Refinement

Modules
	Refinement implementation for hypercubes
	Refinement implementation for triangulating hypercubes
	Refinement implementation for simplices
	Virtual Refinement

Classes
class	Dune::RefinementIntervals
	Holds the number of refined intervals per axis needed for virtual and static refinement. More...
class	Dune::StaticRefinement< topologyId, CoordType, coerceToId, dimension_ >
	Wrap each Refinement implementation to get a consistent interface. More...

Functions
RefinementIntervals	Dune::refinementIntervals (int intervals)
	Creates a RefinementIntervals object. More...
RefinementIntervals	Dune::refinementLevels (int levels)
	Creates a RefinementIntervals object. More...

Detailed Description

General

The Refinement system allows to temporarily refine a grid or single entities without changing the grid itself. You may want to do this because you want to write your data to a file and have to do subsampling, but want to continue the calculation with the unmodified grid afterwards.

What Refinement can do for you

For a given geometry type and number of refined intervals, Refinement will

• assign consecutive integer indices starting at 0 to each subvertex,
• assign consecutive integer indices starting at 0 to each subelement,
• calculate the coordinates of the subvertices for you,
• calculate subvertex-indices of the corners of the subelements for you.

The geometry type of the refined entity and of the subelements may be different, for example you can refine a quadrilateral but get subelements which are triangles.

Currently the following geometry types are supported:

• hypercubes (quadrilaterals, hexahedrons),
• simplices (triangles, tetrahedrons),
• triangulating hypercubes into simplices (quadrilaterals -> triangles, hexahedrons -> tetrahedrons).

What Refinement can't do for you

• Refinement does not actually subsample your data, it only tells you where to subsample your data.
• The geometry types need to be known at compile time. See VirtualRefinement if you need to calculate the right geometry type at run time.
• No Refinement implementations for anything besides hypercubes and simplices have been written yet.

The user interface

template<unsigned topologyId, class CoordType,
         unsigned coerceToId, int dimension_>
class StaticRefinement
{
public:
  constexpr static int dimension = dimension_;
 
  template<int codimension>
  struct codim
  {
    class SubEntityIterator;
  };
  typedef ImplementationDefined VertexIterator;  // These are aliases for codim<codim>::SubEntityIterator
  typedef ImplementationDefined ElementIterator;
 
  typedef ImplementationDefined IndexVector; // These are FieldVectors
  typedef ImplementationDefined CoordVector;
 
  static int nVertices(Dune::RefinementIntervals intervals);
  static VertexIterator vBegin(Dune::RefinementIntervals intervals);
  static VertexIterator vEnd(Dune::RefinementIntervals intervals);
 
  static int nElements(Dune::RefinementIntervals intervals);
  static ElementIterator eBegin(Dune::RefinementIntervals intervals);
  static ElementIterator eEnd(Dune::RefinementIntervals intervals);
}

The Iterators can do all the usual things that Iterators can do, except dereferencing. In addition, to do something useful, they support some additional methods:

template<unsigned topologyId, class CoordType, unsigned coerceToId, int dimension>
class VertexIterator
{
public:
  typedef ImplementationDefined Refinement;
 
  int index() const;
  Refinement::CoordVector coords() const;
}
 
template<unsigned topologyId, class CoordType, unsigned coerceToId, int dimension>
class ElementIterator
{
public:
  typedef ImplementationDefined Refinement;
 
  int index() const;
  // Coords of the center of mass of the element
  Refinement::CoordVector coords() const;
  Refinement::IndexVector vertexIndices() const;
}

How to use it

Either use VirtualRefinement, or if you don't want to do that, read on.

// Include the necessary files
#include <dune/grid/common/refinement.hh>
 
// If you know that you are only ever going to need one refinement
// backend, you can include the corresponding file directly:
//#include <dune/grid/common/refinement/hcube.cc>
 
// Get yourself the Refinement you need:
typedef StaticRefinement<GeometryTypes::cube(2),
                         SGrid<2, 2>::ctype,
                         GeometryTypes::cube(2),
                         2> MyRefinement;
 
int main()
{
  const Dune::RefinementIntervals refinementlevel = Dune::refinementLevels(2); // equivalent: Dune::refinementIntervals(4)
  cout << "Using refinementlevel = " << refinementlevel.intervals() << endl << endl;
 
  // get Number of Vertices
  cout << "Number of Vertices: "
       << MyRefinement::nVertices(refinementlevel)
       << endl;
 
  // Iterate over Vertices
  cout << "Index\tx\ty" << endl;
  MyRefinement::VertexIterator vEnd = MyRefinement::vEnd(refinementlevel);
  for(MyRefinement::VertexIterator i = MyRefinement::vBegin(refinementlevel); i != vEnd; ++i)
    cout << i.index() << "\t" << i.coords()[0] << "\t" << i.coords()[1] << endl;
  cout << endl;
 
  // Iterate over Vertices
  cout << "Index\tEcke0\tEcke1\tEcke2\tEcke3" << endl;
  MyRefinement::ElementIterator eEnd = MyRefinement::eEnd(refinementlevel);
  for(MyRefinement::ElementIterator i = MyRefinement::eBegin(refinementlevel); i != eEnd; ++i)
    cout << i.index() << "\t"
         << i.indexVertices()[0] << "\t" << i.indexVertices()[1] << "\t"
         << i.indexVertices()[2] << "\t" << i.indexVertices()[3] << endl;
  cout << endl;
}

Guarantees

The Refinement system gives this guarantee (besides conforming to the above interface:

• The indices of the subvertices and subelement start at 0 and are consecutive.

Implementing a new Refinement type

If you want to write a Refinement implementation for a particular geometry type, e.g. SquaringTheCircle (or a particular set of geometry types) here is how:

• create a file refinement/squaringthecircle.cc and #include "base.cc". Your file will be included by others, so don't forget to protect against double inclusion.
• implement a class (or template class) RefinementImp conforming exactly to the user interface above.
• put it (and it's helper stuff as appropriate) into it's own namespace Dune::RefinementImp::SquaringTheCircle.
• define the mapping of topologyId, CoordType and coerceToId to your implementation by specialising template struct RefinementImp::Traits. It should look like this:

  namespace Dune::RefinementImp {
    // we're only implementing this for dim=2
    template<class CoordType>
    struct Traits<sphereTopologyId, CoordType,
                  GeometryTypes::cube(2), 2>
  {
      typedef SquaringTheCircle::RefinementImp<CoordType> Imp;
    };
  }

  If you implement a template class, you have to specialise struct RefinementImp::Traits for every possible combination of topologyId and coerceToId that your implementation supports.
• #include "refinement/squaringthecircle.cc" from refinement.hh.

This is enough to integrate your implementation into the Refinement system. You probably want to include it into VirtualRefinement also.

Namespaces

The (non-virtual) Refinement system is organized in the following way into namespaces:

• Only template class StaticRefinement lives directly in namespace Dune.
• Use namespace Dune::RefinementImp for all the Implementation.
• Use namespace Dune::RefinementImp::HCube, namespace Dune::RefinementImp::Simplex, ... for each implementation.

The complete VirtualRefinement stuff is directly in namespace Dune.

Conceptual layers

• Layer 0 declares struct RefinementImp::Traits<topologyId, CoordType, coerceToId, dim>. It's member typedef Imp tells which Refinement implementation to use for a given topologyId (and CoordType). It is located in refinementbase.cc.
• Layer 1 defines RefinementImp::XXX::RefinementImp. It implements the Refinements for each topologyId, coerceToId (and CoordType). Also in this layer are the definitions of struct RefinementImp::Traits. This layer is located in refinementXXX.cc.
• Layer 2 puts it all together. It defines class StaticRefinement<topologyId, CoordType, coerceToId, dim> by deriving from the corresponding RefinementImp. It is located in refinementbase.cc.
• There is a dummy layer 2.5 which simply includes all the refinementXXX.cc files. It is located in refinement.cc.

VirtualRefinement adds two more layers to the ones defined here.

Function Documentation

refinementIntervals()

RefinementIntervals Dune::refinementIntervals ( int  intervals )

inline

Creates a RefinementIntervals object.

Parameters

intervals

Number of refined intervals per axis

refinementLevels()

RefinementIntervals Dune::refinementLevels ( int  levels )

inline

Creates a RefinementIntervals object.

Parameters

levels

Number of refinement levels, translates to \(2^{levels}\) intervals per axis