3. Static condensation

Field \(\delta\) disappears from the overall formulation thanks to static condensation. At each collocation point \({s}_{g}\), we have (according to eq.1.4):

\({t}_{g}\mathrm{=}{\lambda }_{g}+r({g}_{g}\mathrm{-}{\delta }_{g})\mathrm{\in }\mathrm{\partial }\Pi ({\delta }_{g})\)

with \({t}_{g}\mathrm{=}t({s}_{g})\), \({\lambda }_{g}\mathrm{=}\lambda ({s}_{g})\), \({g}_{g}\mathrm{=}g({s}_{g})\), and \({\delta }_{g}\mathrm{=}\delta ({s}_{g})\).

Once the problem is discretized, \({g}_{g}\) and \({\lambda }_{g}\) will be obtained by interpolations with the appropriate form functions of the discretized values at the nodes of \(g\) and \(\lambda\) that are denoted by \(\mathrm{\{}G\mathrm{\}}\) and \(\{\Lambda \}\).

The integration of the constitutive relationship (see below) makes it possible to calculate \({\delta }_{g}\) as a function of \(\mathrm{\{}G\mathrm{\}}\) and \(\mathrm{\{}\Lambda \mathrm{\}}\), which is denoted by \(\delta\):

\({t}_{g}\mathrm{=}{\lambda }_{g}+r({g}_{g}\mathrm{-}{\delta }_{g})\mathrm{\in }\mathrm{\partial }\Pi ({\delta }_{g})\mathrm{\iff }{\delta }_{g}\mathrm{=}\stackrel{ˆ}{\delta }({g}_{g},{\lambda }_{g})\mathrm{=}\delta (\mathrm{\{}G\mathrm{\}},\mathrm{\{}\Lambda \mathrm{\}})\)