polar_navier_stokes_elements.h

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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// Copyright (C) 2006-2024 Matthias Heil and Andrew Hazel
// LIC//
// LIC// This library is free software; you can redistribute it and/or
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// 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// Lesser General Public License for more details.
// LIC//
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// Created (18/03/08) by recopying from my_oomph-codes/polar_navier_stokes.h
// Now I know what I'm doing it needs updating to current oomph conventions.
 
// 17/06/08 - Weakform adjusted to "correct" traction version
 
// "_pnst" added to: - npres()
//                   - pshape()
//                   - u()
//                   - p()
//                   - du_dt()
//                   - interpolated_u()
//                   - interpolated_p()
//                   - interpolated_dudx()
//                   - dshape_and_dtest_eulerian()
//                   - dshape_and_dtest_eulerian_at_knot()
 
// The following generality is necessary for elements with multiple physics.
// That is where we can't just assume that values one and two at the nodes
// are velocities and the third a pressure.
 
// Then renamed "index_of_nodal_pressure_value to p_nodal_index_pnst.
 
// Added u_index_pnst (by default just returns the value you give it)
 
// The internal pressure data in Crouzeix Raviart elements is now stored
// as one data object with three values:
// Added P_pnst_internal_index which stores the index of that internal
// data, specified in the constructor.
// Adjusted:   - p_pnst
//             - fix_pressure
//             - get_load_data
// To reflect this change in internal data storage.
 
// Plus added Pressure_not_stored_at_node, default = -100.
 
// assign_additional_local_eqn_numbers removed in favour of
//  - p_local_eqn
// And we have:
//  - local_eqn = nodal_local_eqn(l,u_nodal_index[i]);
//  - local_eqn = p_local_eqn(l);
// in fill_in_generic_residual_contribution
 
// Also removed the following functions for pinning/unpinning pressures
// as they're only needed in the refineable case:
//----------------------------------------------------------------------
//  - unpin_all_nodal_pressure_dofs()
//  - unpin_all_internal_pressure_dofs()
//  - pin_all_nodal_pressure_dofs()
//  - unpin_proper_nodal_pressure_dofs()
//  - pin_redundant_nodal_pressures()
//  - unpin_all_pressure_dofs()
//  - pressure_dof_is_hanging()
//----------------------------------------------------------------------
 
// strain_rate adapted to return the polar strain components.
 
// interpolated positions / velocities now assembled using nodal_position
// and nodal_value - both of which should cope with hanging nodes.
 
// Also stokes_output() added to compute convergence rates when exact
// (Stokes) solution is known.
 
#ifndef OOMPH_POLAR_NAVIER_STOKES
#define OOMPH_POLAR_NAVIER_STOKES
 
// Config header generated by autoconfig
#ifdef HAVE_CONFIG_H
#include <oomph-lib-config.h>
#endif
 
// OOMPH-LIB headers
#include "../generic/Qelements.h"
 
 
namespace oomph
{
  //=====================================================================
  /// A class for elements that solve the polar Navier--Stokes equations,
  /// This contains the generic maths -- any concrete implementation must
  /// be derived from this.
  //======================================================================
  class PolarNavierStokesEquations : public virtual FiniteElement
  {
  public:
    /// 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);
 
    /// Function pointer to source function fct(t,x)
    /// x is a Vector!
    typedef double (*NavierStokesSourceFctPt)(const double& time,
                                              const Vector<double>& x);
 
  private:
    /// 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:
    // Physical constants
 
    /// Pointer to the viscosity ratio (relative to the
    /// viscosity used in the definition of the Reynolds number)
    double* Viscosity_Ratio_pt;
 
    /// 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 the angle alpha
    double* Alpha_pt;
 
    /// Pointer to global Reynolds number
    double* Re_pt;
 
    /// Pointer to global Reynolds number x Strouhal number (=Womersley)
    double* ReSt_pt;
 
    /// 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;
 
    /// 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) = 0;
 
    /// 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_pnst(const Vector<double>& s,
                                                  Shape& psi,
                                                  DShape& dpsidx,
                                                  Shape& test,
                                                  DShape& dtestdx) const = 0;
 
    /// 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_pnst(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const = 0;
 
    /// Compute the pressure shape functions at local coordinate s
    virtual void pshape_pnst(const Vector<double>& s, Shape& psi) const = 0;
 
    /// Compute the pressure shape and test functions
    /// at local coordinate s
    virtual void pshape_pnst(const Vector<double>& s,
                             Shape& psi,
                             Shape& test) const = 0;
 
 
    /// Calculate the body force at a given time and Eulerian position
    void get_body_force(double time,
                        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 < 2; i++)
        {
          result[i] = 0.0;
        }
      }
      // Otherwise call the function
      else
      {
        (*Body_force_fct_pt)(time, x, result);
      }
    }
 
    /// Calculate the source fct at given time and
    /// Eulerian position
    double get_source_fct(double time, 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);
      }
    }
 
  public:
    /// Constructor: NULL the body force and source function
    PolarNavierStokesEquations() : Body_force_fct_pt(0), Source_fct_pt(0)
    {
      // 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;
    }
 
    /// Alpha
    const double& alpha() const
    {
      return *Alpha_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 Alpha
    double*& alpha_pt()
    {
      return Alpha_pt;
    }
 
    /// Pointer to product of Reynolds and Strouhal number (=Womersley number)
    double*& re_st_pt()
    {
      return ReSt_pt;
    }
 
    /// 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;
    }
 
    /// 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;
    }
 
    /// Function to return number of pressure degrees of freedom
    virtual unsigned npres_pnst() const = 0;
 
    /// Velocity i at local node n. Uses suitably interpolated value
    /// for hanging nodes.
    virtual double u_pnst(const unsigned& n, const unsigned& i) const = 0;
 
    /// Velocity i at local node n at timestep t (t=0: present;
    /// t>0: previous). Uses suitably interpolated value for hanging nodes.
    virtual double u_pnst(const unsigned& t,
                          const unsigned& n,
                          const unsigned& i) const = 0;
 
    /// 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_pnst(const unsigned& i) const
    {
      return i;
    }
 
    /// i-th component of du/dt at local node n.
    /// Uses suitably interpolated value for hanging nodes.
    double du_dt_pnst(const unsigned& n, const unsigned& i) const
    {
      // Get the data's timestepper
      TimeStepper* time_stepper_pt = node_pt(n)->time_stepper_pt();
 
      // Number of timsteps (past & present)
      unsigned n_time = time_stepper_pt->ntstorage();
 
      // Initialise dudt
      double dudt = 0.0;
      // Loop over the timesteps, if there is a non Steady timestepper
      if (time_stepper_pt->type() != "Steady")
      {
        for (unsigned t = 0; t < n_time; t++)
        {
          dudt += time_stepper_pt->weight(1, t) * u_pnst(t, n, i);
        }
      }
 
      return dudt;
    }
 
    /// Pressure at local pressure "node" n_p
    /// Uses suitably interpolated value for hanging nodes.
    virtual double p_pnst(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;
 
    /// Which nodal value represents the pressure? (Default: negative,
    /// indicating that pressure is not based on nodal interpolation).
    virtual int p_nodal_index_pnst()
    {
      return Pressure_not_stored_at_node;
    }
 
    /// Pointer to n_p-th pressure node (Default: NULL,
    /// indicating that pressure is not based on nodal interpolation).
    virtual Node* pressure_node_pt(const unsigned& n_p)
    {
      return NULL;
    }
 
    /// Integral of pressure over element
    double pressure_integral() const;
 
    /// Return integral of dissipation over element
    double dissipation() const;
 
    /// Return dissipation at local coordinate s
    double dissipation(const Vector<double>& s) const;
 
    /// Get integral of kinetic energy over element
    double kin_energy() const;
 
    /// Strain-rate tensor: Now returns polar strain
    void strain_rate(const Vector<double>& s,
                     DenseMatrix<double>& strain_rate) const;
 
    /// Function to return polar strain multiplied by r
    void strain_rate_by_r(const Vector<double>& s,
                          DenseMatrix<double>& strain_rate) const;
 
    /// Compute traction (on the viscous scale) at local coordinate s
    /// for outer unit normal N
    void get_traction(const Vector<double>& s,
                      const Vector<double>& N,
                      Vector<double>& traction);
 
    /// The potential loading on an external SolidElement is always
    /// provided by the traction function
    void get_load(const Vector<double>& s,
                  const Vector<double>& xi,
                  const Vector<double>& x,
                  const Vector<double>& N,
                  Vector<double>& load)
    {
      get_traction(s, N, load);
    }
 
 
    /// Output functionget_vels(const Vector<double>& x_to_get,
    /// Vector<double>& vels): 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);
    }
 
    /// 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);
 
    /// 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);
    }
 
    /// 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);
 
    /// 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);
    }
 
    /// 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);
 
 
    /// 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);
 
    /// 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);
 
    /// 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);
 
    /// 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);
 
    /// 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
      fill_in_generic_residual_contribution(residuals,
                                            GeneralisedElement::Dummy_matrix,
                                            GeneralisedElement::Dummy_matrix,
                                            0);
    }
 
    /// 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(
        residuals, jacobian, GeneralisedElement::Dummy_matrix, 1);
 
      /*
      // Only needed for test purposes
      //Create a new matrix
      unsigned n_dof = ndof();
      DenseMatrix<double> jacobian_fd(n_dof,n_dof,0.0);
      fill_in_jacobian_from_internal_by_fd(residuals,jacobian_fd);
      fill_in_jacobian_from_nodal_by_fd(residuals,jacobian_fd);
 
      if(n_dof==21)
       {
         for(unsigned i=0;i<n_dof;i++)
          {
      for(unsigned j=0;j<n_dof;j++)
       {
         if(std::fabs(jacobian(i,j) - jacobian_fd(i,j)) > 1.0e-3)
          {
            std::cout << i << " " << j << " " << std::fabs(jacobian(i,j) -
      jacobian_fd(i,j)) << std::endl;
          }
       }
          }
       }
      // Only needed for test purposes
 
      // Only needed for test purposes
      //Create a new matrix
      unsigned n_dof = ndof();
      DenseMatrix<double>
      jacobian_new(n_dof,n_dof,0.0),jacobian_old(n_dof,n_dof,0.0);
      Vector<double> residuals_new(n_dof,0.0),residuals_old(n_dof,0.0);
 
      //Call the generic routine with the flag set to 1
      PolarNavierStokesEquations::fill_in_generic_residual_contribution(residuals_old,jacobian_old,
                                            GeneralisedElement::Dummy_matrix,1);
      //Call the generic routine with the flag set to 1
      fill_in_generic_residual_contribution(residuals_new,jacobian_new,
                                            GeneralisedElement::Dummy_matrix,1);
 
      if(n_dof==21)
        {
         for(unsigned i=0;i<n_dof;i++)
          {
      if(std::fabs(residuals_new[i] - residuals_old[i])>1.0e-8) std::cout << i
      << " " << std::fabs(residuals_new[i] - residuals_old[i]) << std::endl;
      //std::cout << residuals_new[i] << std::endl;
      for(unsigned j=0;j<n_dof;j++)
       {
         if(std::fabs(jacobian_new(i,j) - jacobian_old(i,j)) > 1.0e-8)
          {
            std::cout << i << " " << j << " " << std::fabs(jacobian_new(i,j) -
      jacobian_old(i,j)) << std::endl;
          }
       }
          }
        }
      // Only needed for test purposes
      */
    } // End of fill_in_contribution_to_jacobian
 
    /// Compute the element's residual Vector and the jacobian matrix
    /// Plus the mass matrix especially for eigenvalue problems
    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(
        residuals, jacobian, mass_matrix, 2);
    }
 
    /// Compute the residuals for the Navier--Stokes equations;
    /// flag=1(or 0): do (or don't) compute the Jacobian as well.
    virtual void fill_in_generic_residual_contribution(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      DenseMatrix<double>& mass_matrix,
      unsigned flag);
 
    /// Compute vector of FE interpolated velocity u at local coordinate s
    void interpolated_u_pnst(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 < 2; 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] += u_pnst(l, i) * psi[l];
        }
      }
    }
 
    /// Return FE interpolated velocity u[i] at local coordinate s
    double interpolated_u_pnst(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);
 
      // 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 += u_pnst(l, i) * psi[l];
      }
 
      return (interpolated_u);
    }
 
    /// Return FE interpolated pressure at local coordinate s
    double interpolated_p_pnst(const Vector<double>& s) const
    {
      // Find number of nodes
      unsigned n_pres = npres_pnst();
      // Local shape function
      Shape psi(n_pres);
      // Find values of shape function
      pshape_pnst(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_pnst(l) * psi[l];
      }
 
      return (interpolated_p);
    }
 
    /// ////////////////////////////////////////////////////////////////
    /// My own work:                                                ///
    /// Return FE interpolated velocity derivative du[i]/dx[j]      ///
    /// at local coordinate s                                       ///
    /// ////////////////////////////////////////////////////////////////
    double interpolated_dudx_pnst(const Vector<double>& s,
                                  const unsigned& i,
                                  const unsigned& j) const
    {
      // Find number of nodes
      unsigned n_node = nnode();
 
      // Set up memory for the shape and test functions
      Shape psif(n_node);
      DShape dpsifdx(n_node, 2);
 
      // double J =
      dshape_eulerian(s, psif, dpsifdx);
 
      // Initialise to zero
      double interpolated_dudx = 0.0;
 
      // Calculate velocity derivative:
 
      // Loop over nodes
      for (unsigned l = 0; l < n_node; l++)
      {
        interpolated_dudx += u_pnst(l, i) * dpsifdx(l, j);
      }
 
      return (interpolated_dudx);
    }
  };
 
  /// ///////////////////////////////////////////////////////////////////////////
  /// ///////////////////////////////////////////////////////////////////////////
  /// ///////////////////////////////////////////////////////////////////////////
 
 
  //==========================================================================
  /// Crouzeix_Raviart elements are Navier--Stokes elements with quadratic
  /// interpolation for velocities and positions, but a discontinuous linear
  /// pressure interpolation
  //==========================================================================
  class PolarCrouzeixRaviartElement : public virtual QElement<2, 3>,
                                      public virtual PolarNavierStokesEquations
  {
  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_pnst_internal_index;
 
    /// 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_pnst(const Vector<double>& s,
                                                 Shape& psi,
                                                 DShape& dpsidx,
                                                 Shape& test,
                                                 DShape& dtestdx) const;
 
    /// 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_pnst(const unsigned& ipt,
                                                         Shape& psi,
                                                         DShape& dpsidx,
                                                         Shape& test,
                                                         DShape& dtestdx) const;
 
    /// Pressure shape functions at local coordinate s
    inline void pshape_pnst(const Vector<double>& s, Shape& psi) const;
 
    /// Pressure shape and test functions at local coordinte s
    inline void pshape_pnst(const Vector<double>& s,
                            Shape& psi,
                            Shape& test) const;
 
    /// Return the local equation numbers for the pressure values.
    inline int p_local_eqn(const unsigned& n)
    {
      return this->internal_local_eqn(P_pnst_internal_index, n);
    }
 
  public:
    /// Constructor, there are DIM+1 internal values (for the pressure)
    PolarCrouzeixRaviartElement()
      : QElement<2, 3>(), PolarNavierStokesEquations()
    {
      // Allocate and add one Internal data object that stores 3 pressure
      // values;
      P_pnst_internal_index = this->add_internal_data(new Data(3));
    }
 
    /// Number of values (pinned or dofs) required at local node n.
    virtual unsigned required_nvalue(const unsigned& n) const;
 
    /// Return the velocity component u[i] at local node
    /// n. Uses suitably interpolated value for hanging nodes.
    double u_pnst(const unsigned& n, const unsigned& i) const
    {
      return this->nodal_value(n, i);
    }
 
    /// Return the velocity component u[i] at local node
    /// n at timestep t (t=0: present; t>0: previous timestep).
    /// Uses suitably interpolated value for hanging nodes.
    double u_pnst(const unsigned& t, const unsigned& n, const unsigned& i) const
    {
      return this->nodal_value(t, n, i);
    }
 
    /// Return the pressure values at internal dof i_internal
    /// (Discontinous pressure interpolation -- no need to cater for hanging
    /// nodes).
    double p_pnst(const unsigned& i_internal) const
    {
      return *this->internal_data_pt(P_pnst_internal_index)
                ->value_pt(i_internal);
    }
 
    /// Return number of pressure values
    unsigned npres_pnst() const
    {
      return 3;
    }
 
    /// Pin p_dof-th pressure dof and set it to value specified by p_value.
    void fix_pressure(const unsigned& p_dof, const double& p_value)
    {
      this->internal_data_pt(P_pnst_internal_index)->pin(p_dof);
      *this->internal_data_pt(P_pnst_internal_index)->value_pt(p_dof) = p_value;
    }
 
    /// Add to the set paired_load_data
    /// pairs of pointers to data objects and unsigned integers that
    /// index the values in the data object that affect the load (traction),
    /// as specified in the get_load() function.
    void get_load_data(std::set<std::pair<Data*, unsigned>>& paired_load_data);
 
    /// Redirect output to NavierStokesEquations output
    void output(std::ostream& outfile)
    {
      PolarNavierStokesEquations::output(outfile);
    }
 
    /// Redirect output to NavierStokesEquations output
    void output(std::ostream& outfile, const unsigned& Nplot)
    {
      PolarNavierStokesEquations::output(outfile, Nplot);
    }
 
 
    /// Redirect output to NavierStokesEquations output
    void output(FILE* file_pt)
    {
      PolarNavierStokesEquations::output(file_pt);
    }
 
    /// Redirect output to NavierStokesEquations output
    void output(FILE* file_pt, const unsigned& Nplot)
    {
      PolarNavierStokesEquations::output(file_pt, Nplot);
    }
 
 
    /// 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)
    {
      PolarNavierStokesEquations::full_output(outfile);
    }
 
    /// 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)
    {
      PolarNavierStokesEquations::full_output(outfile, nplot);
    }
  };
 
  // Inline functions
 
  //=======================================================================
  /// 2D
  /// Derivatives of the shape functions and test functions w.r.t. to global
  /// (Eulerian) coordinates. Return Jacobian of mapping between
  /// local and global coordinates.
  //=======================================================================
  inline double PolarCrouzeixRaviartElement::dshape_and_dtest_eulerian_pnst(
    const Vector<double>& s,
    Shape& psi,
    DShape& dpsidx,
    Shape& test,
    DShape& dtestdx) const
  {
    // Call the geometrical shape functions and derivatives
    double J = QElement<2, 3>::dshape_eulerian(s, psi, dpsidx);
    // 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];
      dtestdx(i, 0) = dpsidx(i, 0);
      dtestdx(i, 1) = dpsidx(i, 1);
    }
    // Return the jacobian
    return J;
  }
 
 
  //=======================================================================
  /// 2D
  /// Derivatives of the shape functions and test functions w.r.t. to global
  /// (Eulerian) coordinates. Return Jacobian of mapping between
  /// local and global coordinates.
  //=======================================================================
  inline double PolarCrouzeixRaviartElement::
    dshape_and_dtest_eulerian_at_knot_pnst(const unsigned& ipt,
                                           Shape& psi,
                                           DShape& dpsidx,
                                           Shape& test,
                                           DShape& dtestdx) const
  {
    // Call the geometrical shape functions and derivatives
    double J = QElement<2, 3>::dshape_eulerian_at_knot(ipt, psi, dpsidx);
    // 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];
      dtestdx(i, 0) = dpsidx(i, 0);
      dtestdx(i, 1) = dpsidx(i, 1);
    }
    // Return the jacobian
    return J;
  }
 
  //=======================================================================
  /// 2D :
  /// Pressure shape functions
  //=======================================================================
  inline void PolarCrouzeixRaviartElement::pshape_pnst(const Vector<double>& s,
                                                       Shape& psi) const
  {
    psi[0] = 1.0;
    psi[1] = s[0];
    psi[2] = s[1];
  }
 
  /// Define the pressure shape and test functions
  inline void PolarCrouzeixRaviartElement::pshape_pnst(const Vector<double>& s,
                                                       Shape& psi,
                                                       Shape& test) const
  {
    // Call the pressure shape functions
    pshape_pnst(s, psi);
    // Loop over the test functions and set them equal to the shape functions
    for (unsigned i = 0; i < 3; i++) test[i] = psi[i];
  }
 
  //=======================================================================
  /// Face geometry of the 2D Crouzeix_Raviart elements
  //=======================================================================
  template<>
  class FaceGeometry<PolarCrouzeixRaviartElement>
    : 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
  /// continous linear pressure interpolation
  //=======================================================================
  class PolarTaylorHoodElement : public virtual QElement<2, 3>,
                                 public virtual PolarNavierStokesEquations
  {
  private:
    /// Static array of ints to hold number of variables at node
    static const unsigned Initial_Nvalue[];
 
  protected:
    /// Static array of ints to hold conversion from pressure
    /// node numbers to actual node numbers
    static const unsigned Pconv[];
 
    /// 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_pnst(const Vector<double>& s,
                                                 Shape& psi,
                                                 DShape& dpsidx,
                                                 Shape& test,
                                                 DShape& dtestdx) const;
 
    /// 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_pnst(const unsigned& ipt,
                                                         Shape& psi,
                                                         DShape& dpsidx,
                                                         Shape& test,
                                                         DShape& dtestdx) const;
 
    /// Pressure shape functions at local coordinate s
    inline void pshape_pnst(const Vector<double>& s, Shape& psi) const;
 
    /// Pressure shape and test functions at local coordinte s
    inline void pshape_pnst(const Vector<double>& s,
                            Shape& psi,
                            Shape& test) const;
 
    /// Return the local equation numbers for the pressure values.
    inline int p_local_eqn(const unsigned& n)
    {
      return this->nodal_local_eqn(Pconv[n], p_nodal_index_pnst());
    }
 
  public:
    /// Constructor, no internal data points
    PolarTaylorHoodElement() : QElement<2, 3>(), PolarNavierStokesEquations() {}
 
    /// 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];
    }
 
    /// Which nodal value represents the pressure?
    virtual int p_nodal_index_pnst()
    {
      return 2;
    }
 
    /// Pointer to n_p-th pressure node
    Node* pressure_node_pt(const unsigned& n_p)
    {
      return this->node_pt(Pconv[n_p]);
    }
 
    /// Return the velocity component u[i] at local node
    /// n. Uses suitably interpolated value for hanging nodes.
    double u_pnst(const unsigned& n, const unsigned& i) const
    {
      return this->nodal_value(n, i);
    }
 
    /// Return the velocity component u[i] at local node
    /// n at timestep t (t=0: present; t>0: previous timestep).
    /// Uses suitably interpolated value for hanging nodes.
    double u_pnst(const unsigned& t, const unsigned& n, const unsigned& i) const
    {
      return this->nodal_value(t, n, i);
    }
 
    /// Access function for the pressure values at local pressure
    /// node n_p (const version)
    double p_pnst(const unsigned& n_p) const
    {
      return this->nodal_value(Pconv[n_p], 2);
    }
    // {return this->nodal_value(Pconv[n_p],this->p_nodal_index_pnst());}
 
    /// Return number of pressure values
    unsigned npres_pnst() const
    {
      return 4;
    }
 
    /// 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_pnst());
      *this->node_pt(Pconv[p_dof])->value_pt(this->p_nodal_index_pnst()) =
        p_value;
    }
 
 
    /// Add to the set paired_load_data
    /// pairs of pointers to data objects and unsigned integers that
    /// index the values in the data object that affect the load (traction),
    /// as specified in the get_load() function.
    void get_load_data(std::set<std::pair<Data*, unsigned>>& paired_load_data);
 
    /// Redirect output to NavierStokesEquations output
    void output(std::ostream& outfile)
    {
      PolarNavierStokesEquations::output(outfile);
    }
 
    /// Redirect output to NavierStokesEquations output
    void output(std::ostream& outfile, const unsigned& Nplot)
    {
      PolarNavierStokesEquations::output(outfile, Nplot);
    }
 
    /// Redirect output to NavierStokesEquations output
    void output(FILE* file_pt)
    {
      PolarNavierStokesEquations::output(file_pt);
    }
 
    /// Redirect output to NavierStokesEquations output
    void output(FILE* file_pt, const unsigned& Nplot)
    {
      PolarNavierStokesEquations::output(file_pt, Nplot);
    }
  };
 
  // Inline functions
 
  //==========================================================================
  /// 2D :
  /// Derivatives of the shape functions and test functions w.r.t to
  /// global (Eulerian) coordinates. Return Jacobian of mapping between
  /// local and global coordinates.
  //==========================================================================
  inline double PolarTaylorHoodElement::dshape_and_dtest_eulerian_pnst(
    const Vector<double>& s,
    Shape& psi,
    DShape& dpsidx,
    Shape& test,
    DShape& dtestdx) const
  {
    // Call the geometrical shape functions and derivatives
    double J = QElement<2, 3>::dshape_eulerian(s, psi, dpsidx);
    // 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];
      dtestdx(i, 0) = dpsidx(i, 0);
      dtestdx(i, 1) = dpsidx(i, 1);
    }
    // Return the jacobian
    return J;
  }
 
 
  //==========================================================================
  /// 2D :
  /// Derivatives of the shape functions and test functions w.r.t to
  /// global (Eulerian) coordinates. Return Jacobian of mapping between
  /// local and global coordinates.
  //==========================================================================
  inline double PolarTaylorHoodElement::dshape_and_dtest_eulerian_at_knot_pnst(
    const unsigned& ipt,
    Shape& psi,
    DShape& dpsidx,
    Shape& test,
    DShape& dtestdx) const
  {
    // Call the geometrical shape functions and derivatives
    double J = QElement<2, 3>::dshape_eulerian_at_knot(ipt, psi, dpsidx);
    // 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];
      dtestdx(i, 0) = dpsidx(i, 0);
      dtestdx(i, 1) = dpsidx(i, 1);
    }
    // Return the jacobian
    return J;
  }
 
 
  //==========================================================================
  /// 2D :
  /// Pressure shape functions
  //==========================================================================
  inline void PolarTaylorHoodElement::pshape_pnst(const Vector<double>& s,
                                                  Shape& psi) const
  {
    // Call the OneDimensional Shape functions
    // 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];
      }
    }
  }
 
 
  //==========================================================================
  /// 2D :
  /// Pressure shape and test functions
  //==========================================================================
  inline void PolarTaylorHoodElement::pshape_pnst(const Vector<double>& s,
                                                  Shape& psi,
                                                  Shape& test) const
  {
    // Call the pressure shape functions
    pshape_pnst(s, psi);
    // Loop over the test functions and set them equal to the shape functions
    for (unsigned i = 0; i < 4; i++) test[i] = psi[i];
  }
 
  //=======================================================================
  /// Face geometry of the 2D Taylor_Hood elements
  //=======================================================================
  template<>
  class FaceGeometry<PolarTaylorHoodElement> : public virtual QElement<1, 3>
  {
  public:
    FaceGeometry() : QElement<1, 3>() {}
  };
 
} // namespace oomph
#endif