Finding minimal cubature rules for finite elements passing the patch test

Cubature, i.e. multivariate numerical integration, plays a core part in the finite-element method. For a given element geometry and interpolation, it is possible to choose different cubature rules, leading to concepts like full and reduced integration. These cubature rules are usually chosen from a rather small set of existing rules, which were not specifically derived for finite-element applications, and may therefore not be optimal.
We present a novel method to find element-specific cubature rules, based only on the requirement that the element passes the patch test. Starting from the monomial sets generating the displacement and geometry interpolations, the method computes the monomials that must be integrated exactly, and thus the moment equations that generate the required rules.
We use the presented method to compute rules for quadrilateral and hexahedral elements which try to minimise the number of integration points required, and test the resulting elements using a series of standard tests. The results show that, for higher-order interpolation, several of these new rules have an advantage over existing ones.