Is it possible to optimize a 3D Delaunay triangulation with CGAL ?

[ImperatorS79]

I would like to optimize a 3D triangulation with CGAL. More specifically, I would like to be able to, starting for a 3D triangulation object, to optimize the tetrahedrons and respect a size field using Delaunay refinement. Also, only certain tetrahedrons that are "colored" should be optimized, and a certain surface should be preserved.

I know there is a lot of algorithms for this in the different mesh related modules of CGAL. However, they do not seem to be applicable to a simple 3D Delaunay triangulation of points, and I also need to recover the newly added points by CGAL (and to not delete any input points).

[janetournois]

Hello,

you can use the optimizers described in the Tetrahedral mesh generation package. They move vertices, change the connectivity, and keep surfaces valid, but they do not insert or remove any vertices. Both the input and output triangulations and Regular triangulations (weighted Delaunay triangulation).

You can also use the Tetrahedral remeshing package. It does insert and remove vertices, and can be tuned to use a given sizing field, or protect selected vertices to make sure they still are in the output mesh.
The output mesh is not Delaunay anymore, but the surfaces are kept valid too.

What are the criteria you want to fulfill with this optimization?

[ImperatorS79]

In th Particle Finite Element Method (PFEM), a fluid is represented with a cloud of points which is Delaunay tetrahedralized (e.g. using the CGAL::Delaunay_triangulation_3 class). Then various methods are used to "colour" the right fluid tetrahedron and discards the other ones (e.g. it can be an alpha-shape, but it can be other methods too).

What I would like to do is to optimise this mesh (only the coloured tetrahedra), i.e. the circumcenter-to-shortest-edge, to respect a size field, potentially to remove slivers, using some kind of Delaunay refinement or other optimisations, while keeping the same boundary and being as much Delaunay as possible.

I see there is the [Remeshing_triangulation_3](https://doc.cgal.org/latest/Tetrahedral_remeshing/classCGAL_1_1Tetrahedral__remeshing_1_1Remeshing__triangulation__3.html), which can be used with the Tetrahedral Remeshing package. My questions:

1. Can this triangulation be made as most Delaunay as possible ?
2. How can I constraint the surface and colour the mesh ?
3. How can I recover the vertex handles added by the tetrahedral remeshing ?

[janetournois]

To take benefit of all the new features in tetrahedral remeshing, you should use the master branch, see the documentation here

1. Not really. There is no such criterion. However the output surface mesh is very isotropic so it must be close to Delaunay, but without any guarantees. You should have a look at the result.
   2a. Colors are stored in the Subdomain_index of each cell (tetrahedron), in particular in the Remeshing_cell_base.
   2b. The surfaces (i.e. interfaces between subdomains) are resampled but topologically kept valid. If you want to keep them exactly as is, use remesh_boundaries=false and/or facet_is_constrained_map to define more surfaces.
2. use vertex_is_constrained_map to set vertices that you want to be in the output mesh. The newly inserted vertices will have a false value in this property map.

[ImperatorS79]

How to send the nodal size to Adaptive_remeshing_sizing_field ? I guess I have to provide an info field to the vertex, put I already use it to store the index of my points.

Other question: can I convert a Delaunay_Triangulation_3 to a Remeshing_Triangulation_3 ? That way, I would start the optimizations by a Delaunay triangulation.

[janetournois]

Sizing is evaluated automatically by Adaptive_remeshing_sizing_field, no need to provide more information.

For triangulation conversion, have a look at convert_to_triangulation_3() and the provided examples