%INTEGRATION_BND_TIME%

surface and time integration of a vector valued function along pieces of boundary

begin_alias{} "Alias1" = " ... IDENT%BND_wall% ... MAT$MaterialTag$ ... BC$BCindex$ ... POSTPROCESS$PostprocessTag1$ ... " # definition of Alias1 "Alias2" = " ... IDENT%BND_wall% ... MAT$MaterialTag$ ... BC$BCindex$ ... POSTPROCESS$PostprocessTag2$ ... " # definition of Alias2 end_alias INTEGRATION($IntInd$) = ( %INTEGRATION_BND_TIME%, Integrand_x, Integrand_y, Integrand_z, $PostprocessTag1$, $PostprocessTag2$, ... )

This computes the integral with respect to the region $ \partial\Omega$ identified by the POSTPROCESS-flags

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

by a preliminary approximation

\begin{align} I_\text{Bnd} \approx \sum_{i \in P} \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{BndTime}}\left( t_{n+1} \right) = I_{\text{BndTime}}\left( t_{n} \right) + \left(t_{n+1}-t_{n} \right) \cdot I_\text{Bnd}\end{align}

$ {\bf n}$ represents the local boundary normal. The integrand $ {\bf u}$ is given by the vector (Integrand_x, Integrand_y, Integrand_z), whose components are all of type ExpressionOfIntegrand. $ P$ is the set of all boundary points with the given postprocess flags and $ A_i$ is the area of the i-th point. The POSTPROCESS-flags $PostprocessTag1$, $PostprocessTag2$, ... define the IntegrationArea. Their number is not limited. Example:

INTEGRATION($pressure_x$) = ( %INTEGRATION_BND_TIME%, [Y%ind_p%+Y%ind_p_dyn%], [0], [0], $PostprocessTag1$, $PostprocessTag2$, $PostprocessTag3$ )