generalised_newtonian_navier_stokes_elements.h

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//Header file for Navier Stokes elements

#ifndef OOMPH_GENERALISED_NEWTONIAN_NAVIER_STOKES_ELEMENTS_HEADER
#define OOMPH_GENERALISED_NEWTONIAN_NAVIER_STOKES_ELEMENTS_HEADER

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


//OOMPH-LIB headers 
#include "../generic/Qelements.h"
#include "../generic/fsi.h"
#include "../generic/projection.h"
#include "../generic/generalised_newtonian_constitutive_models.h"


namespace oomph
{


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


//======================================================================
/// Template-free base class for Navier-Stokes equations to avoid
/// casting problems
//======================================================================
class GeneralisedNewtonianTemplateFreeNavierStokesEquationsBase : 
public virtual NavierStokesElementWithDiagonalMassMatrices,
 public virtual FiniteElement
{

  public:

 /// Constructor (empty)
 GeneralisedNewtonianTemplateFreeNavierStokesEquationsBase(){};

 /// Virtual destructor (empty)
 virtual ~GeneralisedNewtonianTemplateFreeNavierStokesEquationsBase(){};
 
 
 /// \short Return the index at which the pressure is stored if it is
 /// stored at the nodes. If not stored at the nodes this will return 
 /// a negative number.
 virtual int p_nodal_index_nst() const =0;
 
 /// \short Access function for the local equation number information for
 /// the pressure.
 /// p_local_eqn[n] = local equation number or < 0 if pinned
 virtual int p_local_eqn(const unsigned &n)const=0;

 /// \short Pin all non-pressure dofs and backup eqn numbers of all Data
 virtual void pin_all_non_pressure_dofs(std::map<Data*,std::vector<int> >& 
                                        eqn_number_backup)=0;
 

 /// \short 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)
 virtual void get_pressure_and_velocity_mass_matrix_diagonal(
  Vector<double> &press_mass_diag, Vector<double> &veloc_mass_diag,
  const unsigned& which_one=0)=0;

};

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





//======================================================================
/// A class for elements that solve the cartesian Navier--Stokes equations,
/// templated by the dimension DIM.
/// This contains the generic maths -- any concrete implementation must 
/// be derived from this.
///
/// We're solving:
///
///  \f$ { Re \left( St \frac{\partial u_i}{\partial t} + 
///              (u_j - u_j^{M}) \frac{\partial u_i}{\partial x_j} \right) =
///      - \frac{\partial p}{\partial x_i}  - R_\rho B_i(x_j) -
///       \frac{Re}{Fr} G_i +
///        \frac{\partial }{\partial x_j} \left[  R_\mu \left(
///        \frac{\partial u_i}{\partial x_j} +  
///         \frac{\partial u_j}{\partial x_i} \right) \right] } \f$
///
///  and 
///
///  \f$ { \frac{\partial u_i}{\partial x_i} = Q } \f$
///
/// We also provide all functions required to use this element
/// in FSI problems, by deriving it from the FSIFluidElement base
/// class. 
//======================================================================
template <unsigned DIM>
 class GeneralisedNewtonianNavierStokesEquations : 
 public virtual FSIFluidElement,
 public virtual GeneralisedNewtonianTemplateFreeNavierStokesEquationsBase
{

  public:

 /// \short Function pointer to body force function fct(t,x,f(x))
 /// x is a Vector!
 typedef void (*NavierStokesBodyForceFctPt)(const double& time,
                                            const Vector<double>& x,
                                            Vector<double>& body_force);

 /// \short Function pointer to source function fct(t,x)
 /// x is a Vector!
 typedef double (*NavierStokesSourceFctPt)(const double& time,
                                           const Vector<double>& x);


 /// \short Function pointer to source function fct(x) for the 
 /// pressure advection diffusion equation (only used during
 /// validation!). x is a Vector!
 typedef double (*NavierStokesPressureAdvDiffSourceFctPt)(
  const Vector<double>& x);

  private:

 /// \short Static "magic" number that indicates that the pressure is
 /// not stored at a node
 static int Pressure_not_stored_at_node;

 /// Static default value for the physical constants (all initialised to zero)
 static double Default_Physical_Constant_Value;

 /// Static default value for the physical ratios (all are initialised to one)
 static double Default_Physical_Ratio_Value;

 /// Static default value for the gravity vector
 static Vector<double> Default_Gravity_vector; 

  protected:

 /// Static boolean telling us whether we pre-multiply the pressure and
 /// continuity by the viscosity ratio
 static bool Pre_multiply_by_viscosity_ratio;

 //Physical constants

 /// \short Pointer to the viscosity ratio (relative to the 
 /// viscosity used in the definition of the Reynolds number)
 double *Viscosity_Ratio_pt;
 
 /// \short Pointer to the density ratio (relative to the density used in the 
 /// definition of the Reynolds number)
 double *Density_Ratio_pt;
 
 // Pointers to global physical constants

 /// Pointer to global Reynolds number
 double *Re_pt;
 
 /// Pointer to global Reynolds number x Strouhal number (=Womersley)
 double *ReSt_pt;
 
 /// \short Pointer to global Reynolds number x inverse Froude number
 /// (= Bond number / Capillary number) 
 double *ReInvFr_pt;
 
 /// Pointer to global gravity Vector
 Vector<double> *G_pt;
 
 /// Pointer to body force function
 NavierStokesBodyForceFctPt Body_force_fct_pt;
 
 /// Pointer to volumetric source function
 NavierStokesSourceFctPt Source_fct_pt;

 /// Pointer to the generalised Newtonian constitutive equation
 GeneralisedNewtonianConstitutiveEquation<DIM> *Constitutive_eqn_pt;

 // Boolean that indicates whether we use the extrapolated strain rate
 // for calculation of viscosity or not
 bool Use_extrapolated_strainrate_to_compute_second_invariant;

 /// \short Boolean flag to indicate if ALE formulation is disabled when
 /// time-derivatives are computed. Only set to true if you're sure
 /// that the mesh is stationary.
 bool ALE_is_disabled;

 /// \short Compute the shape functions and derivatives 
 /// w.r.t. global coords at local coordinate s.
 /// Return Jacobian of mapping between local and global coordinates.
 virtual double dshape_and_dtest_eulerian_nst(const Vector<double> &s, 
                                              Shape &psi, 
                                              DShape &dpsidx, Shape &test, 
                                              DShape &dtestdx) const=0;

 /// \short Compute the shape functions and derivatives 
 /// w.r.t. global coords at ipt-th integration point
 /// Return Jacobian of mapping between local and global coordinates.
 virtual double dshape_and_dtest_eulerian_at_knot_nst(const unsigned &ipt, 
                                                      Shape &psi, 
                                                      DShape &dpsidx, 
                                                      Shape &test, 
                                                      DShape &dtestdx) const=0;

 /// \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.
 virtual double dshape_and_dtest_eulerian_at_knot_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=0;
 
 /// \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_nst(const Vector<double> &s, 
                                                Shape &ppsi, 
                                                DShape &dppsidx, 
                                                Shape &ptest, 
                                                DShape &dptestdx) const=0;
  

 /// \short Calculate the body force at a given time and local and/or 
 /// Eulerian position. This function is virtual so that it can be 
 /// overloaded in multi-physics elements where the body force might
 /// depend on another variable.
 virtual void get_body_force_nst(const double& time,
                                 const unsigned& ipt,
                                 const Vector<double> &s,
                                 const Vector<double> &x, 
                                 Vector<double> &result)
  {
   //If the function pointer is zero return zero
   if(Body_force_fct_pt == 0)
    {
     //Loop over dimensions and set body forces to zero
     for(unsigned i=0;i<DIM;i++) {result[i] = 0.0;}
    }
   //Otherwise call the function
   else
    {
     (*Body_force_fct_pt)(time,x,result);
    }
  }

 /// Get gradient of body force term at (Eulerian) position x. This function is
 /// virtual to allow overloading in multi-physics problems where
 /// the strength of the source function might be determined by
 /// another system of equations. Computed via function pointer 
 /// (if set) or by finite differencing (default)
 inline virtual void get_body_force_gradient_nst(
  const double& time,
  const unsigned& ipt,
  const Vector<double>& s,
  const Vector<double>& x, 
  DenseMatrix<double>& d_body_force_dx)
  {
// hierher: Implement function pointer version
/*    //If no gradient function has been set, FD it */
/*    if(Body_force_fct_gradient_pt==0) */
    {
     // Reference value
     Vector<double> body_force(DIM,0.0);
     get_body_force_nst(time,ipt,s,x,body_force);

     // FD it
     double eps_fd=GeneralisedElement::Default_fd_jacobian_step;
     Vector<double> body_force_pls(DIM,0.0);
     Vector<double> x_pls(x);
     for (unsigned i=0;i<DIM;i++)
      {
       x_pls[i]+=eps_fd;
       get_body_force_nst(time,ipt,s,x_pls,body_force_pls);
       for (unsigned j=0;j<DIM;j++)
        {
         d_body_force_dx(j,i)=(body_force_pls[j]-body_force[j])/eps_fd;
        }
       x_pls[i]=x[i];
      }
    }
/*    else */
/*     { */
/*      // Get gradient */
/*      (*Source_fct_gradient_pt)(time,x,gradient); */
/*     } */
  }



 /// \short Calculate the source fct at given time and
 /// Eulerian position 
 virtual double get_source_nst(const double& time, const unsigned& ipt,
                               const Vector<double> &x)
  {
   //If the function pointer is zero return zero
   if (Source_fct_pt == 0) {return 0;}
   //Otherwise call the function
   else {return (*Source_fct_pt)(time,x);}
  }
 

 /// Get gradient of source term at (Eulerian) position x. This function is
 /// virtual to allow overloading in multi-physics problems where
 /// the strength of the source function might be determined by
 /// another system of equations. Computed via function pointer 
 /// (if set) or by finite differencing (default)
 inline virtual void get_source_gradient_nst(
  const double& time,
  const unsigned& ipt,
  const Vector<double>& x, 
  Vector<double>& gradient)
  {
// hierher: Implement function pointer version
/*    //If no gradient function has been set, FD it */
/*    if(Source_fct_gradient_pt==0) */
    {
     // Reference value
     double source=get_source_nst(time,ipt,x);

     // FD it
     double eps_fd=GeneralisedElement::Default_fd_jacobian_step;
     double source_pls=0.0;
     Vector<double> x_pls(x);
     for (unsigned i=0;i<DIM;i++)
      {
       x_pls[i]+=eps_fd;
       source_pls=get_source_nst(time,ipt,x_pls);
       gradient[i]=(source_pls-source)/eps_fd;
       x_pls[i]=x[i];
      }
    }
/*    else */
/*     { */
/*      // Get gradient */
/*      (*Source_fct_gradient_pt)(time,x,gradient); */
/*     } */
  }


 ///\short Compute the residuals for the Navier--Stokes equations.
 /// Flag=1 (or 0): do (or don't) compute the Jacobian as well.
 /// Flag=2: Fill in mass matrix too.
 virtual void fill_in_generic_residual_contribution_nst(
  Vector<double> &residuals, DenseMatrix<double> &jacobian, 
  DenseMatrix<double> &mass_matrix, unsigned flag);

 ///\short Compute the derivatives of the 
 /// residuals for the Navier--Stokes equations with respect to a parameter
 /// Flag=1 (or 0): do (or don't) compute the Jacobian as well.
 /// Flag=2: Fill in mass matrix too.
 virtual void 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);
 
 /// \short Compute the hessian tensor vector products required to
 /// perform continuation of bifurcations analytically
 void fill_in_contribution_to_hessian_vector_products(
  Vector<double> const &Y,
  DenseMatrix<double> const &C,
  DenseMatrix<double> &product);

    
public:

 /// \short Constructor: NULL the body force and source function
 /// and make sure the ALE terms are included by default.
 GeneralisedNewtonianNavierStokesEquations() : 
 Body_force_fct_pt(0), Source_fct_pt(0),
  //Press_adv_diff_source_fct_pt(0),
  Constitutive_eqn_pt(new NewtonianConstitutiveEquation<DIM>), 
  Use_extrapolated_strainrate_to_compute_second_invariant(false),
  ALE_is_disabled(false)
  {
   //Set all the Physical parameter pointers to the default value zero
   Re_pt = &Default_Physical_Constant_Value;
   ReSt_pt = &Default_Physical_Constant_Value;
   ReInvFr_pt = &Default_Physical_Constant_Value;
   G_pt = &Default_Gravity_vector;
   //Set the Physical ratios to the default value of 1
   Viscosity_Ratio_pt = &Default_Physical_Ratio_Value;
   Density_Ratio_pt = &Default_Physical_Ratio_Value;
  }

 /// Vector to decide whether the stress-divergence form is used or not
 // N.B. This needs to be public so that the intel compiler gets things correct
 // somehow the access function messes things up when going to refineable
 // navier--stokes
 static Vector<double> Gamma;

 //Access functions for the physical constants

 /// Reynolds number
 const double &re() const {return *Re_pt;}

 /// Product of Reynolds and Strouhal number (=Womersley number)
 const double &re_st() const {return *ReSt_pt;}

 /// Pointer to Reynolds number
 double* &re_pt() {return Re_pt;}

 /// Pointer to product of Reynolds and Strouhal number (=Womersley number)
 double* &re_st_pt() {return ReSt_pt;}

 /// \short Viscosity ratio for element: Element's viscosity relative to the 
 /// viscosity used in the definition of the Reynolds number
 const double &viscosity_ratio() const {return *Viscosity_Ratio_pt;}

 /// Pointer to Viscosity Ratio
 double* &viscosity_ratio_pt() {return Viscosity_Ratio_pt;}

 /// \short Density ratio for element: Element's density relative to the 
 ///  viscosity used in the definition of the Reynolds number
 const double &density_ratio() const {return *Density_Ratio_pt;}

 /// Pointer to Density ratio
 double* &density_ratio_pt() {return Density_Ratio_pt;}

 /// Global inverse Froude number
 const double &re_invfr() const {return *ReInvFr_pt;}

 /// Pointer to global inverse Froude number
 double* &re_invfr_pt() {return ReInvFr_pt;}
 
 /// Vector of gravitational components
 const Vector<double> &g() const {return *G_pt;}

 /// Pointer to Vector of gravitational components
 Vector<double>* &g_pt() {return G_pt;}

 /// Access function for the body-force pointer
 NavierStokesBodyForceFctPt& body_force_fct_pt() 
  {return Body_force_fct_pt;}

 /// Access function for the body-force pointer. Const version
 NavierStokesBodyForceFctPt body_force_fct_pt() const
  {return Body_force_fct_pt;}
 
 ///Access function for the source-function pointer
 NavierStokesSourceFctPt& source_fct_pt() {return Source_fct_pt;}

 ///Access function for the source-function pointer. Const version
 NavierStokesSourceFctPt source_fct_pt() const {return Source_fct_pt;}

 /// Access function for the constitutive equation pointer
 GeneralisedNewtonianConstitutiveEquation<DIM>* &constitutive_eqn_pt() 
  {return Constitutive_eqn_pt;}
 
 /// Switch to fully implict evaluation of non-Newtonian const eqn
 void use_current_strainrate_to_compute_second_invariant()
 {
  Use_extrapolated_strainrate_to_compute_second_invariant=false;
 }
 
 /// Use extrapolation for non-Newtonian const eqn
 void use_extrapolated_strainrate_to_compute_second_invariant()
 {
  Use_extrapolated_strainrate_to_compute_second_invariant=true;
 }

 ///Function to return number of pressure degrees of freedom
 virtual unsigned npres_nst() const=0;
 
 /// Compute the pressure shape functions at local coordinate s
 virtual void pshape_nst(const Vector<double> &s, Shape &psi) const=0;

 /// \short Compute the pressure shape and test functions 
 /// at local coordinate s
 virtual void pshape_nst(const Vector<double> &s, Shape &psi, 
                         Shape &test) const=0;

 /// \short Velocity i at local node n. Uses suitably interpolated value 
 /// for hanging nodes. The use of u_index_nst() permits the use of this
 /// element as the basis for multi-physics elements. The default
 /// is to assume that the i-th velocity component is stored at the
 /// i-th location of the node
 double u_nst(const unsigned &n, const unsigned &i) const
  {return nodal_value(n,u_index_nst(i));}

 /// \short Velocity i at local node n at timestep t (t=0: present; 
 /// t>0: previous). Uses suitably interpolated value for hanging nodes.
 double u_nst(const unsigned &t, const unsigned &n, 
              const unsigned &i) const
  {return nodal_value(t,n,u_index_nst(i));}
 
  /// \short Return the index at which the i-th unknown velocity component
 /// is stored. The default value, i, is appropriate for single-physics
 /// problems.
 /// In derived multi-physics elements, this function should be overloaded
 /// to reflect the chosen storage scheme. Note that these equations require
 /// that the unknowns are always stored at the same indices at each node.
 virtual inline unsigned u_index_nst(const unsigned &i) const {return i;}

 /// \short Return the number of velocity components
 /// Used in FluidInterfaceElements
 inline unsigned n_u_nst() const {return DIM;}

 /// \short i-th component of du/dt at local node n. 
 /// Uses suitably interpolated value for hanging nodes.
 double du_dt_nst(const unsigned &n, const unsigned &i) const
  {
   // Get the data's timestepper
   TimeStepper* time_stepper_pt = this->node_pt(n)->time_stepper_pt();

   //Initialise dudt
   double dudt=0.0;

   //Loop over the timesteps, if there is a non Steady timestepper
   if (!time_stepper_pt->is_steady())
    {
     //Find the index at which the dof is stored
     const unsigned u_nodal_index = this->u_index_nst(i);
     
     // Number of timsteps (past & present)
     const unsigned n_time = time_stepper_pt->ntstorage();
     // Loop over the timesteps
     for(unsigned t=0;t<n_time;t++)
      {
       dudt+=time_stepper_pt->weight(1,t)*nodal_value(t,n,u_nodal_index);
      }
    }
   
   return dudt;
  }

 /// \short Disable ALE, i.e. assert the mesh is not moving -- you do this
 /// at your own risk!
 void disable_ALE()
  {
   ALE_is_disabled=true;
  }

 /// \short (Re-)enable ALE, i.e. take possible mesh motion into account
 /// when evaluating the time-derivative. Note: By default, ALE is
 /// enabled, at the expense of possibly creating unnecessary work
 /// in problems where the mesh is, in fact, stationary.
 void enable_ALE()
  {
   ALE_is_disabled=false;
  }

 /// \short Pressure at local pressure "node" n_p
 /// Uses suitably interpolated value for hanging nodes.
 virtual double p_nst(const unsigned &n_p)const=0; 

 /// \short Pressure at local pressure "node" n_p at time level t
 virtual double p_nst(const unsigned &t, const unsigned &n_p)const=0;

 /// Pin p_dof-th pressure dof and set it to value specified by p_value.
 virtual void fix_pressure(const unsigned &p_dof, const double &p_value)=0;

 /// \short Return the index at which the pressure is stored if it is
 /// stored at the nodes. If not stored at the nodes this will return 
 /// a negative number.
 virtual int p_nodal_index_nst() const {return Pressure_not_stored_at_node;}
 
 /// Integral of pressure over element
 double pressure_integral() const;

 /// \short Get max. and min. strain rate invariant and visocosity
 /// over all integration points in element 
 void max_and_min_invariant_and_viscosity(double& min_invariant,
                                          double& max_invariant,
                                          double& min_viscosity,
                                          double& max_viscosity) const;

 /// \short Return integral of dissipation over element
 double dissipation() const;
 
 /// \short Return dissipation at local coordinate s
 double dissipation(const Vector<double>& s) const;

 /// \short Compute the vorticity vector at local coordinate s
 void get_vorticity(const Vector<double>& s, Vector<double>& vorticity) const;

 /// \short Get integral of kinetic energy over element
 double kin_energy() const;

 /// \short Get integral of time derivative of kinetic energy over element
 double d_kin_energy_dt() const;

 /// Strain-rate tensor: 1/2 (du_i/dx_j + du_j/dx_i)
 void strain_rate(const Vector<double>& s, 
                  DenseMatrix<double>& strain_rate) const;

 /// \short Strain-rate tensor: 1/2 (du_i/dx_j + du_j/dx_i)
 /// at a specific time history value
 void strain_rate(const unsigned& t, const Vector<double>& s, 
                  DenseMatrix<double>& strain_rate) const;
 
 /// \short Extrapolated strain-rate tensor: 1/2 (du_i/dx_j + du_j/dx_i)
 /// based on the previous time steps evaluated
 /// at local coordinate s
 virtual void extrapolated_strain_rate(const Vector<double>& s, 
                                       DenseMatrix<double>& strain_rate) const;

 /// \short Extrapolated strain-rate tensor: 1/2 (du_i/dx_j + du_j/dx_i)
 /// based on the previous time steps evaluated at
 /// ipt-th integration point
 virtual void extrapolated_strain_rate(const unsigned& ipt,
                                       DenseMatrix<double>& strain_rate) const
 {
  //Set the Vector to hold local coordinates
  Vector<double> s(DIM);
  for(unsigned i=0;i<DIM;i++) s[i] = integral_pt()->knot(ipt,i);
  extrapolated_strain_rate(s,strain_rate);
 }
 
 /// \short Compute traction (on the viscous scale) exerted onto 
 /// the fluid at local coordinate s. N has to be outer unit normal
 /// to the fluid. 
 void get_traction(const Vector<double>& s, const Vector<double>& N, 
                   Vector<double>& traction);

 /// \short 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.
 void get_traction(const Vector<double>& s, const Vector<double>& N,
                   Vector<double>& traction_p,
                   Vector<double>& traction_visc_n,
                   Vector<double>& traction_visc_t);

 /// \short This implements a pure virtual function defined
 /// in the FSIFluidElement class. The function computes
 /// the traction (on the viscous scale), at the element's local 
 /// coordinate s, that the fluid element exerts onto an adjacent
 /// solid element. The number of arguments is imposed by
 /// the interface defined in the FSIFluidElement -- only the 
 /// unit normal N (pointing into the fluid!) is actually used
 /// in the computation. 
 void get_load(const Vector<double> &s, 
               const Vector<double> &N,
               Vector<double> &load)
  {
   // Note: get_traction() computes the traction onto the fluid
   // if N is the outer unit normal onto the fluid; here we're
   // exepcting N to point into the fluid so we're getting the
   // traction onto the adjacent wall instead!
   get_traction(s,N,load);
  }

 /// \short 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)
 void get_pressure_and_velocity_mass_matrix_diagonal(
  Vector<double> &press_mass_diag, Vector<double> &veloc_mass_diag,
  const unsigned& which_one=0);

 /// \short Number of scalars/fields output by this element. Reimplements
 /// broken virtual function in base class.
 unsigned nscalar_paraview() const
  {
   return DIM+1;
  }

 /// \short Write values of the i-th scalar field at the plot points. Needs 
 /// to be implemented for each new specific element type.
 void scalar_value_paraview(std::ofstream& file_out,
                            const unsigned& i,
                            const unsigned& nplot) const
 {
  // Vector of local coordinates
  Vector<double> s(DIM);

 
  // Loop over plot points
  unsigned num_plot_points=nplot_points_paraview(nplot);
  for (unsigned iplot=0;iplot<num_plot_points;iplot++)
   {
    
    // Get local coordinates of plot point
    get_s_plot(iplot,nplot,s);
    
    // Velocities
    if(i<DIM) {file_out << interpolated_u_nst(s,i) << std::endl;}

    // Pressure
    else if(i==DIM) {file_out << interpolated_p_nst(s)  << std::endl;}

    // Never get here
    else
     {
#ifdef PARANOID
     std::stringstream error_stream;
    error_stream 
     << "These Navier Stokes elements only store " << DIM+1 << " fields, "
     << "but i is currently  " << i << std::endl;
    throw OomphLibError(
     error_stream.str(),
     OOMPH_CURRENT_FUNCTION,
     OOMPH_EXCEPTION_LOCATION);
#endif
     }
   }
 }
 
 /// \short Name of the i-th scalar field. Default implementation
 /// returns V1 for the first one, V2 for the second etc. Can (should!) be
 /// overloaded with more meaningful names in specific elements.
 std::string scalar_name_paraview(const unsigned& i) const
  {
    // Velocities
   if(i<DIM)
    {
     return "Velocity "+StringConversion::to_string(i);
    }
    // Preussre
   else if(i==DIM)
    {
     return "Pressure";
    }
    // Never get here
    else
     {
      std::stringstream error_stream;
      error_stream  
       << "These Navier Stokes elements only store " << DIM+1 << "  fields,\n"
       << "but i is currently  " << i << std::endl;
      throw OomphLibError(
       error_stream.str(),
       OOMPH_CURRENT_FUNCTION,
       OOMPH_EXCEPTION_LOCATION);
      // Dummy return
      return " ";
     }
  }

 /// \short Output function: x,y,[z],u,v,[w],p
 /// in tecplot format. Default number of plot points
 void output(std::ostream &outfile)
  {
   unsigned nplot=5;
   output(outfile,nplot);
  }

 /// \short Output function: x,y,[z],u,v,[w],p
 /// in tecplot format. nplot points in each coordinate direction
 void output(std::ostream &outfile, const unsigned &nplot);

 /// \short C-style output function: x,y,[z],u,v,[w],p
 /// in tecplot format. Default number of plot points
 void output(FILE* file_pt)
  {
   unsigned nplot=5;
   output(file_pt,nplot);
  }

 /// \short C-style output function: x,y,[z],u,v,[w],p
 /// in tecplot format. nplot points in each coordinate direction
 void output(FILE* file_pt, const unsigned &nplot);

 /// \short Full output function: 
 /// x,y,[z],u,v,[w],p,du/dt,dv/dt,[dw/dt],dissipation
 /// in tecplot format. Default number of plot points
 void full_output(std::ostream &outfile) 
  {
   unsigned nplot=5;
   full_output(outfile,nplot);
  }

 /// \short Full output function: 
 /// x,y,[z],u,v,[w],p,du/dt,dv/dt,[dw/dt],dissipation
 /// in tecplot format. nplot points in each coordinate direction
 void full_output(std::ostream &outfile, const unsigned &nplot);


 /// \short Output function: x,y,[z],u,v,[w] in tecplot format.
 /// nplot points in each coordinate direction at timestep t
 /// (t=0: present; t>0: previous timestep)
 void output_veloc(std::ostream &outfile, const unsigned &nplot, 
                   const unsigned& t);


 /// \short Output function: x,y,[z], [omega_x,omega_y,[and/or omega_z]] 
 /// in tecplot format. nplot points in each coordinate direction
 void output_vorticity(std::ostream &outfile, 
                       const unsigned &nplot);

 /// \short Output exact solution specified via function pointer
 /// at a given number of plot points. Function prints as
 /// many components as are returned in solution Vector
 void output_fct(std::ostream &outfile, const unsigned &nplot, 
                 FiniteElement::SteadyExactSolutionFctPt exact_soln_pt);

 /// \short Output exact solution specified via function pointer
 /// at a given time and at a given number of plot points.
 /// Function prints as many components as are returned in solution Vector.
 void output_fct(std::ostream &outfile, const unsigned &nplot, 
                 const double& time,
                 FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt);

 /// \short Validate against exact 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
 void compute_error(std::ostream &outfile,
                    FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt,
                    const double& time,
                    double& error, double& norm);

 /// \short Validate against exact 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
 void compute_error(std::ostream &outfile,
                    FiniteElement::SteadyExactSolutionFctPt exact_soln_pt,
                    double& error, double& norm);

 /// Compute the element's residual Vector
 void fill_in_contribution_to_residuals(Vector<double> &residuals)
  {
   //Call the generic residuals function with flag set to 0
   //and using a dummy matrix argument
   fill_in_generic_residual_contribution_nst(
    residuals,GeneralisedElement::Dummy_matrix,
    GeneralisedElement::Dummy_matrix,0);
  }

 ///\short Compute the element's residual Vector and the jacobian matrix
 /// Virtual function can be overloaded by hanging-node version
 void fill_in_contribution_to_jacobian(Vector<double> &residuals,
                                   DenseMatrix<double> &jacobian)
  {
   //Call the generic routine with the flag set to 1
   fill_in_generic_residual_contribution_nst(residuals,jacobian,
                                        GeneralisedElement::Dummy_matrix,1);
   // obacht FD version
   //FiniteElement::fill_in_contribution_to_jacobian(residuals,jacobian);
  }

 /// \short Add the element's contribution to its residuals vector,
 /// jacobian matrix and mass matrix
 void fill_in_contribution_to_jacobian_and_mass_matrix(
  Vector<double> &residuals, DenseMatrix<double> &jacobian, 
  DenseMatrix<double> &mass_matrix)
  {
   //Call the generic routine with the flag set to 2
   fill_in_generic_residual_contribution_nst(residuals,jacobian,mass_matrix,2);
  }

 /// Compute the element's residual Vector
 void fill_in_contribution_to_dresiduals_dparameter(
  double* const &parameter_pt,Vector<double> &dres_dparam)
  {
   //Call the generic residuals function with flag set to 0
   //and using a dummy matrix argument
   fill_in_generic_dresidual_contribution_nst(
    parameter_pt,
    dres_dparam,GeneralisedElement::Dummy_matrix,
    GeneralisedElement::Dummy_matrix,0);
  }

 ///\short Compute the element's residual Vector and the jacobian matrix
 /// Virtual function can be overloaded by hanging-node version
 void fill_in_contribution_to_djacobian_dparameter(
  double* const &parameter_pt,Vector<double> &dres_dparam,
  DenseMatrix<double> &djac_dparam)
  {
   //Call the generic routine with the flag set to 1
   fill_in_generic_dresidual_contribution_nst(
    parameter_pt,
    dres_dparam,djac_dparam,GeneralisedElement::Dummy_matrix,1);
  }
 
 /// Add the element's contribution to its residuals vector,
 /// jacobian matrix and mass matrix
 void fill_in_contribution_to_djacobian_and_dmass_matrix_dparameter(
  double* const &parameter_pt,
  Vector<double> &dres_dparam, 
  DenseMatrix<double> &djac_dparam, 
  DenseMatrix<double> &dmass_matrix_dparam)
  {
   //Call the generic routine with the flag set to 2
   fill_in_generic_dresidual_contribution_nst(
    parameter_pt,dres_dparam,djac_dparam,dmass_matrix_dparam,2);
  }

  
 /// \short Pin all non-pressure dofs and backup eqn numbers
 void pin_all_non_pressure_dofs(std::map<Data*,std::vector<int> >& 
                                eqn_number_backup)
 {
  // Loop over internal data and pin the values (having established that
  // pressure dofs aren't amongst those)
  unsigned nint=this->ninternal_data();
  for (unsigned j=0;j<nint;j++)
   {
    Data* data_pt=this->internal_data_pt(j);
    if (eqn_number_backup[data_pt].size()==0)
     {
      unsigned nvalue=data_pt->nvalue();
      eqn_number_backup[data_pt].resize(nvalue);
      for (unsigned i=0;i<nvalue;i++)
       {
        // Backup
        eqn_number_backup[data_pt][i]=data_pt->eqn_number(i);
        
        // Pin everything
        data_pt->pin(i);
       }
     }
   }
  
  // Now deal with nodal values
  unsigned nnod=this->nnode();
  for (unsigned j=0;j<nnod;j++)
   {

    Node* nod_pt=this->node_pt(j);
    if (eqn_number_backup[nod_pt].size()==0)
     {

      unsigned nvalue=nod_pt->nvalue();
      eqn_number_backup[nod_pt].resize(nvalue);
      for (unsigned i=0;i<nvalue;i++)
       {
        // Pin everything apart from the nodal pressure 
        // value
        if (int(i)!=this->p_nodal_index_nst())
         {
          // Backup
          eqn_number_backup[nod_pt][i]=nod_pt->eqn_number(i);
          
          // Pin
          nod_pt->pin(i);
         }
        // Else it's a pressure value
        else 
         {
          // Exclude non-nodal pressure based elements
          if (this->p_nodal_index_nst()>=0)
           {
            // Backup
            eqn_number_backup[nod_pt][i]=nod_pt->eqn_number(i);
           }
         }
       }
        
      
      // If it's a solid node deal with its positional data too
      SolidNode* solid_nod_pt=dynamic_cast<SolidNode*>(nod_pt);
      if (solid_nod_pt!=0)
       {
        Data* solid_posn_data_pt=solid_nod_pt->variable_position_pt();
        if (eqn_number_backup[solid_posn_data_pt].size()==0)
         {
          unsigned nvalue=solid_posn_data_pt->nvalue();
          eqn_number_backup[solid_posn_data_pt].resize(nvalue);
          for (unsigned i=0;i<nvalue;i++)
           {
            // Backup
            eqn_number_backup[solid_posn_data_pt][i]=
             solid_posn_data_pt->eqn_number(i);
            
            // Pin
            solid_posn_data_pt->pin(i);
           }
         }
       }
     }
   }
 }
 

 /// \short Compute derivatives of elemental residual vector with respect
 /// to nodal coordinates. Overwrites default implementation in 
 /// FiniteElement base class.
 /// dresidual_dnodal_coordinates(l,i,j) = d res(l) / dX_{ij}
 virtual void get_dresidual_dnodal_coordinates(RankThreeTensor<double>&
                                               dresidual_dnodal_coordinates);
  

 /// Compute vector of FE interpolated velocity u at local coordinate s
 void interpolated_u_nst(const Vector<double> &s, Vector<double>& veloc) const
  {
   //Find number of nodes
   unsigned n_node = nnode();
   //Local shape function
   Shape psi(n_node);
   //Find values of shape function
   shape(s,psi);
   
   for (unsigned i=0;i<DIM;i++)
    {
     //Index at which the nodal value is stored
     unsigned u_nodal_index = u_index_nst(i);
     //Initialise value of u
     veloc[i] = 0.0;
     //Loop over the local nodes and sum
     for(unsigned l=0;l<n_node;l++) 
      {
       veloc[i] += nodal_value(l,u_nodal_index)*psi[l];
      }
    }
  }

 /// Return FE interpolated velocity u[i] at local coordinate s
 double interpolated_u_nst(const Vector<double> &s, const unsigned &i) const
  {
   //Find number of nodes
   unsigned n_node = nnode();
   //Local shape function
   Shape psi(n_node);
   //Find values of shape function
   shape(s,psi);
   
   //Get nodal index at which i-th velocity is stored
   unsigned u_nodal_index = u_index_nst(i);

   //Initialise value of u
   double interpolated_u = 0.0;
   //Loop over the local nodes and sum
   for(unsigned l=0;l<n_node;l++) 
    {
     interpolated_u += nodal_value(l,u_nodal_index)*psi[l];
    }
   
   return(interpolated_u);
  }

 /// \short Return FE interpolated velocity u[i] at local coordinate s
 /// at time level t (t=0: present; t>0: history)
 double interpolated_u_nst(const unsigned& t, 
                           const Vector<double> &s, 
                           const unsigned &i) const
 {
  //Find number of nodes
  unsigned n_node = nnode();
  
  //Local shape function
  Shape psi(n_node);
  
  //Find values of shape function
  shape(s,psi);
  
  //Get nodal index at which i-th velocity is stored
  unsigned u_nodal_index = u_index_nst(i);
  
  //Initialise value of u
  double interpolated_u = 0.0;
  //Loop over the local nodes and sum
  for(unsigned l=0;l<n_node;l++) 
   {
    interpolated_u += nodal_value(t,l,u_nodal_index)*psi[l];
   }
  
  return(interpolated_u);
 }
 
 /// \short Compute the derivatives of the i-th component of 
 /// velocity at point s with respect
 /// to all data that can affect its value. In addition, return the global
 /// equation numbers corresponding to the data. The function is virtual 
 /// so that it can be overloaded in the refineable version
 virtual void dinterpolated_u_nst_ddata(const Vector<double> &s,
                                        const unsigned &i,
                                        Vector<double> &du_ddata,
                                        Vector<unsigned> &global_eqn_number)
  {
   //Find number of nodes
   unsigned n_node = nnode();
   //Local shape function
   Shape psi(n_node);
   //Find values of shape function
   shape(s,psi);

   //Find the index at which the velocity component is stored
   const unsigned u_nodal_index = u_index_nst(i);
   
   //Find the number of dofs associated with interpolated u
   unsigned n_u_dof=0;
   for(unsigned l=0;l<n_node;l++)
    {
     int global_eqn = this->node_pt(l)->eqn_number(u_nodal_index);
     //If it's positive add to the count
     if(global_eqn >= 0) {++n_u_dof;}
    }
   
   //Now resize the storage schemes
   du_ddata.resize(n_u_dof,0.0);
   global_eqn_number.resize(n_u_dof,0);
   
   //Loop over th nodes again and set the derivatives
   unsigned count=0;
   //Loop over the local nodes and sum
   for(unsigned l=0;l<n_node;l++) 
    {
     //Get the global equation number
     int global_eqn = this->node_pt(l)->eqn_number(u_nodal_index);
     if (global_eqn >= 0)
      {
       //Set the global equation number
       global_eqn_number[count] = global_eqn;
       //Set the derivative with respect to the unknown
       du_ddata[count] = psi[l];
       //Increase the counter
       ++count;
      }
    }
  }


 /// Return FE interpolated pressure at local coordinate s
 virtual double interpolated_p_nst(const Vector<double> &s) const
  {
   //Find number of nodes
   unsigned n_pres = npres_nst();
   //Local shape function
   Shape psi(n_pres);
   //Find values of shape function
   pshape_nst(s,psi);
   
   //Initialise value of p
   double interpolated_p = 0.0;
   //Loop over the local nodes and sum
   for(unsigned l=0;l<n_pres;l++) 
    {
     interpolated_p += p_nst(l)*psi[l];
    }
   
   return(interpolated_p);
  }


 /// Return FE interpolated pressure at local coordinate s at time level t
 double interpolated_p_nst(const unsigned &t, const Vector<double> &s) const
  {
   //Find number of nodes
   unsigned n_pres = npres_nst();
   //Local shape function
   Shape psi(n_pres);
   //Find values of shape function
   pshape_nst(s,psi);
   
   //Initialise value of p
   double interpolated_p = 0.0;
   //Loop over the local nodes and sum
   for(unsigned l=0;l<n_pres;l++) 
    {
     interpolated_p += p_nst(t,l)*psi[l];
    }
   
   return(interpolated_p);
  }


 /// \short Output solution in data vector at local cordinates s:
 /// x,y [,z], u,v,[w], p
 void point_output_data(const Vector<double> &s, Vector<double>& data)
 {
  // Dimension
  unsigned dim=s.size();
  
  // Resize data for values
  data.resize(2*dim+1);
  
  // Write values in the vector
  for (unsigned i=0; i<dim; i++)
   {
    data[i]=interpolated_x(s,i);
    data[i+dim]=this->interpolated_u_nst(s,i);
   }
  data[2*dim]=this->interpolated_p_nst(s);
 }

}; 

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


//==========================================================================
/// Crouzeix_Raviart elements are Navier--Stokes elements with quadratic
/// interpolation for velocities and positions, but a discontinuous linear
/// pressure interpolation. They can be used within oomph-lib's
/// block preconditioning framework.
//==========================================================================
template <unsigned DIM>
class GeneralisedNewtonianQCrouzeixRaviartElement : 
public virtual QElement<DIM,3>, 
 public virtual GeneralisedNewtonianNavierStokesEquations<DIM>
{
  private:

 /// Static array of ints to hold required number of variables at nodes
 static const unsigned Initial_Nvalue[];
 
  protected:

 /// Internal index that indicates at which internal data the pressure
 /// is stored
 unsigned P_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_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_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_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_nst(const Vector<double> &s, 
                                             Shape &ppsi, 
                                             DShape &dppsidx, 
                                             Shape &ptest, 
                                             DShape &dptestdx) const;
 
public:

 /// Constructor, there are DIM+1 internal values (for the pressure)
 GeneralisedNewtonianQCrouzeixRaviartElement() : QElement<DIM,3>(), 
  GeneralisedNewtonianNavierStokesEquations<DIM>()
  {
   //Allocate and add one Internal data object that stored DIM+1 pressure
   //values;
   P_nst_internal_index = this->add_internal_data(new Data(DIM+1));
  }
 
 /// \short Number of values (pinned or dofs) required at local node n. 
 virtual unsigned required_nvalue(const unsigned &n) const;

 
 /// Pressure shape functions at local coordinate s
 inline void pshape_nst(const Vector<double> &s, Shape &psi) const;
 
 /// Pressure shape and test functions at local coordinte s
 inline void pshape_nst(const Vector<double> &s, Shape &psi, 
                        Shape &test) const;
 
 /// \short Return the i-th pressure value
 /// (Discontinous pressure interpolation -- no need to cater for hanging 
 /// nodes). 
 double p_nst(const unsigned &i) const
  {return this->internal_data_pt(P_nst_internal_index)->value(i);}

 /// \short Return the i-th pressure value
 /// (Discontinous pressure interpolation -- no need to cater for hanging 
 /// nodes). 
 double p_nst(const unsigned &t, const unsigned &i) const
 {return this->internal_data_pt(P_nst_internal_index)->value(t,i);}

 /// Return number of pressure values
 unsigned npres_nst() const {return DIM+1;} 


 /// Return the local equation numbers for the pressure values.
 inline int p_local_eqn(const unsigned &n) const
  {return this->internal_local_eqn(P_nst_internal_index,n);}
 
 /// 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_nst_internal_index)->pin(p_dof);
   this->internal_data_pt(P_nst_internal_index)->set_value(p_dof,p_value);
  }


 /// \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)
  {GeneralisedNewtonianNavierStokesEquations<DIM>::output(outfile);}

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


 /// Redirect output to NavierStokesEquations output
 void output(FILE* file_pt) 
 {GeneralisedNewtonianNavierStokesEquations<DIM>::output(file_pt);}

 /// Redirect output to NavierStokesEquations output
 void output(FILE* file_pt, const unsigned &nplot)
  {GeneralisedNewtonianNavierStokesEquations<DIM>::output(file_pt,nplot);}


 /// \short Full output function: 
 /// x,y,[z],u,v,[w],p,du/dt,dv/dt,[dw/dt],dissipation
 /// in tecplot format. Default number of plot points
 void full_output(std::ostream &outfile)
  {GeneralisedNewtonianNavierStokesEquations<DIM>::full_output(outfile);}

 /// \short Full output function: 
 /// x,y,[z],u,v,[w],p,du/dt,dv/dt,[dw/dt],dissipation
 /// in tecplot format. nplot points in each coordinate direction
 void full_output(std::ostream &outfile, const unsigned &nplot)
  {GeneralisedNewtonianNavierStokesEquations<DIM>::full_output(outfile,nplot);}


 /// \short The number of "DOF types" that degrees of freedom in this element
 /// are sub-divided into: Velocity and pressure.
 unsigned ndof_types() const
  {
   return DIM+1;
  }
 
 /// \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.) Velocity=0; Pressure=1
 void get_dof_numbers_for_unknowns(
  std::list<std::pair<unsigned long,unsigned> >& dof_lookup_list) const;

};

//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.
//=======================================================================
template<unsigned DIM>
inline double GeneralisedNewtonianQCrouzeixRaviartElement<DIM>::
dshape_and_dtest_eulerian_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.
//=======================================================================
template<unsigned DIM>
inline double GeneralisedNewtonianQCrouzeixRaviartElement<DIM>::
dshape_and_dtest_eulerian_at_knot_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 equal to the shape functions
 test = psi;
 dtestdx = dpsidx;
 //Return the jacobian
 return J;
}


//=======================================================================
/// 2D
/// 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
//=======================================================================
template<>
inline double GeneralisedNewtonianQCrouzeixRaviartElement<2>::
dshape_and_dtest_eulerian_at_knot_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);
  
  // Loop over the test functions and derivatives and set them equal to the
  // shape functions
  for(unsigned i=0;i<9;i++)
   {
    test[i] = psi[i];

    for(unsigned k=0;k<2;k++)
     {
      dtestdx(i,k) = dpsidx(i,k);

      for(unsigned p=0;p<2;p++)
       {
        for(unsigned q=0;q<9;q++)
         {
          d_dtestdx_dX(p,q,i,k) = d_dpsidx_dX(p,q,i,k);
         }
       }
     }
   }

  // Return the jacobian
  return J;
 }


//=======================================================================
/// 3D
/// 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
//=======================================================================
template<>
inline double GeneralisedNewtonianQCrouzeixRaviartElement<3>::
dshape_and_dtest_eulerian_at_knot_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);

  // Loop over the test functions and derivatives and set them equal to the
  // shape functions
  for(unsigned i=0;i<27;i++)
   {
    test[i] = psi[i];

    for(unsigned k=0;k<3;k++)
     {
      dtestdx(i,k) = dpsidx(i,k);

      for(unsigned p=0;p<3;p++)
       {
        for(unsigned q=0;q<27;q++)
         {
          d_dtestdx_dX(p,q,i,k) = d_dpsidx_dX(p,q,i,k);
         }
       }
     }
   }

  // Return the jacobian
  return J;
 }



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



//==========================================================================
/// 2D :
/// Pressure shape and test functions and derivs w.r.t. to Eulerian coords.
/// Return Jacobian of mapping between local and global coordinates.
//==========================================================================
template<>
 inline double GeneralisedNewtonianQCrouzeixRaviartElement<2>::
dpshape_and_dptest_eulerian_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(9);
  DShape dpsi(9,2);
  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;

 }


//=======================================================================
/// Ppressure shape and test functions
//=======================================================================
template<unsigned DIM>
 inline void GeneralisedNewtonianQCrouzeixRaviartElement<DIM>::
pshape_nst(const Vector<double> &s, 
           Shape &psi, 
           Shape &test) const
{
 //Call the pressure shape functions
 this->pshape_nst(s,psi);
 //Test functions are equal to shape functions
 test = psi;
}


//=======================================================================
/// 3D :
/// Pressure shape functions
//=======================================================================
template<>
inline void GeneralisedNewtonianQCrouzeixRaviartElement<3>::
pshape_nst(
const Vector<double> &s, 
Shape &psi)
const
{
 psi[0] = 1.0;
 psi[1] = s[0];
 psi[2] = s[1];
 psi[3] = s[2];
}


//==========================================================================
/// 3D :
/// Pressure shape and test functions and derivs w.r.t. to Eulerian coords.
/// Return Jacobian of mapping between local and global coordinates.
//==========================================================================
template<>
 inline double GeneralisedNewtonianQCrouzeixRaviartElement<3>::
dpshape_and_dptest_eulerian_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];
  ppsi[3] = s[2];

  dppsidx(0,0) = 0.0;
  dppsidx(1,0) = 1.0;
  dppsidx(2,0) = 0.0;
  dppsidx(3,0) = 0.0;

  dppsidx(0,1) = 0.0;
  dppsidx(1,1) = 0.0;
  dppsidx(2,1) = 1.0;
  dppsidx(3,1) = 0.0;

  dppsidx(0,2) = 0.0;
  dppsidx(1,2) = 0.0;
  dppsidx(2,2) = 0.0;
  dppsidx(3,2) = 1.0;


  //Get the values of the shape functions and their local derivatives
  Shape psi(27);
  DShape dpsi(27,3);
  dshape_local(s,psi,dpsi);

  // Allocate memory for the inverse 3x3 jacobian
  DenseMatrix<double> inverse_jacobian(3);

  // 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<GeneralisedNewtonianQCrouzeixRaviartElement<2> >: 
public virtual QElement<1,3>
{
  public:
 FaceGeometry() : QElement<1,3>() {}
};

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

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


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



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


//=======================================================================
/// Taylor--Hood elements are Navier--Stokes elements 
/// with quadratic interpolation for velocities and positions and 
/// continuous linear pressure interpolation. They can be used
/// within oomph-lib's block-preconditioning framework.
//=======================================================================
template <unsigned DIM>
class GeneralisedNewtonianQTaylorHoodElement : public virtual QElement<DIM,3>, 
 public virtual GeneralisedNewtonianNavierStokesEquations<DIM>
{
  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_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_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_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_nst(const Vector<double> &s, 
                                               Shape &ppsi, 
                                               DShape &dppsidx, 
                                               Shape &ptest, 
                                               DShape &dptestdx) const;
 
  public:

 /// Constructor, no internal data points
 GeneralisedNewtonianQTaylorHoodElement() : QElement<DIM,3>(), 
  GeneralisedNewtonianNavierStokesEquations<DIM>() {}
 
 /// \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];}


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

 /// Pressure shape and test functions at local coordinte s
 inline void pshape_nst(const Vector<double> &s, Shape &psi, 
                        Shape &test) const;
 
 /// \short Set the value at which the pressure is stored in the nodes
 virtual int p_nodal_index_nst() const {return static_cast<int>(DIM);}

 /// 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],p_nodal_index_nst());}

 /// \short Access function for the pressure values at local pressure 
 /// node n_p (const version)
 double p_nst(const unsigned &n_p) const
  {return this->nodal_value(Pconv[n_p],this->p_nodal_index_nst());}

 /// \short Access function for the pressure values at local pressure 
 /// node n_p (const version)
 double p_nst(const unsigned &t, const unsigned &n_p) const
 {return this->nodal_value(t,Pconv[n_p],this->p_nodal_index_nst());}
 
 /// Return number of pressure values
 unsigned npres_nst() const 
  {return static_cast<unsigned>(pow(2.0,static_cast<int>(DIM)));}

 /// 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(this->p_nodal_index_nst());
   this->node_pt(Pconv[p_dof])->set_value(this->p_nodal_index_nst(),p_value);
  }


 /// \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) 
  {GeneralisedNewtonianNavierStokesEquations<DIM>::output(outfile);}

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

 /// Redirect output to NavierStokesEquations output
 void output(FILE* file_pt) 
 {GeneralisedNewtonianNavierStokesEquations<DIM>::output(file_pt);}

 /// Redirect output to NavierStokesEquations output
 void output(FILE* file_pt, const unsigned &nplot)
  {GeneralisedNewtonianNavierStokesEquations<DIM>::output(file_pt,nplot);}

 
 /// \short Returns the number of "DOF types" that degrees of freedom
 /// in this element are sub-divided into: Velocity and pressure.
 unsigned ndof_types() const
  {
   return DIM+1;
  }
 
 /// \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.) Velocity=0; Pressure=1
 void get_dof_numbers_for_unknowns(
  std::list<std::pair<unsigned long, unsigned> >& dof_lookup_list) const;
 
 
};

//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.
//==========================================================================
template<unsigned DIM>
inline double GeneralisedNewtonianQTaylorHoodElement<DIM>::
dshape_and_dtest_eulerian_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.
//==========================================================================
template<unsigned DIM>
inline double GeneralisedNewtonianQTaylorHoodElement<DIM>::
dshape_and_dtest_eulerian_at_knot_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 equal to the shape functions
 test = psi;
 dtestdx = dpsidx;

 //Return the jacobian
 return J;
}


//==========================================================================
/// 2D :
/// Pressure shape and test functions and derivs w.r.t. to Eulerian coords.
/// Return Jacobian of mapping between local and global coordinates.
//==========================================================================
template<>
 inline double GeneralisedNewtonianQTaylorHoodElement<2>::
dpshape_and_dptest_eulerian_nst(
  const Vector<double> &s, 
  Shape &ppsi, 
  DShape &dppsidx, 
  Shape &ptest, 
  DShape &dptestdx) const
 {
  
  //Local storage
  double psi1[2], psi2[2];
  double dpsi1[2],dpsi2[2];
  
  //Call the OneDimensional Shape functions
  OneDimLagrange::shape<2>(s[0],psi1);
  OneDimLagrange::shape<2>(s[1],psi2);
  OneDimLagrange::dshape<2>(s[0],dpsi1);
  OneDimLagrange::dshape<2>(s[1],dpsi2);
  
  //Now let's loop over the nodal points in the element
  //s1 is the "x" coordinate, s2 the "y" 
  for(unsigned i=0;i<2;i++)
   {
    for(unsigned j=0;j<2;j++)
     {
      /*Multiply the two 1D functions together to get the 2D function*/
      ppsi[2*i+j] = psi2[i]*psi1[j];
      dppsidx(2*i+j,0)= psi2[i]*dpsi1[j];
      dppsidx(2*i+j,1)=dpsi2[i]* psi1[j];
     }
   }
  

  //Get the values of the shape functions and their local derivatives
  Shape psi(9);
  DShape dpsi(9,2);
  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;
    
 }



//==========================================================================
/// 2D : 
/// 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
//==========================================================================
template<>
inline double GeneralisedNewtonianQTaylorHoodElement<2>::
 dshape_and_dtest_eulerian_at_knot_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);

  // Loop over the test functions and derivatives and set them equal to the
  // shape functions
  for(unsigned i=0;i<9;i++)
   {
    test[i] = psi[i];

    for(unsigned k=0;k<2;k++)
     {
      dtestdx(i,k) = dpsidx(i,k);

      for(unsigned p=0;p<2;p++)
       {
        for(unsigned q=0;q<9;q++)
         {
          d_dtestdx_dX(p,q,i,k) = d_dpsidx_dX(p,q,i,k);
         }
       }
     }
   }

  // Return the jacobian
  return J;
 }


//==========================================================================
/// 3D : 
/// 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
//==========================================================================
template<>
 inline double GeneralisedNewtonianQTaylorHoodElement<3>::
 dshape_and_dtest_eulerian_at_knot_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);

  // Loop over the test functions and derivatives and set them equal to the
  // shape functions
  for(unsigned i=0;i<27;i++)
   {
    test[i] = psi[i];

    for(unsigned k=0;k<3;k++)
     {
      dtestdx(i,k) = dpsidx(i,k);

      for(unsigned p=0;p<3;p++)
       {
        for(unsigned q=0;q<27;q++)
         {
          d_dtestdx_dX(p,q,i,k) = d_dpsidx_dX(p,q,i,k);
         }
       }
     }
   }

  // Return the jacobian
  return J;
 }



//==========================================================================
/// 2D :
/// Pressure shape functions
//==========================================================================
template<>
inline void GeneralisedNewtonianQTaylorHoodElement<2>::
pshape_nst(
const Vector<double> &s, 
Shape &psi)
const
{
 //Local storage
 double psi1[2], psi2[2];
 //Call the OneDimensional Shape functions
 OneDimLagrange::shape<2>(s[0],psi1);
 OneDimLagrange::shape<2>(s[1],psi2);

 //Now let's loop over the nodal points in the element
 //s1 is the "x" coordinate, s2 the "y" 
 for(unsigned i=0;i<2;i++)
  {
   for(unsigned j=0;j<2;j++)
    {
     /*Multiply the two 1D functions together to get the 2D function*/
     psi[2*i + j] = psi2[i]*psi1[j];
    }
  }
}


//==========================================================================
/// 3D :
/// Pressure shape and test functions and derivs w.r.t. to Eulerian coords.
/// Return Jacobian of mapping between local and global coordinates.
//==========================================================================
template<>
 inline double GeneralisedNewtonianQTaylorHoodElement<3>::
dpshape_and_dptest_eulerian_nst(
  const Vector<double> &s, 
  Shape &ppsi, 
  DShape &dppsidx, 
  Shape &ptest, 
  DShape &dptestdx) const
 {
  //Local storage
  double psi1[2], psi2[2], psi3[2];
  double dpsi1[2],dpsi2[2],dpsi3[2];
  
  //Call the OneDimensional Shape functions
  OneDimLagrange::shape<2>(s[0],psi1);
  OneDimLagrange::shape<2>(s[1],psi2);
  OneDimLagrange::shape<2>(s[2],psi3);
  OneDimLagrange::dshape<2>(s[0],dpsi1);
  OneDimLagrange::dshape<2>(s[1],dpsi2);
  OneDimLagrange::dshape<2>(s[2],dpsi3);
  
  //Now let's loop over the nodal points in the element
  //s0 is the "x" coordinate, s1 the "y", s2 is the "z"  
  for(unsigned i=0;i<2;i++)
   {
    for(unsigned j=0;j<2;j++)
     {
      for(unsigned k=0;k<2;k++)
       {
        /*Multiply the three 1D functions together to get the 3D function*/
        ppsi[4*i + 2*j + k] = psi3[i]*psi2[j]*psi1[k];
        dppsidx(4*i + 2*j + k,0) =  psi3[i]   * psi2[j]    *dpsi1[k];
        dppsidx(4*i + 2*j + k,1) =  psi3[i]   *dpsi2[j]    * psi1[k];
        dppsidx(4*i + 2*j + k,2) = dpsi3[i]   * psi2[j]    * psi1[k];
       }
     }
   }
  

  //Get the values of the shape functions and their local derivatives
  Shape psi(27);
  DShape dpsi(27,3);
  dshape_local(s,psi,dpsi);

  // Allocate memory for the inverse 3x3 jacobian
  DenseMatrix<double> inverse_jacobian(3);
  
  // 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;

}

//==========================================================================
/// 3D :
/// Pressure shape functions
//==========================================================================
template<>
inline void GeneralisedNewtonianQTaylorHoodElement<3>::
pshape_nst(
const Vector<double> &s, 
Shape &psi)
const
{
 //Local storage
 double psi1[2], psi2[2], psi3[2];

 //Call the OneDimensional Shape functions
 OneDimLagrange::shape<2>(s[0],psi1);
 OneDimLagrange::shape<2>(s[1],psi2);
 OneDimLagrange::shape<2>(s[2],psi3);

 //Now let's loop over the nodal points in the element
 //s0 is the "x" coordinate, s1 the "y", s2 is the "z"  
 for(unsigned i=0;i<2;i++)
  {
   for(unsigned j=0;j<2;j++)
    {
     for(unsigned k=0;k<2;k++)
      {
       /*Multiply the three 1D functions together to get the 3D function*/
       psi[4*i + 2*j + k] = psi3[i]*psi2[j]*psi1[k];
      }
    }
  }
}


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


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

//=======================================================================
/// Face geometry of the 3D Taylor_Hood elements
//=======================================================================
template<>
class FaceGeometry<GeneralisedNewtonianQTaylorHoodElement<3> >: 
public virtual QElement<2,3>
{
 
  public:
 FaceGeometry() : QElement<2,3>() {}
};


//=======================================================================
/// Face geometry of the FaceGeometry of the 2D Taylor Hoodelements
//=======================================================================
template<>
class FaceGeometry<FaceGeometry<GeneralisedNewtonianQTaylorHoodElement<2> > >: 
public virtual PointElement
{
  public:
 FaceGeometry() : PointElement() {}
};


//=======================================================================
/// Face geometry of the FaceGeometry of the 3D Taylor_Hood elements
//=======================================================================
template<>
class FaceGeometry<FaceGeometry<GeneralisedNewtonianQTaylorHoodElement<3> > >: 
public virtual QElement<1,3>
{
  public:
 FaceGeometry() : QElement<1,3>() {}
};







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


//==========================================================
/// Taylor Hood upgraded to become projectable
//==========================================================
 template<class TAYLOR_HOOD_ELEMENT>
 class ProjectableGeneralisedNewtonianTaylorHoodElement : 
  public virtual ProjectableElement<TAYLOR_HOOD_ELEMENT>
 {

 public:

  /// \short Constructor [this was only required explicitly
  /// from gcc 4.5.2 onwards...]
  ProjectableGeneralisedNewtonianTaylorHoodElement(){}
  

  /// \short Specify the values associated with field fld. 
  /// The information is returned in a vector of pairs which comprise 
  /// the Data object and the value within it, that correspond to field fld. 
  /// In the underlying Taylor Hood elements the fld-th velocities are stored
  /// at the fld-th value of the nodes; the pressures (the dim-th 
  /// field) are the dim-th values at the vertex nodes etc. 
  Vector<std::pair<Data*,unsigned> > data_values_of_field(const unsigned& fld)
   {   
    // Create the vector
    Vector<std::pair<Data*,unsigned> > data_values;
   
    // Velocities dofs
    if (fld<this->dim())
     {
      // Loop over all nodes
      unsigned nnod=this->nnode();
      for (unsigned j=0;j<nnod;j++)
       {
        // Add the data value associated with the velocity components
        data_values.push_back(std::make_pair(this->node_pt(j),fld));
       }
     }
    // Pressure
    else
     {
      // Loop over all vertex nodes
      unsigned Pconv_size=this->dim()+1;
      for (unsigned j=0;j<Pconv_size;j++)
       {
        // Add the data value associated with the pressure components
        unsigned vertex_index=this->Pconv[j];
        data_values.push_back(std::make_pair(this->node_pt(vertex_index),fld));
       }
     }

    // Return the vector
    return data_values;

   }

  /// \short Number of fields to be projected: dim+1, corresponding to 
  /// velocity components and  pressure
  unsigned nfields_for_projection()
   {
    return this->dim()+1;
   }
 
  /// \short Number of history values to be stored for fld-th field. Whatever
  /// the timestepper has set up for the velocity components and
  /// none for the pressure field.   
  /// (Note: count includes current value!)
  unsigned nhistory_values_for_projection(const unsigned &fld)
   {
    if (fld==this->dim())
     {
      //pressure doesn't have history values
      return this->node_pt(0)->ntstorage();//1; 
     }
    else 
     {
      return this->node_pt(0)->ntstorage();
     }
   }

  /// \short Number of positional history values 
  /// (Note: count includes current value!)
  unsigned nhistory_values_for_coordinate_projection()
   {
    return this->node_pt(0)->position_time_stepper_pt()->ntstorage();
   }
 
  /// \short Return Jacobian of mapping and shape functions of field fld
  /// at local coordinate s
  double jacobian_and_shape_of_field(const unsigned &fld, 
                                     const Vector<double> &s, 
                                     Shape &psi)
   {
    unsigned n_dim=this->dim();
    unsigned n_node=this->nnode();
   
    if (fld==n_dim) 
     {
      //We are dealing with the pressure
      this->pshape_nst(s,psi);
     
      Shape psif(n_node),testf(n_node); 
      DShape dpsifdx(n_node,n_dim), dtestfdx(n_node,n_dim);
     
      //Domain Shape
      double J=this->dshape_and_dtest_eulerian_nst(s,psif,dpsifdx,
                                                   testf,dtestfdx);    
      return J;
     }
    else 
     {
      Shape testf(n_node); 
      DShape dpsifdx(n_node,n_dim), dtestfdx(n_node,n_dim);
     
      //Domain Shape
      double J=this->dshape_and_dtest_eulerian_nst(s,psi,dpsifdx,
                                                   testf,dtestfdx);
      return J;
     }
   }



  /// \short Return interpolated field fld at local coordinate s, at time level
  /// t (t=0: present; t>0: history values)
  double get_field(const unsigned &t, 
                   const unsigned &fld,
                   const Vector<double>& s)
   {
    unsigned n_dim =this->dim(); 
    unsigned n_node=this->nnode();
   
    //If fld=n_dim, we deal with the pressure
    if (fld==n_dim)
     {
      return this->interpolated_p_nst(t,s);
     }
    // Velocity
    else
     {
      //Find the index at which the variable is stored
      unsigned u_nodal_index = this->u_index_nst(fld);
     
      //Local shape function
      Shape psi(n_node);
     
      //Find values of shape function
      this->shape(s,psi);
     
      //Initialise value of u
      double interpolated_u = 0.0;
     
      //Sum over the local nodes at that time
      for(unsigned l=0;l<n_node;l++) 
       {
        interpolated_u += this->nodal_value(t,l,u_nodal_index)*psi[l];
       }
      return interpolated_u;     
     }
   }



  ///Return number of values in field fld
  unsigned nvalue_of_field(const unsigned &fld)
   {
    if (fld==this->dim())
     {
      return this->npres_nst();
     }
    else
     {
      return this->nnode();
     }
   }

 
  ///Return local equation number of value j in field fld.
  int local_equation(const unsigned &fld,
                     const unsigned &j)
   {
    if (fld==this->dim()) 
     {
      return this->p_local_eqn(j);
     }
    else
     {
      const unsigned u_nodal_index = this->u_index_nst(fld);
      return this->nodal_local_eqn(j,u_nodal_index);
     }
   }
  
 };


//=======================================================================
/// Face geometry for element is the same as that for the underlying
/// wrapped element
//=======================================================================
 template<class ELEMENT>
 class FaceGeometry<ProjectableGeneralisedNewtonianTaylorHoodElement<ELEMENT> > 
  : public virtual FaceGeometry<ELEMENT>
 {
 public:
  FaceGeometry() : FaceGeometry<ELEMENT>() {}
 };


//=======================================================================
/// Face geometry of the Face Geometry for element is the same as 
/// that for the underlying wrapped element
//=======================================================================
 template<class ELEMENT>
 class FaceGeometry<FaceGeometry
 <ProjectableGeneralisedNewtonianTaylorHoodElement<ELEMENT> > >
  : public virtual FaceGeometry<FaceGeometry<ELEMENT> >
 {
 public:
  FaceGeometry() : FaceGeometry<FaceGeometry<ELEMENT> >() {}
 };


//==========================================================
/// Crouzeix Raviart upgraded to become projectable
//==========================================================
 template<class CROUZEIX_RAVIART_ELEMENT>
 class ProjectableGeneralisedNewtonianCrouzeixRaviartElement : 
  public virtual ProjectableElement<CROUZEIX_RAVIART_ELEMENT>
 {

 public:

  /// \short Constructor [this was only required explicitly
  /// from gcc 4.5.2 onwards...]
  ProjectableGeneralisedNewtonianCrouzeixRaviartElement(){}
  
  /// \short Specify the values associated with field fld. 
  /// The information is returned in a vector of pairs which comprise 
  /// the Data object and the value within it, that correspond to field fld. 
  /// In the underlying Crouzeix Raviart elements the 
  /// fld-th velocities are stored
  /// at the fld-th value of the nodes; the pressures are stored internally
  Vector<std::pair<Data*,unsigned> > data_values_of_field(const unsigned& fld)
   {   
    // Create the vector
    Vector<std::pair<Data*,unsigned> > data_values;
   
    // Velocities dofs
    if (fld < this->dim())
     {
      // Loop over all nodes
      const unsigned n_node=this->nnode();
      for (unsigned n=0;n<n_node;n++)
       {
        // Add the data value associated with the velocity components
        data_values.push_back(std::make_pair(this->node_pt(n),fld));
       }
     }
    // Pressure
    else
     {
      //Need to push back the internal data
      const unsigned n_press = this->npres_nst();
      //Loop over all pressure values
      for(unsigned j=0;j<n_press;j++)
       {
        data_values.push_back(
         std::make_pair(
          this->internal_data_pt(this->P_nst_internal_index),j));
       }
     }

    // Return the vector
    return data_values;
   }

  /// \short Number of fields to be projected: dim+1, corresponding to 
  /// velocity components and  pressure
  unsigned nfields_for_projection()
   {
    return this->dim()+1;
   }
 
  /// \short Number of history values to be stored for fld-th field. Whatever
  /// the timestepper has set up for the velocity components and
  /// none for the pressure field.
  /// (Note: count includes current value!)
  unsigned nhistory_values_for_projection(const unsigned &fld)
   {
    if (fld==this->dim())
     {
      //pressure doesn't have history values
      return 1; 
     }
    else 
     {
      return this->node_pt(0)->ntstorage();
     }
   }

  ///\short Number of positional history values.
  /// (Note: count includes current value!)
  unsigned nhistory_values_for_coordinate_projection()
   {
    return this->node_pt(0)->position_time_stepper_pt()->ntstorage();
   }
 
  /// \short Return Jacobian of mapping and shape functions of field fld
  /// at local coordinate s
  double jacobian_and_shape_of_field(const unsigned &fld, 
                                     const Vector<double> &s, 
                                     Shape &psi)
   {
    unsigned n_dim=this->dim();
    unsigned n_node=this->nnode();
   
    if (fld==n_dim) 
     {
      //We are dealing with the pressure
      this->pshape_nst(s,psi);
     
      Shape psif(n_node),testf(n_node); 
      DShape dpsifdx(n_node,n_dim), dtestfdx(n_node,n_dim);
     
      //Domain Shape
      double J=this->dshape_and_dtest_eulerian_nst(s,psif,dpsifdx,
                                                   testf,dtestfdx);    
      return J;
     }
    else 
     {
      Shape testf(n_node); 
      DShape dpsifdx(n_node,n_dim), dtestfdx(n_node,n_dim);
     
      //Domain Shape
      double J=this->dshape_and_dtest_eulerian_nst(s,psi,dpsifdx,
                                                   testf,dtestfdx);
      return J;
     }
   }



  /// \short Return interpolated field fld at local coordinate s, at time level
  /// t (t=0: present; t>0: history values)
  double get_field(const unsigned &t, 
                   const unsigned &fld,
                   const Vector<double>& s)
   {
    unsigned n_dim =this->dim(); 
   
    //If fld=n_dim, we deal with the pressure
    if (fld==n_dim)
     {
      return this->interpolated_p_nst(s);
     }
    // Velocity
    else
     {
      return this->interpolated_u_nst(t,s,fld);
     }
   }


  ///Return number of values in field fld
  unsigned nvalue_of_field(const unsigned &fld)
   {
    if (fld==this->dim())
     {
      return this->npres_nst();
     }
    else
     {
      return this->nnode();
     }
   }

 
  ///Return local equation number of value j in field fld.
  int local_equation(const unsigned &fld,
                     const unsigned &j)
   {
    if (fld==this->dim()) 
     {
      return this->p_local_eqn(j);
     }
    else
     {
      const unsigned u_nodal_index = this->u_index_nst(fld);
      return this->nodal_local_eqn(j,u_nodal_index);
     }
   }
  
 };


//=======================================================================
/// Face geometry for element is the same as that for the underlying
/// wrapped element
//=======================================================================
 template<class ELEMENT>
 class FaceGeometry
 <ProjectableGeneralisedNewtonianCrouzeixRaviartElement<ELEMENT> > 
  : public virtual FaceGeometry<ELEMENT>
 {
 public:
  FaceGeometry() : FaceGeometry<ELEMENT>() {}
 };


//=======================================================================
/// Face geometry of the Face Geometry for element is the same as 
/// that for the underlying wrapped element
//=======================================================================
 template<class ELEMENT>
 class FaceGeometry<FaceGeometry
 <ProjectableGeneralisedNewtonianCrouzeixRaviartElement<ELEMENT> > >
  : public virtual FaceGeometry<FaceGeometry<ELEMENT> >
 {
 public:
  FaceGeometry() : FaceGeometry<FaceGeometry<ELEMENT> >() {}
 };


}

#endif