This section compares how much the mesh element count increases with rectangular PML and spherical PML. The results shown here are not fully generalizable, largely because the proportion of the mesh that's extruded changes dramatically with mesh density, with coarser meshes showing a more dramatic change than finer meshes. Nevertheless, this is a reasonable representative example. The following figure shows the initial mesh for a geometry with a rectangular aperture antenna in a PEC ground plane on the bottom boundary of the domain, with PML applied to all other domain boundaries. In the PML material properties grid, the Level is set to Low. The default mesh settings produce an initial mesh with 1688 elements.
During the PML extrusion process, each element face on the PML boundary is extruded to create new PML elements. Next, the sides of the new PML blocks are extruded to fill in the PML at the corners and edges of the domain, to create a solid block of PML surrounding the domain.
The resulting mesh has 16628 elements, nearly 10 times higher than in the initial mesh. Roughly a third of the new elements are at the corners and edges of the domain.
The following figure shows the same system, where a spherical PML boundary is being used in place of the rectangular boundary. For the purpose of the comparison the sphere radius has been chosen to give the same system volume as in the rectangular case. In addition, the initial mesh parameters have been modified to produce a similar element count. In this case, the initial mesh has 1685 elements.
As before, each face on the PML boundary is extruded to create new PML elements. In this case, however, the domain has no corners or edges, so those do not need to be extruded. The following image shows the extruded mesh.
The resulting mesh has 6275 elements, which is approximately 3.7 times larger than the initial mesh. While this increase is substantial, it is significantly less than the 10x increase we saw with the rectangular PML. In addition, this PML will perform at least as well as the rectangular PML, since the propagating fields approach the spherical PML with angles that are closer to normal incidence over the entire PML surface.
While this example is quite dramatic, it cannot be representative of every mesh. Finer meshes show a much smaller relative element count increase with PML extrusion, for both spherical and rectangular domains. In general, however, you will get a smaller mesh if you use spherical PML instead of using planar PML on multiple adjoining domain boundaries. Spheres have less surface area than boxes of the same volume, resulting in fewer element faces on the PML boundary in general. Further, the spherical domain does not have edges and corners that must be filled with mesh elements, as planar PML does.