%INTEGRATION_INT%

volume integration of a functional with respect to a given material

begin_alias{} "AliasOmega" = " ... IDENT%BND_wall% ... MAT$MaterialTag$ ... BC$BCindex$ ... " # definition of AliasOmega end_alias INTEGRATION($IntInd$) = ( %INTEGRATION_INT%, ExpressionOfIntegrand, $MaterialTag$ )

This computes the integral of a functional $ f$ (ExpressionOfIntegrand) with respect to the region $ \Omega$ identified by the material flag $MaterialTag$

\begin{align} I = \int\limits_{\Omega} f dV\end{align}

by a sum approximation

\begin{align} I \approx \sum_{i \in P} f_i \cdot V_i,\end{align}

where $ P$ is the set of all points with the given material flag and $ V_i$ is the volume of the i-th point. Note: Analogous to %INTEGRATION_BND%, a list of material flags can be used to specify the integration region. The number of flags is not limited. Example:

volume of a material

INTEGRATION($volume$) = ( %INTEGRATION_INT%, [1], $MaterialTag$ )
kinetic energy of a material

INTEGRATION($energy$) = ( %INTEGRATION_INT%, [0.5*Y%ind_r%*(Y%ind_v(1)%^2 + Y%ind_v(2)%^2 + Y%ind_v(3)%^2)], $MaterialTag$ )

DROPLETPHASE

The variable $ V_i$ corresponds to the value stored in %ind_Vi%, so that for DROPLETPHASE points it does not represent a volume estimate from a Delaunay triangulation but the volume of the Droplet according to %ind_d30%. Furthermore, the parcel concept is automatically taken into account, so that

\begin{align} I \approx \sum_{i \in P} f_i \cdot n_i \cdot V_i,\end{align}

where $ n_i$ corresponds to the multiplicity stored in %ind_mult%.