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By finite element type I specifically mean element geometry.

I am aware of Beam/Strut, Quad, Tri, Tet & Hex but are there other types, eg a hexagonal prism, rhombihedral etc.. ?

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  • $\begingroup$ In my experience, no. You have listed every type I've come across. What do you perceive the benefit might be for other types? How does this compare to second (or higher) order elements of the more established types? cdn.comsol.com/cyclopedia/finite-element-method/… $\endgroup$ – Jonathan R Swift Jan 14 '18 at 13:14
  • $\begingroup$ I'm interested in what the shape functions would look like $\endgroup$ – DrBwts Jan 14 '18 at 14:32
  • $\begingroup$ there are only classical continuum mechanics shape functions for 2d quad and triangle shapes. By extension to 3d we get tet and hex(brick) and triangular prizms. A "rhombihedral" element is just a hex isnt it? ...Throwing beams into the question opens another can of worms, there are all knids of structural elements, springs, and so forth.. This question is really too broad. $\endgroup$ – agentp Jan 15 '18 at 14:32
  • $\begingroup$ The question is specifically about different geometries. A rhombi-dodecahedral element would have 12 faces not 6 but would also have the space filling property of hex elements. $\endgroup$ – DrBwts Jan 15 '18 at 17:02
  • $\begingroup$ Surely the need for additional required computing power would quickly outstrip any potential benefits. K.I.S.S. $\endgroup$ – Rick supports Monica Jan 15 '18 at 19:07
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Certain geometries can benefit from polyhedral elements or elements with edge degrees of freedom. I can think of three main developments in that direction:

1) Voronoi cell finite elements, e.g., https://www.sciencedirect.com/science/article/pii/0045794994904359

2) Isogeometric elements, e.g., https://www.sciencedirect.com/science/article/pii/S0045782504005171

3) Finite elements based on exterior calculus, e.g., https://www.cambridge.org/core/journals/acta-numerica/article/finite-element-exterior-calculus-homological-techniques-and-applications/1A2AEB067BCA561D9ED6D674026539B9

These elements are still being actively designed, but attention today is mostly on isogeometric and exterior calculus-based elements; primarily based of meshing issues and the need to solve multiphysics problems and to avoid element locking.

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