8. Option FORC_NODA

At the level of continuous equations, option FORC_NODA corresponds to the calculation of the operator \(R={Q}^{T}\overline{\Sigma }\). At the discrete level, option FORC_NODA is equivalent to calculating the vector \({R}_{g}^{\text{el}}={Q}_{g}^{{\text{el}}^{T}}\text{.}{\overline{\Sigma }}_{g}^{\text{el}}\).

As we already noted that \(\overline{\Sigma }\) depends not only on \(\Sigma\), but also on \(\dot{\Sigma }\), it should come as no surprise to see the time step \(\Delta t\) and the constraints on both time + and time appear.

The Newton-Raphson algorithm in command STAT_NON_LINE uses option FORC_NODA to calculate the prediction at the start of each time step. It is therefore not trivial to correctly calculate all the terms for this option, including those that depend on the time step. We illustrate this question with a simple example corresponding to the hydraulic equation alone.

This is a simplified version of the hydraulic equation:

:math:`-{int }_{Omega }frac{text{dm}}{text{dt}}text{.}{p}^{text{*}}dOmega +{int }_{Omega }Mtext{.}nabla {p}^{text{*}}dOmega ={M}_{text{ext}}text{.}{p}^{text{*}}`**eq 8-1**

After discretization in time:

\(-{\int }_{\Omega }\Delta \text{m}\mathrm{.}{p}^{\text{*}}d\Omega +\Delta t{\int }_{\Omega }(\theta {M}^{+}+(1-\theta ){M}^{-})\text{.}\nabla {p}^{\text{*}}d\Omega ={M}_{\text{ext}}^{\theta }\text{.}{p}^{\text{*}}\)

Bringing up \(\Delta M={M}^{+}-{M}^{-}\) and \(\Delta p={p}^{+}-{p}^{-}\), and writing a law of behavior: \(\Delta m =N\Delta p\), we find:

\(-{\int }_{\Omega }N\Delta \text{p}\text{.}{p}^{\text{*}}d\Omega +\Delta t{\int }_{\Omega }\Delta M\text{.}\nabla {p}^{\text{*}}d\Omega ={M}_{\text{ext}}^{\theta }{p}^{\text{*}}-\Delta t{\int }_{\Omega }{M}^{-}\text{.}\nabla {p}^{\text{*}}d\Omega\)

By definition the prediction phase STAT_NON_LINE is written as:

\({K}_{0}\Delta {u}^{0}={F}_{\text{ext}}^{1}-{Q}^{T}{\sigma }_{0}\)

It is then clear that we must take \(\underset{\Omega }{\int }{Q}^{T}{\sigma }_{0}{p}^{\text{*}}d\Omega =-\Delta t \underset{\Omega }{\int }{M}^{-}\nabla {p}^{\text{*}}d\Omega\)