The details below contain some information regarding coaxial lines and micrstrip lines, but also contains other meshing suggestions.
A solved model is only as good as its mesh. Modelling and meshing guidelines are given in the FEKO User Manual. This how-to gives detailed (but not exhaustive) examples of how to create a good mesh, and also shows comparative results for different meshes.
CEM techniques discretise either the fields (such as FDTD codes) or the currents (such as MoM codes). Discretisation introduces small but controllable errors in the results. To reduce the error, the mesh size can be reduced, however, this increases computational cost. This how-to will show examples of how to create an appropriate mesh given the accuracy vs computational cost consideration. In addition, comparisons are shown between "good" and "bad" meshes.
Example 1: A simple strip dipole in free space
The first example is an ordinary strip dipole. To show the mesh differences more clearly, the dipole length was made 1.5 free space wavelengths.
Figure 1a: Geometry of a strip dipole - dimensions 1.5 x 1/30 wavelegnths
We now consider three different meshes and their results.
In the first case we set the triangle mesh size under the Create Mesh dialog to a Custom size of 1/10 of the free space wavelength. This results in an average size for a mesh triangle on the dipole of around 1/20 of the free space wavelength. The smaller than requested elements are due to the narrow width of the dipole - the mesher tries to create each triangle so that its edges have similar lengths. The total number of triangles are 57 triangles.
Figure 1b: The strip dipole mesh using a triangle edge length of lam0/10
In the second case we use a finer mesh size on the strip dipole. Here we set the triangle edge length to 1/60 of the free space wavelength. The total number of elements are 414 triangles.
Figure 1c: The strip dipole mesh with edge length = lam0/60, zoomed view
In the third case the edges of the dipole are protruded perpendicular to the dipole surface with a distance of 1.25 mm or lam0/800. The total number of elements were 140.
Figure 1d: Strip dipole with edge triangles
Zoomed in view
Zoomed out view
In the latter case the perpendicular protrusion of the edges causes basis functions to be placed all along the edges of the dipole. This provides increased accuracy for modelling the width of the dipole.
Fig. 1e compares the input the resistance and input reactance for the three meshes. It can be seen that the mesh with edge triangles and the lam0/60 mesh provide very similar results, but the "standard" lam0/10 mesh shows a deviation. It is clear that a very accurate answer can still be obtained for a small increase in unknowns with the edge triangles method compared to the very fine lam0/60 mesh.
Figure 1e: Comparing the input resistance (left) and input reactance (right) for the different meshes
Note that if we were mainly interested in the far fields of the dipole and not interested in modelling the impedance of the dipole accurately, then the mesh size of 1/10 of the free space wavelength would have been sufficient.
Example 2: A section of coax
A section of coax can be considered to be geometrically complex - the electrically small radius of curvature requires a fine mesh to accurately represent the curvature. If the mesh is inadequate, the TEM wave that is intended to be launched will not be purely TEM and the impedance of the coax could be inaccurate.
Consider the section of air-filled coax in Fig. 2a. The inner radius is 3 mm and outer radius is 2.31 times larger, resulting in a 50 Ohm line.
Figure 2a: Geometry of a section of air filled 50 Ohm coax
We again consider three different meshes, their unknowns and results.
The first mesh uses a mesh size of lam0/10 in the Create Mesh dialog and results in 1292 triangles. This is depicted in Fig. 2b.
Figure 2b: The coax meshed with a triangle edge length of lam0/10
The second mesh shown in Fig. 2c creates a so-called ruled mesh by imprinting axial lines onto the inner and outer surfaces of the coax every 30 degrees around the surface. These lines force the mesh triangles to be longitudinally shaped. These triangles are more appropriate for the coax as the current flow is mainly along the axis of the coax and not radial. The mesh contains 1710 triangles.
Figure 2c: Ruled mesh of the coax
Outer conductor view
Inner conductor view
The third mesh is also a ruled mesh but here the lines were imprinted every 45 degrees instead of 30 degrees. This mesh contains 1854 triangles.
The results are shown in Fig. 2d. It can be seen that the ratio between the reflected power vs incident power is smallest for the "30 degree" ruled mesh, despite this mesh having fewer triangles than the "45 degree" mesh.
Figure 2d: Input power vs reflected power for different meshes of the coax
Note that ruled meshes are automatically created for curved geometry - it is not necessary to imprint lines on the curvature - the imprinting here was just for demonstration purposes. The ruled mesh can be controlled on the Advanced tab of the Create mesh dialog. It can be enabled/disabled by checking/unchecking the box, "Allow elongated triangles".
Example 3: A box with a narrow aperture
The geometry consists of a box with a wire inside. The wire is fed with a voltage source and the near fields that radiate through the slot are calculated on a sphere about 3m away on the outside of the box. The model is from the article, "EMI from cavity enclosures," IEEE Trans. EMC. Feb. 2000.
The geometry of the box is depicted in Fig. 3a.
Figure 3a: Geometry of the box with aperture
Inside view (slightly zoomed)
We consider two meshes. The first mesh is where we set the triangle edge length to the usual lam0/10 resulting in 1948 triangles. In the second mesh, we set a local mesh size on the edges of the slot of 3x the width of the slot resulting in 2432 triangles.
Specifically, we use the following expression for the local mesh size: min(lam0/10, 3*slot_width). This expression ensures that the slot mesh is parametric - the minimum is taken of lam0/10 and 3*slot_width. Should the frequency increase substantially, the first term in the expression will dominate. Should the slot be made narrower, the second term of the expression will dominate. Fig. 3b shows the two meshes.
Figure 3b: Mesh of the box with aperture
No local meshing applied
With local meshing applied on the slot edges (zoomed view)
Fig. 3c shows the result comparison. We see a significant shift in some of the peaks in the curves of the near fields for the two meshes. Clearly, the local mesh size is required.
Figure 3c: Emitted nearfield from the box for different meshes
Example 4: A section of microstrip line
When designing feed networks usually the impedance of the microstrip lines need to be modelled accurately. Similar to Example 1, we will see that using edge triangles provide high accuracy without adding to the computational resources. A section of 50 Ohm microstrip line is modelled. We consider a few meshes and their performance.
The first mesh uses a mesh size of min(3*h, lambda_d/10) where h is the substrate height and lambda_d is the wavelength in the substrate. The 1st expression is required to ensure that geometry is meshed such that the triangle size is of the order of the spacing between the geometry (here the line itself and the ground plane beneath). The second expression specifies a mesh size of 1/10th of the wavelength in the dielectric. This mesh results in 2 triangles across the width of the line.
The second mesh uses a mesh size of min(h, lambda_d/10). This results in about 4 triangles across the width of the line.
Figure 4a: Meshed microstrip line
Using a mesh size of min(3*0.5, lambda_d/10)
Using a mesh size of min(0.5, lambda_d/10)
The third mesh uses edge triangles by means of a vertical protrusion of the microstrip edges. The fourth mesh is a slight variation of the third mesh in that the edge triangles are still used but they are in the same plane as all other triangles.
Figure 4b: Meshed microstrip line continued
Using edge triangles vertically protruded from the line edge
Using edge triangles in the same plane
Comparing the input resistance and reflection coefficient of the line it is seen that the line with 2 triangles across the width of the line gives the least accurate answer. Using a finer mesh to obtain about 4 triangles across the line provides better accuracy. However, the meshes with edge triangles are the most accurate.
Figure 4c: Input resistance and reflection coefficient for the 50 Ohm microstrip line
Input reflection coefficient
The reason that the input resistance is not exactly 50 Ohm is due to the feed. It represents a discontinuity in the microstrip mode. De-embedding the feed should provide more accuracy. (The same feed was used in all cases).
Example 5: An F5 generic aircraft model - RCS
The monostatic RCS of a generic F5 aircraft model is calculated. We compare the results for different mesh densities. The F5 model is shown in Fig. 5.1
Figure 5.1: Geometry the F5 aircraft
The mesh sizes and number of elements evaluated are as follows:
lam0/3.5 = 44712 triangles
lam0/4.5 = 76111 triangles
lam0/6.8 = 170246 triangles
The RCS comparison is shown in Fig. 5.2.
Figure 5.2: Monostatic RCS from the side and front of the F5, swept from top to bottom
RCS from the front
RCS from the side
It is seen that even for a very coarse mesh, the RCS result is nearly converged. It must be stressed that the above results are very dependent on the geometry. Nearfields computed close to some areas on the surfaces could be inaccurate, or the received power in an antenna attached to the aircraft could also be inaccurate.
The model was solved with the MLFMM. It must be noted that in all cases the residuum for the iterative solution for the MLFMM was set to 1e-5 (the default is 3e-3). This is sometimes required when very small values are expected in the results. Here, for example, the RCS goes down to 40 dB below the maximum.
Example 6: An F5 generic aircraft model - antenna coupling
We use the F5 model again but this time compute the coupling between two monopole antennas. This is to further demonstrate the mesh size is dictated also by the type of problem being solved.
One monopole is located on the top near the nose of the aircraft. The other monopole is located on the bottom near the tail of the aircraft. Fig. 6.1 depicts the F5 with the monopole antennas (only their ports are visible).
Figure 6.1: Geometry of the F5 aircraft showing the ports of the two antennas
Again we expect very low values, so we set the residuum for the MLFMM to 1e-5. To save on computation time, the maximum number of samples for the adaptive frequency sampling was set to 31. This causes some discontinuities in the interpolated results displayed. The coupling for different mesh sizes are shown in Fig. 6.2.
Figure 6.2: Antenna coupling for different mesh sizes
It is seen that there is a 10 to 15 dB difference between the coarsest mesh and the finest mesh. It may even be necessary to use an even smaller than lam0/8 mesh size.
Notes on FDTD meshing
Surface current techniques in general require that the mesh size is reduced in proportion to the frequency. Doubling the frequency requires the mesh size to be halved.
Due to numerical dispersion in the FDTD, this general rule of thumb is not conservative enough. According to Davidson , "it is important to appreciate that phase error accumulates across a domain. The absolute phase error over a fixed length L is approximately (k^3)(h^2)L/36" where k is the phase constant and h is the voxel size.
This "implies that cell size must scale with frequency as (2*pi*f)^-1.5 to keep the error constant and hence the number of cells in each dimension scales with (2*pi*f)^1.5".
For example, if the frequency is doubled, the cell size must be reduced to approximately 0.35 of the cell size at the initial frequency.
This how-to didn't cover all the techniques in FEKO. For example, the ray-launching solver requires that the mesh only represents the geometry accurately. Also the FEM-MoM solver uses higher order basis functions and thus requires mesh sizes of around 1/5 of the wavelength in the dielectric. There are also higher order basis functions for the MoM and physical optics (PO) with large triangles.
However, this how-to shows that coarse meshes are useful to obtain fast "ball-park" results, and in some cases very good results. The mesh sizes used in the examples in this how-to should not be taken as final for any model. It is given as a starting guideline. Mesh convergence tests* should always be done before taking results as truly final.
 David B. Davidson, "Computational Electromagnetics for RF and Microwave Engineering" Second Ed. Cambridge University Press, p 117
* Rerunning the model with 50% more mesh elements and comparing the results with that of the original mesh.