/*!
\page interpkernel Interpolation kernel toolkit

\section InterpKerIntro Introduction

The main purpose of this module is to propose a set of algorithms for
mesh interpolation \b fully \b independant \b of \b mesh \b datastructure to
support several type of format. This component is parameterized as
much as possible using C++ templates.
For the moment only interpolators for unstructured meshes are present in
%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]

\section InterpKerMainArchitecture Main architecture of interpolation kernel.

In %interpolation kernel, the algorithm that computes \f$ W_{ij} \f$ given cell i
in source mesh and cell j in target mesh is called intersector.

As seen in \ref InterpKerTheory "the theory of interpolation", for all interpolation the aim is to
fill the W matrix (which is generally a sparse matrix). Fundamatally for each pair (i,j) \f$ W_{ij} \f$ is obtained
by calling each time the wanted intersector. The problem is that each call to algorithm
is CPU-expensive.
To reduce the computation time a first filtering is done to found a
maximim of (i,j) pairs where \f$ W_{ij} \f$ is obviously equal to 0. Typically it
is the case when a cell in source mesh is too far from an another cell
in target mesh each other.

So for a given type of interpolation, the computation of W is
performed in two steps :

      -# A filtering process reduces the number of pairs of
      elements for which the calculation must be carried out by
      eliminating pairs that do not intersect through a comparison of
      their bounding boxes. It reduces as much as possible call to intersector.
      -# For all remaining pairs calling for each intersector (see \ref interpolation2D, \ref interpolation3Dsurf or \ref interpolation3D).

Each interpolator inherits from INTERP_KERNEL::Interpolation ( whatever
its dimension and its type ) that is a
CRTP class in order to clearly see the main API without useless CPU cost.

\subsection InterpKerMeshType class MeshType

Each Interpolators and Intersectors are parameterized (templated in
C++ langage) with \c class \c MeshType . This type of generalization
has been chosen to reduce at maximum overhead. \n
Thanks to this principle \b intersectors \b and \b interpolators \b are \b usable
\b with \b several \b formats \b without \b preformance \b loss. For example MED, VTK...\n
\c MeshType is a concept that should strictly fulfilled the following
rules :

      - Const values / Types
        - MyConnType : represents type of connectivity index. This is typically \c int or \c long \c int .
        - MY_SPACEDIM : space dimension. Dimension relative to coordinates.
        - MY_MESHDIM : the dimension of all cells in meshes.
        - My_numPol : policy of numbering. C Format ( \f$ [0,n-1] \f$ ) or FORTRAN ( \f$ [1,n] \f$ ).
      - Methods
        -# \code void getBoundingBox(double *boundingBox) const \endcode
        -# \code INTERP_KERNEL::NormalizedCellType getTypeOfElement(MyConnType eltId) const \endcode
        -# \code unsigned char getNumberOfNodesOfElement(MyConnType eltId) const \endcode
        -# \code unsigned long getNumberOfNodes() const \endcode
        -# \code unsigned long getNumberOfElements() const \endcode
        -# \code const MyConnType *getConnectivityPtr() const \endcode
        -# \code const double *getCoordinatesPtr() const \endcode
        -# \code const MyConnType *getConnectivityIndexPtr() const \endcode
        -# \code void ReleaseTempArrays() \endcode
      - Formats of arrays
        - the array returned by \c getCoordinatesPtr must be a \b full \b interlace array.
        - the arrays returned by \c getConnectivityPtr and \c
        - getConnectivityIndexPtr must be with the same principle as it is in \ref medmemConnArrays "medmem". Of course the numbering format may change according to \a My_numPol policy.

\note The arrays formats of connectivity is kept close to MED. It is
close to VTK too but slightly different. So it needs VTK side a copy
on wrap. To avoid this copy of a part of connectivity structure, iterator should be used.

\subsection InterpKerMatrixType class MatrixType

As already said, the matrix returned by interpolator is typically a sparse matrix. Instances of
\c class \c MatrixType are used to stores these results of
interpolation. To be able to be filled by the interpolator the \c MatrixType class has to match the following concept :

      - Methods
        -# \code void resize(uint nbrows) \endcode
        -# \code Row &operator [] (uint irow) \endcode

\c class \c Row has to match at least following concept :

      - Methods
        - \code void insert(const std::pair<int,T>& myPair) \endcode

\note \c std::vector\c < \c std::map<int,double> > is a candidate for
\c MatrixType.

\section InterpKerGenUsage Usage of interpolation kernel.

\subsection InterpKerHighLevUsage high-level usage

This the simplest mode of usage of interpolator. This way is strongly
linked with MED data-structure. All interpolators is completely hidden
to you. Even sparse interpolation matrix is hidden. An exemple of
usage :

\code
...
std::string sourceFileName("source.med");
MEDMEM::MESH med_source_mesh(MED_DRIVER,sourceFileName,"Source_Mesh");
std::string targetFileName("target.med");
MEDMEM::MESH med_target_mesh(MED_DRIVER,targetFileName,"Target_Mesh");
FIELD<double> sourceField(MED_DRIVER,sourceFileName,"Density",0,0);
FIELD<double> targetField;
Remapper mapper;
mapper.prepare(med_source_mesh,med_target_mesh);
mapper.transfer(sourceField,targetField);
//use targetField
...
\endcode

\subsection InterpKerMidLevUsage middle-level usage

This mode is the mode that needs the minimum of prerequisites
(algorithms and the datastructure you intend to use). On the other
hand it is needed to specify precisely nature of interpolator.

As consequence of genericity of interpolators,  they are usable only by
instanciating an underneath mesh data structure. The two following
examples show how to use interpolator at this level.

- The simplest way to use the interpolator with MED datastructure is :

\code
...
std::string sourceFileName("source.med");
MEDMEM::MESH med_source_mesh(MED_DRIVER,sourceFileName,"Source_Mesh");
std::string targetFileName("target.med");
MEDMEM::MESH med_target_mesh(MED_DRIVER,targetFileName,"Target_Mesh");
// Ok at this point we have our mesh in MED-Memory format.
// Go to wrap med_source_mesh and med_target_mesh.
MEDNormalizedUnstructuredMesh<2,2> wrap_source_mesh(&med_source_mesh);
MEDNormalizedUnstructuredMesh<2,2> wrap_target_mesh(&med_target_mesh);
// Go for interpolation...
INTERP_KERNEL::Interpolation2D myInterpolator;
//optionnal call to parametrize your interpolation. First precision, tracelevel, intersector wanted.
myInterpolator.setOptions(1e-7,0,Geometric2D);
INTERP_KERNEL::Matrix<double,ALL_FORTRAN_MODE> resultMatrix;
myInterpolator.interpolateMeshes(wrap_source_mesh,wrap_target_mesh,resultMatrix,"P0P0");
//Ok let's multiply resultMatrix by source field to interpolate to target field.
resultMatrix.multiply(...)
...
\endcode

- Same with VTK datastructure :

\code
...
vtkXMLUnstructuredGridReader *readerSource=vtkXMLUnstructuredGridReader::New();
readerSource->SetFileName("source.vtu");
vtkUnstructuredGrid *vtk_source_mesh=readerSource->GetOutput();
readerSource->Update();
vtkXMLUnstructuredGridReader *readerTarget=vtkXMLUnstructuredGridReader::New();
readerTarget->SetFileName("target.vtu");
vtkUnstructuredGrid *vtk_target_mesh=readerTarget->GetOutput();
readerTarget->Update();
// Ok at this point we have our mesh in VTK format.
// Go to wrap vtk_source_mesh and vtk_target_mesh.
VTKNormalizedUnstructuredMesh<2> wrap_source_mesh(vtk_source_mesh);
VTKNormalizedUnstructuredMesh<2> wrap_target_mesh(vtk_target_mesh);
// Go for interpolation...
INTERP_KERNEL::Interpolation2D myInterpolator;
//optionnal call to parametrize your interpolation. First precision, tracelevel, intersector wanted.
myInterpolator.setOptions(1e-7,0,Geometric2D);
INTERP_KERNEL::Matrix<double,ALL_C_MODE> resultMatrix;
myInterpolator.interpolateMeshes(wrap_source_mesh,wrap_target_mesh,resultMatrix,"P0P0");
//Ok let's multiply resultMatrix by source field to interpolate to target field.
resultMatrix.multiply(...)
//clean-up
readerSource->Delete();
readerTarget->Delete();
...
\endcode

*/