Foreplay
=============

This method is based on the observation [:ref:`bib1 <bib1>`] that there are points where the calculation of constraints, based on displacements in a primal displacement formulation, is more accurate.

In the case of isoparametric finite elements of order 2 (SEG3 in 1D, QUAD8 and QUAD9 in 1D, and in 2D, HEXA20 in 3D), we show that the points of GAUSS of the quadrature formula with :math:`{2}^{n}` points (:math:`n`: dimension of space) are such that one can hope for, without this being formally demonstrated, for the calculation of :math:`\sigma` the same order of precision as for the calculation of the displacement field :math:`u`.

The idea of the method is to calculate for each element the constraints :math:`\stackrel{ˆ}{\sigma }` at the nodes starting from :math:`{\sigma }^{k}` at the points of GAUSS, the latter being calculated on each element by the formula:

:math:`{\sigma }^{k}=D{B}^{k}u=D\sum _{i=1}^{\mathrm{NNO}}{B}_{i}^{k}{U}_{i}`

where:

:math:`D` is the elasticity matrix,

:math:`{B}^{k}` is the matrix relating the deformations to the displacements at the point of GAUSS :math:`k`,

:math:`{U}_{i}` are the nodal displacements (:math:`\mathrm{NNO}` knots)