%INTEGRATION_FS_TIME%

surface and time integration of a vector valued function along the free surface

INTEGRATION($IntInd$) = ( %INTEGRATION_FS_TIME%, Integrand_x, Integrand_y, Integrand_z, $MaterialTag$ )

This computes the integral with respect to the free surface $ \partial\Omega_\text{FS}$ identified by the material flag

\begin{align} I_\text{FSTime} = \int\limits_{t_0}^{t_{n+1}} \int_{\partial\Omega_\text{FS}} {\bf u}(t) \cdot {\bf n}(t) dA dt \end{align}

by a preliminary approximation

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

and a subsequent time integration:

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

$ {\bf n}$ represents the local free surface normal. The integrand $ {\bf u}$ is given by the vector (Integrand_x, Integrand_y, Integrand_z), whose components are all of type ExpressionOfIntegrand. $ 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. The material flag $MaterialTag$ defines the integration area (analogous to the POSTPROCESS-flags for %INTEGRATION_BND%). 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($pressure_x$) = ( %INTEGRATION_FS_TIME%, [Y%ind_p%+Y%ind_p_dyn%], [0], [0], $PostprocessTag1$, $PostprocessTag2$, $PostprocessTag3$ )

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