specify quadrature

One of the things arising from the recent investigation into the poor behaviour of elastic modal analysis with ElementTetP1 #75 was the idea of reduced quadrature; e.g. in discussing ElementHex1, Petyt (1990, Introduction to Finite Element Vibration Analysis, Cambridge University Press, §5.8) recommends a rule with one abscissa at the centroid. Reduced quadrature or ‘selective integration’ is also discussed by

• Benzley, S. E., Perry, E., Merkley, K., Clark, B. & Sjaardema, G. (1995). A comparison of all hexagonal and all tetrahedral finite element meshes for elastic and elasto-plastic analysis. In T. Tautges (ed.), In Proceedings, 4th International Meshing Roundtable (pp. 179–191).

although their practical recommendation is for increasing the order of the element as done here in #322.

Currently I don’t see an easy way to experiment with quadrature in scikit-fem. I think though that enabling this would be simple generalization of the constructor for skfem.assembly.Basis.

https://github.com/kinnala/scikit-fem/blob/89f726808f50bf9773f1e2c7d43248ccbd0714a0/skfem/assembly/basis/basis.py#L26

or maybe

https://github.com/kinnala/scikit-fem/blob/89f726808f50bf9773f1e2c7d43248ccbd0714a0/skfem/assembly/basis/interior_basis.py#L35-L40

(and similarly for FacetBasis) so that either one specifies:

• the actual quadrature rule, i.e. X and W which become attributes of the InteriorBasis, or
• intorder: int as now, and the rule is computed from skfem.quadrature.get_quadrature, or
• nothing, also as now, and intorder is computed from self.elem.maxdeg.

A possible further modification is that the type of the rule returned by get_quadrature is Tuple[ndarray, ndarray] from which it is unclear which is weights and which abscissae. Perhaps a quadrature rule might be represented as in quadpy.nsimplex._helpers.NSimplexScheme as an object with attributes weights and points, or more simply as a NamedTuple (or dataclass) with similarly named attributes; given Python’s duck-typing, reference from within skfem.assembly only to attributes weights and points might enable direct use of one of the many schemes from quadpy while only importing it at the top-level user-script in which the basis is instantiated.

I don’t think that this is urgent—I don’t have any immediate plans to use it myself—but it seems like an easy addition and something that should be present in a general finite element library. I think it’s mostly a matter of deciding on the desired syntax.

[kinnala]

Nevertheless, I still don’t oppose using these representations in skfem.quadrature. I think it pays off in the long term to be compatible with other packages in the ecosystem.

[gdmcbain]

I wonder whether multiple inheritance would help here; e.g. deriving MeshTet from MeshSimplex as well as Mesh3D.