Discussion in 'MATLAB' started by Sarath Ramadurgam, Jul 14, 2008.

Hi,
I am looking for a structured quadrilateral mesh
generator for polygon domains (preferably arbitrary
domains). Most codes always use triangles. Is there a way to

Thank you
Sarath

2. ### Walter RobersonGuest

The merging sounds non-trivial to me, unless the process
is allowed to introduce degenerate quadrilaterals (in which case
just replace all of the triangles with degenerate quadrilaterals
and the problem is solved.)

There is no general solution to merging triangles to form
quadrilaterals, even when the number of triangles is even.
Simple proof: take an equilateral triangle and inscribe another
equilateral triangle within it from the center of each face:

|\
--\
|\|\
----

that's a triangle in the top row (A), and 3 triangles in alternating
directions (B, C, D) on the bottom row.

A cannot be merged with B or D: they share common vertices but
not any common faces. A must therefore be merged with C, the centre
triangle. But once that is done, B and D cannot be merged with
anything, as they share a common vertex but not a common face.

One can construct lattices in which one cannot determine which
direction to start the merging without examining the properties
of the entire lattice: for example, it could be an odd count in
one direction but an even count crosswise, and thus mergible if
one goes one of the directions but not if one goes the other way.
One could imagine unusual connections in which one had to
pretty much "solve a puzzle" to figure out the way in which one
could merge together triangles.

It also isn't clear to me what one should do if the triangles to
be merged are not coplanar: if one preserves the changes in
projection then to my mind the result would not be a "quadrilateral".

Walter Roberson, Jul 14, 2008

3. ### Bruno LuongGuest

You cannot always achieve that.

Example: Let's assume you have a a polygonal with n, a odd
number of edges (e.g., 3). Triangular mesh is simply n
triangles, each has two corners of an edge and third corner
is the center of the polygonal. Because you need exactly 2
triangles for form a quadrilaterals, it is impossible
topologically to replace mesh with odd-triangles by

Bruno

Bruno Luong, Jul 14, 2008

Thank you very much for your replies.

The domain is planar, so that removes one of the problems
merging triangles. I wanted to know about the merging as one
of the mesh generation softwares available on the internet
claim that they implemented a simple merging algorithm.
As you have pointed out that merging is difficult, do you
have any suggestions on how to construct a structured
quadrilateral mesh for polygon domains, if not arbitrary?

Thank you.
Sarath

5. ### Bruno LuongGuest

If the question is asked generally like that, I would said
use Scwharz-Christoffel transformation and its inverse to
map a rectangle to the polygonal. Then the natural

It's not clear why you need quadrilateral mesh and how it
will be used. Depending on the application you might require

Meshing is a complex problem and a comprehensive science by
itself. Better leave this task to people who work on this topic.

Bruno

Bruno Luong, Jul 14, 2008