multilineargeometry.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_GEOMETRY_MULTILINEARGEOMETRY_HH
#define DUNE_GEOMETRY_MULTILINEARGEOMETRY_HH
 
#include <cassert>
#include <functional>
#include <iterator>
#include <limits>
#include <vector>
 
#include <dune/common/fmatrix.hh>
#include <dune/common/fvector.hh>
#include <dune/common/typetraits.hh>
 
#include <dune/geometry/affinegeometry.hh>
#include <dune/geometry/referenceelements.hh>
#include <dune/geometry/type.hh>
 
namespace Dune
{
 
  // MultiLinearGeometryTraits
  // -------------------------
 
  template< class ct >
  struct MultiLinearGeometryTraits
  {
    typedef Impl::FieldMatrixHelper< ct > MatrixHelper;
 
    static ct tolerance () { return ct( 16 ) * std::numeric_limits< ct >::epsilon(); }
 
    template< int mydim, int cdim >
    struct CornerStorage
    {
      typedef std::vector< FieldVector< ct, cdim > > Type;
    };
 
    template< int dim >
    struct hasSingleGeometryType
    {
      static const bool v = false;
      static const unsigned int topologyId = ~0u;
    };
  };
 
 
 
  // MultiLinearGeometry
  // -------------------
 
  template< class ct, int mydim, int cdim, class Traits = MultiLinearGeometryTraits< ct > >
  class MultiLinearGeometry
  {
    typedef MultiLinearGeometry< ct, mydim, cdim, Traits > This;
 
  public:
    typedef ct ctype;
 
    static const int mydimension= mydim;
    static const int coorddimension = cdim;
 
    typedef FieldVector< ctype, mydimension > LocalCoordinate;
    typedef FieldVector< ctype, coorddimension > GlobalCoordinate;
    typedef ctype Volume;
 
    typedef FieldMatrix< ctype, mydimension, coorddimension > JacobianTransposed;
 
    class JacobianInverseTransposed;
 
  protected:
 
    typedef Dune::ReferenceElements< ctype, mydimension > ReferenceElements;
 
  public:
 
    typedef typename ReferenceElements::ReferenceElement ReferenceElement;
 
  private:
    static const bool hasSingleGeometryType = Traits::template hasSingleGeometryType< mydimension >::v;
 
  protected:
    typedef typename Traits::MatrixHelper MatrixHelper;
    typedef typename std::conditional< hasSingleGeometryType, std::integral_constant< unsigned int, Traits::template hasSingleGeometryType< mydimension >::topologyId >, unsigned int >::type TopologyId;
 
  public:
    template< class Corners >
    MultiLinearGeometry ( const ReferenceElement &refElement,
                          const Corners &corners )
      : refElement_( refElement ),
        corners_( corners )
    {}
 
    template< class Corners >
    MultiLinearGeometry ( Dune::GeometryType gt,
                          const Corners &corners )
      : refElement_( ReferenceElements::general( gt ) ),
        corners_( corners )
    {}
 
    bool affine () const
    {
      JacobianTransposed jt;
      return affine( jt );
    }
 
    Dune::GeometryType type () const { return GeometryType( toUnsignedInt(topologyId()), mydimension ); }
 
    int corners () const { return refElement().size( mydimension ); }
 
    GlobalCoordinate corner ( int i ) const
    {
      assert( (i >= 0) && (i < corners()) );
      return std::cref(corners_).get()[ i ];
    }
 
    GlobalCoordinate center () const { return global( refElement().position( 0, 0 ) ); }
 
    GlobalCoordinate global ( const LocalCoordinate &local ) const
    {
      using std::begin;
 
      auto cit = begin(std::cref(corners_).get());
      GlobalCoordinate y;
      global< false >( topologyId(), std::integral_constant< int, mydimension >(), cit, ctype( 1 ), local, ctype( 1 ), y );
      return y;
    }
 
    LocalCoordinate local ( const GlobalCoordinate &globalCoord ) const
    {
      const ctype tolerance = Traits::tolerance();
      LocalCoordinate x = refElement().position( 0, 0 );
      LocalCoordinate dx;
      const bool affineMapping = this->affine();
      do
      {
        // Newton's method: DF^n dx^n = F^n, x^{n+1} -= dx^n
        const GlobalCoordinate dglobal = (*this).global( x ) - globalCoord;
        const bool invertible =
          MatrixHelper::template xTRightInvA< mydimension, coorddimension >( jacobianTransposed( x ), dglobal, dx );
        if( ! invertible )
          return LocalCoordinate( std::numeric_limits< ctype > :: max() );
 
        // update x with correction
        x -= dx;
 
        // for affine mappings only one iteration is needed
        if ( affineMapping ) break;
      } while( dx.two_norm2() > tolerance );
      return x;
    }
 
    ctype integrationElement ( const LocalCoordinate &local ) const
    {
      return MatrixHelper::template sqrtDetAAT< mydimension, coorddimension >( jacobianTransposed( local ) );
    }
 
    Volume volume () const
    {
      return integrationElement( refElement().position( 0, 0 ) ) * refElement().volume();
    }
 
    JacobianTransposed jacobianTransposed ( const LocalCoordinate &local ) const
    {
      using std::begin;
 
      JacobianTransposed jt;
      auto cit = begin(std::cref(corners_).get());
      jacobianTransposed< false >( topologyId(), std::integral_constant< int, mydimension >(), cit, ctype( 1 ), local, ctype( 1 ), jt );
      return jt;
    }
 
    JacobianInverseTransposed jacobianInverseTransposed ( const LocalCoordinate &local ) const;
 
    friend ReferenceElement referenceElement ( const MultiLinearGeometry &geometry )
    {
      return geometry.refElement();
    }
 
  protected:
 
    ReferenceElement refElement () const
    {
      return refElement_;
    }
 
    TopologyId topologyId () const
    {
      return topologyId( std::integral_constant< bool, hasSingleGeometryType >() );
    }
 
    template< bool add, int dim, class CornerIterator >
    static void global ( TopologyId topologyId, std::integral_constant< int, dim >,
                         CornerIterator &cit, const ctype &df, const LocalCoordinate &x,
                         const ctype &rf, GlobalCoordinate &y );
    template< bool add, class CornerIterator >
    static void global ( TopologyId topologyId, std::integral_constant< int, 0 >,
                         CornerIterator &cit, const ctype &df, const LocalCoordinate &x,
                         const ctype &rf, GlobalCoordinate &y );
 
    template< bool add, int rows, int dim, class CornerIterator >
    static void jacobianTransposed ( TopologyId topologyId, std::integral_constant< int, dim >,
                                     CornerIterator &cit, const ctype &df, const LocalCoordinate &x,
                                     const ctype &rf, FieldMatrix< ctype, rows, cdim > &jt );
    template< bool add, int rows, class CornerIterator >
    static void jacobianTransposed ( TopologyId topologyId, std::integral_constant< int, 0 >,
                                     CornerIterator &cit, const ctype &df, const LocalCoordinate &x,
                                     const ctype &rf, FieldMatrix< ctype, rows, cdim > &jt );
 
    template< int dim, class CornerIterator >
    static bool affine ( TopologyId topologyId, std::integral_constant< int, dim >, CornerIterator &cit, JacobianTransposed &jt );
    template< class CornerIterator >
    static bool affine ( TopologyId topologyId, std::integral_constant< int, 0 >, CornerIterator &cit, JacobianTransposed &jt );
 
    bool affine ( JacobianTransposed &jacobianT ) const
    {
      using std::begin;
 
      auto cit = begin(std::cref(corners_).get());
      return affine( topologyId(), std::integral_constant< int, mydimension >(), cit, jacobianT );
    }
 
  private:
    // The following methods are needed to convert the return type of topologyId to
    // unsigned int with g++-4.4. It has problems casting integral_constant to the
    // integral type.
    static unsigned int toUnsignedInt(unsigned int i) { return i; }
    template<unsigned int v>
    static unsigned int toUnsignedInt(std::integral_constant<unsigned int,v> ) { return v; }
    TopologyId topologyId ( std::integral_constant< bool, true > ) const { return TopologyId(); }
    unsigned int topologyId ( std::integral_constant< bool, false > ) const { return refElement().type().id(); }
 
    ReferenceElement refElement_;
    typename Traits::template CornerStorage< mydimension, coorddimension >::Type corners_;
  };
 
 
 
  // MultiLinearGeometry::JacobianInverseTransposed
  // ----------------------------------------------
 
  template< class ct, int mydim, int cdim, class Traits >
  class MultiLinearGeometry< ct, mydim, cdim, Traits >::JacobianInverseTransposed
    : public FieldMatrix< ctype, coorddimension, mydimension >
  {
    typedef FieldMatrix< ctype, coorddimension, mydimension > Base;
 
  public:
    void setup ( const JacobianTransposed &jt )
    {
      detInv_ = MatrixHelper::template rightInvA< mydimension, coorddimension >( jt, static_cast< Base & >( *this ) );
    }
 
    void setupDeterminant ( const JacobianTransposed &jt )
    {
      detInv_ = MatrixHelper::template sqrtDetAAT< mydimension, coorddimension >( jt );
    }
 
    ctype det () const { return ctype( 1 ) / detInv_; }
    ctype detInv () const { return detInv_; }
 
  private:
    ctype detInv_;
  };
 
 
 
  template< class ct, int mydim, int cdim, class Traits = MultiLinearGeometryTraits< ct > >
  class CachedMultiLinearGeometry
    : public MultiLinearGeometry< ct, mydim, cdim, Traits >
  {
    typedef CachedMultiLinearGeometry< ct, mydim, cdim, Traits > This;
    typedef MultiLinearGeometry< ct, mydim, cdim, Traits > Base;
 
  protected:
    typedef typename Base::MatrixHelper MatrixHelper;
 
  public:
    typedef typename Base::ReferenceElement ReferenceElement;
 
    typedef typename Base::ctype ctype;
 
    using Base::mydimension;
    using Base::coorddimension;
 
    typedef typename Base::LocalCoordinate LocalCoordinate;
    typedef typename Base::GlobalCoordinate GlobalCoordinate;
    typedef typename Base::Volume Volume;
 
    typedef typename Base::JacobianTransposed JacobianTransposed;
    typedef typename Base::JacobianInverseTransposed JacobianInverseTransposed;
 
    template< class CornerStorage >
    CachedMultiLinearGeometry ( const ReferenceElement &referenceElement, const CornerStorage &cornerStorage )
      : Base( referenceElement, cornerStorage ),
        affine_( Base::affine( jacobianTransposed_ ) ),
        jacobianInverseTransposedComputed_( false ),
        integrationElementComputed_( false )
    {}
 
    template< class CornerStorage >
    CachedMultiLinearGeometry ( Dune::GeometryType gt, const CornerStorage &cornerStorage )
      : Base( gt, cornerStorage ),
        affine_( Base::affine( jacobianTransposed_ ) ),
        jacobianInverseTransposedComputed_( false ),
        integrationElementComputed_( false )
    {}
 
    bool affine () const { return affine_; }
 
    using Base::corner;
 
    GlobalCoordinate center () const { return global( refElement().position( 0, 0 ) ); }
 
    GlobalCoordinate global ( const LocalCoordinate &local ) const
    {
      if( affine() )
      {
        GlobalCoordinate global( corner( 0 ) );
        jacobianTransposed_.umtv( local, global );
        return global;
      }
      else
        return Base::global( local );
    }
 
    LocalCoordinate local ( const GlobalCoordinate &global ) const
    {
      if( affine() )
      {
        LocalCoordinate local;
        if( jacobianInverseTransposedComputed_ )
          jacobianInverseTransposed_.mtv( global - corner( 0 ), local );
        else
          MatrixHelper::template xTRightInvA< mydimension, coorddimension >( jacobianTransposed_, global - corner( 0 ), local );
        return local;
      }
      else
        return Base::local( global );
    }
 
    ctype integrationElement ( const LocalCoordinate &local ) const
    {
      if( affine() )
      {
        if( !integrationElementComputed_ )
        {
          jacobianInverseTransposed_.setupDeterminant( jacobianTransposed_ );
          integrationElementComputed_ = true;
        }
        return jacobianInverseTransposed_.detInv();
      }
      else
        return Base::integrationElement( local );
    }
 
    Volume volume () const
    {
      if( affine() )
        return integrationElement( refElement().position( 0, 0 ) ) * refElement().volume();
      else
        return Base::volume();
    }
 
    JacobianTransposed jacobianTransposed ( const LocalCoordinate &local ) const
    {
      if( affine() )
        return jacobianTransposed_;
      else
        return Base::jacobianTransposed( local );
    }
 
    JacobianInverseTransposed jacobianInverseTransposed ( const LocalCoordinate &local ) const
    {
      if( affine() )
      {
        if( !jacobianInverseTransposedComputed_ )
        {
          jacobianInverseTransposed_.setup( jacobianTransposed_ );
          jacobianInverseTransposedComputed_ = true;
          integrationElementComputed_ = true;
        }
        return jacobianInverseTransposed_;
      }
      else
        return Base::jacobianInverseTransposed( local );
    }
 
  protected:
    using Base::refElement;
 
  private:
    mutable JacobianTransposed jacobianTransposed_;
    mutable JacobianInverseTransposed jacobianInverseTransposed_;
 
    mutable bool affine_ : 1;
 
    mutable bool jacobianInverseTransposedComputed_ : 1;
    mutable bool integrationElementComputed_ : 1;
  };
 
 
 
  // Implementation of MultiLinearGeometry
  // -------------------------------------
 
  template< class ct, int mydim, int cdim, class Traits >
  inline typename MultiLinearGeometry< ct, mydim, cdim, Traits >::JacobianInverseTransposed
  MultiLinearGeometry< ct, mydim, cdim, Traits >::jacobianInverseTransposed ( const LocalCoordinate &local ) const
  {
    JacobianInverseTransposed jit;
    jit.setup( jacobianTransposed( local ) );
    return jit;
  }
 
 
  template< class ct, int mydim, int cdim, class Traits >
  template< bool add, int dim, class CornerIterator >
  inline void MultiLinearGeometry< ct, mydim, cdim, Traits >
  ::global ( TopologyId topologyId, std::integral_constant< int, dim >,
             CornerIterator &cit, const ctype &df, const LocalCoordinate &x,
             const ctype &rf, GlobalCoordinate &y )
  {
    const ctype xn = df*x[ dim-1 ];
    const ctype cxn = ctype( 1 ) - xn;
 
    if( Impl::isPrism( toUnsignedInt(topologyId), mydimension, mydimension-dim ) )
    {
      // apply (1-xn) times mapping for bottom
      global< add >( topologyId, std::integral_constant< int, dim-1 >(), cit, df, x, rf*cxn, y );
      // apply xn times mapping for top
      global< true >( topologyId, std::integral_constant< int, dim-1 >(), cit, df, x, rf*xn, y );
    }
    else
    {
      assert( Impl::isPyramid( toUnsignedInt(topologyId), mydimension, mydimension-dim ) );
      // apply (1-xn) times mapping for bottom (with argument x/(1-xn))
      if( cxn > Traits::tolerance() || cxn < -Traits::tolerance() )
        global< add >( topologyId, std::integral_constant< int, dim-1 >(), cit, df/cxn, x, rf*cxn, y );
      else
        global< add >( topologyId, std::integral_constant< int, dim-1 >(), cit, df, x, ctype( 0 ), y );
      // apply xn times the tip
      y.axpy( rf*xn, *cit );
      ++cit;
    }
  }
 
  template< class ct, int mydim, int cdim, class Traits >
  template< bool add, class CornerIterator >
  inline void MultiLinearGeometry< ct, mydim, cdim, Traits >
  ::global ( TopologyId, std::integral_constant< int, 0 >,
             CornerIterator &cit, const ctype &, const LocalCoordinate &,
             const ctype &rf, GlobalCoordinate &y )
  {
    const GlobalCoordinate &origin = *cit;
    ++cit;
    for( int i = 0; i < coorddimension; ++i )
      y[ i ] = (add ? y[ i ] + rf*origin[ i ] : rf*origin[ i ]);
  }
 
 
  template< class ct, int mydim, int cdim, class Traits >
  template< bool add, int rows, int dim, class CornerIterator >
  inline void MultiLinearGeometry< ct, mydim, cdim, Traits >
  ::jacobianTransposed ( TopologyId topologyId, std::integral_constant< int, dim >,
                         CornerIterator &cit, const ctype &df, const LocalCoordinate &x,
                         const ctype &rf, FieldMatrix< ctype, rows, cdim > &jt )
  {
    assert( rows >= dim );
 
    const ctype xn = df*x[ dim-1 ];
    const ctype cxn = ctype( 1 ) - xn;
 
    auto cit2( cit );
    if( Impl::isPrism( toUnsignedInt(topologyId), mydimension, mydimension-dim ) )
    {
      // apply (1-xn) times Jacobian for bottom
      jacobianTransposed< add >( topologyId, std::integral_constant< int, dim-1 >(), cit2, df, x, rf*cxn, jt );
      // apply xn times Jacobian for top
      jacobianTransposed< true >( topologyId, std::integral_constant< int, dim-1 >(), cit2, df, x, rf*xn, jt );
      // compute last row as difference between top value and bottom value
      global< add >( topologyId, std::integral_constant< int, dim-1 >(), cit, df, x, -rf, jt[ dim-1 ] );
      global< true >( topologyId, std::integral_constant< int, dim-1 >(), cit, df, x, rf, jt[ dim-1 ] );
    }
    else
    {
      assert( Impl::isPyramid( toUnsignedInt(topologyId), mydimension, mydimension-dim ) );
      /*
       * In the pyramid case, we need a transformation Tb: B -> R^n for the
       * base B \subset R^{n-1}. The pyramid transformation is then defined as
       *   T: P \subset R^n  -> R^n
       *      (x, xn)        |-> (1-xn) Tb(x*) + xn t  (x \in R^{n-1}, xn \in R)
       * with the tip of the pyramid mapped to t and x* = x/(1-xn)
       * the projection of (x,xn) onto the base.
       *
       * For the Jacobi matrix DT we get
       *   DT = ( A | b )
       * with A = DTb(x*)   (n x n-1 matrix)
       *  and b = dT/dxn    (n-dim column vector).
       * Furthermore
       *   b = -Tb(x*) + t + \sum_i dTb/dx_i(x^*) x_i/(1-xn)
       *
       * Note that both A and b are not defined in the pyramid tip (x=0, xn=1)!
       * Indeed for B the unit square, Tb mapping B to the quadrilateral given
       * by the vertices (0,0,0), (2,0,0), (0,1,0), (1,1,0) and t=(0,0,1), we get
       *
       *   T(x,y,xn) = ( x(2-y/(1-xn)), y, xn )
       *               / 2-y/(1-xn)  -x   0 \
       *  DT(x,y,xn) = |    0         1   0 |
       *               \    0         0   1 /
       * which is not continuous for xn -> 1, choose for example
       *   x=0,    y=1-xn, xn -> 1   --> DT -> diag(1,1,1)
       *   x=1-xn, y=0,    xn -> 1   --> DT -> diag(2,1,1)
       *
       * However, for Tb affine-linear, Tb(y) = My + y0, DTb = M:
       *   A = M
       *   b = -M x* - y0 + t + \sum_i M_i x_i/(1-xn)
       *     = -M x* - y0 + t + M x*
       *     = -y0 + t
       * which is continuous for xn -> 1. Note that this b is also given by
       *   b = -Tb(0) + t + \sum_i dTb/dx_i(0) x_i/1
       * that is replacing x* by 1 and 1-xn by 1 in the formular above.
       *
       * For xn -> 1, we can thus set x*=0, "1-xn"=1 (or anything != 0) and get
       * the right result in case Tb is affine-linear.
       */
 
      /* The second case effectively results in x* = 0 */
      ctype dfcxn = (cxn > Traits::tolerance() || cxn < -Traits::tolerance()) ? ctype(df / cxn) : ctype(0);
 
      // initialize last row
      // b =  -Tb(x*)
      // (b = -Tb(0) = -y0 in case xn -> 1 and Tb affine-linear)
      global< add >( topologyId, std::integral_constant< int, dim-1 >(), cit, dfcxn, x, -rf, jt[ dim-1 ] );
      // b += t
      jt[ dim-1 ].axpy( rf, *cit );
      ++cit;
      // apply Jacobian for bottom (with argument x/(1-xn)) and correct last row
      if( add )
      {
        FieldMatrix< ctype, dim-1, coorddimension > jt2;
        // jt2 = dTb/dx_i(x*)
        jacobianTransposed< false >( topologyId, std::integral_constant< int, dim-1 >(), cit2, dfcxn, x, rf, jt2 );
        // A = dTb/dx_i(x*)                      (jt[j], j=0..dim-1)
        // b += \sum_i dTb/dx_i(x*) x_i/(1-xn)   (jt[dim-1])
        // (b += 0 in case xn -> 1)
        for( int j = 0; j < dim-1; ++j )
        {
          jt[ j ] += jt2[ j ];
          jt[ dim-1 ].axpy( dfcxn*x[ j ], jt2[ j ] );
        }
      }
      else
      {
        // jt = dTb/dx_i(x*)
        jacobianTransposed< false >( topologyId, std::integral_constant< int, dim-1 >(), cit2, dfcxn, x, rf, jt );
        // b += \sum_i dTb/dx_i(x*) x_i/(1-xn)
        for( int j = 0; j < dim-1; ++j )
          jt[ dim-1 ].axpy( dfcxn*x[ j ], jt[ j ] );
      }
    }
  }
 
  template< class ct, int mydim, int cdim, class Traits >
  template< bool add, int rows, class CornerIterator >
  inline void MultiLinearGeometry< ct, mydim, cdim, Traits >
  ::jacobianTransposed ( TopologyId, std::integral_constant< int, 0 >,
                         CornerIterator &cit, const ctype &, const LocalCoordinate &,
                         const ctype &, FieldMatrix< ctype, rows, cdim > & )
  {
    ++cit;
  }
 
 
 
  template< class ct, int mydim, int cdim, class Traits >
  template< int dim, class CornerIterator >
  inline bool MultiLinearGeometry< ct, mydim, cdim, Traits >
  ::affine ( TopologyId topologyId, std::integral_constant< int, dim >, CornerIterator &cit, JacobianTransposed &jt )
  {
    const GlobalCoordinate &orgBottom = *cit;
    if( !affine( topologyId, std::integral_constant< int, dim-1 >(), cit, jt ) )
      return false;
    const GlobalCoordinate &orgTop = *cit;
 
    if( Impl::isPrism( toUnsignedInt(topologyId), mydimension, mydimension-dim ) )
    {
      JacobianTransposed jtTop;
      if( !affine( topologyId, std::integral_constant< int, dim-1 >(), cit, jtTop ) )
        return false;
 
      // check whether both jacobians are identical
      ctype norm( 0 );
      for( int i = 0; i < dim-1; ++i )
        norm += (jtTop[ i ] - jt[ i ]).two_norm2();
      if( norm >= Traits::tolerance() )
        return false;
    }
    else
      ++cit;
    jt[ dim-1 ] = orgTop - orgBottom;
    return true;
  }
 
  template< class ct, int mydim, int cdim, class Traits >
  template< class CornerIterator >
  inline bool MultiLinearGeometry< ct, mydim, cdim, Traits >
  ::affine ( TopologyId, std::integral_constant< int, 0 >, CornerIterator &cit, JacobianTransposed & )
  {
    ++cit;
    return true;
  }
 
} // namespace Dune
 
#endif // #ifndef DUNE_GEOMETRY_MULTILINEARGEOMETRY_HH