%INTEGRATION_BND_DIRECT_TIME%

surface and time integration of a scalar value 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_DIRECT_TIME%, ExpressionOfIntegrand, $PostprocessTag1$, $PostprocessTag2$, ... )

The POSTPROCESS-flags $PostprocessTag1$, $PostprocessTag2$, ... define the IntegrationArea. Their number is not limited. This computes the integral of a functional $ f$ (ExpressionOfIntegrand) with respect to the region $ \partial\Omega$ identified by the POSTPROCESS-flags

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

by a preliminary approximation

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

$ P_{Bnd}$ is the set of all boundary points with the given postprocess flags and $ A_i$ is the area of the i-th point. Example:

INTEGRATION($IntInd1$) = ( %INTEGRATION_BND_DIRECT_TIME%, [Y%ind_p%+Y%ind_p_dyn%], $PostprocessTag1$, $PostprocessTag2$, $PostprocessTag3$ ) INTEGRATION($IntInd2$) = ( %INTEGRATION_BND_DIRECT_TIME%, equn{$EqnName$}, $PostprocessTag1$, $PostprocessTag2$, $PostprocessTag3$ ) INTEGRATION($IntInd3$) = ( %INTEGRATION_BND_DIRECT_TIME%, curve{$CrvName$}depvar{%ind_DepVar%}, $PostprocessTag1$, $PostprocessTag2$, $PostprocessTag3$ )