4.3     Automatic Meshing

QuickWave enables the mechanism for automatic mesh generation, which is called Automatic Meshing Intelligent Generation Option (AMIGO). AMIGO serves one main purpose, which is optimising the meshing so as to provide requested wavelength resolution in all media, while avoiding unnecessarily small cells. The three main parameters set for AMIGO are:

• Minimum number of cells per wavelength.  AMIGO will prepare the meshing so that the wavelength-to-cell size ratio does not drop below this value in any material within the project. In typical applications, this number should be set to 10-15 but precise calculations may require refined meshing with 20 cells per wavelength or more.

• Frequency range of analysis (together with frequency resolution df).

When calculating the required cell size (to assure given cells per wavelength ratio) in the particular scenario area, AMIGO performs the calculations for the upper frequency of the given range and takes into account the parameters of the medium filling this area. The parameters that are taken for the calculations are: relative permittivity (e_r), relative permeability (μ_r), conductivity (s), and magnetic loss (s_M).

• Avoid cells below lmin. It restricts automatic mesh refinements by AMIGO, guiding it to avoid generating too small FDTD cells. Very small cells may have a strong effect on computing time. Not only the total number of cells increases but also the FDTD step in time is automatically adjusted. This value can be set to a programme default value, which is equal to half of the maximum cell size allowed in the medium of smallest wavelength (highest medium parameters values) within the project. The user may set a different value of lmin: lower to allow better mesh refinements or higher to further economise on the mesh. Attention: setting lmin higher than amax (the cell size in air, corresponding to the chosen minimum number of cells per wavelength, at the highest frequency in the chosen range) is inconsistent and may produce confused reaction of AMIGO.

What should be mentioned at this point is that the software displays information about three other parameters, which depend on the above user's choices. They are:

• amax - the cell size in air, corresponding to the chosen minimum number of cells per wavelength, at the highest frequency in the chosen range (f2),

• the current number of FDTD cells in the project,

• amin - the smallest cell size in the currently existing FDTD mesh.

There are subtle relationships between amax, amin, and lmin that when obeyed assure correct AMIGO functioning. In general, the smallest cell size should not be smaller than half of the maximum cell allowed for the upper frequency in the medium of the shortest wavelength, and not smaller than lmin set by the user (this explains the default value of lmin described above). If amin obeys these conditions, it is displayed in green. Otherwise it is displayed in red suggesting that some details of the geometry or mesh snapping planes explicitly enforced by the user make the cell size smaller. The user is encouraged to press the button Inspect to investigate the cause of generating such a small cell. The obtained window shows the mesh snapping planes generated by the software. They are classified into four groups of priorities (Master, Hard, Soft and Weak). The meaning of those priorities is described later in this Section.

Additionally, AMIGO automatically enforces mesh snapping planes of electric type. In most applications, the electric type of mesh snapping planes are most preferable because they provide the best conditions for accurate analysis of the most critical parts - edges of metal elements in the structure -  since snapping the metal edges to the mesh enables the singularity corrections algorithms. Thus AMIGO tries to generate mesh snapping planes of the electric type along edges of geometrical elements and passing through their vertices. However, in many practical projects such a process (if unrestrained) would result in generation of many small FDTD cells. This would cause a radical drop in the smallest cell size amin, and as a consequence - in the basic time step dt. This in turn would cause a drastic increase of computing time. To avoid such an effect, AMIGO sets priorities to the mesh snapping planes and, in case of conflicts, eliminates those less important in order to try and keep the cell size above the declared value of lmin. There are four priorities of mesh snapping planes.

• weak - generated by AMIGO and eliminated in the first place when the condition for lmin cannot be met,

• soft - generated by AMIGO and eliminated in the second place when the conditions for lmin cannot be met,

• hard - generated by AMIGO upon the user's declaration of a particular element as hard or of hard edges. They are not automatically eliminated by AMIGO when the condition for lmin cannot be met. Instead, they can be reviewed by the user using Inspect command in the Mesh Control group of AMIGO dialogue and possibly deleted upon the user's specific request. Hard planes are mostly used for metal edges (like for example edges of a strip of a microstrip line). At metal edges, the fields are highly singular. When a metal edge is aligned with edges of FDTD cells, QW-3D automatically introduces special mathematical models of field singularities to enhance the modelling accuracy. Because of fast changing fields near such edges, it is also advisable to refine the FDTD cells near the edge. That is why the hard mesh snapping planes generated by AMIGO are accompanied by weak planes separated from the hard planes by the distance lmin.

• master - are mesh snapping planes explicitly declared by the user outside the AMIGO system. They can also be viewed using Inspect command in the Mesh Control group of Amigo dialogue.

From the AMIGO viewpoint, all the geometrical elements of the considered structure can be classified into four groups:

• disabled - no action by AMIGO,

• soft (default) - AMIGO generates soft planes along edges of these elements and at their limits, as well as weak planes passing through their vertices,

• hard edges - AMIGO generates hard planes along edges, soft planes along the limits and weak planes passing through the vertices,

• hard - AMIGO generates hard planes along edges and hard planes passing through the vertices.

The properties of each element can be changed manually in the Element Change dialogue.

AMIGO shows also useful information about: details of structure definition that cannot be modelled within requested mesh constraints, time step forced by the current mesh (indirectly), the minimum number of FDTD iterations needed to obtain well converged results, and expected duration of the analysis. The minimum number of FDTD iterations is determined by the frequency resolution df and the smallest cell size amin. From the Fourier transform properties, resolution df requires simulation time T=1 /(2 df). On the other hand, T=N*dt where N is the number of FDTD iterations and dt is the time step of the FDTD leap frog algorithm. The software automatically chooses dt so that it is proportional to the smallest cell size (amin) to obey the Courant stability criterion. AMIGO shows also the corresponding requested physical time T expressed in [ns] and as the number of wave travels across the structure (assuming that the structure is filled with air).

It is worth noting at this point that the value of df /5 will be used as a default frequency step for all active post-processings. AMIGO also allows setting automatic stop criteria, which suspends the simulation after calculated minimum number of FDTD iterations needed to obtain the required frequency resolution.

The AMIGO algorithm can be enabled for X and/or Y and/or Z axes. If any of the axes is excluded from the AMIGO option, the manual mesh settings apply to meshing along that axis.

AMIGO can be used in all projects but it is especially highly recommended for optimisation tasks because it automatically generates meshing close to optimal one and avoids creating small FDTD cells.