spherical_navier_stokes_elements.h

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// LIC// ====================================================================
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// LIC// multi-physics finite-element library, available
// LIC// at http://www.oomph-lib.org.
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// LIC// Copyright (C) 2006-2025 Matthias Heil and Andrew Hazel
// LIC//
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// Header file for Navier Stokes elements
 
#ifndef OOMPH_SPHERICAL_NAVIER_STOKES_ELEMENTS_HEADER
#define OOMPH_SPHERICAL_NAVIER_STOKES_ELEMENTS_HEADER
 
// Config header
#ifdef HAVE_CONFIG_H
#include <oomph-lib-config.h>
#endif
 
 
// OOMPH-LIB headers
#include "generic/Qelements.h"
#include "generic/fsi.h"
// #include "generic/block_preconditioner.h"
 
namespace oomph
{
  //======================================================================
  /// A class for elements that solve the Navier--Stokes equations,
  /// in axisymmetric spherical polar coordinates.
  /// This contains the generic maths -- any concrete implementation must
  /// be derived from this.
  ///
  /// We also provide all functions required to use this element
  /// in FSI problems, by deriving it from the FSIFluidElement base
  /// class.
  //======================================================================
  class SphericalNavierStokesEquations
    : public virtual FSIFluidElement,
      public virtual NavierStokesElementWithDiagonalMassMatrices
  {
  public:
    /// Function pointer to body force function fct(t,x,f(x))
    /// x is a Vector!
    typedef void (*SphericalNavierStokesBodyForceFctPt)(
      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 (*SphericalNavierStokesSourceFctPt)(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 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 Reynolds number x inverse Rossby number
    /// (used when in a rotating frame)
    double* ReInvRo_pt;
 
    /// Pointer to global gravity Vector
    Vector<double>* G_pt;
 
    /// Pointer to body force function
    SphericalNavierStokesBodyForceFctPt Body_force_fct_pt;
 
    /// Pointer to volumetric source function
    SphericalNavierStokesSourceFctPt Source_fct_pt;
 
    /// 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;
 
    /// 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;
 
    /// 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_spherical_nst(
      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_spherical_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_spherical_nst(const Vector<double>& s,
                                      Shape& psi) const = 0;
 
    /// Compute the pressure shape and test functions
    /// at local coordinate s
    virtual void pshape_spherical_nst(const Vector<double>& s,
                                      Shape& psi,
                                      Shape& test) const = 0;
 
 
    /// 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_spherical_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 three spatial dimensions and set body forces to zero
        for (unsigned i = 0; i < 3; 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
    virtual double get_source_spherical_nst(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);
      }
    }
 
    /// 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_spherical_nst(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      DenseMatrix<double>& mass_matrix,
      unsigned flag);
 
 
  public:
    /// Include a cot function to simplify the
    /// NS momentum and jacobian expressions
    inline double cot(const double& th) const
    {
      return (1 / tan(th));
    }
 
 
    /// Constructor: NULL the body force and source function
    /// and make sure the ALE terms are included by default.
    SphericalNavierStokesEquations()
      : Body_force_fct_pt(0), Source_fct_pt(0), 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;
      ReInvRo_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;
    }
 
    /// 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;
    }
 
    /// Global Reynolds number multiplied by inverse Rossby number
    const double& re_invro() const
    {
      return *ReInvRo_pt;
    }
 
    /// Pointer to global inverse Froude number
    double*& re_invro_pt()
    {
      return ReInvRo_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
    SphericalNavierStokesBodyForceFctPt& body_force_fct_pt()
    {
      return Body_force_fct_pt;
    }
 
    /// Access function for the body-force pointer. Const version
    SphericalNavierStokesBodyForceFctPt body_force_fct_pt() const
    {
      return Body_force_fct_pt;
    }
 
    /// Access function for the source-function pointer
    SphericalNavierStokesSourceFctPt& source_fct_pt()
    {
      return Source_fct_pt;
    }
 
    /// Access function for the source-function pointer. Const version
    SphericalNavierStokesSourceFctPt source_fct_pt() const
    {
      return Source_fct_pt;
    }
 
    /// Function to return number of pressure degrees of freedom
    virtual unsigned npres_spherical_nst() const = 0;
 
    /// Velocity i at local node n. Uses suitably interpolated value
    /// for hanging nodes.
    /// The use of u_index_spherical_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_spherical_nst(const unsigned& n, const unsigned& i) const
    {
      return nodal_value(n, u_index_spherical_nst(i));
    }
 
    /// Velocity i at local node n at timestep t (t=0: present;
    /// t>0: previous). Uses suitably interpolated value for hanging nodes.
    double u_spherical_nst(const unsigned& t,
                           const unsigned& n,
                           const unsigned& i) const
    {
      return nodal_value(t, n, u_index_spherical_nst(i));
    }
 
    /// 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_spherical_nst(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_spherical_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->type() != "Steady")
      {
        // Find the index at which the dof is stored
        const unsigned u_nodal_index = this->u_index_spherical_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;
    }
 
    /// 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;
    }
 
    /// Pressure at local pressure "node" n_p
    /// Uses suitably interpolated value for hanging nodes.
    virtual double p_spherical_nst(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;
 
    /// 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_spherical_nst() const
    {
      return Pressure_not_stored_at_node;
    }
 
    /// 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;
 
    /// Compute the vorticity vector at local coordinate s
    void get_vorticity(const Vector<double>& s,
                       Vector<double>& vorticity) const;
 
    /// Get integral of kinetic energy over element
    double kin_energy() const;
 
    /// 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;
 
    /// 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);
 
 
    /// 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);
    }
 
    /// 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)
    /// NOTE: pressure versions isn't implemented yet because this
    ///       element has never been tried with Fp preconditoner.
    void get_pressure_and_velocity_mass_matrix_diagonal(
      Vector<double>& press_mass_diag,
      Vector<double>& veloc_mass_diag,
      const unsigned& which_one = 0);
 
 
    /// 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);
    }
 
    /// 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 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);
 
    /// 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);
 
    /// Validate against exact solution.
    /// Solution is provided direct from exact_soln function.
    /// Plot at a given number of plot points and compute the energy error
    /// and energy norm of the velocity solution over the element.
    void compute_error_e(
      std::ostream& outfile,
      FiniteElement::SteadyExactSolutionFctPt exact_soln_pt,
      FiniteElement::SteadyExactSolutionFctPt exact_soln_dr_pt,
      FiniteElement::SteadyExactSolutionFctPt exact_soln_dtheta_pt,
      double& u_error,
      double& u_norm,
      double& p_error,
      double& p_norm);
 
    // Listing for the shear stress function
    void compute_shear_stress(std::ostream& outfile);
 
    // Listing for the velocity extraction function
    void extract_velocity(std::ostream& outfile);
 
    // Calculate the analytic solution at the point x and output a vector
    Vector<double> actual(const Vector<double>& x)
    {
      const double Re = this->re();
 
      double r = x[0];
      double theta = x[1];
      Vector<double> ans(4, 0.0);
 
      ans[2] = r * sin(theta);
      ans[3] = 0.5 * Re * r * r * sin(theta) * sin(theta);
 
      return (ans);
    }
 
    // Calculate the r-derivatives of the analytic solution at the point x and
    // output a vector
    Vector<double> actual_dr(const Vector<double>& x)
    {
      const double Re = this->re();
 
      double r = x[0];
      double theta = x[1];
      Vector<double> ans(4, 0.0);
 
      ans[2] = sin(theta);
      ans[3] = Re * r * sin(theta) * sin(theta);
 
      return (ans);
    }
 
    // Calculate the theta-derivatives of the analytic solution at the point x
    // and output a vector
    Vector<double> actual_dth(const Vector<double>& x)
    {
      const double Re = this->re();
 
      double r = x[0];
      double theta = x[1];
      Vector<double> ans(4, 0.0);
 
      ans[2] = r * cos(theta);
      ans[3] = Re * r * r * sin(theta) * cos(theta);
 
      return (ans);
    }
 
    /// 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_spherical_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_spherical_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_spherical_nst(
        residuals, jacobian, mass_matrix, 2);
    }
 
    /// Compute vector of FE interpolated velocity u at local coordinate s
    void interpolated_u_spherical_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 < 3; i++)
      {
        // Index at which the nodal value is stored
        unsigned u_nodal_index = u_index_spherical_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_spherical_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_spherical_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 matrix entry dudx(i,j) of the FE interpolated velocity derivative
    /// at local coordinate s
    double interpolated_dudx_spherical_nst(const Vector<double>& s,
                                           const unsigned& i,
                                           const unsigned& j) const
    {
      // Find number of nodes
      unsigned n_node = nnode();
      // Local shape function
      Shape psi(n_node);
      DShape dpsidx(n_node, 2);
      // Find values of shape function
      (void)dshape_eulerian(s, psi, dpsidx);
 
      // Get nodal index at which i-th velocity is stored
      unsigned u_nodal_index = u_index_spherical_nst(i);
 
      // Initialise value of u
      double interpolated_dudx = 0.0;
      // Loop over the local nodes and sum
      for (unsigned l = 0; l < n_node; l++)
      {
        interpolated_dudx += nodal_value(l, u_nodal_index) * dpsidx(l, j);
      }
 
      return (interpolated_dudx);
    }
 
    /// Return FE interpolated pressure at local coordinate s
    double interpolated_p_spherical_nst(const Vector<double>& s) const
    {
      // Find number of nodes
      unsigned n_pres = npres_spherical_nst();
      // Local shape function
      Shape psi(n_pres);
      // Find values of shape function
      pshape_spherical_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_spherical_nst(l) * psi[l];
      }
 
      return (interpolated_p);
    }
  };
 
  //////////////////////////////////////////////////////////////////////////////
  //////////////////////////////////////////////////////////////////////////////
  //////////////////////////////////////////////////////////////////////////////
 
 
  //==========================================================================
  /// 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.
  //==========================================================================
  class QSphericalCrouzeixRaviartElement
    : public virtual QElement<2, 3>,
      public virtual SphericalNavierStokesEquations
  {
  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_spherical_nst_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_spherical_nst(
      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_spherical_nst(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const;
 
    /// Pressure shape functions at local coordinate s
    inline void pshape_spherical_nst(const Vector<double>& s, Shape& psi) const;
 
    /// Pressure shape and test functions at local coordinte s
    inline void pshape_spherical_nst(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) const
    {
      return this->internal_local_eqn(P_spherical_nst_internal_index, n);
    }
 
  public:
    /// Constructor, there are 3 internal values (for the pressure)
    QSphericalCrouzeixRaviartElement()
      : QElement<2, 3>(), SphericalNavierStokesEquations()
    {
      // Allocate and add one Internal data object that stored 2+1 pressure
      // values;
      P_spherical_nst_internal_index = this->add_internal_data(new Data(3));
    }
 
    /// Number of values (pinned or dofs) required at local node n.
    inline unsigned required_nvalue(const unsigned& n) const
    {
      return 3;
    }
 
 
    /// Return the i-th pressure value
    /// (Discontinous pressure interpolation -- no need to cater for hanging
    /// nodes).
    double p_spherical_nst(const unsigned& i) const
    {
      return this->internal_data_pt(P_spherical_nst_internal_index)->value(i);
    }
 
    /// Return number of pressure values
    unsigned npres_spherical_nst() const
    {
      return 3;
    }
 
    /// Pin p_dof-th pressure dof and set it to value specified by p_value.
    void fix_pressure(const unsigned& p_dof, const double& p_value)
    {
      this->internal_data_pt(P_spherical_nst_internal_index)->pin(p_dof);
      this->internal_data_pt(P_spherical_nst_internal_index)
        ->set_value(p_dof, p_value);
    }
 
    ///  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);
 
    ///  Add to the set \c paired_pressure_data pairs containing
    /// - the pointer to a Data object
    /// and
    /// - the index of the value in that Data object
    /// .
    /// for pressure values that affect the
    /// load computed in the \c get_load(...) function.
    void identify_pressure_data(
      std::set<std::pair<Data*, unsigned>>& paired_pressure_data);
 
    /// Redirect output to SphericalNavierStokesEquations output
    void output(std::ostream& outfile)
    {
      SphericalNavierStokesEquations::output(outfile);
    }
 
    /// Redirect output to SphericalNavierStokesEquations output
    void output(std::ostream& outfile, const unsigned& nplot)
    {
      SphericalNavierStokesEquations::output(outfile, nplot);
    }
 
 
    /// Redirect output to SphericalNavierStokesEquations output
    void output(FILE* file_pt)
    {
      SphericalNavierStokesEquations::output(file_pt);
    }
 
    /// Redirect output to SphericalNavierStokesEquations output
    void output(FILE* file_pt, const unsigned& nplot)
    {
      SphericalNavierStokesEquations::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)
    {
      SphericalNavierStokesEquations::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)
    {
      SphericalNavierStokesEquations::full_output(outfile, nplot);
    }
 
 
    /// The number of "DOF types" that degrees of freedom in this element
    /// are sub-divided into: Velocity (three comp) and pressure.
    unsigned ndof_types() const
    {
      return 4;
    }
 
    /// Create a list of pairs for all unknowns in this element,
    /// so that the first entry in each pair contains the global equation
    /// number of the unknown, while the second one contains the number
    /// of the "DOF type" that this unknown is associated with.
    /// (Function can obviously only be called if the equation numbering
    /// scheme has been set up.)
    void get_dof_numbers_for_unknowns(
      std::list<std::pair<unsigned long, unsigned>>& dof_lookup_list) const;
  };
 
  // Inline functions
 
  //=======================================================================
  /// Derivatives of the shape functions and test functions w.r.t. to global
  /// (Eulerian) coordinates. Return Jacobian of mapping between
  /// local and global coordinates.
  //=======================================================================
  inline double QSphericalCrouzeixRaviartElement::
    dshape_and_dtest_eulerian_spherical_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);
    // 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. to global
  /// (Eulerian) coordinates. Return Jacobian of mapping between
  /// local and global coordinates.
  //=======================================================================
  inline double QSphericalCrouzeixRaviartElement::
    dshape_and_dtest_eulerian_at_knot_spherical_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);
    // 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 QSphericalCrouzeixRaviartElement::pshape_spherical_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 QSphericalCrouzeixRaviartElement::pshape_spherical_nst(
    const Vector<double>& s, Shape& psi, Shape& test) const
  {
    // Call the pressure shape functions
    pshape_spherical_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 Spherical Crouzeix_Raviart elements
  //=======================================================================
  template<>
  class FaceGeometry<QSphericalCrouzeixRaviartElement>
    : public virtual QElement<1, 3>
  {
  public:
    FaceGeometry() : QElement<1, 3>() {}
  };
 
  //=======================================================================
  /// Face geometry of the FaceGeometry of the Spherical
  /// Crouzeix_Raviart elements
  //=======================================================================
  template<>
  class FaceGeometry<FaceGeometry<QSphericalCrouzeixRaviartElement>>
    : public virtual PointElement
  {
  public:
    FaceGeometry() : PointElement() {}
  };
 
 
  ////////////////////////////////////////////////////////////////////////////
  ////////////////////////////////////////////////////////////////////////////
 
 
  //=======================================================================
  /// Taylor--Hood elements are Navier--Stokes elements
  /// with quadratic interpolation for velocities and positions and
  /// continous linear pressure interpolation. They can be used
  /// within oomph-lib's block-preconditioning framework.
  //=======================================================================
  class QSphericalTaylorHoodElement
    : public virtual QElement<2, 3>,
      public virtual SphericalNavierStokesEquations
  {
  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_spherical_nst(
      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_spherical_nst(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const;
 
    /// Pressure shape functions at local coordinate s
    inline void pshape_spherical_nst(const Vector<double>& s, Shape& psi) const;
 
    /// Pressure shape and test functions at local coordinte s
    inline void pshape_spherical_nst(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) const
    {
      return this->nodal_local_eqn(Pconv[n], p_nodal_index_spherical_nst());
    }
 
  public:
    /// Constructor, no internal data points
    QSphericalTaylorHoodElement()
      : QElement<2, 3>(), SphericalNavierStokesEquations()
    {
    }
 
    /// 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];
    }
 
    /// Set the value at which the pressure is stored in the nodes
    /// In this case the third index because there are three velocity components
    int p_nodal_index_spherical_nst() const
    {
      return 3;
    }
 
    /// Access function for the pressure values at local pressure
    /// node n_p (const version)
    double p_spherical_nst(const unsigned& n_p) const
    {
      return this->nodal_value(Pconv[n_p], this->p_nodal_index_spherical_nst());
    }
 
    /// Return number of pressure values
    unsigned npres_spherical_nst() 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_spherical_nst());
      this->node_pt(Pconv[p_dof])
        ->set_value(this->p_nodal_index_spherical_nst(), p_value);
    }
 
 
    ///  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);
 
 
    ///  Add to the set \c paired_pressure_data pairs containing
    /// - the pointer to a Data object
    /// and
    /// - the index of the value in that Data object
    /// .
    /// for pressure values that affect the
    /// load computed in the \c get_load(...) function.
    void identify_pressure_data(
      std::set<std::pair<Data*, unsigned>>& paired_pressure_data);
 
    /// Redirect output to SphericalNavierStokesEquations output
    void output(std::ostream& outfile)
    {
      SphericalNavierStokesEquations::output(outfile);
    }
 
    /// Redirect output to SphericalNavierStokesEquations output
    void output(std::ostream& outfile, const unsigned& nplot)
    {
      SphericalNavierStokesEquations::output(outfile, nplot);
    }
 
    /// Redirect output to SphericalNavierStokesEquations output
    void output(FILE* file_pt)
    {
      SphericalNavierStokesEquations::output(file_pt);
    }
 
    /// Redirect output to SphericalNavierStokesEquations output
    void output(FILE* file_pt, const unsigned& nplot)
    {
      SphericalNavierStokesEquations::output(file_pt, nplot);
    }
 
 
    /// The number of "DOF types" that degrees of freedom in this element
    /// are sub-divided into: Velocity (3 components) and pressure.
    unsigned ndof_types() const
    {
      return 4;
    }
 
    /// Create a list of pairs for all unknowns in this element,
    /// so that the first entry in each pair contains the global equation
    /// number of the unknown, while the second one contains the number
    /// of the "Dof type" that this unknown is associated with.
    /// (Function can obviously only be called if the equation numbering
    /// scheme has been set up.)
    void get_dof_numbers_for_unknowns(
      std::list<std::pair<unsigned long, unsigned>>& dof_lookup_list) const;
  };
 
  // 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 QSphericalTaylorHoodElement::
    dshape_and_dtest_eulerian_spherical_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);
    // 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 to
  /// global (Eulerian) coordinates. Return Jacobian of mapping between
  /// local and global coordinates.
  //==========================================================================
  inline double QSphericalTaylorHoodElement::
    dshape_and_dtest_eulerian_at_knot_spherical_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);
    // 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 QSphericalTaylorHoodElement::pshape_spherical_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];
      }
    }
  }
 
 
  //==========================================================================
  /// Pressure shape and test functions
  //==========================================================================
  inline void QSphericalTaylorHoodElement::pshape_spherical_nst(
    const Vector<double>& s, Shape& psi, Shape& test) const
  {
    // Call the pressure shape functions
    pshape_spherical_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 Spherical Taylor_Hood elements
  //=======================================================================
  template<>
  class FaceGeometry<QSphericalTaylorHoodElement>
    : public virtual QElement<1, 3>
  {
  public:
    FaceGeometry() : QElement<1, 3>() {}
  };
 
  //=======================================================================
  /// Face geometry of the FaceGeometry of the 2D Taylor Hoodelements
  //=======================================================================
  template<>
  class FaceGeometry<FaceGeometry<QSphericalTaylorHoodElement>>
    : public virtual PointElement
  {
  public:
    FaceGeometry() : PointElement() {}
  };
 
 
} // namespace oomph
 
#endif