3D to 2D surface mesh parameterisation for the purposes of unstructured transmission line modelling method simulations

Small scale fabrication processes have led to the advent of very thin flexible devices such as RFID tags, flexible PCBs and smart clothing. In a geometrical sense, these present themselves as curved two dimensional surfaces embedded in a three dimensional domain. When simulating electromagnetic behaviour on these surfaces at low frequencies, a full 3D field model is not always necessary. Using 3D algorithms to solve these problems can result in a large portion of the computer memory and runtime being used to mesh and simulate areas of the domain that present little electromagnetic activity. The theme of this thesis is concerned with the improvement of the runtime and memory consumption of electromagnetic simulations of these surfaces. The main contributions of this work are presented as an investigation into the feasibility of applying a 2D Unstructured Transmission Line Modelling method (UTLM) simulation to open, curved surfaces embedded in 3D space, by providing a one-to-one mapping of the geometry to a 2D flat plane. First, an investigation into the various methods of how a computer represents unstructured meshes in its memory is presented, and how this affects the runtime of the simulation. The underlying mesh data structures used to represent the geometrical problem space can have a huge impact on the efficiency and memory consumption of the simulation. This investigation served to demonstrate that it is not just simply the optimisation of the simulation algorithms that facilitate improvements to the runtime and memory consumption of a simulation. How a computer understands the connectivity of the mesh can have far greater impacts to the computational resources available. The concepts of surface parameterisation are then introduced; a process of mapping curved surfaces embedded in a three dimensional domain to a flat two dimensional plane. By providing a one-to-one mapping of the geometry from the 3D domain to the 2D flat plane, a low frequency 2D unstructured TLM simulation can be applied, negating the need for 3D algorithms. Because this mapping is one-to-one, the results of the simulation can then be mapped back to 3D space for visualisation. Parameterisations will almost always introduce distortion to angle and area, and minimising this distortion is paramount to maintaining an accurate simulation. Test cases were used to measure the extent of this distortion, and the investigation concluded that Angle Based Flattening (ABF) and Least Squares Conformal Mapping (LSCM) methods resulted in the best quality parameterisations. Simulations were then conducted on these test cases as a demonstration of how UTLM can be performed on 2D surfaces, embedded in a 3D domain.