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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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.