Generating quadrilateral mesh from Mathematica surface mesh

Question

I am trying to make a quadrilateral mesh from a surface mesh (which is mostly triangular) generated by Mathematica. I am not looking for high quality mesher but a simple work around algorithm. I use GMSH for doing it externally. We can make use of Mathematic's CAD import capabilities to generate 3D geometries that are understood by the Mathematica kernel.

We can see the imported Geometry3D objects and the plots of number of sides in each polygons they consist of. It become visible that the polygons that form the mesh are not always triangles.

Name3D=RandomChoice[ExampleData["Geometry3D"][[All,2]],6];
AllPic=
  Table[
  Vertex=ExampleData[{"Geometry3D",Name3D[[i]]},"VertexData"];
  Polygons=ExampleData[{"Geometry3D",Name3D[[i]]},"PolygonData"];
  GraphicsGrid[
  {{ListPlot[#,Frame-> True,PlotLabel->Name3D[[i]] ]&@(Length[#]&/@Polygons),
  Graphics3D[GraphicsComplex[Vertex,Polygon[Polygons]],Boxed-> False]}}
  ,ImageSize-> 300,Spacings-> {0,0}],
  {i,1,Length@Name3D}];
GraphicsGrid[Partition[AllPic,2],Spacings-> {0,0}]

enter image description here

Now what I am looking for is an algorithm to form a quadrilateral mesh from that polygon information available to MMA. Any easy solution is very much welcome. By easy solution I mean which is not going to work in a very general setting (where mesh constitutes of polygons with sides more than 5 or 6) and which might be quite inefficient compared to commercial software. But one can see that there are not many quadrilateral surface mesh generator available other than few expensive commercial one.

BR

Are you looking for a full mesh (hexahedra/bricks) or a surface mesh (quads)? How do you want to deal with triangles that look like this: Graphics3D[ GraphicsComplex[{{0, 0, 0.5}, {1, 0, 0}, {1, 1, 0}, {0, 1, 0}}, Polygon[{{1, 2, 3}, {3, 4, 1}}]]] - propably you'd need a normal test. And what to do with intersecting triangles like shown at the bottom of reference.wolfram.com/mathematica/TetGenLink/tutorial/…. The "SpaceShuttle" is watertight.. Could you upload a picture of a result done with GMSH of, say, the cow? — user1054186
If you want the quad surface the qmorph algorithm could be a basis: pages.cs.wisc.edu/~csverma/CS899_09/qmorph.pdf — user1054186
Are you considering solutions like removing the shared side from a two-triangle set? — Dr. belisarius
@belisarius Yes this is one of the crudest way to go. As ruebenko pointed out qmorph is a well know algorithm that requires quite lot of work in MMA. I want something simple likewise you said. I will update the question once I have a better example with the cow and how to use GMSH to generate quadmesh. I need little more time as I currently busy with something else. I hope you find the problem interesting. — PlatoManiac

atb atb · Accepted Answer · 2012-06-13T19:26:35

this will produce quads regardless of the input topology:

insert one vertex in the center of each face
insert one vertex at the midpoint of each edge
insert edges connecting each face's center vertex with it's edges' midpoint vertices

Generating quadrilateral mesh from Mathematica surface mesh

1 Answers