%INTEGRATION_BE_DIRECT_TIME%

surface and time integration of a scalar value on boundary elements

INTEGRATION($IntInd$) = ( %INTEGRATION_BE_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{BEDirectTime} = \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{BEDirect} \approx \sum_{i \in BE} 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{BEDirectTime} \left( t_{n+1} \right) = I_{\text{BEDirectTime}}\left( t_{n} \right) + \left(t_{n+1}-t_{n} \right) \cdot I_\text{BEDirect}\end{align}

$ BE$ is the set of all boundary elements with the given postprocess flags. $ f_i$ is the function value and $ A_i$ is the area of the i-th boundary element. Example:

INTEGRATION($time_area_PostprocessTag1$) = ( %INTEGRATION_BE_DIRECT_TIME%, [1.0], $PostprocessTag1$ )

Note: In contrast to %INTEGRATION_BND_DIRECT_TIME%, ExpressionOfIntegrand is defined and evaluated on the boundary elements and not on the MESHFREE point cloud!