Core: spatial gradient for arbitrary lattices

[GPMueller]

This is related to #64.

If the basis is irregular and does not produce a simple lattice structure for which the neighbours for the spatial gradient are known, we should use tetrahedra from qhull.

Even when some tetrahedra are chose sub-optimally, the fact that other neighbouring tetrahedra will likely be fine makes large errors for single spins quite unlikely.

[GPMueller]

Related to #224, the neighbouring spins for the gradient could, in 2D, be determined by Delaunay triangulation and in 3D by dividing the lattice into tetrahedra. Both could be achieved using QHull.

[GPMueller]

Since cf99c840f63e5154b61a97a4f8dbb2d1ab6fbae0, for 2D, the Delaunay triangulation of the lattice could be used. Both triangulation and tetrahedra are available, so this could now be implemented.

[GPMueller]

Note that the Delaunay triangulation and the tetrahedra do not include periodical neighbours, so they need to be extended with boundaries in order to give physically correct results (see also #292).

[GPMueller]

This seems to be a non-trivial thing to do...

CGAL has this functionality:

• http://doc.cgal.org/latest/Periodic_2_triangulation_2/index.html#Chapter_2D_Periodic_Triangulations
• http://doc.cgal.org/latest/Periodic_3_triangulation_3/index.html

but it is only for periodicity in all directions.

Implementing this by hand would mean adding an additional layer of points around the positions. The resulting triangles/tetrahedra could be checked, where an index>nos would mean that it is an artificial triangle. If at least one index is not artificial, the triangle/tetrahedron is valid.

[GPMueller]

Started work on better triangulations and tetrahedra with ca42835b3a1d6c6ecb1fe606fb9f0ad1e6b52f45 on feature-triangulations. Open questions:

• how to use the new periodical triangulation appropriately for the gradient
• does the topological charge still correspond to before this change when a skyrmion sits on the boundary?
• should "periodical" triangles contain translations instead of indices?
• should the topological charge require periodical boundaries?
• what are the right conditions to get periodical tetrahedra?

[GPMueller]

A good starting point is the implementation done with 8b7f3e8416ad964face02adf2ec6e5f3821aa8ae. Using the given approach, the gradient could be calculated as a weighted average over the finite differences to the atoms in the same cell and the neighbouring cells. That approach should work adequately in almost any case.