%INTEGRATION_FS_DIRECT_TIME%

surface and time integration of a scalar value along the free surface

INTEGRATION($IntInd$) = ( %INTEGRATION_FS_DIRECT_TIME%, ExpressionOfIntegrand, $MaterialTag$ )

The material flag $MaterialTag$ defines the integration area (analogous to the POSTPROCESS-flags for %INTEGRATION_BND_DIRECT_TIME%). This computes the integral of a functional $ f$ (ExpressionOfIntegrand) with respect to the free surface $ \partial\Omega_\text{FS}$ identified by the material flag

\begin{align} I_\text{FSDirectTime} = \int\limits_{t_0}^{t_{n+1}} \int\limits_{\partial\Omega_\text{FS}} f(t) dA dt\end{align}

by a preliminary approximation

\begin{align} I_\text{FSDirect} \approx \sum_{i \in P_\text{FS}} f_i \left( t_{n+1} \right) \cdot A_i\left( t_{n+1} \right)\end{align}

and a subsequent time integration:

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

$ P_\text{FS}$ is the set of all free surface points with the given material flag and $ A_i$ is the area of the i-th point. Note: Analogous to %INTEGRATION_INT_TIME%, a list of material flags can be used to specify the integration area. The number of flags is not limited. Example:

INTEGRATION($IntInd1$) = ( %INTEGRATION_FS_DIRECT_TIME%, [Y%ind_p%+Y%ind_p_dyn%], $MaterialTag$ ) INTEGRATION($IntInd2$) = ( %INTEGRATION_FS_DIRECT_TIME%, equn{$EqnName$}, $MaterialTag1$, $MaterialTag2$, $MaterialTag3$ ) INTEGRATION($IntInd3$) = ( %INTEGRATION_FS_DIRECT_TIME%, curve{$CrvName$}depvar{%ind_DepVar%}, $MaterialTag$ )

Note: In case of multiphase simulations with detection of interface connections (see PHASE_distinction), the interface points are treated like free surface points.