Taxisym_navier_stokes_elements.h

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//Header file for triangular/tetrahedaral Axisymmetric Navier Stokes elements

#ifndef OOMPH_TAXISYM_NAVIER_STOKES_ELEMENTS_HEADER
#define OOMPH_TAXISYM_NAVIER_STOKES_ELEMENTS_HEADER

// Config header generated by autoconfig
#ifdef HAVE_CONFIG_H
#include <oomph-lib-config.h>
#endif


//OOMPH-LIB headers 
#include "../generic/Telements.h"
#include "axisym_navier_stokes_elements.h"
#include "../generic/error_estimator.h"

namespace oomph
{

//////////////////////////////////////////////////////////////////////////////
//////////////////////////////////////////////////////////////////////////////
// NOTE: TRI/TET CROZIER RAVIARTS REQUIRE BUBBLE FUNCTIONS! THEY'RE NOT
// STRAIGHTFORWARD GENERALISATIONS OF THE Q-EQUIVALENTS (WHICH ARE
// LBB UNSTABLE!)
//////////////////////////////////////////////////////////////////////////////
//////////////////////////////////////////////////////////////////////////////


//==========================================================================
///AxisymmetricTCrouzeix_Raviart elements are Navier--Stokes elements with quadratic
///interpolation for velocities and positions enriched by a single cubic
///bubble function, but a discontinuous linear
///pressure interpolation
//==========================================================================
class AxisymmetricTCrouzeixRaviartElement : 
 public virtual TBubbleEnrichedElement<2,3>,
 public virtual AxisymmetricNavierStokesEquations,
 public virtual ElementWithZ2ErrorEstimator
{
 protected:


 /// Internal index that indicates at which internal datum the pressure is
 /// stored
 unsigned P_axi_nst_internal_index;

 
 /// \short Velocity shape and test functions and their derivs
 /// w.r.t. to global coords  at local coordinate s (taken from geometry)
 ///Return Jacobian of mapping between local and global coordinates.
 inline double dshape_and_dtest_eulerian_axi_nst(const Vector<double> &s, 
                                                 Shape &psi,
                                                 DShape &dpsidx, Shape &test,
                                                 DShape &dtestdx) const;
 
 /// \short Velocity shape and test functions and their derivs
 /// w.r.t. to global coords at ipt-th integation point (taken from geometry)
 ///Return Jacobian of mapping between local and global coordinates.
 inline double dshape_and_dtest_eulerian_at_knot_axi_nst(const unsigned &ipt,
                                                         Shape &psi,
                                                         DShape &dpsidx, 
                                                         Shape &test,
                                                         DShape &dtestdx) 
  const;
 
 /// \short Shape/test functions and derivs w.r.t. to global coords at 
 /// integration point ipt; return Jacobian of mapping (J). Also compute
 /// derivatives of dpsidx, dtestdx and J w.r.t. nodal coordinates.
 inline double dshape_and_dtest_eulerian_at_knot_axi_nst(
  const unsigned &ipt, 
  Shape &psi, 
  DShape &dpsidx,
  RankFourTensor<double> &d_dpsidx_dX,
  Shape &test, 
  DShape &dtestdx,
  RankFourTensor<double> &d_dtestdx_dX,
  DenseMatrix<double> &djacobian_dX) const;

 /// \short Pressure shape and test functions and their derivs 
 /// w.r.t. to global coords  at local coordinate s (taken from geometry)
 /// Return Jacobian of mapping between local and global coordinates.
 inline double dpshape_and_dptest_eulerian_axi_nst(const Vector<double> &s, 
                                                   Shape &ppsi, 
                                                   DShape &dppsidx, 
                                                   Shape &ptest, 
                                                   DShape &dptestdx) const;
 
public:

 /// Pressure shape functions at local coordinate s
 inline void pshape_axi_nst(const Vector<double> &s, Shape &psi) const;

 /// Pressure shape and test functions at local coordinte s
 inline void pshape_axi_nst(const Vector<double> &s, Shape &psi, Shape &test) 
  const;
 
 /// Unpin all internal pressure dofs
 void unpin_all_internal_pressure_dofs();

 /// Return the local equation numbers for the pressure values.
 inline int p_local_eqn(const unsigned &n) const
  {return this->internal_local_eqn(P_axi_nst_internal_index,n);}
 
public:
 
 /// Constructor, there are 3 internal values (for the pressure)
 AxisymmetricTCrouzeixRaviartElement() : TBubbleEnrichedElement<2,3>(), 
  AxisymmetricNavierStokesEquations()
  {
   //Allocate and a single internal datum with 3 entries for the
   //pressure
   P_axi_nst_internal_index = this->add_internal_data(new Data(3));
  }

 /// Broken copy constructor
 AxisymmetricTCrouzeixRaviartElement(
  const 
  AxisymmetricTCrouzeixRaviartElement& dummy) 
  { 
   BrokenCopy::broken_copy("AxisymmetricTCrouzeixRaviartElement");
  } 
 
 /// Broken assignment operator
//Commented out broken assignment operator because this can lead to a conflict warning
//when used in the virtual inheritence hierarchy. Essentially the compiler doesn't
//realise that two separate implementations of the broken function are the same and so,
//quite rightly, it shouts.
 /*void operator=(const AxisymmetricTCrouzeixRaviartElement&) 
  {
   BrokenCopy::broken_assign("AxisymmetricTCrouzeixRaviartElement");
   }*/


 /// \short Number of values (pinned or dofs) required at local node n.
 inline virtual unsigned required_nvalue(const unsigned &n) const
  {return 3;}
  
 
 /// \short Return the pressure values at internal dof i_internal
 /// (Discontinous pressure interpolation -- no need to cater for hanging
 /// nodes).
 double p_axi_nst(const unsigned &i) const
  {return this->internal_data_pt(P_axi_nst_internal_index)->value(i);}

 /// Return number of pressure values
 unsigned npres_axi_nst() const {return 3;}

 /// Pin p_dof-th pressure dof and set it to value specified by p_value.
 void fix_pressure(const unsigned &p_dof, const double &p_value)
  {
   this->internal_data_pt(P_axi_nst_internal_index)->pin(p_dof);
   this->internal_data_pt(P_axi_nst_internal_index)->set_value(p_dof, p_value);
  }

 /// \short Build FaceElements that apply the Robin boundary condition
 /// to the pressure advection diffusion problem required by 
 /// Fp preconditioner
 //void build_fp_press_adv_diff_robin_bc_element(const unsigned& 
 //                                              face_index)
 // {
 // this->Pressure_advection_diffusion_robin_element_pt.push_back(
 //  new FpPressureAdvDiffRobinBCElement<AxisymmetricTCrouzeixRaviartElement<DIM> >(
 //   this, face_index));
 // }
 
 /// \short Add to the set paired_load_data
 /// pairs of pointers to data objects and unsignedegers that
 /// index the values in the data object that affect the load (traction),
 /// as specified in the get_load() function.
 void identify_load_data(std::set<std::pair<Data*,unsigned> > 
                         &paired_load_data);

 /// \short  Add to the set \c paired_pressure_data pairs 
 /// containing
 /// - the pointer to a Data object
 /// and
 /// - the index of the value in that Data object
 /// .
 /// for all pressure values that affect the
 /// load computed in the \c get_load(...) function.
 void identify_pressure_data(
  std::set<std::pair<Data*,unsigned> > &paired_pressure_data);

 /// Redirect output to NavierStokesEquations output
 void output(std::ostream &outfile) 
  {AxisymmetricNavierStokesEquations::output(outfile);}
 
 /// Redirect output to NavierStokesEquations output
 void output(std::ostream &outfile, const unsigned &nplot)
  {AxisymmetricNavierStokesEquations::output(outfile,nplot);}
 
 /// Redirect output to NavierStokesEquations output
 void output(FILE* file_pt) 
  {AxisymmetricNavierStokesEquations::output(file_pt);}
 
 /// Redirect output to NavierStokesEquations output
 void output(FILE* file_pt, const unsigned &n_plot)
  {AxisymmetricNavierStokesEquations::output(file_pt,n_plot);}
 

 /// \short Order of recovery shape functions for Z2 error estimation:
 /// Same order as unenriched shape functions.
 unsigned nrecovery_order() {return 2;}
 
 /// \short Number of vertex nodes in the element
 unsigned nvertex_node() const
 {return 3;}
 
 /// \short Pointer to the j-th vertex node in the element
 Node* vertex_node_pt(const unsigned& j) const
 {return node_pt(j);}
  
 /// Number of 'flux' terms for Z2 error estimation 
 unsigned num_Z2_flux_terms()
  {
   // 3 diagonal strain rates, 3 off diagonal
   return 6;
  }
 
 /// \short Get 'flux' for Z2 error recovery:   Upper triangular entries
 /// in strain rate tensor.
 void get_Z2_flux(const Vector<double>& s, Vector<double>& flux)
  {
#ifdef PARANOID
   unsigned num_entries=6;
    if (flux.size() < num_entries)
     {
      std::ostringstream error_message;
      error_message << "The flux vector has the wrong number of entries, " 
      << flux.size() << ", whereas it should be at least " 
      << num_entries << std::endl;
      throw OomphLibError(error_message.str(),
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
     }
#endif
   
   // Get strain rate matrix
   DenseMatrix<double> strainrate(3);
   this->strain_rate(s,strainrate);
   
  // Pack into flux Vector
  unsigned icount=0;
  
  // Start with diagonal terms
  for(unsigned i=0;i<3;i++)
   {
    flux[icount]=strainrate(i,i);
    icount++;
   }
  
  //Off diagonals row by row
  for(unsigned i=0;i<3;i++)
   {
    for(unsigned j=i+1;j<3;j++)
     {
      flux[icount]=strainrate(i,j);
      icount++;
     }
   }
 }
 

 /// \short The number of "DOF types" that degrees of freedom in this element
 /// are sub-divided into: Velocity (3 components) and pressure.
 unsigned ndof_types() const
  {
   return 4;
  }
 
 /// \short Create a list of pairs for all unknowns in this element,
 /// so that the first entry in each pair contains the global equation
 /// number of the unknown, while the second one contains the number
 /// of the "DOF type" that this unknown is associated with.
 /// (Function can obviously only be called if the equation numbering
 /// scheme has been set up.) 
 void get_dof_numbers_for_unknowns(
  std::list<std::pair<unsigned long,unsigned> >& dof_lookup_list) const
  {
   // number of nodes
   unsigned n_node = this->nnode();
   
   // number of pressure values
   unsigned n_press = this->npres_axi_nst();
   
   // temporary pair (used to store dof lookup prior to being added to list)
   std::pair<unsigned,unsigned> dof_lookup;
   
   // pressure dof number
   unsigned pressure_dof_number = 3;
   
   // loop over the pressure values
   for (unsigned n = 0; n < n_press; n++)
    {
     // determine local eqn number
     int local_eqn_number = this->p_local_eqn(n);
     
     // ignore pinned values - far away degrees of freedom resulting 
     // from hanging nodes can be ignored since these are be dealt
     // with by the element containing their master nodes
     if (local_eqn_number >= 0)
      {
       // store dof lookup in temporary pair: First entry in pair
       // is global equation number; second entry is dof type
       dof_lookup.first = this->eqn_number(local_eqn_number);
       dof_lookup.second = pressure_dof_number;
       
       // add to list
       dof_lookup_list.push_front(dof_lookup);
      }
    }
   
   // loop over the nodes
   for (unsigned n = 0; n < n_node; n++)
    {
     // find the number of values at this node
     unsigned nv = this->node_pt(n)->nvalue();
     
     //loop over these values
     for (unsigned v = 0; v < nv; v++)
      {
       // determine local eqn number
       int local_eqn_number = this->nodal_local_eqn(n, v);
       
       // ignore pinned values
       if (local_eqn_number >= 0)
        {
         // store dof lookup in temporary pair: First entry in pair
         // is global equation number; second entry is dof type
         dof_lookup.first = this->eqn_number(local_eqn_number);
         dof_lookup.second = v;
         
         // add to list
         dof_lookup_list.push_front(dof_lookup);
         
        }
      }
    }
  }


};

//Inline functions

//=======================================================================
/// Derivatives of the shape functions and test functions w.r.t. to global
/// (Eulerian) coordinates. Return Jacobian of mapping between
/// local and global coordinates.
//=======================================================================
inline double AxisymmetricTCrouzeixRaviartElement::
 dshape_and_dtest_eulerian_axi_nst(
  const Vector<double> &s, Shape &psi,
  DShape &dpsidx, Shape &test,
  DShape &dtestdx) const
 {
  //Call the geometrical shape functions and derivatives
  double J = this->dshape_eulerian(s,psi,dpsidx);
  //The test functions are equal to the shape functions
  test = psi;
  dtestdx = dpsidx;
  //Return the jacobian
  return J;
 }
 
 
//=======================================================================
/// Derivatives of the shape functions and test functions w.r.t. to global
/// (Eulerian) coordinates. Return Jacobian of mapping between
/// local and global coordinates.
//=======================================================================
 inline double AxisymmetricTCrouzeixRaviartElement::
  dshape_and_dtest_eulerian_at_knot_axi_nst(
   const unsigned &ipt, Shape &psi,
   DShape &dpsidx, Shape &test,
   DShape &dtestdx) const
  {
   //Call the geometrical shape functions and derivatives
   double J = this->dshape_eulerian_at_knot(ipt,psi,dpsidx);
   //The test functions are the shape functions
   test = psi;
   dtestdx = dpsidx;
   //Return the jacobian
   return J;
  }
 

//=======================================================================
/// Define the shape functions (psi) and test functions (test) and
/// their derivatives w.r.t. global coordinates (dpsidx and dtestdx)
/// and return Jacobian of mapping (J). Additionally compute the
/// derivatives of dpsidx, dtestdx and J w.r.t. nodal coordinates.
///
/// Galerkin: Test functions = shape functions
//=======================================================================
inline double AxisymmetricTCrouzeixRaviartElement::
 dshape_and_dtest_eulerian_at_knot_axi_nst(
  const unsigned &ipt, Shape &psi, DShape &dpsidx,
  RankFourTensor<double> &d_dpsidx_dX,
  Shape &test, DShape &dtestdx,
  RankFourTensor<double> &d_dtestdx_dX,
  DenseMatrix<double> &djacobian_dX) const
 {
  // Call the geometrical shape functions and derivatives  
  const double J = this->dshape_eulerian_at_knot(ipt,psi,dpsidx,
                                                 djacobian_dX,d_dpsidx_dX);
  
  // Set the test functions equal to the shape functions
  test = psi;
  dtestdx = dpsidx;
  d_dtestdx_dX = d_dpsidx_dX;

  // Return the jacobian
  return J;
 }


//=======================================================================
/// Pressure shape functions
//=======================================================================
inline void AxisymmetricTCrouzeixRaviartElement::pshape_axi_nst(
 const Vector<double> &s,
 Shape &psi) const
 {
  psi[0] = 1.0;
  psi[1] = s[0];
  psi[2] = s[1];
 }
 
//=======================================================================
/// Pressure shape and test functions
//=======================================================================
 inline void AxisymmetricTCrouzeixRaviartElement::pshape_axi_nst(
  const Vector<double> &s,
  Shape &psi,
  Shape &test) const
  {
   //Call the pressure shape functions
   this->pshape_axi_nst(s,psi);
   //The test functions are the shape functions
   test = psi;
  }
 
//==========================================================================
/// 2D :
/// Pressure shape and test functions and derivs w.r.t. to Eulerian coords.
/// Return Jacobian of mapping between local and global coordinates.
//==========================================================================
 inline double AxisymmetricTCrouzeixRaviartElement::
  dpshape_and_dptest_eulerian_axi_nst(
   const Vector<double> &s, 
   Shape &ppsi, 
   DShape &dppsidx, 
   Shape &ptest, 
   DShape &dptestdx) const
  {
   
   // Initalise with shape fcts and derivs. w.r.t. to local coordinates
   ppsi[0] = 1.0;
   ppsi[1] = s[0];
   ppsi[2] = s[1];
   
   dppsidx(0,0) = 0.0;
   dppsidx(1,0) = 1.0;
   dppsidx(2,0) = 0.0;
   
   dppsidx(0,1) = 0.0;
   dppsidx(1,1) = 0.0;
   dppsidx(2,1) = 1.0;
   
   
   //Get the values of the shape functions and their local derivatives
   Shape psi(7);
   DShape dpsi(7,2);
   dshape_local(s,psi,dpsi);
   
   //Allocate memory for the inverse 2x2 jacobian
   DenseMatrix<double> inverse_jacobian(2);
   
   //Now calculate the inverse jacobian
   const double det = local_to_eulerian_mapping(dpsi,inverse_jacobian);
   
   //Now set the values of the derivatives to be derivs w.r.t. to the
   // Eulerian coordinates
   transform_derivatives(inverse_jacobian,dppsidx);
   
   //The test functions are equal to the shape functions
  ptest = ppsi;
  dptestdx = dppsidx;
  
  //Return the determinant of the jacobian
  return det;
 }


//=======================================================================
/// Face geometry of the 2D Crouzeix_Raviart elements
//=======================================================================
template<>
class FaceGeometry<AxisymmetricTCrouzeixRaviartElement>: 
 public virtual TElement<1,3>
 {
   public:
  FaceGeometry() : TElement<1,3>() {}
 };


//=======================================================================
/// Face geometry of the FaceGeometry of the 2D CrouzeixRaviart elements
//=======================================================================
template<>
class FaceGeometry<FaceGeometry<AxisymmetricTCrouzeixRaviartElement> >: 
public virtual PointElement
{
  public:
 FaceGeometry() : PointElement() {}
};


////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////



//=======================================================================
/// Taylor--Hood elements are Navier--Stokes elements
/// with quadratic interpolation for velocities and positions and
/// continous linear pressure interpolation
//=======================================================================
class AxisymmetricTTaylorHoodElement : 
public virtual TElement<2,3>,
public virtual AxisymmetricNavierStokesEquations,
public virtual ElementWithZ2ErrorEstimator

{
  private:
 
 /// Static array of ints to hold number of variables at node
 static const unsigned Initial_Nvalue[];
 
  protected:

 /// \short Static array of ints to hold conversion from pressure
 /// node numbers to actual node numbers
 static const unsigned Pconv[];
 
 /// \short Velocity shape and test functions and their derivs
 /// w.r.t. to global coords  at local coordinate s (taken from geometry)
 /// Return Jacobian of mapping between local and global coordinates.
 inline double dshape_and_dtest_eulerian_axi_nst(const Vector<double> &s, 
                                                 Shape &psi,
                                                 DShape &dpsidx, Shape &test,
                                                 DShape &dtestdx) const;

 /// \short Velocity shape and test functions and their derivs
 /// w.r.t. to global coords  at local coordinate s (taken from geometry)
 /// Return Jacobian of mapping between local and global coordinates.
 inline double dshape_and_dtest_eulerian_at_knot_axi_nst(const unsigned &ipt,
                                                         Shape &psi,
                                                         DShape &dpsidx, 
                                                         Shape &test,
                                                         DShape &dtestdx) 
  const;
 
 /// \short Shape/test functions and derivs w.r.t. to global coords at 
 /// integration point ipt; return Jacobian of mapping (J). Also compute
 /// derivatives of dpsidx, dtestdx and J w.r.t. nodal coordinates.
 inline double dshape_and_dtest_eulerian_at_knot_axi_nst(
  const unsigned &ipt, 
  Shape &psi, 
  DShape &dpsidx,
  RankFourTensor<double> &d_dpsidx_dX,
  Shape &test, 
  DShape &dtestdx,
  RankFourTensor<double> &d_dtestdx_dX,
  DenseMatrix<double> &djacobian_dX) const;

 /// \short Compute the pressure shape and test functions and derivatives 
 /// w.r.t. global coords at local coordinate s.
 /// Return Jacobian of mapping between local and global coordinates.
 virtual double dpshape_and_dptest_eulerian_axi_nst(const Vector<double> &s, 
                                                    Shape &ppsi, 
                                                    DShape &dppsidx, 
                                                    Shape &ptest, 
                                                    DShape &dptestdx) const;

 /// Unpin all pressure dofs
 void unpin_all_nodal_pressure_dofs();

 ///  Pin all nodal pressure dofs
 void pin_all_nodal_pressure_dofs();
   
 ///  Unpin the proper nodal pressure dofs
 void unpin_proper_nodal_pressure_dofs();


public:

 /// Constructor, no internal data points
 AxisymmetricTTaylorHoodElement() : 
  TElement<2,3>(),  AxisymmetricNavierStokesEquations() {}
 

 /// Broken copy constructor
 AxisymmetricTTaylorHoodElement(const AxisymmetricTTaylorHoodElement& dummy) 
  { 
   BrokenCopy::broken_copy("AxisymmetricTTaylorHoodElement");
  } 
 
 /// Broken assignment operator
 /*void operator=(const AxisymmetricTTaylorHoodElement&) 
  {
   BrokenCopy::broken_assign("AxisymmetricTTaylorHoodElement");
   }*/

 /// \short Number of values (pinned or dofs) required at node n. Can
 /// be overwritten for hanging node version
 inline virtual unsigned required_nvalue(const unsigned &n) const
  {return Initial_Nvalue[n];}

 /// Test whether the pressure dof p_dof hanging or not?
 //bool pressure_dof_is_hanging(const unsigned& p_dof)
 // {return this->node_pt(Pconv[p_dof])->is_hanging(DIM);}


 /// Pressure shape functions at local coordinate s
 inline void pshape_axi_nst(const Vector<double> &s, Shape &psi) const;

 /// Pressure shape and test functions at local coordinte s
 inline void pshape_axi_nst(const Vector<double> &s, Shape &psi, 
                        Shape &test) const;

 /// \short Which nodal value represents the pressure?
 unsigned p_index_axi_nst() {return 3;}

 /// \short Pointer to n_p-th pressure node
 //Node* pressure_node_pt(const unsigned &n_p)
 //{return this->Node_pt[Pconv[n_p]];}

 /// Return the local equation numbers for the pressure values.
 inline int p_local_eqn(const unsigned &n) const
  {return this->nodal_local_eqn(Pconv[n],3);}

 /// \short Access function for the pressure values at local pressure
 /// node n_p (const version)
 double p_axi_nst(const unsigned &n_p) const
  {return this->nodal_value(Pconv[n_p],3);}
 
 /// \short Set the value at which the pressure is stored in the nodes
 int p_nodal_index_axi_nst() const {return static_cast<int>(3);}
 
 /// Return number of pressure values
 unsigned npres_axi_nst() const {return 3;}

 /// Pin p_dof-th pressure dof and set it to value specified by p_value.
 void fix_pressure(const unsigned &p_dof, const double &p_value)
  {
   this->node_pt(Pconv[p_dof])->pin(3);
   this->node_pt(Pconv[p_dof])->set_value(3,p_value);
  }


 /// \short Build FaceElements that apply the Robin boundary condition
 /// to the pressure advection diffusion problem required by 
 /// Fp preconditioner
// void build_fp_press_adv_diff_robin_bc_element(const unsigned& 
//                                               face_index)
// {
//  this->Pressure_advection_diffusion_robin_element_pt.push_back(
//   new FpPressureAdvDiffRobinBCElement<AxisymmetricTTaylorHoodElement<DIM> >(
//    this, face_index));
// }

 /// \short Add to the set \c paired_load_data pairs containing
 /// - the pointer to a Data object
 /// and
 /// - the index of the value in that Data object
 /// .
 /// for all values (pressures, velocities) that affect the
 /// load computed in the \c get_load(...) function. 
 void identify_load_data(
  std::set<std::pair<Data*,unsigned> > &paired_load_data);

 /// \short  Add to the set \c paired_pressure_data pairs 
 /// containing
 /// - the pointer to a Data object
 /// and
 /// - the index of the value in that Data object
 /// .
 /// for all pressure values that affect the
 /// load computed in the \c get_load(...) function.
 void identify_pressure_data(
  std::set<std::pair<Data*,unsigned> > &paired_pressure_data);

 /// Redirect output to NavierStokesEquations output
 void output(std::ostream &outfile) 
  {AxisymmetricNavierStokesEquations::output(outfile);}

 /// Redirect output to NavierStokesEquations output
 void output(std::ostream &outfile, const unsigned &nplot)
  {AxisymmetricNavierStokesEquations::output(outfile,nplot);}

 /// Redirect output to NavierStokesEquations output
 void output(FILE* file_pt) 
  {AxisymmetricNavierStokesEquations::output(file_pt);}

 /// Redirect output to NavierStokesEquations output
 void output(FILE* file_pt, const unsigned &n_plot)
  {AxisymmetricNavierStokesEquations::output(file_pt,n_plot);}
 
 /// \short Order of recovery shape functions for Z2 error estimation:
 /// Same order as shape functions.
 unsigned nrecovery_order() {return 2;}
 
 /// \short Number of vertex nodes in the element
 unsigned nvertex_node() const
 {return 3;}
 
 /// \short Pointer to the j-th vertex node in the element
 Node* vertex_node_pt(const unsigned& j) const
 {return node_pt(j);}
 
 
 /// Number of 'flux' terms for Z2 error estimation 
 unsigned num_Z2_flux_terms()
 {
  // 3 diagonal strain rates, 3 off diagonal rates
  return 6;
 }
 
 /// \short Get 'flux' for Z2 error recovery:   Upper triangular entries
 /// in strain rate tensor.
 void get_Z2_flux(const Vector<double>& s, Vector<double>& flux)
 {
#ifdef PARANOID
  unsigned num_entries=6;
  if (flux.size() < num_entries)
   {
    std::ostringstream error_message;
    error_message << "The flux vector has the wrong number of entries, " 
                  << flux.size() << ", whereas it should be at least " 
                  << num_entries << std::endl;
    throw OomphLibError(error_message.str(),
                        OOMPH_CURRENT_FUNCTION,
                        OOMPH_EXCEPTION_LOCATION);
   }
#endif
  
  // Get strain rate matrix
  DenseMatrix<double> strainrate(3);
  this->strain_rate(s,strainrate);
  
  // Pack into flux Vector
  unsigned icount=0;
  
  // Start with diagonal terms
  for(unsigned i=0;i<3;i++)
   {
    flux[icount]=strainrate(i,i);
    icount++;
   }
  
  //Off diagonals row by row
  for(unsigned i=0;i<3;i++)
   {
    for(unsigned j=i+1;j<3;j++)
     {
      flux[icount]=strainrate(i,j);
      icount++;
     }
   }
 }
 
 /// \short The number of "DOF types" that degrees of freedom in this element
 /// are sub-divided into: Velocities (3  components) and pressure.
 unsigned ndof_types() const
 {
  return 4;
 }
 
 /// \short Create a list of pairs for all unknowns in this element,
 /// so that the first entry in each pair contains the global equation
 /// number of the unknown, while the second one contains the number
 /// of the "DOF type" that this unknown is associated with.
 /// (Function can obviously only be called if the equation numbering
 /// scheme has been set up.) 
 void get_dof_numbers_for_unknowns(
  std::list<std::pair<unsigned long,unsigned> >& dof_lookup_list) const
  {
   // number of nodes
   unsigned n_node = this->nnode();
   
   // temporary pair (used to store dof lookup prior to being added to list)
   std::pair<unsigned,unsigned> dof_lookup;
   
   // loop over the nodes
   for (unsigned n = 0; n < n_node; n++)
    {
     // find the number of Navier Stokes values at this node
     unsigned nv = this->required_nvalue(n); 
     
     //loop over these values
     for (unsigned v = 0; v < nv; v++)
      {
       // determine local eqn number
       int local_eqn_number = this->nodal_local_eqn(n, v);
       
       // ignore pinned values - far away degrees of freedom resulting 
       // from hanging nodes can be ignored since these are be dealt
       // with by the element containing their master nodes
       if (local_eqn_number >= 0)
        {
         // store dof lookup in temporary pair: Global equation number
         // is the first entry in pair
         dof_lookup.first = this->eqn_number(local_eqn_number);
         
         // set dof numbers: Dof number is the second entry in pair
         dof_lookup.second = v;
         
         // add to list
         dof_lookup_list.push_front(dof_lookup);
        }
      }
    }
  }
};




//Inline functions

//==========================================================================
/// Derivatives of the shape functions and test functions w.r.t to
/// global (Eulerian) coordinates. Return Jacobian of mapping between
/// local and global coordinates.
//==========================================================================
inline double AxisymmetricTTaylorHoodElement::
dshape_and_dtest_eulerian_axi_nst(
 const Vector<double> &s,
 Shape &psi,
 DShape &dpsidx, Shape &test,
 DShape &dtestdx) const
{
 //Call the geometrical shape functions and derivatives
 double J = this->dshape_eulerian(s,psi,dpsidx);
 //Test functions are the shape functions
 test = psi;
 dtestdx = dpsidx;
 //Return the jacobian
 return J;
}


//==========================================================================
/// Derivatives of the shape functions and test functions w.r.t to
/// global (Eulerian) coordinates. Return Jacobian of mapping between
/// local and global coordinates.
//==========================================================================
inline double AxisymmetricTTaylorHoodElement::
dshape_and_dtest_eulerian_at_knot_axi_nst(
 const unsigned &ipt,Shape &psi, DShape &dpsidx, Shape &test,
 DShape &dtestdx) const
{
 //Call the geometrical shape functions and derivatives
 double J = this->dshape_eulerian_at_knot(ipt,psi,dpsidx);
 //Test functions are the shape functions
 test = psi;
 dtestdx = dpsidx;
 //Return the jacobian
 return J;
}

//==========================================================================
/// Define the shape functions (psi) and test functions (test) and
/// their derivatives w.r.t. global coordinates (dpsidx and dtestdx)
/// and return Jacobian of mapping (J). Additionally compute the
/// derivatives of dpsidx, dtestdx and J w.r.t. nodal coordinates.
///
/// Galerkin: Test functions = shape functions
//==========================================================================
inline double AxisymmetricTTaylorHoodElement::
 dshape_and_dtest_eulerian_at_knot_axi_nst(
  const unsigned &ipt, Shape &psi, DShape &dpsidx,
  RankFourTensor<double> &d_dpsidx_dX,
  Shape &test, DShape &dtestdx,
  RankFourTensor<double> &d_dtestdx_dX,
  DenseMatrix<double> &djacobian_dX) const
 {
  // Call the geometrical shape functions and derivatives  
  const double J = this->dshape_eulerian_at_knot(ipt,psi,dpsidx,
                                                 djacobian_dX,d_dpsidx_dX);

  // Set the test functions equal to the shape functions
  test = psi;
  dtestdx = dpsidx;
  d_dtestdx_dX = d_dpsidx_dX;

  // Return the jacobian
  return J;
 }

//==========================================================================
/// Pressure shape and test functions and derivs w.r.t. to Eulerian coords.
/// Return Jacobian of mapping between local and global coordinates.
//==========================================================================
 inline double AxisymmetricTTaylorHoodElement::
 dpshape_and_dptest_eulerian_axi_nst(
  const Vector<double> &s, 
  Shape &ppsi, 
  DShape &dppsidx, 
  Shape &ptest, 
  DShape &dptestdx) const
 {
  
  ppsi[0] = s[0];
  ppsi[1] = s[1];
  ppsi[2] = 1.0-s[0]-s[1];
  
  dppsidx(0,0)=1.0;
  dppsidx(0,1)=0.0;

  dppsidx(1,0)=0.0;
  dppsidx(1,1)=1.0;

  dppsidx(2,0)=-1.0;
  dppsidx(2,1)=-1.0;

    // Allocate memory for the inverse 2x2 jacobian
  DenseMatrix<double> inverse_jacobian(2);


  //Get the values of the shape functions and their local derivatives
  Shape psi(6);
  DShape dpsi(6,2);
  dshape_local(s,psi,dpsi);
    
  // Now calculate the inverse jacobian
  const double det = local_to_eulerian_mapping(dpsi,inverse_jacobian);
  
  // Now set the values of the derivatives to be derivs w.r.t. to the
  // Eulerian coordinates
  transform_derivatives(inverse_jacobian,dppsidx);

  //Test functions are shape functions
  ptest = ppsi;
  dptestdx = dppsidx;
  
  // Return the determinant of the jacobian
  return det;
    
 }


//==========================================================================
/// Pressure shape functions
//==========================================================================
inline void AxisymmetricTTaylorHoodElement::
pshape_axi_nst(const Vector<double> &s, Shape &psi)
const
{
 psi[0] = s[0];
 psi[1] = s[1];
 psi[2] = 1.0-s[0]-s[1];
}

//==========================================================================
/// Pressure shape and test functions
//==========================================================================
inline void AxisymmetricTTaylorHoodElement::
pshape_axi_nst(const Vector<double> &s, 
               Shape &psi,
               Shape &test) const
{
 //Call the pressure shape functions
 this->pshape_axi_nst(s,psi);
 //Test functions are shape functions
 test = psi;
}


//=======================================================================
/// Face geometry of the Axisymmetric Taylor_Hood elements
//=======================================================================
template<>
class FaceGeometry<AxisymmetricTTaylorHoodElement>: 
public virtual TElement<1,3>
{
  public:

  /// Constructor: Call constructor of base
  FaceGeometry() : TElement<1,3>() {}
};



//=======================================================================
/// Face geometry of the FaceGeometry of the Axisymmetric TaylorHood elements
//=======================================================================
template<>
class FaceGeometry<FaceGeometry<AxisymmetricTTaylorHoodElement> >: 
public virtual PointElement
{
  public:
 FaceGeometry() : PointElement() {}
};


}

#endif