to Maxwell’s equations (or of equations derived from them) which satisfy specific boundary and initial conditions. A common approach taken by most methods involves a discretization over the function domain, which creates both space and time intervals upon which the field values can be computed. Unlike numerical methods which directly approximate the mathematics of Maxwell’s equations, the TLM method models the physical propagation of fields using a wave scattering approach derived from Huygens principle which is implemented by discretizing the spatial domain using a suitably selected network of interconnecting transmission lines. The basis for such a physical discretization is a derived equivalence between the field variables in space and voltage impulses propagating on transmission lines. The field solutions are therefore obtained by mapping Maxwell’s equations to a network of equivalent transmission line equations for which exact solutions can be obtained.

As with all differential equation based numerical methods, open boundary problems 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. The computational domain must therefore be terminated with either the simple matched boundary, an analytical absorbing boundary condition (ABC) or the Perfectly Matched Layer (PML). However, due to the low absorption capability of the matched boundary and analytical ABC a significant distance must exist between the boundary and the features of the problem in order to ensure an accurate solution is obtained. Therefore, substantially increasing the overall computational burden. On the other hand, the PML demonstrates a superior absorption performance over a wider frequency range and for wider angles of incidence. Owing to the high accuracy and computational efficiency attainable the PML is generally considered as the preferred absorbing boundary technique.

Developing a stable and efficient PML formulation presents a unique challenge for TLM developers. Unlike the finite difference method where differential operators are straightforwardly discretized by central-differencing approximations, the TLM method relies on the derivation of a suitable circuit to field equivalence in the PML medium which, in a similar fashion to the classical TLM development, ensures an unconditionally stable algorithm.

In light of the huge benefits attainable (i.e. in terms of the higher accuracy and lower computational demand gained) my PhD research is focused on developing a framework for implementing an efficient stretched coordinate PML suitable for truncating TLM grids. A vital aspect of the development presented is the implementation approach taken to ensure the unconditional stability of TLM is preserved.

Many mathematical equations reveal new interesting insights and capabilities on the underlying physical phenomena upon a transformation of coordinate systems. For this reason coordinate transformation techniques have long been explored in the mathematical analysis of a variety of engineering problems. (1), (2), (3). In computational electromagnetics, complex domain displacements are widely utilized as a method to manipulate electromagnetic fields. An example application of this methodology, one which is of interest in achieving my research aims, is the stretched coordinate PML formulation which achieves the reflectionless attenuation of propagating waves through an analytical continuation of Maxwell’s equations solutions from real to complex space. Compared to the split field and anisotropic PML formulations, the stretched coordinate PML theory offers a more mathematically rigorous definition of the PML theory which has motivated its easy extension to curvilinear coordinates, conformal mesh terminations. Further motivated by its suitability to general media the stretched coordinate PML theory forms the basis for the TLM-PML formulation developed in this thesis.

The choice of the approach I have taken in implementing the PML is strongly influenced by the particular nature of the numerical method in which it is being employed. Therefore, in order to maintain the efficacy of the TLM algorithm, the PML equation to be discretized must lend itself well to being physically realized, i.e. field solutions must satisfy Maxwell’s equations (or the PML equation must be Maxwellian in its form). Ensuring this enables the required PML constitutive parameters to be obtained which result in the derivations of suitable equivalent field to circuit representations for the PML medium. This approach to a TLM-PML implementation has been demonstrated to avoid a direct discretization of the PML equations and in a similar fashion to the classical TLM development, ensures an unconditionally stable algorithm.

1. Designed / Developed a scalable high performance C++ TLM numerical modelling research software. https://github.com/jojusimz/GEMINI

2. Various scientific papers have been submitted and are under review and others have been published.

3. Collaborations with industry scientists.

4. PhD thesis documenting the research carried out and reviewing the state of the art computational electromagnetic techniques.