linearised_axisym_navier_stokes_elements.h

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// Header file for linearised axisymmetric Navier-Stokes elements
 
#ifndef OOMPH_LINEARISED_AXISYM_NAVIER_STOKES_ELEMENTS_HEADER
#define OOMPH_LINEARISED_AXISYM_NAVIER_STOKES_ELEMENTS_HEADER
 
// Config header
#ifdef HAVE_CONFIG_H
#include <oomph-lib-config.h>
#endif
 
// oomph-lib includes
#include "generic/Qelements.h"
#include "generic/fsi.h"
 
namespace oomph
{
  //=======================================================================
  /// A class for elements that solve the linearised version of the
  /// unsteady Navier--Stokes equations in cylindrical polar coordinates,
  /// where we have Fourier-decomposed in the azimuthal direction so that
  /// the theta-dependance is replaced by an azimuthal mode number.
  //=======================================================================
  class LinearisedAxisymmetricNavierStokesEquations
    : public virtual FiniteElement
  {
  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 azimuthal mode number (zero)
    static int Default_Azimuthal_Mode_Number_Value;
 
    /// Static default value for the physical ratios (all initialised to one)
    static double Default_Physical_Ratio_Value;
 
  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;
 
    /// Pointer to global Reynolds number
    double* Re_pt;
 
    /// Pointer to global Reynolds number x Strouhal number (=Womersley)
    double* ReSt_pt;
 
    /// Pointer to azimuthal mode number k in e^ik(theta) decomposition
    int* Azimuthal_Mode_Number_pt;
 
    /// Pointer to base flow solution (velocity components) function
    void (*Base_flow_u_fct_pt)(const double& time,
                               const Vector<double>& x,
                               Vector<double>& result);
 
    /// Pointer to derivatives of base flow solution velocity
    /// components w.r.t. global coordinates (r and z) function
    void (*Base_flow_dudx_fct_pt)(const double& time,
                                  const Vector<double>& x,
                                  DenseMatrix<double>& result);
 
    /// Boolean flag to indicate if ALE formulation is disabled when
    /// the time-derivatives are computed. Only set to true if you're sure
    /// that the mesh is stationary.
    bool ALE_is_disabled;
 
    /// Access function for the local equation number
    /// information for the i-th component of the pressure.
    /// p_local_eqn[n,i] = local equation number or < 0 if pinned.
    virtual int p_local_eqn(const unsigned& n, const unsigned& i) = 0;
 
    /// Compute the shape functions and their derivatives
    /// w.r.t. global coordinates at local coordinate s.
    /// Return Jacobian of mapping between local and global coordinates.
    virtual double dshape_and_dtest_eulerian_linearised_axi_nst(
      const Vector<double>& s,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const = 0;
 
    /// Compute the shape functions and their derivatives
    /// w.r.t. global coordinates at the ipt-th integration point.
    /// Return Jacobian of mapping between local and global coordinates.
    virtual double dshape_and_dtest_eulerian_at_knot_linearised_axi_nst(
      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_linearised_axi_nst(const Vector<double>& s,
                                           Shape& psi) const = 0;
 
    /// Compute the pressure shape and test functions at local coordinate s
    virtual void pshape_linearised_axi_nst(const Vector<double>& s,
                                           Shape& psi,
                                           Shape& test) const = 0;
 
    /// Calculate the velocity components of the base flow solution
    /// at a given time and Eulerian position
    virtual void get_base_flow_u(const double& time,
                                 const unsigned& ipt,
                                 const Vector<double>& x,
                                 Vector<double>& result) const
    {
      // If the function pointer is zero return zero
      if (Base_flow_u_fct_pt == 0)
      {
        // Loop over velocity components and set base flow solution to zero
        for (unsigned i = 0; i < 3; i++)
        {
          result[i] = 0.0;
        }
      }
      // Otherwise call the function
      else
      {
        (*Base_flow_u_fct_pt)(time, x, result);
      }
    }
 
    /// Calculate the derivatives of the velocity components of the
    /// base flow solution w.r.t. global coordinates (r and z) at a given
    /// time and Eulerian position
    virtual void get_base_flow_dudx(const double& time,
                                    const unsigned& ipt,
                                    const Vector<double>& x,
                                    DenseMatrix<double>& result) const
    {
      // If the function pointer is zero return zero
      if (Base_flow_dudx_fct_pt == 0)
      {
        // Loop over velocity components
        for (unsigned i = 0; i < 3; i++)
        {
          // Loop over coordinate directions and set to zero
          for (unsigned j = 0; j < 2; j++)
          {
            result(i, j) = 0.0;
          }
        }
      }
      // Otherwise call the function
      else
      {
        (*Base_flow_dudx_fct_pt)(time, x, result);
      }
    }
 
    /// 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_linearised_axi_nst(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      DenseMatrix<double>& mass_matrix,
      unsigned flag);
 
  public:
    /// Constructor: NULL the base flow solution and the
    /// derivatives of the base flow function
    LinearisedAxisymmetricNavierStokesEquations()
      : Base_flow_u_fct_pt(0), Base_flow_dudx_fct_pt(0), ALE_is_disabled(false)
    {
      // Set all the physical parameter pointers to the default value of zero
      Re_pt = &Default_Physical_Constant_Value;
      ReSt_pt = &Default_Physical_Constant_Value;
 
      // Set the azimuthal mode number to default (zero)
      Azimuthal_Mode_Number_pt = &Default_Azimuthal_Mode_Number_Value;
 
      // Set the physical ratios to the default value of one
      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;
    }
 
    /// Azimuthal mode number k in e^ik(theta) decomposition
    const int& azimuthal_mode_number() const
    {
      return *Azimuthal_Mode_Number_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;
    }
 
    /// Pointer to azimuthal mode number k in e^ik(theta) decomposition
    int*& azimuthal_mode_number_pt()
    {
      return Azimuthal_Mode_Number_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 the 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 the density ratio
    double*& density_ratio_pt()
    {
      return Density_Ratio_pt;
    }
 
    /// Access function for the base flow solution pointer
    void (*&base_flow_u_fct_pt())(const double& time,
                                  const Vector<double>& x,
                                  Vector<double>& f)
    {
      return Base_flow_u_fct_pt;
    }
 
    /// Access function for the derivatives of the base flow
    /// w.r.t. global coordinates solution pointer
    void (*&base_flow_dudx_fct_pt())(const double& time,
                                     const Vector<double>& x,
                                     DenseMatrix<double>& f)
    {
      return Base_flow_dudx_fct_pt;
    }
 
    /// Return the number of pressure degrees of freedom
    /// associated with a single pressure component in the element
    virtual unsigned npres_linearised_axi_nst() 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_linearised_axi_nst(const unsigned& i) const
    {
      return i;
    }
 
    /// Return the i-th component of du/dt at local node n.
    /// Uses suitably interpolated value for hanging nodes.
    double du_dt_linearised_axi_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())
      {
        // Get the index at which the velocity is stored
        const unsigned u_nodal_index = u_index_linearised_axi_nst(i);
 
        // Determine number of timsteps (past & present)
        const unsigned n_time = time_stepper_pt->ntstorage();
 
        // Add the contributions to the time derivative
        for (unsigned t = 0; t < n_time; t++)
        {
          dudt +=
            time_stepper_pt->weight(1, t) * nodal_value(t, n, u_nodal_index);
        }
      }
 
      return dudt;
    }
 
    /// Disable ALE, i.e. assert the mesh is not moving -- you do this
    /// at your own risk!
    void disable_ALE()
    {
      ALE_is_disabled = true;
    }
 
    /// (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;
    }
 
    /// Return the i-th pressure value at local pressure "node" n_p.
    /// Uses suitably interpolated value for hanging nodes.
    virtual double p_linearised_axi_nst(const unsigned& n_p,
                                        const unsigned& i) const = 0;
 
    /// Which nodal value represents the pressure?
    //  N.B. This function has return type "int" (rather than "unsigned"
    //  as in the u_index case) so that we can return the "magic" number
    //  "Pressure_not_stored_at_node" ( = -100 )
    virtual inline int p_index_linearised_axi_nst(const unsigned& i) const
    {
      return Pressure_not_stored_at_node;
    }
 
    /// Strain-rate tensor: \f$ e_{ij} \f$
    /// where \f$ i,j = r,z,\theta \f$ (in that order)
    void strain_rate(const Vector<double>& s,
                     DenseMatrix<double>& strain_rate) const;
 
    /// Output function: r, z, U^C, U^S, V^C, V^S, W^C, W^S, P^C, P^S
    /// in tecplot format. Default number of plot points
    void output(std::ostream& outfile)
    {
      const unsigned nplot = 5;
      output(outfile, nplot);
    }
 
    /// Output function: r, z, U^C, U^S, V^C, V^S, W^C, W^S, P^C, P^S
    /// in tecplot format. nplot points in each coordinate direction
    void output(std::ostream& outfile, const unsigned& nplot);
 
    /// Output function: r, z, U^C, U^S, V^C, V^S, W^C, W^S, P^C, P^S
    /// in tecplot format. Default number of plot points
    void output(FILE* file_pt)
    {
      const unsigned nplot = 5;
      output(file_pt, nplot);
    }
 
    /// Output function: r, z, U^C, U^S, V^C, V^S, W^C, W^S, P^C, P^S
    /// in tecplot format. nplot points in each coordinate direction
    void output(FILE* file_pt, const unsigned& nplot);
 
    /// Output function: r, z, U^C, U^S, V^C, V^S, W^C, W^S,
    /// 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);
 
    /// 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_linearised_axi_nst(
        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_linearised_axi_nst(
        residuals, jacobian, GeneralisedElement::Dummy_matrix, 1);
    }
 
    /// 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_linearised_axi_nst(
        residuals, jacobian, mass_matrix, 2);
    }
 
    /// Return the i-th component of the FE interpolated velocity
    /// u[i] at local coordinate s
    double interpolated_u_linearised_axi_nst(const Vector<double>& s,
                                             const unsigned& i) const
    {
      // Determine number of nodes in the element
      const unsigned n_node = nnode();
 
      // Provide storage for local shape functions
      Shape psi(n_node);
 
      // Find values of shape functions
      shape(s, psi);
 
      // Get the index at which the velocity is stored
      const unsigned u_nodal_index = u_index_linearised_axi_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);
    }
 
    /// Return the i-th component of the FE interpolated pressure
    /// p[i] at local coordinate s
    double interpolated_p_linearised_axi_nst(const Vector<double>& s,
                                             const unsigned& i) const
    {
      // Determine number of pressure nodes in the element
      const unsigned n_pressure_nodes = npres_linearised_axi_nst();
 
      // Provide storage for local shape functions
      Shape psi(n_pressure_nodes);
 
      // Find values of shape functions
      pshape_linearised_axi_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_pressure_nodes; l++)
      {
        // N.B. The pure virtual function p_linearised_axi_nst(...)
        // automatically calculates the index at which the pressure value
        // is stored, so we don't need to worry about this here
        interpolated_p += p_linearised_axi_nst(l, i) * psi[l];
      }
 
      return (interpolated_p);
    }
 
  }; // End of LinearisedAxisymmetricNavierStokesEquations class definition
 
 
  //////////////////////////////////////////////////////////////////////////////
  //////////////////////////////////////////////////////////////////////////////
  //////////////////////////////////////////////////////////////////////////////
 
 
  //=======================================================================
  /// Crouzeix-Raviart elements are Navier-Stokes elements with quadratic
  /// interpolation for velocities and positions, but a discontinuous
  /// linear pressure interpolation
  //=======================================================================
  class LinearisedAxisymmetricQCrouzeixRaviartElement
    : public virtual QElement<2, 3>,
      public virtual LinearisedAxisymmetricNavierStokesEquations
  {
  private:
    /// Static array of ints to hold required number of variables at nodes
    static const unsigned Initial_Nvalue[];
 
  protected:
    /// Internal indices that indicate at which internal data the
    /// pressure values are stored. We note that there are two pressure
    /// values, corresponding to the functions P^C(r,z,t) and P^S(r,z,t)
    /// which multiply the cosine and sine terms respectively.
    Vector<unsigned> P_linearised_axi_nst_internal_index;
 
    /// Velocity shape and test functions and their derivatives
    /// w.r.t. global coordinates at local coordinate s (taken from geometry).
    /// Return Jacobian of mapping between local and global coordinates.
    inline double dshape_and_dtest_eulerian_linearised_axi_nst(
      const Vector<double>& s,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const;
 
    /// Velocity shape and test functions and their derivatives
    /// w.r.t. global coordinates at the ipt-th integation point
    /// (taken from geometry).
    /// Return Jacobian of mapping between local and global coordinates.
    inline double dshape_and_dtest_eulerian_at_knot_linearised_axi_nst(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const;
 
    /// Compute the pressure shape functions at local coordinate s
    inline void pshape_linearised_axi_nst(const Vector<double>& s,
                                          Shape& psi) const;
 
    /// Compute the pressure shape and test functions at local coordinate s
    inline void pshape_linearised_axi_nst(const Vector<double>& s,
                                          Shape& psi,
                                          Shape& test) const;
 
  public:
    /// Constructor: there are three internal values for each
    /// of the two pressure components
    LinearisedAxisymmetricQCrouzeixRaviartElement()
      : QElement<2, 3>(),
        LinearisedAxisymmetricNavierStokesEquations(),
        P_linearised_axi_nst_internal_index(2)
    {
      // Loop over the two pressure components
      for (unsigned i = 0; i < 2; i++)
      {
        // Allocate and add one internal data object for each of the two
        // pressure components that store the three pressure values
        P_linearised_axi_nst_internal_index[i] =
          this->add_internal_data(new Data(3));
      }
    }
 
    /// Return number of values (pinned or dofs) required at local node n
    virtual unsigned required_nvalue(const unsigned& n) const;
 
    /// Return the pressure value i at internal dof i_internal
    /// (Discontinous pressure interpolation -- no need to cater for
    /// hanging nodes)
    double p_linearised_axi_nst(const unsigned& i_internal,
                                const unsigned& i) const
    {
      return internal_data_pt(P_linearised_axi_nst_internal_index[i])
        ->value(i_internal);
    }
 
    /// Return number of pressure values corresponding to a
    /// single pressure component
    unsigned npres_linearised_axi_nst() const
    {
      return 3;
    }
 
    /// Fix both components of the internal pressure degrees
    /// of freedom p_dof to pvalue
    void fix_pressure(const unsigned& p_dof, const double& pvalue)
    {
      // Loop over the two pressure components
      for (unsigned i = 0; i < 2; i++)
      {
        this->internal_data_pt(P_linearised_axi_nst_internal_index[i])
          ->pin(p_dof);
        internal_data_pt(P_linearised_axi_nst_internal_index[i])
          ->set_value(p_dof, pvalue);
      }
    }
 
    /// Overload the access function for the i-th component of the
    /// pressure's local equation numbers
    inline int p_local_eqn(const unsigned& n, const unsigned& i)
    {
      return internal_local_eqn(P_linearised_axi_nst_internal_index[i], n);
    }
 
    /// Redirect output to NavierStokesEquations output
    void output(std::ostream& outfile)
    {
      LinearisedAxisymmetricNavierStokesEquations::output(outfile);
    }
 
    /// Redirect output to NavierStokesEquations output
    void output(std::ostream& outfile, const unsigned& n_plot)
    {
      LinearisedAxisymmetricNavierStokesEquations::output(outfile, n_plot);
    }
 
    /// Redirect output to NavierStokesEquations output
    void output(FILE* file_pt)
    {
      LinearisedAxisymmetricNavierStokesEquations::output(file_pt);
    }
 
    /// Redirect output to NavierStokesEquations output
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      LinearisedAxisymmetricNavierStokesEquations::output(file_pt, n_plot);
    }
 
    /// The number of "dof-blocks" that degrees of freedom in this
    /// element are sub-divided into: Velocity and pressure.
    unsigned ndof_types() const
    {
      return 8;
    }
 
  }; // End of LinearisedAxisymmetricQCrouzeixRaviartElement class definition
 
 
  // Inline functions
  // ----------------
 
  //=======================================================================
  /// Derivatives of the shape functions and test functions w.r.t.
  /// global (Eulerian) coordinates at local coordinate s.
  /// Return Jacobian of mapping between local and global coordinates.
  //=======================================================================
  inline double LinearisedAxisymmetricQCrouzeixRaviartElement::
    dshape_and_dtest_eulerian_linearised_axi_nst(const Vector<double>& s,
                                                 Shape& psi,
                                                 DShape& dpsidx,
                                                 Shape& test,
                                                 DShape& dtestdx) const
  {
    // Call the geometrical shape functions and derivatives
    const double J = this->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;
  }
 
  //=======================================================================
  /// Derivatives of the shape functions and test functions w.r.t.
  /// global (Eulerian) coordinates at the ipt-th integration point.
  /// Return Jacobian of mapping between local and global coordinates.
  //=======================================================================
  inline double LinearisedAxisymmetricQCrouzeixRaviartElement::
    dshape_and_dtest_eulerian_at_knot_linearised_axi_nst(const unsigned& ipt,
                                                         Shape& psi,
                                                         DShape& dpsidx,
                                                         Shape& test,
                                                         DShape& dtestdx) const
  {
    // Call the geometrical shape functions and derivatives
    const double J = this->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;
  }
 
  //=======================================================================
  /// Pressure shape functions
  //=======================================================================
  inline void LinearisedAxisymmetricQCrouzeixRaviartElement::
    pshape_linearised_axi_nst(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 LinearisedAxisymmetricQCrouzeixRaviartElement::
    pshape_linearised_axi_nst(const Vector<double>& s,
                              Shape& psi,
                              Shape& test) const
  {
    // Call the pressure shape functions
    pshape_linearised_axi_nst(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 linearised axisym Crouzeix-Raviart elements
  //=======================================================================
  template<>
  class FaceGeometry<LinearisedAxisymmetricQCrouzeixRaviartElement>
    : public virtual QElement<1, 3>
  {
  public:
    FaceGeometry() : QElement<1, 3>() {}
  };
 
  //=======================================================================
  /// Face geometry of face geometry of the linearised axisymmetric
  /// Crouzeix Raviart elements
  //=======================================================================
  template<>
  class FaceGeometry<
    FaceGeometry<LinearisedAxisymmetricQCrouzeixRaviartElement>>
    : public virtual PointElement
  {
  public:
    FaceGeometry() : PointElement() {}
  };
 
 
  //////////////////////////////////////////////////////////////////////////////
  //////////////////////////////////////////////////////////////////////////////
  //////////////////////////////////////////////////////////////////////////////
 
 
  //=======================================================================
  /// Taylor--Hood elements are Navier--Stokes elements with quadratic
  /// interpolation for velocities and positions and continuous linear
  /// pressure interpolation
  //=======================================================================
  class LinearisedAxisymmetricQTaylorHoodElement
    : public virtual QElement<2, 3>,
      public virtual LinearisedAxisymmetricNavierStokesEquations
  {
  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 derivatives
    /// w.r.t. global coordinates  at local coordinate s (taken from geometry).
    /// Return Jacobian of mapping between local and global coordinates.
    inline double dshape_and_dtest_eulerian_linearised_axi_nst(
      const Vector<double>& s,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const;
 
    /// Velocity shape and test functions and their derivatives
    /// w.r.t. global coordinates the ipt-th integation point
    /// (taken from geometry).
    /// Return Jacobian of mapping between local and global coordinates.
    inline double dshape_and_dtest_eulerian_at_knot_linearised_axi_nst(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const;
 
    /// Compute the pressure shape functions at local coordinate s
    inline void pshape_linearised_axi_nst(const Vector<double>& s,
                                          Shape& psi) const;
 
    /// Compute the pressure shape and test functions at local coordinte s
    inline void pshape_linearised_axi_nst(const Vector<double>& s,
                                          Shape& psi,
                                          Shape& test) const;
 
  public:
    /// Constructor, no internal data points
    LinearisedAxisymmetricQTaylorHoodElement()
      : QElement<2, 3>(), LinearisedAxisymmetricNavierStokesEquations()
    {
    }
 
    /// 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? Overload version
    /// in base class which returns static int "Pressure_not_stored_at_node"
    virtual int p_index_linearised_axi_nst(const unsigned& i) const
    {
      return (6 + i);
    }
 
    /// Access function for the i-th component of pressure
    /// at local pressure node n_p (const version).
    double p_linearised_axi_nst(const unsigned& n_p, const unsigned& i) const
    {
      return nodal_value(Pconv[n_p], p_index_linearised_axi_nst(i));
    }
 
    /// Return number of pressure values corresponding to a
    /// single pressure component
    unsigned npres_linearised_axi_nst() const
    {
      return 4;
    }
 
    /// Fix both components of the pressure at local pressure
    /// node n_p to pvalue
    void fix_pressure(const unsigned& n_p, const double& pvalue)
    {
      // Loop over the two pressure components
      for (unsigned i = 0; i < 2; i++)
      {
        this->node_pt(Pconv[n_p])->pin(p_index_linearised_axi_nst(i));
        this->node_pt(Pconv[n_p])
          ->set_value(p_index_linearised_axi_nst(i), pvalue);
      }
    }
 
    /// Overload the access function for the i-th component of the
    /// pressure's local equation numbers
    inline int p_local_eqn(const unsigned& n, const unsigned& i)
    {
      return nodal_local_eqn(Pconv[n], p_index_linearised_axi_nst(i));
    }
 
    /// Redirect output to NavierStokesEquations output
    void output(std::ostream& outfile)
    {
      LinearisedAxisymmetricNavierStokesEquations::output(outfile);
    }
 
    /// Redirect output to NavierStokesEquations output
    void output(std::ostream& outfile, const unsigned& n_plot)
    {
      LinearisedAxisymmetricNavierStokesEquations::output(outfile, n_plot);
    }
 
    /// Redirect output to NavierStokesEquations output
    void output(FILE* file_pt)
    {
      LinearisedAxisymmetricNavierStokesEquations::output(file_pt);
    }
 
    /// Redirect output to NavierStokesEquations output
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      LinearisedAxisymmetricNavierStokesEquations::output(file_pt, n_plot);
    }
 
    /// Returns the number of "dof-blocks" that degrees of freedom
    /// in this element are sub-divided into: Velocity and pressure.
    unsigned ndof_types() const
    {
      return 8;
    }
 
  }; // End of LinearisedAxisymmetricQTaylorHoodElement class definition
 
 
  // Inline functions
  // ----------------
 
  //=======================================================================
  /// Derivatives of the shape functions and test functions w.r.t
  /// global (Eulerian) coordinates at local coordinate s.
  /// Return Jacobian of mapping between local and global coordinates.
  //=======================================================================
  inline double LinearisedAxisymmetricQTaylorHoodElement::
    dshape_and_dtest_eulerian_linearised_axi_nst(const Vector<double>& s,
                                                 Shape& psi,
                                                 DShape& dpsidx,
                                                 Shape& test,
                                                 DShape& dtestdx) const
  {
    // Call the geometrical shape functions and derivatives
    const double J = this->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;
  }
 
  //=======================================================================
  /// Derivatives of the shape functions and test functions w.r.t
  /// global (Eulerian) coordinates at the ipt-th integration point.
  /// Return Jacobian of mapping between local and global coordinates.
  //=======================================================================
  inline double LinearisedAxisymmetricQTaylorHoodElement::
    dshape_and_dtest_eulerian_at_knot_linearised_axi_nst(const unsigned& ipt,
                                                         Shape& psi,
                                                         DShape& dpsidx,
                                                         Shape& test,
                                                         DShape& dtestdx) const
  {
    // Call the geometrical shape functions and derivatives
    const double J = this->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;
  }
 
  //=======================================================================
  /// Pressure shape functions
  //=======================================================================
  inline void LinearisedAxisymmetricQTaylorHoodElement::
    pshape_linearised_axi_nst(const Vector<double>& s, Shape& psi) const
  {
    // Allocate local storage for the pressure shape functions
    double psi1[2], psi2[2];
 
    // Call the one-dimensional 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 "r" coordinate, s2 the "z"
    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];
      }
    }
  }
 
  //=======================================================================
  /// Pressure shape and test functions
  //=======================================================================
  inline void LinearisedAxisymmetricQTaylorHoodElement::
    pshape_linearised_axi_nst(const Vector<double>& s,
                              Shape& psi,
                              Shape& test) const
  {
    // Call the pressure shape functions
    pshape_linearised_axi_nst(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 linearised axisymmetric Taylor Hood elements
  //=======================================================================
  template<>
  class FaceGeometry<LinearisedAxisymmetricQTaylorHoodElement>
    : public virtual QElement<1, 3>
  {
  public:
    FaceGeometry() : QElement<1, 3>() {}
  };
 
  //=======================================================================
  /// Face geometry of the face geometry of the linearised
  /// axisymmetric Taylor Hood elements
  //=======================================================================
  template<>
  class FaceGeometry<FaceGeometry<LinearisedAxisymmetricQTaylorHoodElement>>
    : public virtual PointElement
  {
  public:
    FaceGeometry() : PointElement() {}
  };
 
 
} // namespace oomph
 
#endif