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How quadrilateral elements represented in most FEA solvers for structural mechanics problems?

I know that Nastran-95 represent quadrilaterals for membrane and bending elements as a set of triangles: enter image description here

At the same time there are a lot of literature where stiffness matrix for quadrilaterals is computed using ξ-η coordinates:enter image description here

The question is which one of both methods is used in modern FEA packages?

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    $\begingroup$ Most modern FEA packages still use the infrastructure developed at the time Nastran was developed. However, over time people have started to separate mesh topolgies from element types. See scicomp.stackexchange.com/questions/36981/… for an example. $\endgroup$ Apr 2, 2021 at 17:22
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    $\begingroup$ The Nastran 95 "overlapping triangles" method is totally obsolete. (In fact it was totally obsolete long before 1995, but Nastran tends to retain all sorts of historical junk for backwards compatibility. If you work on projects with very long lifetimes (e.g. 50 years) sometimes you need to reproduce the exact same numbers that were calculated 50 years ago, even if by modern standards the model is very crude and inaccurate. $\endgroup$
    – alephzero
    Apr 3, 2021 at 0:14
  • $\begingroup$ I have used just three modern FEA software, COMSOL, ANSYS and PETSc (in its PETIGA variant). The first two meshed everything using tetrahedra. This is common as tetrahedrons are better at representing curved or complex geometries. PETIGA used quadrilateral meshes, integrating on the \xi \eta space. $\endgroup$
    – Zegpi
    3 hours ago

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