navier_stokes_elements.cc

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//LIC// ====================================================================
//LIC// This file forms part of oomph-lib, the object-oriented, 
//LIC// multi-physics finite-element library, available 
//LIC// at http://www.oomph-lib.org.
//LIC// 
//LIC//    Version 1.0; svn revision $LastChangedRevision$
//LIC//
//LIC// $LastChangedDate$
//LIC// 
//LIC// Copyright (C) 2006-2016 Matthias Heil and Andrew Hazel
//LIC// 
//LIC// This library is free software; you can redistribute it and/or
//LIC// modify it under the terms of the GNU Lesser General Public
//LIC// License as published by the Free Software Foundation; either
//LIC// version 2.1 of the License, or (at your option) any later version.
//LIC// 
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//LIC// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
//LIC// Lesser General Public License for more details.
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//LIC//====================================================================
//Non-inline functions for NS elements

#include "navier_stokes_elements.h"


namespace oomph
{







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


/// Navier--Stokes equations static data
template<unsigned DIM>
Vector<double> NavierStokesEquations<DIM>::Gamma(DIM,1.0);

//=================================================================
/// "Magic" negative number that indicates that the pressure is
/// not stored at a node. This cannot be -1 because that represents
/// the positional hanging scheme in the hanging_pt object of nodes
//=================================================================
template<unsigned DIM>
int NavierStokesEquations<DIM>::Pressure_not_stored_at_node = -100;

/// Navier--Stokes equations static data
template<unsigned DIM>
double NavierStokesEquations<DIM>::Default_Physical_Constant_Value = 0.0;

/// Navier--Stokes equations static data
template<unsigned DIM>
double NavierStokesEquations<DIM>::Default_Physical_Ratio_Value = 1.0;

/// Navier-Stokes equations default gravity vector
template<unsigned DIM>
Vector<double> NavierStokesEquations<DIM>::Default_Gravity_vector(DIM,0.0);




//===================================================================
 /// Compute the diagonal of the velocity/pressure mass matrices.
 /// If which one=0, both are computed, otherwise only the pressure 
 /// (which_one=1) or the velocity mass matrix (which_one=2 -- the 
 /// LSC version of the preconditioner only needs that one)
//===================================================================
 template<unsigned DIM>
 void NavierStokesEquations<DIM>::
 get_pressure_and_velocity_mass_matrix_diagonal(
  Vector<double> &press_mass_diag, Vector<double> &veloc_mass_diag,
  const unsigned& which_one)
 {
  // Resize and initialise
  unsigned n_dof=ndof();

  if ((which_one==0)||(which_one==1)) press_mass_diag.assign(n_dof,0.0);   
  if ((which_one==0)||(which_one==2)) veloc_mass_diag.assign(n_dof,0.0);
  
  // find out how many nodes there are
  unsigned n_node = nnode();
  
  // find number of spatial dimensions
  unsigned n_dim = this->dim();
  
  // Local coordinates
  Vector<double> s(n_dim);

  // find the indices at which the local velocities are stored
  Vector<unsigned> u_nodal_index(n_dim);
  for(unsigned i=0; i<n_dim; i++)
   {
    u_nodal_index[i] = this->u_index_nst(i);
   }

  //Set up memory for veloc shape functions
  Shape psi(n_node);
  
  //Find number of pressure dofs
  unsigned n_pres = npres_nst();

  // Pressure shape function
  Shape psi_p(n_pres);
  
  //Number of integration points
  unsigned n_intpt = integral_pt()->nweight();
  
  //Integer to store the local equations no
  int local_eqn=0;
  
  //Loop over the integration points
  for(unsigned ipt=0; ipt<n_intpt; ipt++)
   {
    
    //Get the integral weight
    double w = integral_pt()->weight(ipt);

    //Get determinant of Jacobian of the mapping
    double J = J_eulerian_at_knot(ipt);
    
    //Assign values of s
    for(unsigned i=0;i<n_dim;i++)
     {
      s[i] = integral_pt()->knot(ipt,i);
     }

    //Premultiply weights and Jacobian
    double W = w*J;



    // Do we want the velocity one?
    if ((which_one==0)||(which_one==2))
     {
      
      //Get the velocity shape functions
      shape_at_knot(ipt,psi);
      
      //Loop over the veclocity shape functions
      for(unsigned l=0; l<n_node; l++)
       {
        //Loop over the velocity components
        for(unsigned i=0; i<n_dim; i++)
         {
          local_eqn = nodal_local_eqn(l,u_nodal_index[i]);
          
          //If not a boundary condition
          if(local_eqn >= 0)
           {
            //Add the contribution
            veloc_mass_diag[local_eqn] += pow(psi[l],2) * W;
           } 
         }
       } 
     }
    
    // Do we want the pressure one?
    if ((which_one==0)||(which_one==1))
     {
      //Get the pressure shape functions
      pshape_nst(s,psi_p);
      
      //Loop over the veclocity shape functions
      for(unsigned l=0; l<n_pres; l++)
       {
        // Get equation number
        local_eqn = p_local_eqn(l);
        
        //If not a boundary condition
        if(local_eqn >= 0)
         {
          //Add the contribution
          press_mass_diag[local_eqn] += pow(psi_p[l],2) * W;
         } 
       }
     }
 
   }
 }



//=======================================================================
/// Compute norm of the solution
//=======================================================================
template <unsigned DIM>
void NavierStokesEquations<DIM>::compute_norm(double& norm)
{
 
 // Initialise
 norm=0.0;
 
 //Vector of local coordinates
 Vector<double> s(DIM);
 
 // Solution
 double u=0.0;
 
 //Find out how many nodes there are in the element
 unsigned n_node = this->nnode();
 
 Shape psi(n_node);
 
 //Set the value of n_intpt
 unsigned n_intpt = this->integral_pt()->nweight();
 
 //Loop over the integration points
 for(unsigned ipt=0;ipt<n_intpt;ipt++)
  {
   
   //Assign values of s
   for(unsigned i=0;i<DIM;i++)
    {
     s[i] = this->integral_pt()->knot(ipt,i);
    }
   
   //Get the integral weight
   double w = this->integral_pt()->weight(ipt);
   
   // Get jacobian of mapping
   double J=this->J_eulerian(s);
   
   //Premultiply the weights and the Jacobian
   double W = w*J;
   
   for (unsigned i=0;i<DIM;i++)
    {
     // Get FE function value
     u=this->interpolated_u_nst(s,i);
     
     // Add to  norm
     norm+=u*u*W;
    }

  }
}


//======================================================================
/// Validate against exact velocity solution at given time.
/// Solution is provided via function pointer.
/// Plot at a given number of plot points and compute L2 error
/// and L2 norm of velocity solution over element.
//=======================================================================
template<unsigned DIM>
void NavierStokesEquations<DIM>::
compute_error(std::ostream &outfile,
              FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt,
              const double& time,
              double& error, double& norm)
{
 error=0.0;
 norm=0.0;

 //Vector of local coordinates
 Vector<double> s(DIM);

 // Vector for coordintes
 Vector<double> x(DIM);

 //Set the value of n_intpt
 unsigned n_intpt = integral_pt()->nweight();
   
 outfile << "ZONE" << std::endl;

 // Exact solution Vector (u,v,[w],p)
 Vector<double> exact_soln(DIM+1);
   
 //Loop over the integration points
 for(unsigned ipt=0;ipt<n_intpt;ipt++)
  {

   //Assign values of s
   for(unsigned i=0;i<DIM;i++)
    {
     s[i] = integral_pt()->knot(ipt,i);
    }

   //Get the integral weight
   double w = integral_pt()->weight(ipt);

   // Get jacobian of mapping
   double J=J_eulerian(s);

   //Premultiply the weights and the Jacobian
   double W = w*J;

   // Get x position as Vector
   interpolated_x(s,x);

   // Get exact solution at this point
   (*exact_soln_pt)(time,x,exact_soln);

   // Velocity error
   for(unsigned i=0;i<DIM;i++)
    {
     norm+=exact_soln[i]*exact_soln[i]*W;
     error+=(exact_soln[i]-interpolated_u_nst(s,i))*
      (exact_soln[i]-interpolated_u_nst(s,i))*W;
    }

   //Output x,y,...,u_exact
   for(unsigned i=0;i<DIM;i++)
    {
     outfile << x[i] << " ";
    }

   //Output x,y,[z],u_error,v_error,[w_error]
   for(unsigned i=0;i<DIM;i++)
    {
     outfile << exact_soln[i]-interpolated_u_nst(s,i) << " ";
    }

   outfile << std::endl;
    
  }
}

//======================================================================
/// Validate against exact velocity solution
/// Solution is provided via function pointer.
/// Plot at a given number of plot points and compute L2 error
/// and L2 norm of velocity solution over element.
//=======================================================================
template<unsigned DIM>
void NavierStokesEquations<DIM>::compute_error(
 std::ostream &outfile,
 FiniteElement::SteadyExactSolutionFctPt exact_soln_pt,
 double& error, double& norm)
{
 
 error=0.0;
 norm=0.0;

 //Vector of local coordinates
 Vector<double> s(DIM);

 // Vector for coordintes
 Vector<double> x(DIM);

 //Set the value of n_intpt
 unsigned n_intpt = integral_pt()->nweight();
   

 outfile << "ZONE" << std::endl;
 
 // Exact solution Vector (u,v,[w],p)
 Vector<double> exact_soln(DIM+1);
   
 //Loop over the integration points
 for(unsigned ipt=0;ipt<n_intpt;ipt++)
  {

   //Assign values of s
   for(unsigned i=0;i<DIM;i++)
    {
     s[i] = integral_pt()->knot(ipt,i);
    }

   //Get the integral weight
   double w = integral_pt()->weight(ipt);

   // Get jacobian of mapping
   double J=J_eulerian(s);

   //Premultiply the weights and the Jacobian
   double W = w*J;

   // Get x position as Vector
   interpolated_x(s,x);

   // Get exact solution at this point
   (*exact_soln_pt)(x,exact_soln);

   // Velocity error
   for(unsigned i=0;i<DIM;i++)
    {
     norm+=exact_soln[i]*exact_soln[i]*W;
     error+=(exact_soln[i]-interpolated_u_nst(s,i))*
      (exact_soln[i]-interpolated_u_nst(s,i))*W;
    }

   //Output x,y,...,u_exact
   for(unsigned i=0;i<DIM;i++)
    {
     outfile << x[i] << " ";
    }

   //Output x,y,[z],u_error,v_error,[w_error]
   for(unsigned i=0;i<DIM;i++)
    {
     outfile << exact_soln[i]-interpolated_u_nst(s,i) << " ";
    }
   outfile << std::endl;   
  }
}

//======================================================================
/// Output "exact" solution
/// Solution is provided via function pointer.
/// Plot at a given number of plot points.
/// Function prints as many components as are returned in solution Vector.
//=======================================================================
template<unsigned DIM>
void NavierStokesEquations<DIM>::output_fct(std::ostream &outfile, 
                                   const unsigned &nplot, 
                                   FiniteElement::SteadyExactSolutionFctPt exact_soln_pt)
{
 
 //Vector of local coordinates
 Vector<double> s(DIM);
 
 // Vector for coordintes
 Vector<double> x(DIM);
  
 // Tecplot header info
 outfile << tecplot_zone_string(nplot);
 
 // Exact solution Vector
 Vector<double> exact_soln;
 
 // Loop over plot points
 unsigned num_plot_points=nplot_points(nplot);
 for (unsigned iplot=0;iplot<num_plot_points;iplot++)
  {
   
   // Get local coordinates of plot point
   get_s_plot(iplot,nplot,s);
   
   // Get x position as Vector
   interpolated_x(s,x);
   
   // Get exact solution at this point
   (*exact_soln_pt)(x,exact_soln);
   
   //Output x,y,...
   for(unsigned i=0;i<DIM;i++)
    {
     outfile << x[i] << " ";
    }
   
   //Output "exact solution"
   for(unsigned i=0;i<exact_soln.size();i++)
    {
     outfile << exact_soln[i] << " ";
    }
   
   outfile << std::endl;
   
  }

 // Write tecplot footer (e.g. FE connectivity lists)
 write_tecplot_zone_footer(outfile,nplot);

}

//======================================================================
/// Output "exact" solution at a given time
/// Solution is provided via function pointer.
/// Plot at a given number of plot points.
/// Function prints as many components as are returned in solution Vector.
//=======================================================================
template<unsigned DIM>
void NavierStokesEquations<DIM>::output_fct(std::ostream &outfile,
                                   const unsigned &nplot, 
                                   const double& time,
                                   FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt)
{
 
 //Vector of local coordinates
 Vector<double> s(DIM);
 
 // Vector for coordintes
 Vector<double> x(DIM);
  
 // Tecplot header info
 outfile << tecplot_zone_string(nplot);
 
 // Exact solution Vector
 Vector<double> exact_soln;
 
 // Loop over plot points
 unsigned num_plot_points=nplot_points(nplot);
 for (unsigned iplot=0;iplot<num_plot_points;iplot++)
  {
   
   // Get local coordinates of plot point
   get_s_plot(iplot,nplot,s);
   
   // Get x position as Vector
   interpolated_x(s,x);
   
   // Get exact solution at this point
   (*exact_soln_pt)(time,x,exact_soln);
   
   //Output x,y,...
   for(unsigned i=0;i<DIM;i++)
    {
     outfile << x[i] << " ";
    }
   
   //Output "exact solution"
   for(unsigned i=0;i<exact_soln.size();i++)
    {
     outfile << exact_soln[i] << " ";
    }
   
   outfile << std::endl;
  }

 // Write tecplot footer (e.g. FE connectivity lists)
 write_tecplot_zone_footer(outfile,nplot);

}

//==============================================================
/// Output function: Velocities only  
/// x,y,[z],u,v,[w]
/// in tecplot format at specified previous timestep (t=0: present;
/// t>0: previous timestep). Specified number of plot points in each
/// coordinate direction.
//==============================================================
template<unsigned DIM>
void NavierStokesEquations<DIM>::output_veloc(std::ostream& outfile, 
                                              const unsigned& nplot, 
                                              const unsigned& t)
{

 //Find number of nodes
 unsigned n_node = nnode();
 
 //Local shape function
 Shape psi(n_node);

 //Vectors of local coordinates and coords and velocities
 Vector<double> s(DIM);
 Vector<double> interpolated_x(DIM);
 Vector<double> interpolated_u(DIM);

 // Tecplot header info
 outfile << tecplot_zone_string(nplot);
 
 // Loop over plot points
 unsigned num_plot_points=nplot_points(nplot);
 for (unsigned iplot=0;iplot<num_plot_points;iplot++)
  {
   
   // Get local coordinates of plot point
   get_s_plot(iplot,nplot,s);
  
   // Get shape functions
   shape(s,psi);

   // Loop over directions
   for(unsigned i=0;i<DIM;i++) 
    {
     interpolated_x[i]=0.0;
     interpolated_u[i]=0.0;
     //Get the index at which velocity i is stored
     unsigned u_nodal_index = u_index_nst(i);
     //Loop over the local nodes and sum
     for(unsigned l=0;l<n_node;l++) 
      {
       interpolated_u[i] += nodal_value(t,l,u_nodal_index)*psi[l];
       interpolated_x[i] += nodal_position(t,l,i)*psi[l];
      }
    }
   
   // Coordinates
   for(unsigned i=0;i<DIM;i++) 
    {
     outfile << interpolated_x[i] << " ";
    }
   
   // Velocities
   for(unsigned i=0;i<DIM;i++) 
    {
     outfile << interpolated_u[i] << " ";
    }
   
   outfile << std::endl;   
  }

 // Write tecplot footer (e.g. FE connectivity lists)
 write_tecplot_zone_footer(outfile,nplot);

}

//==============================================================
/// Output function: 
/// x,y,[z],u,v,[w],p
/// in tecplot format. Specified number of plot points in each
/// coordinate direction.
//==============================================================
template<unsigned DIM>
void NavierStokesEquations<DIM>::output(std::ostream &outfile, 
                                        const unsigned &nplot)
{

 //Vector of local coordinates
 Vector<double> s(DIM);

 // Tecplot header info
 outfile << tecplot_zone_string(nplot);
 
 // Loop over plot points
 unsigned num_plot_points=nplot_points(nplot);
 for (unsigned iplot=0;iplot<num_plot_points;iplot++)
  {
   
   // Get local coordinates of plot point
   get_s_plot(iplot,nplot,s);
  
   // Coordinates
   for(unsigned i=0;i<DIM;i++) 
    {
     outfile << interpolated_x(s,i) << " ";
    }
   
   // Velocities
   for(unsigned i=0;i<DIM;i++) 
    {
     outfile << interpolated_u_nst(s,i) << " ";
    }
   
   // Pressure
   outfile << interpolated_p_nst(s)  << " ";
  
   outfile << std::endl;   
  }
 outfile << std::endl;

 // Write tecplot footer (e.g. FE connectivity lists)
 write_tecplot_zone_footer(outfile,nplot);

}


//==============================================================
/// C-style output function: 
/// x,y,[z],u,v,[w],p
/// in tecplot format. Specified number of plot points in each
/// coordinate direction.
//==============================================================
template<unsigned DIM>
void NavierStokesEquations<DIM>::output(FILE* file_pt,
                                        const unsigned &nplot)
{

 //Vector of local coordinates
 Vector<double> s(DIM);
 
 // Tecplot header info
  fprintf(file_pt,"%s",tecplot_zone_string(nplot).c_str());

 // Loop over plot points
 unsigned num_plot_points=nplot_points(nplot);
 for (unsigned iplot=0;iplot<num_plot_points;iplot++)
  {
   
   // Get local coordinates of plot point
   get_s_plot(iplot,nplot,s);
  
   // Coordinates
   for(unsigned i=0;i<DIM;i++) 
    {
     fprintf(file_pt,"%g ",interpolated_x(s,i));
    }
   
   // Velocities
   for(unsigned i=0;i<DIM;i++) 
    {
     fprintf(file_pt,"%g ",interpolated_u_nst(s,i));
    }
   
   // Pressure
   fprintf(file_pt,"%g \n",interpolated_p_nst(s));
  }
 fprintf(file_pt,"\n");

 // Write tecplot footer (e.g. FE connectivity lists)
 write_tecplot_zone_footer(file_pt,nplot);

}


//==============================================================
/// Full output function: 
/// x,y,[z],u,v,[w],p,du/dt,dv/dt,[dw/dt],dissipation
/// in tecplot format. Specified number of plot points in each
/// coordinate direction 
//==============================================================
template<unsigned DIM>
void NavierStokesEquations<DIM>::full_output(std::ostream &outfile, 
                                             const unsigned &nplot)
{

 //Vector of local coordinates
 Vector<double> s(DIM);
 
 // Acceleration
 Vector<double> dudt(DIM);
 
 // Mesh elocity
 Vector<double> mesh_veloc(DIM,0.0);

 // Tecplot header info
 outfile << tecplot_zone_string(nplot);
  
 //Find out how many nodes there are
 unsigned n_node = nnode();
 
 //Set up memory for the shape functions
 Shape psif(n_node);
 DShape dpsifdx(n_node,DIM);

 // Loop over plot points
 unsigned num_plot_points=nplot_points(nplot);
 for (unsigned iplot=0;iplot<num_plot_points;iplot++)
  {
   
   // Get local coordinates of plot point
   get_s_plot(iplot,nplot,s);
   
   //Call the derivatives of the shape and test functions
   dshape_eulerian(s,psif,dpsifdx);
      
   //Allocate storage
   Vector<double> mesh_veloc(DIM);
   Vector<double> dudt(DIM);
   Vector<double> dudt_ALE(DIM);
   DenseMatrix<double> interpolated_dudx(DIM,DIM);

   //Initialise everything to zero
   for(unsigned i=0;i<DIM;i++)
    {
     mesh_veloc[i]=0.0;
     dudt[i]=0.0;
     dudt_ALE[i]=0.0;
     for(unsigned j=0;j<DIM;j++)
      {
       interpolated_dudx(i,j) = 0.0;
      }
    }
   
   //Calculate velocities and derivatives
   

   //Loop over directions
   for(unsigned i=0;i<DIM;i++)
    {
     //Get the index at which velocity i is stored
     unsigned u_nodal_index = u_index_nst(i);
     // Loop over nodes
     for(unsigned l=0;l<n_node;l++) 
      {
       dudt[i]+=du_dt_nst(l,u_nodal_index)*psif[l];
       mesh_veloc[i]+=dnodal_position_dt(l,i)*psif[l];

       //Loop over derivative directions for velocity gradients
       for(unsigned j=0;j<DIM;j++)
        {                               
         interpolated_dudx(i,j) += nodal_value(l,u_nodal_index)*dpsifdx(l,j);
        }
      }
    }


   // Get dudt in ALE form (incl mesh veloc)
   for(unsigned i=0;i<DIM;i++)
    {
     dudt_ALE[i]=dudt[i];
     for (unsigned k=0;k<DIM;k++)
      {
       dudt_ALE[i]-=mesh_veloc[k]*interpolated_dudx(i,k);
      }
    }
   
  
   // Coordinates
   for(unsigned i=0;i<DIM;i++) 
    {
     outfile << interpolated_x(s,i) << " ";
    }
   
   // Velocities
   for(unsigned i=0;i<DIM;i++) 
    {
     outfile << interpolated_u_nst(s,i) << " ";
    }
   
   // Pressure
   outfile << interpolated_p_nst(s)  << " ";
   
   // Accelerations
   for(unsigned i=0;i<DIM;i++) 
    {
     outfile << dudt_ALE[i] << " ";
    }
   
//    // Mesh velocity
//    for(unsigned i=0;i<DIM;i++) 
//     {
//      outfile << mesh_veloc[i] << " ";
//     }
   
   // Dissipation 
   outfile << dissipation(s) << " ";


   outfile << std::endl;   

  }

 // Write tecplot footer (e.g. FE connectivity lists)
 write_tecplot_zone_footer(outfile,nplot); 
}


//==============================================================
/// Output function for vorticity.
/// x,y,[z],[omega_x,omega_y,[and/or omega_z]]
/// in tecplot format. Specified number of plot points in each
/// coordinate direction.
//==============================================================
template<unsigned DIM>
void NavierStokesEquations<DIM>::output_vorticity(std::ostream &outfile, 
                                                  const unsigned &nplot) 
{

 //Vector of local coordinates
 Vector<double> s(DIM);

 // Create vorticity vector of the required size
 Vector<double> vorticity;
 if (DIM==2) 
  {
   vorticity.resize(1);
  }
 else if (DIM==3)
  {
   vorticity.resize(3);
  }
 else
  {
   std::string error_message =
    "Can't output vorticity in 1D - in fact, what do you mean\n";
   error_message += "by the 1D Navier-Stokes equations?\n";

   throw OomphLibError(error_message,
                       OOMPH_CURRENT_FUNCTION,
                       OOMPH_EXCEPTION_LOCATION);
  }

 // Tecplot header info
 outfile << tecplot_zone_string(nplot);
 
 // Loop over plot points
 unsigned num_plot_points=nplot_points(nplot);
 for (unsigned iplot=0;iplot<num_plot_points;iplot++)
  {
   
   // Get local coordinates of plot point
   get_s_plot(iplot,nplot,s);
  
   // Coordinates
   for(unsigned i=0;i<DIM;i++) 
    {
     outfile << interpolated_x(s,i) << " ";
    }
   
   // Get vorticity
   get_vorticity(s,vorticity);

   if (DIM==2)
    {
     outfile << vorticity[0];
    }
   else 
    {
     outfile << vorticity[0] << " "
             << vorticity[1] << " "
             << vorticity[2] << " ";
    }
  
   outfile << std::endl;   
  }
 outfile << std::endl;

 // Write tecplot footer (e.g. FE connectivity lists)
 write_tecplot_zone_footer(outfile,nplot);

}



//==============================================================
/// Return integral of dissipation over element
//==============================================================
template<unsigned DIM>
double NavierStokesEquations<DIM>::dissipation() const
{  

 // Initialise
 double diss=0.0;

 //Set the value of n_intpt
 unsigned n_intpt = integral_pt()->nweight();
 
 //Set the Vector to hold local coordinates
 Vector<double> s(DIM);
 
 //Loop over the integration points
 for(unsigned ipt=0;ipt<n_intpt;ipt++)
  {
   
   //Assign values of s
   for(unsigned i=0;i<DIM;i++)
    {
     s[i] = integral_pt()->knot(ipt,i);
    }
   
   //Get the integral weight
   double w = integral_pt()->weight(ipt);
   
   // Get Jacobian of mapping
   double J = J_eulerian(s);
   
   // Get strain rate matrix
   DenseMatrix<double> strainrate(DIM,DIM);
   strain_rate(s,strainrate);

   // Initialise
   double local_diss=0.0;
   for(unsigned i=0;i<DIM;i++) 
    {
     for(unsigned j=0;j<DIM;j++) 
      {    
       local_diss+=2.0*strainrate(i,j)*strainrate(i,j);
      }
    }
   
   diss+=local_diss*w*J;
  }

 return diss;

}

//==============================================================
/// Compute traction (on the viscous scale) exerted onto 
/// the fluid at local coordinate s. N has to be outer unit normal
/// to the fluid. 
//==============================================================
template<unsigned DIM>
 void  NavierStokesEquations<DIM>::get_traction(const Vector<double>& s,
                                                const Vector<double>& N, 
                                                Vector<double>& traction)
{
 
 // Get velocity gradients
 DenseMatrix<double> strainrate(DIM,DIM);
 strain_rate(s,strainrate);
 
 // Get pressure
 double press=interpolated_p_nst(s);
 
 // Loop over traction components
 for (unsigned i=0;i<DIM;i++)
  {
   traction[i]=-press*N[i];
   for (unsigned j=0;j<DIM;j++)
    {
     traction[i]+=2.0*strainrate(i,j)*N[j];
    }
  }

}

//==============================================================
/// Compute traction (on the viscous scale) exerted onto 
/// the fluid at local coordinate s, decomposed into pressure and
/// normal and tangential viscous components.
/// N has to be outer unit normal to the fluid.
//==============================================================
template<unsigned DIM>
 void  NavierStokesEquations<DIM>::get_traction(const Vector<double>& s,
                                                const Vector<double>& N,
                                                Vector<double>& traction_p,
                                                Vector<double>& traction_visc_n,
                                                Vector<double>& traction_visc_t)
{
 Vector<double> traction_visc(DIM);

 // Get velocity gradients
 DenseMatrix<double> strainrate(DIM,DIM);
 strain_rate(s,strainrate);

 // Get pressure
 double press=interpolated_p_nst(s);

 // Loop over traction components
 for (unsigned i=0;i<DIM;i++)
  {
   // pressure
   traction_p[i]=-press*N[i];
   for (unsigned j=0;j<DIM;j++)
    {
     // viscous stress
     traction_visc[i]+=2.0*strainrate(i,j)*N[j];
    }
   // Decompose viscous stress into normal and tangential components
   traction_visc_n[i]=traction_visc[i]*N[i];
   traction_visc_t[i]=traction_visc[i]-traction_visc_n[i];
  }

}

//==============================================================
/// Return dissipation at local coordinate s
//==============================================================
template<unsigned DIM>
double NavierStokesEquations<DIM>::dissipation(const Vector<double>& s) const
{  
 // Get strain rate matrix
 DenseMatrix<double> strainrate(DIM,DIM);
 strain_rate(s,strainrate);
 
 // Initialise
 double local_diss=0.0;
 for(unsigned i=0;i<DIM;i++) 
  {
   for(unsigned j=0;j<DIM;j++) 
    {    
     local_diss+=2.0*strainrate(i,j)*strainrate(i,j);
    }
  }
 
 return local_diss;
}

//==============================================================
/// Get strain-rate tensor: 1/2 (du_i/dx_j + du_j/dx_i)
//==============================================================
template<unsigned DIM>
void NavierStokesEquations<DIM>::strain_rate(const Vector<double>& s, 
                                        DenseMatrix<double>& strainrate) const
  {

#ifdef PARANOID
   if ((strainrate.ncol()!=DIM)||(strainrate.nrow()!=DIM))
    {
     std::ostringstream error_message;
     error_message  << "The strain rate has incorrect dimensions " 
                    << strainrate.ncol() << " x " 
                    << strainrate.nrow() << " Not " << DIM << std::endl;

     throw OomphLibError(error_message.str(),
                         OOMPH_CURRENT_FUNCTION,
                         OOMPH_EXCEPTION_LOCATION);
    }
#endif

   // Velocity gradient matrix
   DenseMatrix<double> dudx(DIM);

   //Find out how many nodes there are in the element
   unsigned n_node = nnode();

   //Set up memory for the shape and test functions
   Shape psi(n_node);
   DShape dpsidx(n_node,DIM);
 
   //Call the derivatives of the shape functions
   dshape_eulerian(s,psi,dpsidx);
     
   //Initialise to zero
   for(unsigned i=0;i<DIM;i++)
    {
     for(unsigned j=0;j<DIM;j++)
      {
       dudx(i,j) = 0.0;
      }
    }

     //Loop over veclocity components
     for(unsigned i=0;i<DIM;i++)
      {
       //Get the index at which the i-th velocity is stored
       unsigned u_nodal_index = u_index_nst(i);
       //Loop over derivative directions
       for(unsigned j=0;j<DIM;j++)
        {                               
         // Loop over nodes
         for(unsigned l=0;l<n_node;l++) 
          {
           dudx(i,j) += nodal_value(l,u_nodal_index)*dpsidx(l,j);
          }
        }
      }

   //Loop over veclocity components
   for(unsigned i=0;i<DIM;i++)
    {
     //Loop over derivative directions
     for(unsigned j=0;j<DIM;j++)
      {                               
       strainrate(i,j)=0.5*(dudx(i,j)+dudx(j,i));
      }
    }

  }



//==============================================================
/// Compute 2D vorticity vector at local coordinate s (return in
/// one and only component of vorticity vector
//==============================================================
template<>
void NavierStokesEquations<2>::get_vorticity(const Vector<double>& s, 
                                             Vector<double>& vorticity) const
{

#ifdef PARANOID
   if (vorticity.size()!=1)
    {
     std::ostringstream error_message;
     error_message 
      << "Computation of vorticity in 2D requires a 1D vector\n"
      << "which contains the only non-zero component of the\n"
      << "vorticity vector. You've passed a vector of size "
      << vorticity.size() << std::endl;
     
     throw OomphLibError(error_message.str(),
                         OOMPH_CURRENT_FUNCTION,
                         OOMPH_EXCEPTION_LOCATION);
    }
#endif

 // Specify spatial dimension
 unsigned DIM=2;
 
 // Velocity gradient matrix
 DenseMatrix<double> dudx(DIM);
 
 //Find out how many nodes there are in the element
 unsigned n_node = nnode();
 
 //Set up memory for the shape and test functions
 Shape psi(n_node);
 DShape dpsidx(n_node,DIM);
 
 //Call the derivatives of the shape functions
 dshape_eulerian(s,psi,dpsidx);
 
 //Initialise to zero
 for(unsigned i=0;i<DIM;i++)
  {
   for(unsigned j=0;j<DIM;j++)
    {
     dudx(i,j) = 0.0;
    }
  }
 
   //Loop over veclocity components
   for(unsigned i=0;i<DIM;i++)
    {
     //Get the index at which the i-th velocity is stored
     unsigned u_nodal_index = u_index_nst(i);
     //Loop over derivative directions
     for(unsigned j=0;j<DIM;j++)
      {                               
       // Loop over nodes
       for(unsigned l=0;l<n_node;l++) 
        {
         dudx(i,j) += nodal_value(l,u_nodal_index)*dpsidx(l,j);
        }
      }
    }

 // Z-component of vorticity
 vorticity[0]=dudx(1,0)-dudx(0,1);

}




//==============================================================
/// Compute 3D vorticity vector at local coordinate s
//==============================================================
template<>
void NavierStokesEquations<3>::get_vorticity(const Vector<double>& s, 
                                             Vector<double>& vorticity) const
{

#ifdef PARANOID
   if (vorticity.size()!=3)
    {
     std::ostringstream error_message;
     error_message 
      << "Computation of vorticity in 3D requires a 3D vector\n"
      << "which contains the only non-zero component of the\n"
      << "vorticity vector. You've passed a vector of size "
      << vorticity.size() << std::endl;
     
     throw OomphLibError(error_message.str(),
                         OOMPH_CURRENT_FUNCTION,
                         OOMPH_EXCEPTION_LOCATION);
    }
#endif

 // Specify spatial dimension
 unsigned DIM=3;
 
 // Velocity gradient matrix
 DenseMatrix<double> dudx(DIM);
 
 //Find out how many nodes there are in the element
 unsigned n_node = nnode();
 
 //Set up memory for the shape and test functions
 Shape psi(n_node);
 DShape dpsidx(n_node,DIM);
 
 //Call the derivatives of the shape functions
 dshape_eulerian(s,psi,dpsidx);
 
 //Initialise to zero
 for(unsigned i=0;i<DIM;i++)
  {
   for(unsigned j=0;j<DIM;j++)
    {
     dudx(i,j) = 0.0;
    }
  }
 
   //Loop over veclocity components
   for(unsigned i=0;i<DIM;i++)
    {
     //Get the index at which the i-th velocity component is stored
     unsigned u_nodal_index = u_index_nst(i);
     //Loop over derivative directions
     for(unsigned j=0;j<DIM;j++)
      {                               
       // Loop over nodes
       for(unsigned l=0;l<n_node;l++) 
        {
         dudx(i,j) += nodal_value(l,u_nodal_index)*dpsidx(l,j);
        }
      }
    }

 vorticity[0]=dudx(2,1)-dudx(1,2);
 vorticity[1]=dudx(0,2)-dudx(2,0);
 vorticity[2]=dudx(1,0)-dudx(0,1);
  
}


//==============================================================
///  \short Get integral of kinetic energy over element:
/// Note that this is the "raw" kinetic energy in the sense
/// that the density ratio has not been included. In problems
/// with two or more fluids the user will have to remember
/// to premultiply certain elements by the appropriate density
/// ratio.
//==============================================================
template<unsigned DIM>
double NavierStokesEquations<DIM>::kin_energy() const
{  

 // Initialise
 double kin_en=0.0;

 //Set the value of n_intpt
 unsigned n_intpt = integral_pt()->nweight();
 
 //Set the Vector to hold local coordinates
 Vector<double> s(DIM);
 
 //Loop over the integration points
 for(unsigned ipt=0;ipt<n_intpt;ipt++)
  {   
   //Assign values of s
   for(unsigned i=0;i<DIM;i++)
    {
     s[i] = integral_pt()->knot(ipt,i);
    }
   
   //Get the integral weight
   double w = integral_pt()->weight(ipt);
   
   //Get Jacobian of mapping
   double J = J_eulerian(s);
   
   // Loop over directions
   double veloc_squared=0.0;
   for(unsigned i=0;i<DIM;i++) 
    {
     veloc_squared+=interpolated_u_nst(s,i)*interpolated_u_nst(s,i);
    }
   
   kin_en+=0.5*veloc_squared*w*J;
  }

 return kin_en;

}


//==========================================================================
///  \short Get integral of time derivative of kinetic energy over element:
//==========================================================================
template<unsigned DIM>
double NavierStokesEquations<DIM>::d_kin_energy_dt() const
{  
 // Initialise
 double d_kin_en_dt=0.0;

 //Set the value of n_intpt
 const unsigned n_intpt = integral_pt()->nweight();
 
 //Set the Vector to hold local coordinates
 Vector<double> s(DIM);

 //Get the number of nodes
 const unsigned n_node = this->nnode();

 //Storage for the shape function
 Shape psi(n_node);
 DShape dpsidx(n_node,DIM);

 //Get the value at which the velocities are stored
 unsigned u_index[DIM];
 for(unsigned i=0;i<DIM;i++) {u_index[i] = this->u_index_nst(i);}

 //Loop over the integration points
 for(unsigned ipt=0;ipt<n_intpt;ipt++)
  {   
   //Get the jacobian of the mapping
   double J = dshape_eulerian_at_knot(ipt,psi,dpsidx);

   //Get the integral weight
   double w = integral_pt()->weight(ipt);

   //Now work out the velocity and the time derivative
   Vector<double> interpolated_u(DIM,0.0), interpolated_dudt(DIM,0.0);

   //Loop over the shape functions
   for(unsigned l=0;l<n_node;l++)
    {
     //Loop over the dimensions
     for(unsigned i=0;i<DIM;i++)
      {
       interpolated_u[i] += nodal_value(l,u_index[i])*psi(l);
       interpolated_dudt[i] += du_dt_nst(l,u_index[i])*psi(l);
      }
    }
   
   //Get mesh velocity bit
   if (!ALE_is_disabled)
    {
     Vector<double> mesh_velocity(DIM,0.0);
     DenseMatrix<double> interpolated_dudx(DIM,DIM,0.0);

     // Loop over nodes again
     for(unsigned l=0;l<n_node;l++) 
      {
       for(unsigned i=0;i<DIM;i++)
        {
         mesh_velocity[i] += this->dnodal_position_dt(l,i)*psi(l);

         for(unsigned j=0;j<DIM;j++)
          {
           interpolated_dudx(i,j) += 
            this->nodal_value(l,u_index[i])*dpsidx(l,j);
          }
        }
      }

     //Subtract mesh velocity from du_dt
     for(unsigned i=0;i<DIM;i++)
      {
       for (unsigned k=0;k<DIM;k++)
        {
         interpolated_dudt[i] -= mesh_velocity[k]*interpolated_dudx(i,k);
        }
      }
    }
   

   // Loop over directions and add up u du/dt  terms
   double sum=0.0;
   for(unsigned i=0;i<DIM;i++) 
    { sum += interpolated_u[i]*interpolated_dudt[i];}
   
   d_kin_en_dt += sum*w*J;
  }
 
 return d_kin_en_dt;

}


//==============================================================
/// Return pressure integrated over the element
//==============================================================
template<unsigned DIM>
double NavierStokesEquations<DIM>::pressure_integral() const
{

 // Initialise 
 double press_int=0;

 //Set the value of n_intpt
 unsigned n_intpt = integral_pt()->nweight();
 
 //Set the Vector to hold local coordinates
 Vector<double> s(DIM);
 
 //Loop over the integration points
 for(unsigned ipt=0;ipt<n_intpt;ipt++)
  {
   
   //Assign values of s
   for(unsigned i=0;i<DIM;i++)
    {
     s[i] = integral_pt()->knot(ipt,i);
    }
   
   //Get the integral weight
   double w = integral_pt()->weight(ipt);
   
   //Get Jacobian of mapping
   double J = J_eulerian(s);
   
   //Premultiply the weights and the Jacobian
   double W = w*J;
   
   // Get pressure
   double press=interpolated_p_nst(s);
   
   // Add
   press_int+=press*W;
   
  }
 
 return press_int;

}


//==============================================================
/// Compute the residuals for the associated pressure advection 
/// diffusion problem. Used by the Fp preconditioner.
/// flag=1(or 0): do (or don't) compute the Jacobian as well. 
//==============================================================
template<unsigned DIM>
void NavierStokesEquations<DIM>::
fill_in_generic_pressure_advection_diffusion_contribution_nst(
 Vector<double> &residuals, 
 DenseMatrix<double> &jacobian, 
 unsigned flag)
{
 // Return immediately if there are no dofs
 if (ndof()==0) return;

 //Find out how many nodes there are
 unsigned n_node = nnode();
 
 //Find out how many pressure dofs there are
 unsigned n_pres = npres_nst();

 //Find the indices at which the local velocities are stored
 unsigned u_nodal_index[DIM];
 for(unsigned i=0;i<DIM;i++) {u_nodal_index[i] = u_index_nst(i);}

 //Set up memory for the velocity shape fcts
 Shape psif(n_node);
 DShape dpsidx(n_node,DIM);

 //Set up memory for pressure shape and test functions
 Shape psip(n_pres), testp(n_pres);
 DShape dpsip(n_pres,DIM);
 DShape dtestp(n_pres,DIM);

 //Number of integration points
 unsigned n_intpt = integral_pt()->nweight();
   
 //Set the Vector to hold local coordinates
 Vector<double> s(DIM);

 //Get Physical Variables from Element
 //Reynolds number must be multiplied by the density ratio
 double scaled_re = re()*density_ratio();
 
 //Integers to store the local equations and unknowns
 int local_eqn=0, local_unknown=0;

 //Loop over the integration points
 for(unsigned ipt=0;ipt<n_intpt;ipt++)
  {
   //Assign values of s
   for(unsigned i=0;i<DIM;i++) s[i] = integral_pt()->knot(ipt,i);

   //Get the integral weight
   double w = integral_pt()->weight(ipt);
   
   // Call the derivatives of the veloc shape functions
   // (Derivs not needed but they are free)
   double J = this->dshape_eulerian_at_knot(ipt,psif,dpsidx);
   
   //Call the pressure shape and test functions
   this->dpshape_and_dptest_eulerian_nst(s,psip,dpsip,testp,dtestp);
   
   //Premultiply the weights and the Jacobian
   double W = w*J;
   
   //Calculate local values of the pressure and velocity components
   //Allocate
   Vector<double> interpolated_u(DIM,0.0);
   Vector<double> interpolated_x(DIM,0.0);
   Vector<double> interpolated_dpdx(DIM,0.0);
   
   //Calculate pressure gradient
   for(unsigned l=0;l<n_pres;l++) 
    {
     for (unsigned i=0;i<DIM;i++)
      {
       interpolated_dpdx[i] += p_nst(l)*dpsip(l,i);
      }
    }

   //Calculate velocities 

   // Loop over nodes
   for(unsigned l=0;l<n_node;l++) 
    {
     //Loop over directions
     for(unsigned i=0;i<DIM;i++)
      {
       //Get the nodal value
       double u_value = raw_nodal_value(l,u_nodal_index[i]);
       interpolated_u[i] += u_value*psif[l];
       interpolated_x[i] += raw_nodal_position(l,i)*psif[l];
      }
    }

   // Source function (for validaton only)
   double source=0.0;
   if (Press_adv_diff_source_fct_pt!=0)
    {
     source=Press_adv_diff_source_fct_pt(interpolated_x);
    }

   //Loop over the shape functions
   for(unsigned l=0;l<n_pres;l++)
    {
     local_eqn = p_local_eqn(l);

     //If not a boundary conditions
     if(local_eqn >= 0)
      {
       residuals[local_eqn] -= source*testp[l]*W;
       for(unsigned k=0;k<DIM;k++)
        {
         residuals[local_eqn] += interpolated_dpdx[k]*
          (scaled_re*interpolated_u[k]*testp[l]+dtestp(l,k))*W;
        }
       
       // Jacobian too?
       if(flag)
        {
         //Loop over the shape functions
         for(unsigned l2=0;l2<n_pres;l2++)
          {
           local_unknown = p_local_eqn(l2);
           
           //If not a boundary conditions
           if(local_unknown >= 0)
            {          
             if ((int(eqn_number(local_eqn))!=Pinned_fp_pressure_eqn)&&
                 (int(eqn_number(local_unknown))!=Pinned_fp_pressure_eqn))
              {
               for(unsigned k=0;k<DIM;k++)
                {
                 jacobian(local_eqn,local_unknown)+=dtestp(l2,k)*
                  (scaled_re*interpolated_u[k]*testp[l]+dtestp(l,k))*W;
                }
              }
             else
              {
               if ((int(eqn_number(local_eqn))==Pinned_fp_pressure_eqn)&&
                    (int(eqn_number(local_unknown))==Pinned_fp_pressure_eqn))
                {
                 jacobian(local_eqn,local_unknown)=1.0;
                }
              }
            }
          } 
        } /*End of Jacobian calculation*/
      } //End of if not boundary condition
    }//End of loop over l
  }

 // Now add boundary contributions from Robin BCs
 unsigned nrobin=Pressure_advection_diffusion_robin_element_pt.size();
 for (unsigned e=0;e<nrobin;e++)
  {
   Pressure_advection_diffusion_robin_element_pt[e]->
    fill_in_generic_residual_contribution_fp_press_adv_diff_robin_bc(
     residuals,jacobian,flag);
  }
}


//==============================================================
///  Compute the residuals for the Navier--Stokes 
///  equations; flag=1(or 0): do (or don't) compute the 
///  Jacobian as well. 
//==============================================================
template<unsigned DIM>
void NavierStokesEquations<DIM>::
fill_in_generic_residual_contribution_nst(Vector<double> &residuals, 
                                          DenseMatrix<double> &jacobian, 
                                          DenseMatrix<double> &mass_matrix,
                                          unsigned flag)
{
 // Return immediately if there are no dofs
 if (ndof()==0) return;

 //Find out how many nodes there are
 unsigned n_node = nnode();
 
 // Get continuous time from timestepper of first node
 double time=node_pt(0)->time_stepper_pt()->time_pt()->time();
  
 //Find out how many pressure dofs there are
 unsigned n_pres = npres_nst();

 //Find the indices at which the local velocities are stored
 unsigned u_nodal_index[DIM];
 for(unsigned i=0;i<DIM;i++) {u_nodal_index[i] = u_index_nst(i);}

 //Set up memory for the shape and test functions
 Shape psif(n_node), testf(n_node);
 DShape dpsifdx(n_node,DIM), dtestfdx(n_node,DIM);
  
 //Set up memory for pressure shape and test functions
 Shape psip(n_pres), testp(n_pres);

 //Number of integration points
 unsigned n_intpt = integral_pt()->nweight();
   
 //Set the Vector to hold local coordinates
 Vector<double> s(DIM);

 //Get Physical Variables from Element
 //Reynolds number must be multiplied by the density ratio
 double scaled_re = re()*density_ratio();
 double scaled_re_st = re_st()*density_ratio();
 double scaled_re_inv_fr = re_invfr()*density_ratio();
 double visc_ratio = viscosity_ratio();
 Vector<double> G = g();
 
 //Integers to store the local equations and unknowns
 int local_eqn=0, local_unknown=0;

 //Loop over the integration points
 for(unsigned ipt=0;ipt<n_intpt;ipt++)
  {
   //Assign values of s
   for(unsigned i=0;i<DIM;i++) s[i] = integral_pt()->knot(ipt,i);
   //Get the integral weight
   double w = integral_pt()->weight(ipt);
   
   //Call the derivatives of the shape and test functions
   double J = 
    dshape_and_dtest_eulerian_at_knot_nst(ipt,psif,dpsifdx,testf,dtestfdx);
   
   //Call the pressure shape and test functions
   pshape_nst(s,psip,testp);
   
   //Premultiply the weights and the Jacobian
   double W = w*J;
   
   //Calculate local values of the pressure and velocity components
   //Allocate
   double interpolated_p=0.0;
   Vector<double> interpolated_u(DIM,0.0);
   Vector<double> interpolated_x(DIM,0.0);
   Vector<double> mesh_velocity(DIM,0.0);
   Vector<double> dudt(DIM,0.0);
   DenseMatrix<double> interpolated_dudx(DIM,DIM,0.0);    
   
   //Calculate pressure
   for(unsigned l=0;l<n_pres;l++) interpolated_p += p_nst(l)*psip[l];
   
   //Calculate velocities and derivatives:

   // Loop over nodes
   for(unsigned l=0;l<n_node;l++) 
    {
     //Loop over directions
     for(unsigned i=0;i<DIM;i++)
      {
       //Get the nodal value
       double u_value = raw_nodal_value(l,u_nodal_index[i]);
       interpolated_u[i] += u_value*psif[l];
       interpolated_x[i] += raw_nodal_position(l,i)*psif[l];
       dudt[i] += du_dt_nst(l,i)*psif[l];
       
       //Loop over derivative directions
       for(unsigned j=0;j<DIM;j++)
        {                               
         interpolated_dudx(i,j) += u_value*dpsifdx(l,j);
        }
      }
    }

   if (!ALE_is_disabled)
    {
     // Loop over nodes
     for(unsigned l=0;l<n_node;l++) 
      {
       //Loop over directions
       for(unsigned i=0;i<DIM;i++)
        {
         mesh_velocity[i] += this->raw_dnodal_position_dt(l,i)*psif[l];
        }
      }
    }
   
   //Get the user-defined body force terms
   Vector<double> body_force(DIM);
   get_body_force_nst(time,ipt,s,interpolated_x,body_force);
   
   //Get the user-defined source function
   double source = get_source_nst(time,ipt,interpolated_x);


   //MOMENTUM EQUATIONS
   //------------------
   
   //Loop over the test functions
   for(unsigned l=0;l<n_node;l++)
    {
     //Loop over the velocity components
     for(unsigned i=0;i<DIM;i++)
      {
       /*IF it's not a boundary condition*/
       local_eqn = nodal_local_eqn(l,u_nodal_index[i]);
       if(local_eqn >= 0)
        {
         //Add the user-defined body force terms
         residuals[local_eqn] += 
          body_force[i]*testf[l]*W;

         //Add the gravitational body force term
         residuals[local_eqn] += scaled_re_inv_fr*testf[l]*G[i]*W;
         
         //Add the pressure gradient term
         residuals[local_eqn]  += interpolated_p*dtestfdx(l,i)*W;
         
         //Add in the stress tensor terms
         //The viscosity ratio needs to go in here to ensure
         //continuity of normal stress is satisfied even in flows
         //with zero pressure gradient!
         for(unsigned k=0;k<DIM;k++)
          {
           residuals[local_eqn] -= visc_ratio*
            (interpolated_dudx(i,k) + Gamma[i]*interpolated_dudx(k,i))
            *dtestfdx(l,k)*W;
          }
         
         //Add in the inertial terms
         //du/dt term
         residuals[local_eqn] -= scaled_re_st*dudt[i]*testf[l]*W;
         
         
         //Convective terms, including mesh velocity
         for(unsigned k=0;k<DIM;k++)
          {
           double tmp=scaled_re*interpolated_u[k];
           if (!ALE_is_disabled) tmp-=scaled_re_st*mesh_velocity[k];
           residuals[local_eqn] -= tmp*interpolated_dudx(i,k)*testf[l]*W;

          }
         
         //CALCULATE THE JACOBIAN
         if(flag)
          {
           //Loop over the velocity shape functions again
           for(unsigned l2=0;l2<n_node;l2++)
            { 
             //Loop over the velocity components again
             for(unsigned i2=0;i2<DIM;i2++)
              {
               //If at a non-zero degree of freedom add in the entry
               local_unknown = nodal_local_eqn(l2,u_nodal_index[i2]);
               if(local_unknown >= 0)
                {
                 //Add contribution to Elemental Matrix
                 jacobian(local_eqn,local_unknown) 
                  -= visc_ratio*Gamma[i]*dpsifdx(l2,i)*dtestfdx(l,i2)*W;
                 
                 //Extra component if i2 = i
                 if(i2 == i)
                  {      
                   /*Loop over velocity components*/
                   for(unsigned k=0;k<DIM;k++)
                    { 
                     jacobian(local_eqn,local_unknown)
                      -= visc_ratio*dpsifdx(l2,k)*dtestfdx(l,k)*W;
                    }
                  }
                 
                 //Now add in the inertial terms
                 jacobian(local_eqn,local_unknown)
                  -= scaled_re*psif[l2]*interpolated_dudx(i,i2)*testf[l]*W;
                 
                 //Extra component if i2=i
                 if(i2 == i)
                  {
                   //Add the mass matrix term (only diagonal entries)
                   //Note that this is positive because the mass matrix
                   //is taken to the other side of the equation when
                   //formulating the generalised eigenproblem.
                   if(flag==2)
                    {
                     mass_matrix(local_eqn,local_unknown) +=
                      scaled_re_st*psif[l2]*testf[l]*W;
                    }
                   
                   //du/dt term
                   jacobian(local_eqn,local_unknown)
                    -= scaled_re_st*
                    node_pt(l2)->time_stepper_pt()->weight(1,0)*
                    psif[l2]*testf[l]*W;  
                   
                   //Loop over the velocity components
                   for(unsigned k=0;k<DIM;k++)
                    {
                     double tmp=scaled_re*interpolated_u[k];
                     if (!ALE_is_disabled) tmp-=scaled_re_st*mesh_velocity[k];
                     jacobian(local_eqn,local_unknown) -=
                      tmp*dpsifdx(l2,k)*testf[l]*W;
                    }
                  }
                 
                }
              }
            }
           
           /*Now loop over pressure shape functions*/
           /*This is the contribution from pressure gradient*/
           for(unsigned l2=0;l2<n_pres;l2++)
            {
             /*If we are at a non-zero degree of freedom in the entry*/
             local_unknown = p_local_eqn(l2);
             if(local_unknown >= 0)
              {
               jacobian(local_eqn,local_unknown)
                += psip[l2]*dtestfdx(l,i)*W;
              }
            }
          } /*End of Jacobian calculation*/
         
        } //End of if not boundary condition statement
       
      } //End of loop over dimension
    } //End of loop over shape functions
   
   
   
   //CONTINUITY EQUATION
   //-------------------
   
   //Loop over the shape functions
   for(unsigned l=0;l<n_pres;l++)
    {
     local_eqn = p_local_eqn(l);
     //If not a boundary conditions
     if(local_eqn >= 0)
      {

       // Source term
       //residuals[local_eqn] -=source*testp[l]*W; 
       double aux=-source;

       //Loop over velocity components
       for(unsigned k=0;k<DIM;k++)
        {
         //residuals[local_eqn] += interpolated_dudx(k,k)*testp[l]*W;
         aux += interpolated_dudx(k,k);
        }

       residuals[local_eqn]+=aux*testp[l]*W;
       
       /*CALCULATE THE JACOBIAN*/
       if(flag)
        {
         /*Loop over the velocity shape functions*/
         for(unsigned l2=0;l2<n_node;l2++)
          { 
           /*Loop over velocity components*/
           for(unsigned i2=0;i2<DIM;i2++)
            {
             /*If we're at a non-zero degree of freedom add it in*/ 
             local_unknown = nodal_local_eqn(l2,u_nodal_index[i2]);
             if(local_unknown >= 0)
              {
               jacobian(local_eqn,local_unknown)
                += dpsifdx(l2,i2)*testp[l]*W;
              }
            } /*End of loop over i2*/
          } /*End of loop over l2*/
        } /*End of Jacobian calculation*/
       
      } //End of if not boundary condition
    } //End of loop over l
  }
 
}

//==============================================================
///  Compute the derivatives of the residuals for the Navier--Stokes 
///  equations with respect to a parameter; 
/// flag=2 or 1 or 0: do (or don't) compute the 
///  Jacobian and mass matrix as well  
//==============================================================
template<unsigned DIM>
void NavierStokesEquations<DIM>::
fill_in_generic_dresidual_contribution_nst(
 double* const &parameter_pt,
 Vector<double> &dres_dparam, 
 DenseMatrix<double> &djac_dparam, 
 DenseMatrix<double> &dmass_matrix_dparam,
 unsigned flag)
{
 throw OomphLibError("Not yet implemented\n",
                     OOMPH_CURRENT_FUNCTION,
                     OOMPH_EXCEPTION_LOCATION);
}

//==================================================================
/// \short Compute the hessian tensor vector products required to
/// perform continuation of bifurcations analytically
//================================================================ 
template<unsigned DIM>
void NavierStokesEquations<DIM>::
fill_in_contribution_to_hessian_vector_products(
 Vector<double> const &Y,
 DenseMatrix<double> const &C,
 DenseMatrix<double> &product)
{
 throw OomphLibError(
  "Not yet implemented\n",
  OOMPH_CURRENT_FUNCTION,
  OOMPH_EXCEPTION_LOCATION);
}


//======================================================================
/// Compute derivatives of elemental residual vector with respect
/// to nodal coordinates. 
/// dresidual_dnodal_coordinates(l,i,j) = d res(l) / dX_{ij}
/// Overloads the FD-based version in the FE base class.
//======================================================================
template <unsigned DIM>
void NavierStokesEquations<DIM>::get_dresidual_dnodal_coordinates(
 RankThreeTensor<double>&
 dresidual_dnodal_coordinates)
{
 // Return immediately if there are no dofs
 if(ndof()==0) { return; }

 // Determine number of nodes in element
 const unsigned n_node = nnode();
 
 // Get continuous time from timestepper of first node
 double time=node_pt(0)->time_stepper_pt()->time_pt()->time();
  
 // Determine number of pressure dofs in element
 const unsigned n_pres = npres_nst();

 // Find the indices at which the local velocities are stored
 unsigned u_nodal_index[DIM];
 for(unsigned i=0;i<DIM;i++) { u_nodal_index[i] = u_index_nst(i); }

 // Set up memory for the shape and test functions
 Shape psif(n_node), testf(n_node);
 DShape dpsifdx(n_node,DIM), dtestfdx(n_node,DIM);

 // Set up memory for pressure shape and test functions
 Shape psip(n_pres), testp(n_pres);

 // Deriatives of shape fct derivatives w.r.t. nodal coords
 RankFourTensor<double> d_dpsifdx_dX(DIM,n_node,n_node,DIM);
 RankFourTensor<double> d_dtestfdx_dX(DIM,n_node,n_node,DIM);

 // Derivative of Jacobian of mapping w.r.t. to nodal coords
 DenseMatrix<double> dJ_dX(DIM,n_node);

 // Derivatives of derivative of u w.r.t. nodal coords
 RankFourTensor<double> d_dudx_dX(DIM,n_node,DIM,DIM);

 // Derivatives of nodal velocities w.r.t. nodal coords:
 // Assumption: Interaction only local via no-slip so 
 // X_ij only affects U_ij.
 DenseMatrix<double> d_U_dX(DIM,n_node,0.0);

 // Determine the number of integration points
 const unsigned n_intpt = integral_pt()->nweight();
   
 // Vector to hold local coordinates
 Vector<double> s(DIM);

 // Get physical variables from element
 // (Reynolds number must be multiplied by the density ratio)
 double scaled_re = re()*density_ratio();
 double scaled_re_st = re_st()*density_ratio();
 double scaled_re_inv_fr = re_invfr()*density_ratio();
 double visc_ratio = viscosity_ratio();
 Vector<double> G = g();
 
 // FD step 
 double eps_fd = GeneralisedElement::Default_fd_jacobian_step;
   
 // Pre-compute derivatives of nodal velocities w.r.t. nodal coords:
 // Assumption: Interaction only local via no-slip so 
 // X_ij only affects U_ij.
 bool element_has_node_with_aux_node_update_fct=false;
 for (unsigned q=0;q<n_node;q++)
  {
   Node* nod_pt=node_pt(q);

   // Only compute if there's a node-update fct involved
   if (nod_pt->has_auxiliary_node_update_fct_pt())
    {
     element_has_node_with_aux_node_update_fct=true;

     // Current nodal velocity
     Vector<double> u_ref(DIM);
     for (unsigned i=0;i<DIM;i++)
      {
       u_ref[i]=*(nod_pt->value_pt(u_nodal_index[i]));
      }
     
     // FD
     for (unsigned p=0;p<DIM;p++)
      {
       // Make backup
       double backup=nod_pt->x(p);
       
       // Do FD step. No node update required as we're
       // attacking the coordinate directly...
       nod_pt->x(p)+=eps_fd;
       
       // Do auxiliary node update (to apply no slip)
       nod_pt->perform_auxiliary_node_update_fct();
       
       // Evaluate
       d_U_dX(p,q)=(*(nod_pt->value_pt(u_nodal_index[p]))-u_ref[p])/eps_fd;
       
       // Reset 
       nod_pt->x(p)=backup;
       
       // Do auxiliary node update (to apply no slip)
       nod_pt->perform_auxiliary_node_update_fct();
      }
    }
  }

 // Integer to store the local equation number
 int local_eqn=0;

 // Loop over the integration points
 for(unsigned ipt=0;ipt<n_intpt;ipt++)
  {
   // Assign values of s
   for(unsigned i=0;i<DIM;i++) { s[i] = integral_pt()->knot(ipt,i); }

   // Get the integral weight
   const double w = integral_pt()->weight(ipt);
   
   // Call the derivatives of the shape and test functions
   const double J = dshape_and_dtest_eulerian_at_knot_nst(ipt,psif,dpsifdx,
                                                          d_dpsifdx_dX,
                                                          testf,dtestfdx,
                                                          d_dtestfdx_dX,dJ_dX);
   
   // Call the pressure shape and test functions
   pshape_nst(s,psip,testp);
   
   // Calculate local values of the pressure and velocity components
   // Allocate
   double interpolated_p=0.0;
   Vector<double> interpolated_u(DIM,0.0);
   Vector<double> interpolated_x(DIM,0.0);
   Vector<double> mesh_velocity(DIM,0.0);
   Vector<double> dudt(DIM,0.0);
   DenseMatrix<double> interpolated_dudx(DIM,DIM,0.0);    

   // Calculate pressure
   for(unsigned l=0;l<n_pres;l++) { interpolated_p += p_nst(l)*psip[l]; }
   
   // Calculate velocities and derivatives:

   // Loop over nodes
   for(unsigned l=0;l<n_node;l++) 
    {
     // Loop over directions
     for(unsigned i=0;i<DIM;i++)
      {
       // Get the nodal value
       double u_value = raw_nodal_value(l,u_nodal_index[i]);
       interpolated_u[i] += u_value*psif[l];
       interpolated_x[i] += raw_nodal_position(l,i)*psif[l];
       dudt[i] += du_dt_nst(l,i)*psif[l];
       
       // Loop over derivative directions
       for(unsigned j=0;j<DIM;j++)
        {                               
         interpolated_dudx(i,j) += u_value*dpsifdx(l,j);
        }
      }
    }

   if(!ALE_is_disabled)
    {
     // Loop over nodes
     for(unsigned l=0;l<n_node;l++) 
      {
       // Loop over directions
       for(unsigned i=0;i<DIM;i++)
        {
         mesh_velocity[i] += this->raw_dnodal_position_dt(l,i)*psif[l];
        }
      }
    }

   // Calculate derivative of du_i/dx_k w.r.t. nodal positions X_{pq}
   for(unsigned q=0;q<n_node;q++)
    {
     // Loop over coordinate directions
     for(unsigned p=0;p<DIM;p++)
      {
       for(unsigned i=0;i<DIM;i++)
        {
         for(unsigned k=0;k<DIM;k++)
          {
           double aux=0.0;
           for(unsigned j=0;j<n_node;j++)
            {
             aux += 
              raw_nodal_value(j,u_nodal_index[i])*d_dpsifdx_dX(p,q,j,k);
            }
           d_dudx_dX(p,q,i,k) = aux;
          }
        }
      }
    }

   // Get weight of actual nodal position/value in computation of mesh
   // velocity from positional/value time stepper
   const double pos_time_weight
    = node_pt(0)->position_time_stepper_pt()->weight(1,0);
   const double val_time_weight = node_pt(0)->time_stepper_pt()->weight(1,0);

   // Get the user-defined body force terms
   Vector<double> body_force(DIM);
   get_body_force_nst(time,ipt,s,interpolated_x,body_force);
   
   // Get the user-defined source function
   const double source = get_source_nst(time,ipt,interpolated_x);

   // Note: Can use raw values and avoid bypassing hanging information
   // because this is the non-refineable version! 

   // Get gradient of body force function
   DenseMatrix<double> d_body_force_dx(DIM,DIM,0.0);
   get_body_force_gradient_nst(
    time,ipt,s,interpolated_x,d_body_force_dx);

   // Get gradient of source function
   Vector<double> source_gradient(DIM,0.0);
   get_source_gradient_nst(time,ipt,interpolated_x,source_gradient);


   // Assemble shape derivatives
   // --------------------------

   // MOMENTUM EQUATIONS
   // ------------------
   
   // Loop over the test functions
   for(unsigned l=0;l<n_node;l++)
    {

     // Loop over coordinate directions
     for(unsigned i=0;i<DIM;i++)
      {
       //Get the local equation
       local_eqn = nodal_local_eqn(l,u_nodal_index[i]);;

       // IF it's not a boundary condition
       if(local_eqn >= 0)
        {
         // Loop over coordinate directions
         for (unsigned p=0;p<DIM;p++)
          {              
           // Loop over nodes
           for (unsigned q=0;q<n_node;q++)
            {       

             // Residual x deriv of Jacobian
             //-----------------------------
             
             //Add the user-defined body force terms
             double sum = body_force[i]*testf[l];

             //Add the gravitational body force term
             sum += scaled_re_inv_fr*testf[l]*G[i];
             
             //Add the pressure gradient term
             sum  += interpolated_p*dtestfdx(l,i);
         
             //Add in the stress tensor terms
             //The viscosity ratio needs to go in here to ensure
             //continuity of normal stress is satisfied even in flows
             //with zero pressure gradient!
             for(unsigned k=0;k<DIM;k++)
              {
               sum -= visc_ratio*
                (interpolated_dudx(i,k) + Gamma[i]*interpolated_dudx(k,i))
                *dtestfdx(l,k);
              }
             
             // Add in the inertial terms

             // du/dt term
             sum -= scaled_re_st*dudt[i]*testf[l];
             
             // Convective terms, including mesh velocity
             for(unsigned k=0;k<DIM;k++)
              {
               double tmp=scaled_re*interpolated_u[k];
               if (!ALE_is_disabled) { tmp-=scaled_re_st*mesh_velocity[k]; }
               sum -= tmp*interpolated_dudx(i,k)*testf[l];
              }

             // Multiply through by deriv of Jacobian and integration weight
             dresidual_dnodal_coordinates(local_eqn,p,q)+=sum*dJ_dX(p,q)*w;
             
             // Derivative of residual x Jacobian
             //----------------------------------

             // Body force
             sum=d_body_force_dx(i,p)*psif(q)*testf(l);

             // Pressure gradient term
             sum += interpolated_p*d_dtestfdx_dX(p,q,l,i);

             // Viscous term
             for (unsigned k=0;k<DIM;k++)
              {
               sum -= visc_ratio*(
                (interpolated_dudx(i,k) + Gamma[i]*interpolated_dudx(k,i))
                *d_dtestfdx_dX(p,q,l,k)+                
                (d_dudx_dX(p,q,i,k) + Gamma[i]*d_dudx_dX(p,q,k,i))
                *dtestfdx(l,k));
              }

             // Convective terms, including mesh velocity
             for(unsigned k=0;k<DIM;k++)
              {
               double tmp=scaled_re*interpolated_u[k];
               if(!ALE_is_disabled) { tmp-=scaled_re_st*mesh_velocity[k]; }
               sum -= tmp*d_dudx_dX(p,q,i,k)*testf(l);
              }
              if(!ALE_is_disabled)
               {
                sum+=scaled_re_st*pos_time_weight*
                 psif(q)*interpolated_dudx(i,p)*testf(l);
               }

             // Multiply through by Jacobian and integration weight
             dresidual_dnodal_coordinates(local_eqn,p,q)+=sum*J*w;

             // Derivs w.r.t. to nodal velocities
             // ---------------------------------
             if(element_has_node_with_aux_node_update_fct)
              {
               sum=-visc_ratio*Gamma[i]*dpsifdx(q,i)*dtestfdx(l,p)
                -scaled_re*psif(q)*interpolated_dudx(i,p)*testf(l);
               if (i==p)
                {
                 sum-=scaled_re_st*val_time_weight*psif(q)*testf(l);
                 for (unsigned k=0;k<DIM;k++)
                  {
                   sum-=visc_ratio*dpsifdx(q,k)*dtestfdx(l,k);
                   double tmp=scaled_re*interpolated_u[k];
                   if (!ALE_is_disabled) tmp-=scaled_re_st*mesh_velocity[k];
                   sum-=tmp*dpsifdx(q,k)*testf(l); 
                  }
                }
               dresidual_dnodal_coordinates(local_eqn,p,q)+=
                sum*d_U_dX(p,q)*J*w; 
              }
            }
          }
        }
      }
    } // End of loop over test functions
  
   
   // CONTINUITY EQUATION
   // -------------------
   
   //Loop over the shape functions
   for(unsigned l=0;l<n_pres;l++)
    {
     local_eqn = p_local_eqn(l);
  
     //If not a boundary conditions
     if(local_eqn >= 0)
      {

       // Loop over coordinate directions
       for (unsigned p=0;p<DIM;p++)
        {              
         // Loop over nodes
         for (unsigned q=0;q<n_node;q++)
          {       
           
           // Residual x deriv of Jacobian
           //-----------------------------
           
           // Source term
           double aux=-source;
           
           // Loop over velocity components
           for(unsigned k=0;k<DIM;k++)
            {
             aux += interpolated_dudx(k,k);
            }
           
           // Multiply through by deriv of Jacobian and integration weight
           dresidual_dnodal_coordinates(local_eqn,p,q)+=
            aux*dJ_dX(p,q)*testp[l]*w;

           // Derivative of residual x Jacobian
           //----------------------------------          
           
           // Loop over velocity components
           aux=-source_gradient[p]*psif(q);
           for(unsigned k=0;k<DIM;k++)
            {
             aux += d_dudx_dX(p,q,k,k);
            }
           // Multiply through by Jacobian and integration weight
           dresidual_dnodal_coordinates(local_eqn,p,q)+=
            aux*testp[l]*J*w;

           // Derivs w.r.t. to nodal velocities           
           //---------------------------------
           if(element_has_node_with_aux_node_update_fct)
            {
             aux=dpsifdx(q,p)*testp(l);
             dresidual_dnodal_coordinates(local_eqn,p,q)+=
              aux*d_U_dX(p,q)*J*w;
            }
          }
        }
      }
    }
  } // End of loop over integration points

}   



     




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

///2D Crouzeix-Raviart elements
//Set the data for the number of Variables at each node
template<>
const unsigned QCrouzeixRaviartElement<2>::Initial_Nvalue[9]
={2,2,2,2,2,2,2,2,2};

///3D Crouzeix-Raviart elements
//Set the data for the number of Variables at each node
template<>
const unsigned QCrouzeixRaviartElement<3>::Initial_Nvalue[27]
={3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3};


//========================================================================
/// Number of values (pinned or dofs) required at node n.
//========================================================================
template<unsigned DIM>
unsigned QCrouzeixRaviartElement<DIM>::required_nvalue(const unsigned& n) const
 {return Initial_Nvalue[n];}


//=========================================================================
///  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.
//=========================================================================
template<unsigned DIM>
void QCrouzeixRaviartElement<DIM>::
identify_load_data(std::set<std::pair<Data*,unsigned> > &paired_load_data)
{
 //Find the index at which the velocity is stored
 unsigned u_index[DIM];
 for(unsigned i=0;i<DIM;i++) {u_index[i] = this->u_index_nst(i);}
 
 //Loop over the nodes
 unsigned n_node = this->nnode();
 for(unsigned n=0;n<n_node;n++)
  {
   //Loop over the velocity components and add pointer to their data
   //and indices to the vectors
   for(unsigned i=0;i<DIM;i++)
    {
     paired_load_data.insert(std::make_pair(this->node_pt(n),u_index[i]));
    }
  }

 // Identify the pressure data
 identify_pressure_data(paired_load_data);

}


//=========================================================================
///  Add to the set \c paired_pressue_data pairs containing
/// - the pointer to a Data object
/// and
/// - the index of the value in that Data object
/// .
/// for all pressures values that affect the
/// load computed in the \c get_load(...) function.
//=========================================================================
template<unsigned DIM>
void QCrouzeixRaviartElement<DIM>::
identify_pressure_data(std::set<std::pair<Data*,unsigned> > &paired_pressure_data)
{
 //Loop over the internal data
 unsigned n_internal = this->ninternal_data();
 for(unsigned l=0;l<n_internal;l++)
  {
   unsigned nval=this->internal_data_pt(l)->nvalue();
   //Add internal data
   for (unsigned j=0;j<nval;j++)
    {
     paired_pressure_data.insert(std::make_pair(this->internal_data_pt(l),j));
    }
  }
}      


//=============================================================================
/// 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 that this unknown is associated with.
/// (Function can obviously only be called if the equation numbering
/// scheme has been set up.)
//=============================================================================
template<unsigned DIM>
void QCrouzeixRaviartElement<DIM>::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_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 = DIM;

 // 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);
       
      }
    }
  }
}




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



//2D Taylor--Hood
//Set the data for the number of Variables at each node
template<>
const unsigned QTaylorHoodElement<2>::Initial_Nvalue[9]={3,2,3,2,2,2,3,2,3};

//Set the data for the pressure conversion array
template<>
const unsigned QTaylorHoodElement<2>::Pconv[4]={0,2,6,8};

//3D Taylor--Hood
//Set the data for the number of Variables at each node
template<>
const unsigned QTaylorHoodElement<3>::Initial_Nvalue[27]={4,3,4,3,3,3,4,3,4,3,3,3,3,3,3,3,3,3,4,3,4,3,3,3,4,3,4};

//Set the data for the pressure conversion array
template<>
const unsigned QTaylorHoodElement<3>::Pconv[8]={0,2,6,8,18,20,24,26};

//=========================================================================
///  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.
//=========================================================================
template<unsigned DIM>
void QTaylorHoodElement<DIM>::
identify_load_data(std::set<std::pair<Data*,unsigned> > &paired_load_data)
{
 //Find the index at which the velocity is stored
 unsigned u_index[DIM];
 for(unsigned i=0;i<DIM;i++) {u_index[i] = this->u_index_nst(i);}
 
 //Loop over the nodes
 unsigned n_node = this->nnode();
 for(unsigned n=0;n<n_node;n++)
  {
   //Loop over the velocity components and add pointer to their data
   //and indices to the vectors
   for(unsigned i=0;i<DIM;i++)
    {
     paired_load_data.insert(std::make_pair(this->node_pt(n),u_index[i]));
    }
  }

 //Identify the pressure data
 this->identify_pressure_data(paired_load_data);

}

//=========================================================================
///  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 pressure values that affect the
/// load computed in the \c get_load(...) function., 
//=========================================================================
template<unsigned DIM>
void QTaylorHoodElement<DIM>::
identify_pressure_data(std::set<std::pair<Data*,unsigned> > &paired_pressure_data)
{
 //Find the index at which the pressure is stored
 unsigned p_index = static_cast<unsigned>(this->p_nodal_index_nst());

 //Loop over the pressure data
 unsigned n_pres= npres_nst();
 for(unsigned l=0;l<n_pres;l++)
  {
   //The DIMth entry in each nodal data is the pressure, which
   //affects the traction
   paired_pressure_data.insert(std::make_pair(this->node_pt(Pconv[l]),p_index));
  }
}


//============================================================================
/// Create a list of pairs for all unknowns in this element,
/// so 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.)
//============================================================================
template<unsigned DIM>
void QTaylorHoodElement<DIM>::get_dof_numbers_for_unknowns(
 std::list<std::pair<unsigned long,
 unsigned> >& dof_lookup_list) const
{
   // number of nodes
   unsigned n_node = this->nnode();
   
   // local eqn no for pressure unknown
// unsigned p_index = this->p_nodal_index_nst();
   
   // 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 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);
        }
      }
    }
  }


//====================================================================
//// Force build of templates
//====================================================================
template class NavierStokesEquations<2>;
template class QCrouzeixRaviartElement<2>;
template class QTaylorHoodElement<2>;

template class NavierStokesEquations<3>;
template class QCrouzeixRaviartElement<3>;
template class QTaylorHoodElement<3>;

}