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Volume of a Non-Coplanar Hexahedron

  1. Sep 29, 2010 #1

    I'm trying to calculate the volume of a hexahedron. I know how to do this for any arbitrary hexahedron as long as the 4 points of each face are coplanar (by using shape functions to calculate the Jacobian in a parametric space or using 5 tetrahedrons). However, the catch is that two of the faces are not coplanar. Two of the faces opposite of each other are not coplanar while the 4 surrounding sides are. If it is MUCH easier to do this computation with only one side non-coplanar, then I'd still be interested in that as well.

    I know the location of all 8 vertices, but I can't figure out how to get the area of the non-coplanar hexahedron.

    The non-coplanar surfaces would be doubly ruled surfaces as the one shown below:
    http://upload.wikimedia.org/wikipedia/commons/0/01/Hyperbolic-paraboloid.svg" [Broken]

    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Oct 1, 2010 #2
    hm...I've found the solution. For anyone that stumbles upon this in the future...

    The Jacobian DOES work, even for those non-coplanar faces. My mistake was that I was integrating the Jacobian evaluated at the center of the parametric element. For a 2D element using the shape functions I use, this works. But for a 3D element, this does not work. I now get the correct answer.

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