The above discussion leads towards an intermediate discretisation method in order to accurately model non-Cartesian geometries using structured TLM meshes. The most obvious solutions that spring to mind would be graded meshing [8] and multi meshing [9] in order to use finer meshes to describe boundary regions without compromising on computational demand.
The graded meshing technique enables higher resolution and rectangular cell sizes to represent finer details of the geometry as illustrated in figure 4.8a. The graded meshing maintains one-to-one links to neighbouring cells, by applying the grading along vertical and/or horizontal lines. In order to match
0.01 0.1 1 160 180 200 220 240 Magnitude (normalised) Frequency(THz) inside outside data
Figure 4.6: The energy spectrum of an infinite cylinder excited by a line source inside the resonator of E type waves for inside and outside discretisation along with the analytical resonant frequencies for step size of 0.025µm.
the Courant criteria, the time step is reduced to match the smallest space step in the mesh, demanding more computational effort. Hence, for a given total simulation time the number of time steps will be doubled when the smallest step size is halved, but this approach does not result in a significant increase in the total number of cells. Therefore, the grading operation results in a linear increase in computational complexity, whereas a usual mesh refinement on the entire spatial domain results in a cubic increase in complexity of a 2D simulation.
A similar but a computationally less intensive method is to incorporate multi-grid technique where mesh refinement is applied to areas with fine de-
0.01 0.1 1 160 180 200 220 240 Magnitude (normalised) Frequency(THz) inside outside data
Figure 4.7: The energy spectrum of an infinite cylinder excited by a line source inside the resonator of E type waves for inside and outside discretisation along with the analytical resonant frequencies for step size of 0.0125µm.
to be refined. The compromise of this method is that an approximation has to be made in connecting multiple links to multiple links of neighbouring cells. This disconnection applies to the time-step as well. The time-steps of smaller cells are smaller and that of the larger cells are larger and are multiples of powers of 2. These two approximations if handled properly lead to an efficient mesh refinement method.
CST Microwave Studio [10], a commercial implementation of TLM with a further optimised multi mesh (as of 2012), was use to observe the accuracy improvements obtainable by partially refining the mesh as mentioned above. CST simulation parameters were the same as the parameters given above except the space step and time step. The space step was allowed to vary
(a) Graded Mesh (b) Multi Mesh (c) Same Area Figure 4.8: Alternative Meshing Techniques
between λmin/10 and λmin/100, where λmin = 1.2µm. Since the CST is a
3D TLM implementation, the time step is at least √2 times smaller than that used in the 2D simulation throughout this study. However, the time complexity of the this CST simulation was similar to that of the 2D simulation for the step size of 0.025µm (≈ λmin/20). The energy spectrum of the infinite
cylinder obtained from the time domain CST simulation is given in figure4.9. The resonances were extracted from the time signals with a duration of 18000f s using difference Prony method. The medians of the resonant fre- quencies and Q factors obtained for each resonance are given in table 4.4. It should be noted that, when the resonances are extracted using matrix pen- cil and harmonic inversion, the resonant Q factors varied from that given in table 4.4. No two methods agreed with each other indicating that the ap- proximations used in CST simulation imposes irregular noise affecting each resonant extraction method differently. Since the results from the difference Prony method were similar for time duration of 4000fs, 8000fs and 18000fs its results were chosen to be presented here.
The error in the obtained resonant frequencies and Q factors compared to the analytical results presented in table 4.1 are plotted against theoretical values in figures 4.10 and 4.11 respectively. In table 4.1, WGM stands for whispering gallery modes and the mode number pair represent the azimuthal order and the radial order respectively.
0.01 0.1 1 10 150 160 170 180 190 200 210 220 230 240 250 Magnitude (normalised) Frequency(THz) CST multi/graded mesh Analytical
Figure 4.9: The spectrum of an infinite cylinder (n = 2.82, r = 1.35µm) ex- cited by a line source inside the resonantor of E type waves as sim- ulated by CST Microwave Studio with an optimised multi mesh with varying stepsize between 0.04µm and 0.004µm.
ror less than 0.2% for all resonances of concern. However, a pattern for the resonance shifts cannot readily be observed. Hence, it is difficult to charac- terise the observed resonance shift and provide an explanation for cause of this error. This non uniform behaviour is mainly due to the non uniformity of the mesh. Therefore, it is difficult to predict the accuracy improvement obtainable by refining the mesh further.
On the other hand, the Q factor has a general trend of increasing error with the increasing theoretical Q factor. As observed, Q factors up to 105 could be
obtained to within 10% error. In figure 4.11, the general trend of the error in Q factor of WGM(x,1) modes (i.e. 1st radial order modes) is lower than the general error trend in Q factor of WGM(x,2) modes (2nd radial order modes). The error trend of WGM(x,2) is lower than that of WGM(x,3), though this is not that clearly visible. This is opposite to what one would expect from
Table 4.4: The resonances obtained (using difference Prony method) from time responses of an infinite cylinder of n = 2.82 and r = 1.35µm of a length of 18000fs simulated with CST Microwave Studio.
Resonance Frequency Q factor
WGM(9,1) 151.72 38496 WGM(6,2) 151.90 137 WGM(10,1) 165.938 217218 WGM(7,2) 167.871 305 WGM(11,1) 180.054 66684 WGM(8,2) 183.529 757 WGM(12,1) 194.551 25606 WGM(9,2) 199.09 1994 WGM(13,1) 208.047 4640860 WGM(10,2) 214.361 5681 WGM(14,1) 221.97 2713 WGM(8,3) 226.313 169 WGM(11,2) 229.489 15163 WGM(15,1) 235.797 443718 WGM(9,3) 242.666 324 WGM(12,2) 244.393 13022 WGM(16,1) 249.549 180252
a uniform mesh where WGM(x,1) Q factors are underestimated more than WGM(x,2) Q factors, etc since the stair-step approximation affects the modal volume of WGM(x,1) more. In multi mesh techniques, the opposite is true. The meshing becomes finer away from the centre towards the boundary. The TLM errors are then higher for coarse meshes further inside but lower for finer meshes closer to the boundary. This is a possible explaination of the results obtained using CST MW studio.
In conclusion, these mesh refinement techniques can be thought of as ef- ficient alternatives to complete mesh refinement. Despite being able to ob- tain a resonably accurate result with comparatively small computational de-
0.001 0.01 0.1 1 150 160 170 180 190 200 210 220 230 240 250 Frequency Error % Frequency(THz) WGM (x,1) WGM (x,2) WGM (x,3)
Figure 4.10: The percentage error of the resonant frequencies of the E wave modes supported by an infinite cylinder (n = 2.82, r = 1.35µm) modelled by CST Microwave Studio with a multi mesh with space step size varying between 0.04µm and 0.004µm. 1st ra- dial order modes are deonted by WGM(x,1), 2nd order modes by WGM(x,2) and 3rd order modes by WGM(x,3).
Hence, one cannot deduce the amount of refinement required to obtain a given accuracy level.
Therefore, in the next section some alternative discretisation techniques are explored that can be used to model non-Cartesian geometries using a Cartesian grid. It is shown that they behave slightly more predictively for the illustrative example of the infinite dielectric cylinder.