1. Foreplay

This method is based on the observation [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 \({2}^{n}\) points (\(n\): dimension of space) are such that one can hope for, without this being formally demonstrated, for the calculation of \(\sigma\) the same order of precision as for the calculation of the displacement field \(u\).

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

\({\sigma }^{k}=D{B}^{k}u=D\sum _{i=1}^{\mathrm{NNO}}{B}_{i}^{k}{U}_{i}\)

where:

\(D\) is the elasticity matrix,

\({B}^{k}\) is the matrix relating the deformations to the displacements at the point of GAUSS \(k\),

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