-%interpolation kernel.
-
-\section InterpKerTheory Theory of interpolation
-
-\subsection InterpKerPerfOverl Mesh overlapping
-
-When interpolation is performed between a source mesh S and a target
-mesh T the aspect of overlapping is important. In fact if any cell of
-of S is fully overlapped by cells of T and inversely any cell of T is
-fully overlapped by cells of S the meshes S and T are said \b
-coincidant and some general formulae in next sub section are simpler.
-As far as possible in the next sub sections the formulae for
-coincidant and non-coincidant meshes will be given.
-
-\subsection InterpKerRemapGlobal Global conservative remapping
-
-For fields with polynomial representation on each cell, the components of the discretized field \f$ \phi_s \f$ on the source side can be expressed as linear combinations of the components of the discretized field \f$ \phi_t \f$ on the target side, in terms of a matrix-vector product :
-
-\f[
- \phi_t=W.\phi_s.
-\f]
-
-All the aim of interpolators is to compute W depending on a physical
-quantities and the type of interpolation wanted (P0, P1, P1d etc...).
-
-For the \b intensive \b field \f$ \phi \f$ from the
-source mesh \f$ S \f$ to the target mesh \f$ T \f$ the interpolation should preserve the
-integral of \f$ \phi \f$ on any domain. At the discrete level, for any
-target cell \f$ T_i \f$, the following \b general \b interpolation \b
-equation \anchor InterpKerGenralEq has to
-be always verified :
-
-\f[
-\int_{T_i} \phi = \sum_{S_j\cap T_i \neq \emptyset} \int_{T_i\cap S_j} \phi.
-\f]
-
-To compute \f$ W_{ij} \f$
-this equation is used. The evaluation of integrals depends on the source and target meshes and on the nature of interpolation chosen : P0, P1, P1d etc... For the moment it is only possible to
-remap fields with P0 representations.
-
-\subsection InterpKerRemapInt Conservative remapping of intensive physical quantities
-
-At the basis of many CFD numerical schemes is the fact that physical
-quantities such as density, momentum per unit volume or energy per
-unit volume obey some balance laws that should be preserved at the
-discrete level on every cell. This property is critical for example to
-accurately capture shockwaves.
-
-\subsection InterpKerP0P0Int cell-cell (P0->P0) conservative remapping of intensive physical quantities
-
-In the \ref InterpKerGenralEq "general interpolation equation" the
-left hand side becomes :
-
-\f[
-\int_{T_i} \phi = (\sum_{S_j} Vol(T_i\cap S_j)).\phi_{T_i}.
-\f]
-
-\note \f$ \sum_{S_j} Vol(T_i\cap S_j) = Vol(T_i) \f$ \ref InterpKerPerfOverl "in case of perfect overlapping".
-
-In the \ref InterpKerGenralEq "general interpolation equation" the
-right hand side becomes :
-
-\f[
-\sum_{S_j\cap T_i \neq \emptyset} \int_{T_i\cap S_j} \phi = \sum_{S_j\cap T_i \neq \emptyset} {Vol(T_i\cap S_j)}.\phi_{S_j}.
-\f]
-
-In the case where the \b intensive field values are constant on each
-cell, the coefficients of the linear remapping matrix \f$ W \f$ are
-given by the formula :
-
-\f[
- W_{ij}=\frac{Vol(T_i\cap S_j)}{ \sum_{S_j} Vol(T_i\cap S_j) }.
-\f]
-
-and \ref InterpKerPerfOverl "in case of perfect overlapping" :
-
-\f[
- W_{ij}=\frac{Vol(T_i\cap S_j)}{ Vol(T_i) }.
-\f]
-
-Where Vol represents the volume with mesh dimension of interpolation equals to 3, the
-area when mesh dimension equals to 2, and length when mesh dimension equals to 1.
-
-\subsection InterpKerRemapExt Conservative remapping of extensive physical quantities
-
-In code coupling from neutronics to hydraulic code \b extensive field
-of power is exchanged and the all power as to be kept the same. The
-principle is to 'intensify' the field to move on from extensive field
-\e P to an intensive one \f$ \phi \f$.
-
-\subsection InterpKerP0P0Ext cell-cell (P0->P0) conservative remapping of extensive physical quantities
-
-In the \ref InterpKerGenralEq "general interpolation equation" the
-left hand side becomes :
-
-\f[
-\int_{T_i} \phi = P_{T_i}.
-\f]
-
-In the \ref InterpKerGenralEq "general interpolation equation" the
-right hand side becomes :
-
-\f[
-\sum_{S_j\cap T_i \neq \emptyset} \int_{T_i\cap S_j} \phi =
-\sum_{S_j\cap T_i \neq \emptyset} \frac{Vol(T_i\cap S_j)}{\sum_{T_i} Vol(T_i \cap S_j)}.P_{S_j}.
-\f]
-
-\note \f$ \sum_{T_i} Vol(T_i \cap S_j) = Vol(S_j) \f$ \ref InterpKerPerfOverl "in case of perfect overlapping".
-
-In the case where the \b extensive field values are constant on each
-cell, the coefficients of the linear remapping matrix \f$ W \f$ are
-given by the formula :
-
-\f[
- W_{ij}=\frac{Vol(T_i\cap S_j)}{ \sum_{T_i} Vol(T_i \cap S_j) }.
-\f]
-
-and \ref InterpKerPerfOverl "in case of perfect overlapping" :
-
-\f[
- W_{ij}=\frac{Vol(T_i\cap S_j)}{ Vol(S_j) }.
-\f]