Algorithm Outline
- The basic idea of the algorithm implemented in the SimplicialVIEW software package,
is to allow the user to select a predefined Nyquist map and to choose a
domain of definition and the mesh of the Q-triangulation that will be
generated for it
- Then, the Nyquist map is applied for every vertex of the
Q-triangulation, generating a set of image points
- The Voronoi Diagram and the Delaunay Triangulation are computed
for this set of image points
- We then check if the Nyquist map is simplicial, checking
whether all simplexes of the domain of definition are mapped to
simplexes of the image
- If the map is simplicial, then the inverse image of the origin
will give an approximation of the stability crossover
- If the map is not simplicial, we have to refine the initial mesh
of the Q-triangulation. However, we will keep the image freezed, that
is, we will not recompute the Voronoi Diagram and the Delaunay
Triangulation for the new set of domain points
- Instead, we will apply the simplicial approximation
theorem, as follows:
- For each new vertex of the refined Q-triangulation, we will determine in which
Voronoi cell its image falls, and then we will approximate its image to be
the site associated with that Voronoi cell
- This defines the simplicial approximated Nyquist map
- We continue refining the domain of definition until the
approximated Nyquist map is simplicial. This refinement can be global
(isotropic) or local (anisotropic)
