yaspgridentity.hh

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// -*- tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 2 -*-
// vi: set et ts=4 sw=2 sts=2:
#ifndef DUNE_GRID_YASPGRIDENTITY_HH
#define DUNE_GRID_YASPGRIDENTITY_HH
 
//========================================================================
 
 
 
 
namespace Dune {
 
  namespace Yasp {
 
#ifndef DOXYGEN
 
    // table for quick evaluation of binomial coefficients
    template<int n>
    struct BinomialTable
    {
      // evaluation - note that in general d!=n, n is only the
      // maximum value of d (in our case dimworld)
      static constexpr int evaluate(int d, int c)
      {
        return _values[_offsets[d] + c];
      }
 
      // the actual implementation
      static constexpr int binomial(int d, int c)
      {
        long binomial=1;
        for (int i=d-c+1; i<=d; i++)
          binomial *= i;
        for (long i=2; i<=c; i++)
          binomial /= i;
        return binomial;
      }
 
    private:
      // prevent construction
      BinomialTable() = delete;
 
      // compute binomial(r, c) and advance row `r` and column `c`
      static constexpr int nextValue(int& r, int& c)
        {
          const auto result = binomial(r, c);
 
          c += 1;
          if (c > r) {
            r += 1;
            c = 0;
          }
 
          return result;
        }
 
      template<std::size_t... I>
      static constexpr std::array<int, sizeof...(I)> computeValues(std::index_sequence<I...>)
        {
          int r = 0, c = 0;
          return {{ ((void)I, nextValue(r, c))... }};
        }
 
      template<std::size_t... I>
      static constexpr std::array<int, sizeof...(I)> computeOffsets(std::index_sequence<I...>)
        { return {{ (I*(I+1)/2)... }}; }
 
      static constexpr std::array<int,(n+1)*(n+2)/2> _values = computeValues(std::make_index_sequence<(n+1)*(n+2)/2>{});
      static constexpr std::array<int,n+1> _offsets = computeOffsets(std::make_index_sequence<n+1>{});
    };
 
#if __cplusplus < 201703L
    template<int n>
    constexpr std::array<int,(n+1)*(n+2)/2> BinomialTable<n>::_values;
    template<int n>
    constexpr std::array<int,n+1> BinomialTable<n>::_offsets;
#endif
 
    template<int dimworld>
    constexpr int subEnt(int d, int c)
    {
      return (d < c ? 0 : BinomialTable<dimworld>::evaluate(d,c) << c);
    }
 
    // Make a table mapping all subentities of a codim 0 entity to a value.
    // F is the functor to be evaluated.
    template<typename F, int dim>
    struct EntityShiftTable
    {
      typedef std::bitset<dim> value_type;
 
      static value_type evaluate(int i, int codim)
      {
        return {_values[_offsets[codim] + i]};
      }
 
    private:
 
      // prevent construction
      EntityShiftTable() = delete;
 
      // compute offset of codimension `codim` entities and advance `offset`
      static constexpr int nextOffset(int& offset, int codim)
        {
          if (codim == 0) {
            offset = 0;
            return 0;
          }
 
          offset += subEnt<dim>(dim, codim-1);
          return offset;
        }
 
      template<std::size_t... I>
      static constexpr std::array<int, sizeof...(I)> computeOffsets(std::index_sequence<I...>)
        {
          int offset = 0;
          return {{ (nextOffset(offset, I))... }};
        }
 
      // compute shift table entry for (`codim`, `i`) and advance `codim`, `i`
      static constexpr unsigned char nextValue(int& codim, int& i)
        {
          const auto result = F::evaluate(i, codim);
 
          i += 1;
          if (i >= subEnt<dim>(dim, codim)) {
            codim += 1;
            i = 0;
          }
 
          return result;
        }
 
      template<std::size_t... I>
      static constexpr std::array<unsigned char, sizeof...(I)> computeValues(std::index_sequence<I...>)
        {
          int codim = 0, i = 0;
          return {{ ((void)I, nextValue(codim, i))... }};
        }
 
      static constexpr std::array<int,dim+1> _offsets = computeOffsets(std::make_index_sequence<dim+1>{});
      static constexpr std::array<unsigned char,StaticPower<3,dim>::power> _values = computeValues(std::make_index_sequence<StaticPower<3,dim>::power>{});
 
    };
 
#if __cplusplus < 201703L
    template<typename F, int dim>
    constexpr std::array<int,dim+1> EntityShiftTable<F, dim>::_offsets;
    template<typename F, int dim>
    constexpr std::array<unsigned char,StaticPower<3,dim>::power> EntityShiftTable<F, dim>::_values;
#endif
 
    // functor for doing the actual entity shift calculation
    template<int dim>
    struct calculate_entity_shift
    {
      static constexpr unsigned long long evaluate(int index, int cc)
      {
        auto result = 0ull;
        for (int d = dim; d>0; d--)
          {
            if (cc == d)
              return result;
            if (index < subEnt<dim>(d-1,cc))
              result |= 1ull << (d-1);
            else
              {
                index = (index - subEnt<dim>(d-1, cc)) % subEnt<dim>(d-1,cc-1);
                cc--;
              }
          }
        return result;
      }
    };
 
    template<int dim>
    std::bitset<dim> entityShift(int index, int cc)
    {
      return EntityShiftTable<calculate_entity_shift<dim>,dim>::evaluate(index,cc);
    }
 
    // functor for doing the actual entity move calculation
    template<int dim>
    struct calculate_entity_move
    {
      static constexpr unsigned long long evaluate(int index, int cc)
      {
        auto result = 0ull;
        for (int d = dim; d>0; d--)
          {
            if (d == cc)
              {
                // result[d-1] = index & (1<<(d-1));
                result &= ~(1ull << (d-1));
                result |= index & (1ull << (d-1));
 
                index &= ~(1<<(d-1));
              }
            if (index >= subEnt<dim>(d-1,cc))
              {
                if ((index - subEnt<dim>(d-1,cc)) / subEnt<dim>(d-1,cc-1) == 1)
                  {
                    result |= 1ull << (d-1);
                  }
                index = (index - subEnt<dim>(d-1, cc)) % subEnt<dim>(d-1,cc-1);
                cc--;
              }
          }
        return result;
      }
 
    };
 
    template<int dim>
    std::bitset<dim> entityMove(int index, int cc)
    {
      return EntityShiftTable<calculate_entity_move<dim>,dim>::evaluate(index,cc);
    }
 
#endif //DOXYGEN
 
  } // namespace Yasp.
 
  template<int codim, int dim, class GridImp>
  class YaspEntity
    :  public EntityDefaultImplementation <codim,dim,GridImp,YaspEntity>
  {
 
    template<int, PartitionIteratorType, typename>
    friend class YaspLevelIterator;
 
  public:
    typedef typename GridImp::ctype ctype;
 
    typedef typename GridImp::template Codim<codim>::Geometry Geometry;
    typedef typename GridImp::Traits::template Codim<codim>::GeometryImpl GeometryImpl;
 
    typedef typename GridImp::template Codim<codim>::EntitySeed EntitySeed;
 
    int level () const
    {
      return _g->level();
    }
 
    EntitySeed seed() const
    {
      return EntitySeed(YaspEntitySeed<codim,GridImp>(_g->level(), _it.coord(), _it.which()));
    }
 
    Geometry geometry () const
    {
      GeometryImpl _geometry(_it.lowerleft(),_it.upperright(),_it.shift());
      return Geometry(_geometry);
    }
 
    unsigned int subEntities (unsigned int cc) const
    {
      return Dune::Yasp::subEnt<dim>(dim-codim,cc-codim);
    }
 
    PartitionType partitionType () const
    {
      if (_g->interior[codim].inside(_it.coord(),_it.shift()))
        return InteriorEntity;
      if (_g->interiorborder[codim].inside(_it.coord(),_it.shift()))
        return BorderEntity;
      if (_g->overlap[codim].inside(_it.coord(),_it.shift()))
        return OverlapEntity;
      if (_g->overlapfront[codim].inside(_it.coord(),_it.shift()))
        return FrontEntity;
      return GhostEntity;
    }
 
    typedef typename GridImp::YGridLevelIterator YGLI;
    typedef typename GridImp::YGrid::Iterator I;
    YaspEntity ()
    {}
 
    YaspEntity (const YGLI& g, const I& it)
      : _it(it), _g(g)
    {}
 
    YaspEntity (YGLI&& g, const I&& it)
      : _it(std::move(it)), _g(std::move(g))
    {}
 
    bool equals (const YaspEntity& e) const
    {
      return _it == e._it && _g == e._g;
    }
 
    // IndexSets needs access to the private index methods
    friend class Dune::YaspIndexSet<GridImp,true>;
    friend class Dune::YaspIndexSet<GridImp,false>;
    friend class Dune::YaspGlobalIdSet<GridImp>;
    typedef typename GridImp::PersistentIndexType PersistentIndexType;
 
    PersistentIndexType persistentIndex () const
    {
      // get size of global grid (in elements)
      std::array<int,dim> size;
 
      for (int i=0; i<dim; i++)
      {
        // correct size according to shift
        size[i] = _g->mg->levelSize(_g->level(), i);
        if (!_it.shift(i))
          size[i]++;
      }
 
      // encode codim
      PersistentIndexType id(_it.shift().to_ulong());
 
      // encode level
      id = id << yaspgrid_level_bits;
      id = id+PersistentIndexType(_g->level());
 
      // encode coordinates
      for (int i=dim-1; i>=0; i--)
      {
        id = id << yaspgrid_dim_bits;
        id = id+PersistentIndexType(_it.coord(i));
      }
 
      return id;
    }
 
    int compressedIndex () const
    {
      return _it.superindex();
    }
 
    int subCompressedIndex (int i, unsigned int cc) const
    {
      // get the shift of the entity and the subentity
      // the subentity shift is only available in the space spanned by the entity
      std::bitset<dim-codim> subent_shift = Dune::Yasp::entityShift<dim-codim>(i,cc-codim);
      std::bitset<dim-codim> subent_move = Dune::Yasp::entityMove<dim-codim>(i,cc-codim);
 
      std::bitset<dim> shift = _it.shift();
      std::array<int, dim> coord = _it.coord();
      for (int j=0, k=0; j<dim; j++)
      {
        if (!shift[j])
          continue;
 
        coord[j] += subent_move[k];
        shift[j] = subent_shift[k];
        k++;
      }
 
      int which = _g->overlapfront[cc].shiftmapping(shift);
      return _g->overlapfront[cc].superindex(coord,which);
    }
    public:
    const I& transformingsubiterator() const { return _it; }
    const YGLI& gridlevel() const { return _g; }
    I& transformingsubiterator() { return _it; }
    YGLI& gridlevel() { return _g; }
    const GridImp * yaspgrid() const { return _g->mg; }
    protected:
    I _it;               // position in the grid level
    YGLI _g;               // access to grid level
  };
 
 
  // specialization for codim=0
  template<int dim, class GridImp>
  class YaspEntity<0,dim,GridImp>
    : public EntityDefaultImplementation <0,dim,GridImp,YaspEntity>
  {
    enum { dimworld = GridImp::dimensionworld };
 
    typedef typename GridImp::Traits::template Codim< 0 >::GeometryImpl GeometryImpl;
 
    template<int, PartitionIteratorType, typename>
    friend class YaspLevelIterator;
 
    template<typename>
    friend class YaspHierarchicIterator;
 
  public:
    typedef typename GridImp::ctype ctype;
 
    typedef typename GridImp::YGridLevelIterator YGLI;
    typedef typename GridImp::YGrid::Iterator I;
 
    typedef typename GridImp::template Codim< 0 >::Geometry Geometry;
    typedef typename GridImp::template Codim< 0 >::LocalGeometry LocalGeometry;
 
    template <int cd>
    struct Codim
    {
      typedef typename GridImp::template Codim<cd>::Entity Entity;
    };
 
    typedef typename GridImp::template Codim<0>::Entity Entity;
    typedef typename GridImp::template Codim<0>::EntitySeed EntitySeed;
    typedef typename GridImp::LevelIntersectionIterator IntersectionIterator;
    typedef typename GridImp::LevelIntersectionIterator LevelIntersectionIterator;
    typedef typename GridImp::LeafIntersectionIterator LeafIntersectionIterator;
    typedef typename GridImp::HierarchicIterator HierarchicIterator;
 
    typedef typename GridImp::PersistentIndexType PersistentIndexType;
 
    typedef typename GridImp::YGrid::iTupel iTupel;
 
    // constructor
    YaspEntity ()
    {}
 
    YaspEntity (const YGLI& g, const I& it)
      : _it(it), _g(g)
    {}
 
    YaspEntity (const YGLI& g, I&& it)
      : _it(std::move(it)), _g(g)
    {}
 
    YaspEntity (YGLI&& g, I&& it)
      : _it(std::move(it)), _g(std::move(g))
    {}
 
    bool equals (const YaspEntity& e) const
    {
      return _it == e._it && _g == e._g;
    }
 
    int level () const { return _g->level(); }
 
    EntitySeed seed () const {
      return EntitySeed(YaspEntitySeed<0,GridImp>(_g->level(), _it.coord()));
    }
 
    PartitionType partitionType () const
    {
      if (_g->interior[0].inside(_it.coord(),_it.shift()))
        return InteriorEntity;
      if (_g->overlap[0].inside(_it.coord(),_it.shift()))
        return OverlapEntity;
      DUNE_THROW(GridError, "Impossible GhostEntity");
      return GhostEntity;
    }
 
    Geometry geometry () const {
      // the element geometry
      auto ll = _it.lowerleft();
      auto ur = _it.upperright();
 
      // If on periodic overlap, transform coordinates by domain size
      for (int i=0; i<dimworld; i++) {
        if (gridlevel()->mg->isPeriodic(i)) {
          int coord = transformingsubiterator().coord(i);
          if (coord < 0) {
            auto size = _g->mg->domainSize()[i];
            ll[i] += size;
            ur[i] += size;
          } else if (coord + 1 > gridlevel()->mg->levelSize(gridlevel()->level(),i)) {
            auto size = _g->mg->domainSize()[i];
            ll[i] -= size;
            ur[i] -= size;
          }
        }
      }
 
      GeometryImpl _geometry(ll,ur);
      return Geometry( _geometry );
    }
 
    template<int cc> int count () const
    {
      return Dune::Yasp::subEnt<dim>(dim,cc);
    }
 
    unsigned int subEntities (unsigned int codim) const
    {
      return Dune::Yasp::subEnt<dim>(dim,codim);
    }
 
    template<int cc>
    typename Codim<cc>::Entity subEntity (int i) const
    {
      // calculate move bitset
      std::bitset<dim> move = Dune::Yasp::entityMove<dim>(i,cc);
 
      // get the coordinate and modify it
      iTupel coord = _it.coord();
      for (int j=0; j<dim; j++)
        if (move[j])
          coord[j]++;
 
      int which = _g->overlapfront[cc].shiftmapping(Dune::Yasp::entityShift<dim>(i,cc));
      return typename Codim<cc>::Entity(YaspEntity<cc,GridImp::dimension,GridImp>(_g,_g->overlapfront[cc].begin(coord, which)));
    }
 
    Entity father () const
    {
      // check if coarse level exists
      if (_g->level()<=0)
        DUNE_THROW(GridError, "tried to call father on level 0");
 
      // yes, get iterator to it
      YGLI cg(_g);
      --cg;
 
      // coordinates of the cell
      iTupel coord = _it.coord();
 
      // get coordinates on next coarser level
      for (int k=0; k<dim; k++) coord[k] = coord[k]/2;
 
      return Entity(YaspEntity<0,GridImp::dimension,GridImp>(cg,cg->overlap[0].begin(coord)));
    }
 
    bool hasFather () const
    {
      return (_g->level()>0);
    }
 
    LocalGeometry geometryInFather () const
    {
      // configure one of the 2^dim transformations
      FieldVector<ctype,dim> ll(0.0),ur(0.5);
 
      for (int k=0; k<dim; k++)
      {
        if (_it.coord(k)%2)
        {
          ll[k] = 0.5;
          ur[k] = 1.0;
        }
      }
 
      return LocalGeometry( YaspGeometry<dim,dim,GridImp>(ll,ur) );
    }
 
    const I& transformingsubiterator () const { return _it; }
    const YGLI& gridlevel () const { return _g; }
    I& transformingsubiterator() { return _it; }
    YGLI& gridlevel() { return _g; }
    const GridImp* yaspgrid () const { return _g->mg; }
 
    bool isLeaf() const
    {
      return (_g->level() == yaspgrid()->maxLevel());
    }
 
    bool isNew () const { return yaspgrid()->adaptRefCount > 0 && yaspgrid()->maxLevel() < _g->level() + yaspgrid()->adaptRefCount; }
 
    bool mightVanish () const { return false; }
 
    IntersectionIterator ibegin () const
    {
      return YaspIntersectionIterator<GridImp>(*this,false);
    }
 
    LeafIntersectionIterator ileafbegin () const
    {
      // only if entity is leaf this iterator delivers intersections
      return YaspIntersectionIterator<GridImp>(*this, ! isLeaf() );
    }
 
    LevelIntersectionIterator ilevelbegin () const
    {
      return ibegin();
    }
 
    IntersectionIterator iend () const
    {
      return YaspIntersectionIterator<GridImp>(*this,true);
    }
 
    LeafIntersectionIterator ileafend () const
    {
      return iend();
    }
 
    LevelIntersectionIterator ilevelend () const
    {
      return iend();
    }
 
    HierarchicIterator hbegin (int maxlevel) const
    {
      return YaspHierarchicIterator<GridImp>(_g,_it,maxlevel);
    }
 
    HierarchicIterator hend (int /* maxlevel */) const
    {
      return YaspHierarchicIterator<GridImp>(_g,_it,_g->level());
    }
 
  private:
    // IndexSets needs access to the private index methods
    friend class Dune::YaspIndexSet<GridImp,true>;
    friend class Dune::YaspIndexSet<GridImp,false>;
    friend class Dune::YaspGlobalIdSet<GridImp>;
 
    PersistentIndexType persistentIndex () const
    {
      // encode codim
      PersistentIndexType id(_it.shift().to_ulong());
 
      // encode level
      id = id << yaspgrid_level_bits;
      id = id+PersistentIndexType(_g->level());
 
 
      // encode coordinates
      for (int i=dim-1; i>=0; i--)
      {
        id = id << yaspgrid_dim_bits;
        id = id+PersistentIndexType(_it.coord(i));
      }
 
      return id;
    }
 
    int compressedIndex () const
    {
      return _it.superindex();
    }
 
    PersistentIndexType subPersistentIndex (int i, int cc) const
    {
      // calculate shift and move bitsets
      std::bitset<dim> shift = Dune::Yasp::entityShift<dim>(i,cc);
      std::bitset<dim> move = Dune::Yasp::entityMove<dim>(i,cc);
 
      int trailing = (cc == dim) ? 1000 : 0;
 
      std::array<int,dim> size = _g->mg->levelSize(_g->level());
      std::array<int, dim> coord = _it.coord();
      for (int j=0; j<dim; j++)
      {
        // correct size according to shift
        if (!shift[j])
          size[j]++;
 
        // move the coordinates to the cell on which the entity lives
        if (move[j])
          coord[j]++;
      }
 
      for (int j=0; j<dim; j++)
      {
        // in the codim==dim case, count trailing zeroes.
        if (cc == dim)
        {
          int zeroes = 0;
          for (int k=0; k<_g->level(); k++)
            if (coord[j] & (1<<k))
              break;
            else
              zeroes++;
          trailing = std::min(trailing,zeroes);
        }
      }
 
      // encode codim
      PersistentIndexType id(shift.to_ulong());
 
      // encode level
      id = id << yaspgrid_level_bits;
      id = id+PersistentIndexType(_g->level()-trailing);
 
      // encode coordinates
      for (int j=dim-1; j>=0; j--)
      {
        id = id << yaspgrid_dim_bits;
        id = id+PersistentIndexType(coord[j]>>trailing);
      }
 
      return id;
    }
 
    int subCompressedIndex (int i, int cc) const
    {
      // get shift and move of the subentity in question
      std::bitset<dim> shift = Dune::Yasp::entityShift<dim>(i,cc);
      std::bitset<dim> move = Dune::Yasp::entityMove<dim>(i,cc);
 
      std::array<int, dim> coord = _it.coord();
      for (int j=0; j<dim; j++)
        coord[j] += move[j];
 
      int which = _g->overlapfront[cc].shiftmapping(shift);
      return _g->overlapfront[cc].superindex(coord,which);
    }
 
    I _it;         // position in the grid level
    YGLI _g;         // access to grid level
  };
 
 
  // specialization for codim=dim (vertex)
  template<int dim, class GridImp>
  class YaspEntity<dim,dim,GridImp>
    : public EntityDefaultImplementation <dim,dim,GridImp,YaspEntity>
  {
    enum { dimworld = GridImp::dimensionworld };
 
    template<int, PartitionIteratorType, typename>
    friend class YaspLevelIterator;
 
    typedef typename GridImp::Traits::template Codim<dim>::GeometryImpl GeometryImpl;
 
  public:
    typedef typename GridImp::ctype ctype;
 
    typedef typename GridImp::YGridLevelIterator YGLI;
    typedef typename GridImp::YGrid::Iterator I;
 
    typedef typename GridImp::template Codim<dim>::Geometry Geometry;
 
    typedef typename GridImp::template Codim<dim>::EntitySeed EntitySeed;
 
    typedef typename GridImp::PersistentIndexType PersistentIndexType;
 
    typedef typename GridImp::YGrid::iTupel iTupel;
 
    // constructor
    YaspEntity ()
    {}
 
    YaspEntity (const YGLI& g, const I& it)
      : _it(it), _g(g)
    {}
 
    YaspEntity (YGLI&& g, I&& it)
      : _it(std::move(it)), _g(std::move(g))
    {}
 
    bool equals (const YaspEntity& e) const
    {
      return _it == e._it && _g == e._g;
    }
 
    int level () const {return _g->level();}
 
    EntitySeed seed () const {
      return EntitySeed(YaspEntitySeed<dim,GridImp>(_g->level(), _it.coord(), _it.which()));
    }
 
    unsigned int subEntities (unsigned int cc) const
    {
      return Dune::Yasp::subEnt<dim>(dim-dim,cc-dim);
    }
 
    Geometry geometry () const {
      GeometryImpl _geometry((_it).lowerleft());
      return Geometry( _geometry );
    }
 
    PartitionType partitionType () const
    {
      if (_g->interior[dim].inside(_it.coord(),_it.shift()))
        return InteriorEntity;
      if (_g->interiorborder[dim].inside(_it.coord(),_it.shift()))
        return BorderEntity;
      if (_g->overlap[dim].inside(_it.coord(),_it.shift()))
        return OverlapEntity;
      if (_g->overlapfront[dim].inside(_it.coord(),_it.shift()))
        return FrontEntity;
      return GhostEntity;
    }
 
    int subCompressedIndex (int, unsigned int ) const
    {
      return compressedIndex();
    }
 
  private:
    // IndexSets needs access to the private index methods
    friend class Dune::YaspIndexSet<GridImp,true>;
    friend class Dune::YaspIndexSet<GridImp,false>;
    friend class Dune::YaspGlobalIdSet<GridImp>;
 
    PersistentIndexType persistentIndex () const
    {
      // get coordinate and size of global grid
      iTupel size = _g->mg->levelSize(_g->level());
 
      for (int i=0; i<dim; i++)
      {
        // we have vertices, add 1 size to all directions
        size[i]++;
      }
 
      // determine min number of trailing zeroes
      int trailing = 1000;
      for (int i=0; i<dim; i++)
      {
        // count trailing zeros
        int zeros = 0;
        for (int j=0; j<_g->level(); j++)
          if (_it.coord(i)&(1<<j))
            break;
          else
            zeros++;
        trailing = std::min(trailing,zeros);
      }
 
      // determine the level of this vertex
      int level = _g->level()-trailing;
 
      // encode codim: shift vector of vertices is 0.
      PersistentIndexType id(0);
 
      // encode level
      id = id << yaspgrid_level_bits;
      id = id+PersistentIndexType(level);
 
      // encode coordinates
      for (int i=dim-1; i>=0; i--)
      {
        id = id << yaspgrid_dim_bits;
        id = id+PersistentIndexType(_it.coord(i)>>trailing);
      }
 
      return id;
    }
 
    int compressedIndex () const { return _it.superindex();}
 
  public:
    const I& transformingsubiterator() const { return _it; }
    const YGLI& gridlevel() const { return _g; }
    I& transformingsubiterator() { return _it; }
    YGLI& gridlevel() { return _g; }
 
    const GridImp * yaspgrid() const { return _g->mg; }
  protected:
    I _it;               // position in the grid level
    YGLI _g;               // access to grid level
  };
 
}   // namespace Dune
 
#endif  // DUNE_GRID_YASPGRIDENTITY_HH