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I am hoping someone could point me in the right direction in finding resources for a project I am doing. I need to derive a formula that will calculate the volume of 4 child hexahedrals that will form one parent tetrahedron. This is for a metric in hex meshing. Then I am to put this derived formula into python code. Any guidance is greatly appreciated. Thank you.

Adam
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  • Do you mean this type of decomposition : https://cubit.sandia.gov/public/13.2/help_manual/WebHelp/mesh_generation/meshing_schemes/conversion/thex.htm ? – Jean Marie Feb 11 '20 at 18:09
  • Be conscious that the term "hexahedron", unless you describe it in a precise way is ambiguous. You have to give more details in order that somebody understands exactly what are the "inputs" and which outputs you desire. Another perfectly ambiguous term in this context is "Jacobian" (assuming a change of coordinates, but which coordinates ?) – Jean Marie Feb 11 '20 at 18:22
  • Hi Jean. I believe that if my understanding of the project is correct I am supposed to the opposite of what the link you provided suggests. I will be provided a spreadsheet of coordinates for cuboids. I am to take those coordinates and calculate the volume of each. Then the sum of the volume of 4 cuboids need to equal the volume of one tetrahedral. Not sure if I am explaining myself clearly. – Adam Feb 11 '20 at 19:47
  • Is your fist concern the way the volume of a cuboid can be computed ? In all cases, give the way the data is organized in your spreadsheet file, and what you want to get with it by taking the simplest example. – Jean Marie Feb 11 '20 at 21:22
  • I worked out how to calculate the volume (and CG, inertia) of any tetrahedron from its corners some years ago for a project at work. Unfortunately, I do not trust my memory to get the fomula right, and due to a drive crash, I no longer have it at home. I'll try to get to you tomorrow. Somewhat ironically, to use it to calculate your cuboids will require breaking up the cuboids into smaller tetrahedra, but that is easy enough to do. – Paul Sinclair Feb 12 '20 at 03:06
  • Thank you Paul. I actually have looked at that method but I was told not use that but instead use shape functions, integration, and a jacobian matrix. Like the formula in my comment to Jean below. – Adam Feb 12 '20 at 04:29
  • Jean, yes that is my first concern. I believe the formula should look like this: $I = \iiint f(x, y, z) dx dy dz = \sum_{H_i = 1}^4 \iiint f(\xi , \eta , \zeta ) |J| d\xi d\eta d\zeta$ , the intergration is from -1 to 1. I believe this is derived from shape functions (which have been a little confusing to me). So I will be given a list of 8 node coordinates which I will have to use to calculate the volume of each 8 node cuboid in the list. – Adam Feb 12 '20 at 04:32
  • I wasn't aware of your answer because you have used "Jean" instead of "Jean Marie" for my pseudo. 2) When you say "cuboid" do we agree that it is a volume given by its vertices $(x_1,y_1,z_1)...(x_8,y_8,z_8)$ having the global topology of a cube with 6 (planar) faces, without any assumption of parallelism of a face with its opposite face ?
    • – Jean Marie Feb 13 '20 at 20:48
  • I'm sorry Jean Marie. Yes that is correct. – Adam Feb 14 '20 at 07:21