%INTEGRATION_INT_TIME%

volume and time integration of a functional with respect to a given material

begin_alias{} "AliasOmega" = " ... IDENT%BND_wall% ... MAT$MaterialTag$ ... BC$BCindex$ ... " # definition of AliasOmega end_alias INTEGRATION($IntInd$) = ( %INTEGRATION_INT_TIME%, ExpressionOfIntegrand, $MaterialTag$ )

This computes the integral of a functional $ f$ (ExpressionOfIntegrand) with respect to the region $ \Omega$ identified by the material flag $MaterialTag$

\begin{align} I_{\text{Time}} = \int\limits_{t_0}^{t_{n+1}} \int\limits_{\Omega} f(t) dV dt\end{align}

by a preliminary approximation

\begin{align} I \approx \sum_{i \in P} f_i \left( t_{n+1} \right) \cdot V_i\left( t_{n+1} \right)\end{align}

and a subsequent time integration:

\begin{align} I_{\text{Time}}\left( t_{n+1} \right) = I_{\text{Time}}\left( t_{n} \right) + \left(t_{n+1}-t_{n} \right) \cdot I\end{align}

$ P$ is the set of all points with the given material flag and $ V_i$ is the volume of the i-th point. Note: Analogous to %INTEGRATION_BND_TIME%, a list of material flags can be used to specify the integration region. The number of flags is not limited. Example: total turbulent dissipation of some material

INTEGRATION($dissipation$) = ( %INTEGRATION_INT_TIME%, [Y%ind_eps%], $MaterialTag$ )

DROPLETPHASE

For DROPLETPHASE points the same remarks as for %INTEGRATION_INT% apply and we obtain

\begin{align} I \approx \sum_{i \in P} f_i \left( t_{n+1} \right) \cdot n_i \left( t_{n+1} \right) \cdot V_i\left( t_{n+1} \right)\end{align}

where $ n_i$ corresponds to the multiplicity stored in %ind_mult%.