Interest of Quadrature element

Hello,

I have some questions about the use of the “Quadrature element”. According the FEniCS book, this specific element allows to assess quantities on the integration points only.
By default, I thought that the quantities like stress or strain were already evaluated at quadrature points.
In page 512, they make the difference between an exact linearized formulation where the stresses are evaluted at the degree of freedom alpha and the non-exact linearized formulation where the stresses are evaluted at the quadrature points instead.
So, If I don’t use the “Quadrature element” to define a TensorSpace for example but I use “DG” space, what FFC do by default ? An integration around facet of each cell using nodes located on the edge ?

To visualize stress component evaluated on quadrature, do I necessarily need to project on Quadrature space defined with “Quadrature element” ?

Besides, in the book, the use of “Quadrature element” is illustrated with nonlinear constitutive model but in linear case it remains the same, right ?

I am asking because I hesitate on which quantities I manipule on paraview.

Thanks for the indications !

Lamia