Structured Quadrilateral Mesh

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

  1. 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
    merge the triangles into quadrilaterals? Please help me.

    Thank you
    Sarath Ramadurgam, Jul 14, 2008
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  2. 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
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  3. Sarath Ramadurgam

    Bruno Luong Guest

    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
    quadrilateral tessellation.

    Bruno Luong, Jul 14, 2008
  4. 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 Ramadurgam, Jul 14, 2008
  5. Sarath Ramadurgam

    Bruno Luong Guest

    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
    quadrilateral mesh of your rectangle will provide you a
    quadrilateral mesh of your polygonal.

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

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

    Bruno Luong, Jul 14, 2008
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