Utility/Use of Quadrature Elements

I am looking to understand “Quadrature” finite elements as well as their purpose. ufl documentation states that they represent artificial “finite elements” with degrees of freedom being function evaluations at quadrature points.

Here is a link to a recent post on the computational stack exchange, which is related and useful for this discussion as it shows how a mass matrix can be decomposed via quadrature weights, basis function evaluations at quadrature points, element Jacobians and global to local mappings.

If you have insight into exactly what the Quadrature finite elements are I would greatly appreciate it. Thanks!

The main applied use case that jumps into my mind is for representing internal state or history variables in solid constitutive models. A great example of this would be in @bleyerj’s tutorial on elastoplastic analysis here, where he uses a "Quadrature" element for the plastic strain. In my own research work, I’ve also used "Quadrature" elements to represent a strain-history field for enforcing crack irreversibility in phase-field fracture.