In 2D there is a unique way of combing triangles into quads but is there an equivalent technique available to do the same in 3D, i.e. combine two or more tetrahedral to form unstructured hexahedra?
This is a good question.
Briefly, the answer is Yes.
An indirect method to achieve quadrilateral mesh generation by combining adjacent triangles is given in
S.J. Owen, Non-Simplical Unstructured Mesh Generation, Ph.D. thesis, Carnegie Mellon University, 1999:
www.imr.sandia.gov/papers/other/owen1-50.ps.gz
See Chapter 2, starting on page 8 for a good overview of trends in unstructured mesh generation (see, especially, Section 2.4, starting on page 16).
See Section 2.4.2.1, starting on page 18 (see Fig. 2.6 for a quad mesh generated by splitting each triangle into several quads). For the 3D case, provide a sold can be tet-meshed, each tetrahedron can be subdivided into four hexahedra (see Fig. 2.9, page 22). Direct 3D surface mesh generation is described in Section 2.5.2, starting on page 29.
For more about this, see
https://www.researchgate.net/profile/Steven_Owen3
Are we considering degenerated quads, i.e. three sided with one side shrinks to a point. If we are talking about quads with four non-zero length sides, then we can't always combine triangles into quads. Here is a counter example. Consider a triangular mesh generated with an odd number of sides, 2n+1, and the central point. By deleting every other radial edges, we have a mesh of all quads but one. So both uniqueness and existence are problems.
Transforming tetrahedra to hexahedra may always have some problems in mesh quality or the insufficient tetrahedra elements. In such cases polyhedra may be used.
Thank you for your answers, it seems it is not straight forward to do what I want, due to this both uniqueness and existence problems. Never the less, I wonder if I could combine some “rules” to the conversion routine, as it is quite easy to go from Hexa to tetrahedral, after all this is how mesh refinement works, hence, we should be able to go the opposite way (i.e. using the afrementioned rules).
Forgot to say, I am not worried about the mesh quality at the moment but a vlaid arrays of all Hex's as there are a number of ways to smooth the mesh afterwards. Also not all solvers accept a polyhedra mesh.
by splitting the tetrahedral into four hexahedrons, you can always have an all hex mesh, however, the mesh quality will be a problem.
- Badges
- Science topic
- Similar topics
- Computer Science and Engineering
- Computer Graphics and Animation
- 3D