refineable_polar_navier_stokes_elements.h

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// LIC// ====================================================================
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// LIC//
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// Header file for refineable 2D Polar Navier Stokes elements
// Created (approx 12/03/08) by copying and combining oomph-lib's existing:
// - refineable_navier_stokes_elements.h
// - refineable_navier_stokes_elements.cc
 
// 13/06/08 - Weakform adjusted to "correct" traction version
 
#ifndef OOMPH_REFINEABLE_POLAR_NAVIER_STOKES_HEADER
#define OOMPH_REFINEABLE_POLAR_NAVIER_STOKES_HEADER
 
// Config header
#ifdef HAVE_CONFIG_H
#include <oomph-lib-config.h>
#endif
 
// Oomph-lib headers
// Should already be looking in build/include/ for generic.h
#include "generic/refineable_quad_element.h"
#include "generic/error_estimator.h"
#include "polar_navier_stokes_elements.h"
 
namespace oomph
{
  //======================================================================
  /// Refineable version of my Polar Navier--Stokes equations
  ///
  ///
  //======================================================================
  class RefineablePolarNavierStokesEquations
    : public virtual PolarNavierStokesEquations,
      public virtual RefineableElement,
      public virtual ElementWithZ2ErrorEstimator
  {
  protected:
    /// 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;
    }
 
    /// Unpin all pressure dofs in the element
    virtual void unpin_elemental_pressure_dofs() = 0;
 
    /// Pin unused nodal pressure dofs (empty by default, because
    /// by default pressure dofs are not associated with nodes)
    virtual void pin_elemental_redundant_nodal_pressure_dofs() {}
 
  public:
    /// Constructor
    RefineablePolarNavierStokesEquations()
      : PolarNavierStokesEquations(),
        RefineableElement(),
        ElementWithZ2ErrorEstimator()
    {
    }
 
 
    ///  Loop over all elements in Vector (which typically contains
    /// all the elements in a fluid mesh) and pin the nodal pressure degrees
    /// of freedom that are not being used. Function uses
    /// the member function
    /// - \c RefineablePolarNavierStokesEquations::
    ///      pin_elemental_redundant_nodal_pressure_dofs()
    /// .
    /// which is empty by default and should be implemented for
    /// elements with nodal pressure degrees of freedom
    /// (e.g. for refineable Taylor-Hood.)
    static void pin_redundant_nodal_pressures(
      const Vector<GeneralisedElement*>& element_pt)
    {
      // Loop over all elements and call the function that pins their
      // unused nodal pressure data
      unsigned n_element = element_pt.size();
      for (unsigned e = 0; e < n_element; e++)
      {
        dynamic_cast<RefineablePolarNavierStokesEquations*>(element_pt[e])
          ->pin_elemental_redundant_nodal_pressure_dofs();
      }
    }
 
    /// Unpin all pressure dofs in elements listed in vector.
    static void unpin_all_pressure_dofs(
      const Vector<GeneralisedElement*>& element_pt)
    {
      // Loop over all elements to brutally unpin all nodal pressure degrees of
      // freedom and internal pressure degrees of freedom
      unsigned n_element = element_pt.size();
      for (unsigned e = 0; e < n_element; e++)
      {
        dynamic_cast<RefineablePolarNavierStokesEquations*>(element_pt[e])
          ->unpin_elemental_pressure_dofs();
      }
    }
 
 
    /// Number of 'flux' terms for Z2 error estimation
    unsigned num_Z2_flux_terms()
    {
      // DIM diagonal strain rates, DIM(DIM -1) /2 off diagonal rates
      return 3;
    }
 
    /// Get 'flux' for Z2 error recovery:   Upper triangular entries
    /// in strain rate tensor.
    void get_Z2_flux(const Vector<double>& s, Vector<double>& flux)
    {
#ifdef PARANOID
      unsigned num_entries = 3;
      if (flux.size() != num_entries)
      {
        std::ostringstream error_message;
        error_message << "The flux vector has the wrong number of entries, "
                      << flux.size() << ", whereas it should be " << num_entries
                      << std::endl;
        throw OomphLibError(error_message.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // Get strain rate matrix
      DenseMatrix<double> strainrate(2);
      // this->strain_rate(s,strainrate);
      this->strain_rate_by_r(s, strainrate);
 
      // Pack into flux Vector
      unsigned icount = 0;
 
      // Start with diagonal terms
      for (unsigned i = 0; i < 2; i++)
      {
        flux[icount] = strainrate(i, i);
        icount++;
      }
 
      // Off diagonals row by row
      for (unsigned i = 0; i < 2; i++)
      {
        for (unsigned j = i + 1; j < 2; j++)
        {
          flux[icount] = strainrate(i, j);
          icount++;
        }
      }
    }
 
    ///  Further build, pass the pointers down to the sons
    void further_build()
    {
      // Find the father element
      RefineablePolarNavierStokesEquations* cast_father_element_pt =
        dynamic_cast<RefineablePolarNavierStokesEquations*>(
          this->father_element_pt());
 
      // Set the viscosity ratio pointer
      this->Viscosity_Ratio_pt = cast_father_element_pt->viscosity_ratio_pt();
      // Set the density ratio pointer
      this->Density_Ratio_pt = cast_father_element_pt->density_ratio_pt();
      // Set pointer to global Reynolds number
      this->Re_pt = cast_father_element_pt->re_pt();
      // Set pointer to global Reynolds number x Strouhal number (=Womersley)
      this->ReSt_pt = cast_father_element_pt->re_st_pt();
      // Set pointer to global Reynolds number x inverse Froude number
      this->ReInvFr_pt = cast_father_element_pt->re_invfr_pt();
      // Set pointer to global gravity Vector
      this->G_pt = cast_father_element_pt->g_pt();
      // Set pointer to alpha
      this->Alpha_pt = cast_father_element_pt->alpha_pt();
 
      // Set pointer to body force function
      this->Body_force_fct_pt = cast_father_element_pt->body_force_fct_pt();
 
      // Set pointer to volumetric source function
      this->Source_fct_pt = cast_father_element_pt->source_fct_pt();
    }
 
  protected:
    /// Add element's contribution to elemental residual vector and/or
    /// Jacobian matrix
    /// flag=1: compute both
    /// flag=0: compute only residual vector
    virtual void fill_in_generic_residual_contribution(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      DenseMatrix<double>& mass_matrix,
      unsigned flag);
  };
 
 
  //======================================================================
  /// Refineable version of Polar Taylor Hood elements. These classes
  /// can be written in total generality.
  //======================================================================
  class RefineablePolarTaylorHoodElement
    : public PolarTaylorHoodElement,
      public virtual RefineablePolarNavierStokesEquations,
      public virtual RefineableQElement<2>
  {
  private:
    /// Pointer to n_p-th pressure node
    Node* pressure_node_pt(const unsigned& n_p)
    {
      return this->node_pt(this->Pconv[n_p]);
    }
 
    /// Unpin all pressure dofs
    void unpin_elemental_pressure_dofs()
    {
      // find the index at which the pressure is stored
      int p_index = this->p_nodal_index_pnst();
      unsigned n_node = this->nnode();
      // loop over nodes
      for (unsigned n = 0; n < n_node; n++)
      {
        this->node_pt(n)->unpin(p_index);
      }
    }
 
    ///  Pin all nodal pressure dofs that are not required
    void pin_elemental_redundant_nodal_pressure_dofs()
    {
      // Find the pressure index
      int p_index = this->p_nodal_index_pnst();
      // Loop over all nodes
      unsigned n_node = this->nnode();
      // loop over all nodes and pin all  the nodal pressures
      for (unsigned n = 0; n < n_node; n++)
      {
        this->node_pt(n)->pin(p_index);
      }
 
      // Loop over all actual pressure nodes and unpin if they're not hanging
      unsigned n_pres = this->npres_pnst();
      for (unsigned l = 0; l < n_pres; l++)
      {
        Node* nod_pt = this->pressure_node_pt(l);
        if (!nod_pt->is_hanging(p_index))
        {
          nod_pt->unpin(p_index);
        }
      }
    }
 
  public:
    /// Constructor
    RefineablePolarTaylorHoodElement()
      : RefineableElement(),
        RefineablePolarNavierStokesEquations(),
        RefineableQElement<2>(),
        PolarTaylorHoodElement()
    {
    }
 
    /// Number of values required at local node n. In order to simplify
    /// matters, we allocate storage for pressure variables at all the nodes
    /// and then pin those that are not used.
    unsigned required_nvalue(const unsigned& n) const
    {
      return 3;
    }
 
    /// Number of continuously interpolated values: (DIM velocities + 1
    /// pressure)
    unsigned ncont_interpolated_values() const
    {
      return 3;
    }
 
    /// Rebuild from sons: empty
    void rebuild_from_sons(Mesh*& mesh_pt) {}
 
    /// Order of recovery shape functions for Z2 error estimation:
    /// Same order as shape functions.
    unsigned nrecovery_order()
    {
      return 2;
    }
 
    /// Number of vertex nodes in the element
    unsigned nvertex_node() const
    {
      return PolarTaylorHoodElement::nvertex_node();
    }
 
    /// Pointer to the j-th vertex node in the element
    Node* vertex_node_pt(const unsigned& j) const
    {
      return PolarTaylorHoodElement::vertex_node_pt(j);
    }
 
    /// Get the function value u in Vector.
    /// Note: Given the generality of the interface (this function
    /// is usually called from black-box documentation or interpolation
    /// routines), the values Vector sets its own size in here.
    void get_interpolated_values(const Vector<double>& s,
                                 Vector<double>& values)
    {
      // Set size of Vector: u,v,p and initialise to zero
      values.resize(3, 0.0);
 
      // Calculate velocities: values[0],...
      for (unsigned i = 0; i < 2; i++)
      {
        values[i] = this->interpolated_u_pnst(s, i);
      }
 
      // Calculate pressure: values[DIM]
      values[2] = this->interpolated_p_pnst(s);
    }
 
    /// Get the function value u in Vector.
    /// Note: Given the generality of the interface (this function
    /// is usually called from black-box documentation or interpolation
    /// routines), the values Vector sets its own size in here.
    void get_interpolated_values(const unsigned& t,
                                 const Vector<double>& s,
                                 Vector<double>& values)
    {
#ifdef PARANOID
      // Find out the number of timesteps (present & previous)
      // (the present element might not have been initialised yet but
      // its root must know about the time integrator)
      RefineableElement* root_el_pt = this->tree_pt()->root_pt()->object_pt();
 
      unsigned N_prev_time =
        root_el_pt->node_pt(0)->time_stepper_pt()->nprev_values();
 
      if (t > N_prev_time)
      {
        std::ostringstream error_message;
        error_message
          << "The value of t in get_interpolated_values(...), " << t
          << std::endl
          << "is greater than the number of previous stored timesteps";
 
        throw OomphLibError(error_message.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // Set size of Vector: u,v,p
      values.resize(2 + 1);
 
      // Initialise
      for (unsigned i = 0; i < 2 + 1; i++)
      {
        values[i] = 0.0;
      }
 
      // Find out how many nodes there are
      unsigned n_node = this->nnode();
 
      // Shape functions
      Shape psif(n_node);
      this->shape(s, psif);
 
      // Calculate velocities: values[0],...
      for (unsigned i = 0; i < 2; i++)
      {
        // Get the index at which the i-th velocity is stored
        unsigned u_nodal_index = this->u_index_pnst(i);
        for (unsigned l = 0; l < n_node; l++)
        {
          values[i] += this->nodal_value(t, l, u_nodal_index) * psif[l];
        }
      }
 
      // Calculate pressure: values[DIM]
      //(no history is carried in the pressure)
      values[2] = this->interpolated_p_pnst(s);
    }
 
    ///  Perform additional hanging node procedures for variables
    /// that are not interpolated by all nodes. The pressures are stored
    /// at the p_nodal_index_pnst-th location in each node
    void further_setup_hanging_nodes()
    {
      this->setup_hang_for_value(this->p_nodal_index_pnst());
    }
 
    /// The velocities are isoparametric and so the "nodes" interpolating
    /// the velocities are the geometric nodes. The pressure "nodes" are a
    /// subset of the nodes, so when value_id==DIM, the n-th pressure
    /// node is returned.
    Node* interpolating_node_pt(const unsigned& n, const int& value_id)
 
    {
      // The only different nodes are the pressure nodes
      if (value_id == 2)
      {
        return this->pressure_node_pt(n);
      }
      // The other variables are interpolated via the usual nodes
      else
      {
        return this->node_pt(n);
      }
    }
 
    /// The pressure nodes are the corner nodes, so when n_value==DIM,
    /// the fraction is the same as the 1d node number, 0 or 1.
    double local_one_d_fraction_of_interpolating_node(const unsigned& n1d,
                                                      const unsigned& i,
                                                      const int& value_id)
    {
      if (value_id == 2)
      {
        // The pressure nodes are just located on the boundaries at 0 or 1
        return double(n1d);
      }
      // Otherwise the velocity nodes are the same as the geometric ones
      else
      {
        return this->local_one_d_fraction_of_node(n1d, i);
      }
    }
 
    /// The velocity nodes are the same as the geometric nodes. The
    /// pressure nodes must be calculated by using the same methods as
    /// the geometric nodes, but by recalling that there are only two pressure
    /// nodes per edge.
    Node* get_interpolating_node_at_local_coordinate(const Vector<double>& s,
                                                     const int& value_id)
    {
      // If we are calculating pressure nodes
      if (value_id == 2)
      {
        // Storage for the index of the pressure node
        unsigned total_index = 0;
        // The number of nodes along each 1d edge is 2.
        unsigned NNODE_1D = 2;
        // Storage for the index along each boundary
        Vector<int> index(2);
        // Loop over the coordinates
        for (unsigned i = 0; i < 2; i++)
        {
          // If we are at the lower limit, the index is zero
          if (s[i] == -1.0)
          {
            index[i] = 0;
          }
          // If we are at the upper limit, the index is the number of nodes
          // minus 1
          else if (s[i] == 1.0)
          {
            index[i] = NNODE_1D - 1;
          }
          // Otherwise, we have to calculate the index in general
          else
          {
            // For uniformly spaced nodes the 0th node number would be
            double float_index = 0.5 * (1.0 + s[i]) * (NNODE_1D - 1);
            index[i] = int(float_index);
            // What is the excess. This should be safe because the
            // taking the integer part rounds down
            double excess = float_index - index[i];
            // If the excess is bigger than our tolerance there is no node,
            // return null
            if ((excess > FiniteElement::Node_location_tolerance) &&
                ((1.0 - excess) > FiniteElement::Node_location_tolerance))
            {
              return 0;
            }
          }
          /// Construct the general pressure index from the components.
          total_index +=
            index[i] * static_cast<unsigned>(pow(static_cast<float>(NNODE_1D),
                                                 static_cast<int>(i)));
        }
        // If we've got here we have a node, so let's return a pointer to it
        return this->pressure_node_pt(total_index);
      }
      // Otherwise velocity nodes are the same as pressure nodes
      else
      {
        return this->get_node_at_local_coordinate(s);
      }
    }
 
 
    /// The number of 1d pressure nodes is 2, the number of 1d velocity
    /// nodes is the same as the number of 1d geometric nodes.
    unsigned ninterpolating_node_1d(const int& value_id)
    {
      if (value_id == 2)
      {
        return 2;
      }
      else
      {
        return this->nnode_1d();
      }
    }
 
    /// The number of pressure nodes is 2^DIM. The number of
    /// velocity nodes is the same as the number of geometric nodes.
    unsigned ninterpolating_node(const int& value_id)
    {
      if (value_id == 2)
      {
        return static_cast<unsigned>(pow(2.0, static_cast<int>(2)));
      }
      else
      {
        return this->nnode();
      }
    }
 
    /// The basis interpolating the pressure is given by pshape().
    /// / The basis interpolating the velocity is shape().
    void interpolating_basis(const Vector<double>& s,
                             Shape& psi,
                             const int& value_id) const
    {
      if (value_id == 2)
      {
        return this->pshape_pnst(s, psi);
      }
      else
      {
        return this->shape(s, psi);
      }
    }
 
 
    /// 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.
    /// (Overloads non-refineable version and takes hanging nodes
    /// into account)
    void insert_load_data(
      std::set<std::pair<Data*, unsigned>>& paired_load_data)
    {
      // Get the nodal indices at which the velocities are stored
      unsigned u_index[2];
      for (unsigned i = 0; i < 2; i++)
      {
        u_index[i] = this->u_index_pnst(i);
      }
 
      // Loop over the nodes
      unsigned n_node = this->nnode();
      for (unsigned n = 0; n < n_node; n++)
      {
        // Pointer to current node
        Node* nod_pt = this->node_pt(n);
 
        // Check if it's hanging:
        if (nod_pt->is_hanging())
        {
          // It's hanging -- get number of master nodes
          unsigned nmaster = nod_pt->hanging_pt()->nmaster();
 
          // Loop over masters
          for (unsigned j = 0; j < nmaster; j++)
          {
            Node* master_nod_pt = nod_pt->hanging_pt()->master_node_pt(j);
 
            // Loop over the velocity components and add pointer to their data
            // and indices to the vectors
            for (unsigned i = 0; i < 2; i++)
            {
              paired_load_data.insert(
                std::make_pair(master_nod_pt, u_index[i]));
            }
          }
        }
        // Not hanging
        else
        {
          // Loop over the velocity components and add pointer to their data
          // and indices to the vectors
          for (unsigned i = 0; i < 2; i++)
          {
            paired_load_data.insert(
              std::make_pair(this->node_pt(n), u_index[i]));
          }
        }
      }
 
      // Get the nodal index at which the pressure is stored
      int p_index = this->p_nodal_index_pnst();
 
      // Loop over the pressure data
      unsigned n_pres = this->npres_pnst();
      for (unsigned l = 0; l < n_pres; l++)
      {
        // Get the pointer to the nodal pressure
        Node* pres_node_pt = this->pressure_node_pt(l);
        // Check if the pressure dof is hanging
        if (pres_node_pt->is_hanging(p_index))
        {
          // Get the pointer to the hang info object
          // (pressure is stored as p_index--th nodal dof).
          HangInfo* hang_info_pt = pres_node_pt->hanging_pt(p_index);
 
          // Get number of pressure master nodes (pressure is stored
          unsigned nmaster = hang_info_pt->nmaster();
 
          // Loop over pressure master nodes
          for (unsigned m = 0; m < nmaster; m++)
          {
            // The p_index-th entry in each nodal data is the pressure, which
            // affects the traction
            paired_load_data.insert(
              std::make_pair(hang_info_pt->master_node_pt(m), p_index));
          }
        }
        // It's not hanging
        else
        {
          // The p_index-th entry in each nodal data is the pressure, which
          // affects the traction
          paired_load_data.insert(std::make_pair(pres_node_pt, p_index));
        }
      }
    }
  };
 
 
  //=======================================================================
  /// Face geometry of the RefineablePolarTaylorHoodElements is the
  /// same as the Face geometry of the PolarTaylorHoodElements.
  //=======================================================================
  template<>
  class FaceGeometry<RefineablePolarTaylorHoodElement>
    : public virtual FaceGeometry<PolarTaylorHoodElement>
  {
  public:
    FaceGeometry() : FaceGeometry<PolarTaylorHoodElement>() {}
  };
 
 
  ///////////////////////////////////////////////////////////////////////////
  ///////////////////////////////////////////////////////////////////////////
  ///////////////////////////////////////////////////////////////////////////
 
 
  //======================================================================
  /// Refineable version of Crouzeix Raviart elements. Generic class definitions
  //======================================================================
  class RefineablePolarCrouzeixRaviartElement
    : public PolarCrouzeixRaviartElement,
      public virtual RefineablePolarNavierStokesEquations,
      public virtual RefineableQElement<2>
  {
  private:
    /// Unpin all internal pressure dofs
    void unpin_elemental_pressure_dofs()
    {
      unsigned n_pres = this->npres_pnst();
      // loop over pressure dofs and unpin them
      for (unsigned l = 0; l < n_pres; l++)
      {
        this->internal_data_pt(this->P_pnst_internal_index)->unpin(l);
      }
    }
 
  public:
    /// Constructor
    RefineablePolarCrouzeixRaviartElement()
      : RefineableElement(),
        RefineablePolarNavierStokesEquations(),
        RefineableQElement<2>(),
        PolarCrouzeixRaviartElement()
    {
    }
 
    /// Number of continuously interpolated values: DIM (velocities)
    unsigned ncont_interpolated_values() const
    {
      return 2;
    }
 
    /// Rebuild from sons: Reconstruct pressure from the (merged) sons
    /// This must be specialised for each dimension.
    inline void rebuild_from_sons(Mesh*& mesh_pt);
 
    /// Order of recovery shape functions for Z2 error estimation:
    /// Same order as shape functions.
    unsigned nrecovery_order()
    {
      return 2;
    }
 
    /// Number of vertex nodes in the element
    unsigned nvertex_node() const
    {
      return PolarCrouzeixRaviartElement::nvertex_node();
    }
 
    /// Pointer to the j-th vertex node in the element
    Node* vertex_node_pt(const unsigned& j) const
    {
      return PolarCrouzeixRaviartElement::vertex_node_pt(j);
    }
 
    /// Get the function value u in Vector.
    /// Note: Given the generality of the interface (this function
    /// is usually called from black-box documentation or interpolation
    /// routines), the values Vector sets its own size in here.
    void get_interpolated_values(const Vector<double>& s,
                                 Vector<double>& values)
    {
      // Set size of Vector: u,v,p and initialise to zero
      values.resize(2, 0.0);
 
      // Calculate velocities: values[0],...
      for (unsigned i = 0; i < 2; i++)
      {
        values[i] = this->interpolated_u_pnst(s, i);
      }
    }
 
    /// Get all function values [u,v..,p] at previous timestep t
    /// (t=0: present; t>0: previous timestep).
    /// \n
    /// Note: Given the generality of the interface (this function
    /// is usually called from black-box documentation or interpolation
    /// routines), the values Vector sets its own size in here. \n Note: No
    /// pressure history is kept, so pressure is always the current value.
    void get_interpolated_values(const unsigned& t,
                                 const Vector<double>& s,
                                 Vector<double>& values)
    {
#ifdef PARANOID
 
      // Find out the number of timesteps (present & previous)
      // (the present element might not have been initialised yet but
      // its root must know about the time integrator)
      RefineableElement* root_el_pt = this->tree_pt()->root_pt()->object_pt();
 
      unsigned N_prev_time =
        root_el_pt->node_pt(0)->time_stepper_pt()->nprev_values();
 
      if (t > N_prev_time)
      {
        std::ostringstream error_message;
        error_message
          << "The value of t in get_interpolated_values(...), " << t
          << std::endl
          << "is greater than the number of previous stored timesteps";
 
        throw OomphLibError(error_message.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // Set size of Vector: u,v,p
      values.resize(2);
 
      // Initialise
      for (unsigned i = 0; i < 2; i++)
      {
        values[i] = 0.0;
      }
 
      // Find out how many nodes there are
      unsigned n_node = this->nnode();
 
      // Shape functions
      Shape psif(n_node);
      this->shape(s, psif);
 
      // Calculate velocities: values[0],...
      for (unsigned i = 0; i < 2; i++)
      {
        // Get the nodal index at which the i-th velocity component is stored
        unsigned u_nodal_index = this->u_index_pnst(i);
        for (unsigned l = 0; l < n_node; l++)
        {
          values[i] += this->nodal_value(t, l, u_nodal_index) * psif[l];
        }
      }
    }
 
    ///  Perform additional hanging node procedures for variables
    /// that are not interpolated by all nodes. Empty
    void further_setup_hanging_nodes() {}
 
    /// Further build for Crouzeix_Raviart interpolates the internal
    /// pressure dofs from father element: Make sure pressure values and
    /// dp/ds agree between fathers and sons at the midpoints of the son
    /// elements. This must be specialised for each dimension.
    inline void further_build();
 
 
    /// 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.
    /// (Overloads non-refineable version and takes hanging nodes
    /// into account)
    void insert_load_data(
      std::set<std::pair<Data*, unsigned>>& paired_load_data)
    {
      // Get the nodal indices at which the velocities are stored
      unsigned u_index[2];
      for (unsigned i = 0; i < 2; i++)
      {
        u_index[i] = this->u_index_pnst(i);
      }
 
      // Loop over the nodes
      unsigned n_node = this->nnode();
      for (unsigned n = 0; n < n_node; n++)
      {
        // Pointer to current node
        Node* nod_pt = this->node_pt(n);
 
        // Check if it's hanging:
        if (nod_pt->is_hanging())
        {
          // It's hanging -- get number of master nodes
          unsigned nmaster = nod_pt->hanging_pt()->nmaster();
 
          // Loop over masters
          for (unsigned j = 0; j < nmaster; j++)
          {
            Node* master_nod_pt = nod_pt->hanging_pt()->master_node_pt(j);
 
            // Loop over the velocity components and add pointer to their data
            // and indices to the vectors
            for (unsigned i = 0; i < 2; i++)
            {
              paired_load_data.insert(
                std::make_pair(master_nod_pt, u_index[i]));
            }
          }
        }
        // Not hanging
        else
        {
          // Loop over the velocity components and add pointer to their data
          // and indices to the vectors
          for (unsigned i = 0; i < 2; i++)
          {
            paired_load_data.insert(
              std::make_pair(this->node_pt(n), u_index[i]));
          }
        }
      }
 
 
      // Loop over the pressure data (can't be hanging!)
      unsigned n_pres = this->npres_pnst();
      for (unsigned l = 0; l < n_pres; l++)
      {
        // The entries in the internal data at P_pnst_internal_index
        // are the pressures, which affect the traction
        paired_load_data.insert(std::make_pair(
          this->internal_data_pt(this->P_pnst_internal_index), l));
      }
    }
  };
 
 
  //=======================================================================
  /// Face geometry of the RefineableQuadQCrouzeixRaviartElements
  //=======================================================================
  template<>
  class FaceGeometry<RefineablePolarCrouzeixRaviartElement>
    : public virtual FaceGeometry<PolarCrouzeixRaviartElement>
  {
  public:
    FaceGeometry() : FaceGeometry<PolarCrouzeixRaviartElement>() {}
  };
 
 
  //=====================================================================
  /// 2D Rebuild from sons: Reconstruct pressure from the (merged) sons
  //=====================================================================
  inline void RefineablePolarCrouzeixRaviartElement::rebuild_from_sons(
    Mesh*& mesh_pt)
  {
    using namespace QuadTreeNames;
 
    // Central pressure value:
    //-----------------------
 
    // Use average of the sons central pressure values
    // Other options: Take average of the four (discontinuous)
    // pressure values at the father's midpoint]
 
    double av_press = 0.0;
 
    // Loop over the sons
    for (unsigned ison = 0; ison < 4; ison++)
    {
      // Add the sons midnode pressure
      // Note that we can assume that the pressure is stored at the same
      // location because these are EXACTLY the same type of elements
      av_press += quadtree_pt()
                    ->son_pt(ison)
                    ->object_pt()
                    ->internal_data_pt(this->P_pnst_internal_index)
                    ->value(0);
    }
 
    // Use the average
    internal_data_pt(this->P_pnst_internal_index)
      ->set_value(0, 0.25 * av_press);
 
 
    // Slope in s_0 direction
    //----------------------
 
    // Use average of the 2 FD approximations based on the
    // elements central pressure values
    // [Other options: Take average of the four
    // pressure derivatives]
 
    double slope1 = quadtree_pt()
                      ->son_pt(SE)
                      ->object_pt()
                      ->internal_data_pt(this->P_pnst_internal_index)
                      ->value(0) -
                    quadtree_pt()
                      ->son_pt(SW)
                      ->object_pt()
                      ->internal_data_pt(this->P_pnst_internal_index)
                      ->value(0);
 
    double slope2 = quadtree_pt()
                      ->son_pt(NE)
                      ->object_pt()
                      ->internal_data_pt(this->P_pnst_internal_index)
                      ->value(0) -
                    quadtree_pt()
                      ->son_pt(NW)
                      ->object_pt()
                      ->internal_data_pt(this->P_pnst_internal_index)
                      ->value(0);
 
 
    // Use the average
    internal_data_pt(this->P_pnst_internal_index)
      ->set_value(1, 0.5 * (slope1 + slope2));
 
 
    // Slope in s_1 direction
    //----------------------
 
    // Use average of the 2 FD approximations based on the
    // elements central pressure values
    // [Other options: Take average of the four
    // pressure derivatives]
 
    slope1 = quadtree_pt()
               ->son_pt(NE)
               ->object_pt()
               ->internal_data_pt(this->P_pnst_internal_index)
               ->value(0) -
             quadtree_pt()
               ->son_pt(SE)
               ->object_pt()
               ->internal_data_pt(this->P_pnst_internal_index)
               ->value(0);
 
    slope2 = quadtree_pt()
               ->son_pt(NW)
               ->object_pt()
               ->internal_data_pt(this->P_pnst_internal_index)
               ->value(0) -
             quadtree_pt()
               ->son_pt(SW)
               ->object_pt()
               ->internal_data_pt(this->P_pnst_internal_index)
               ->value(0);
 
 
    // Use the average
    internal_data_pt(this->P_pnst_internal_index)
      ->set_value(2, 0.5 * (slope1 + slope2));
  }
 
  //======================================================================
  /// 2D Further build for Crouzeix_Raviart interpolates the internal
  /// pressure dofs from father element: Make sure pressure values and
  /// dp/ds agree between fathers and sons at the midpoints of the son
  /// elements.
  //======================================================================
  inline void RefineablePolarCrouzeixRaviartElement::further_build()
  {
    // Call the generic further build
    RefineablePolarNavierStokesEquations::further_build();
 
    using namespace QuadTreeNames;
 
    // What type of son am I? Ask my quadtree representation...
    int son_type = quadtree_pt()->son_type();
 
    // Pointer to my father (in element impersonation)
    RefineableElement* father_el_pt = quadtree_pt()->father_pt()->object_pt();
 
    Vector<double> s_father(2);
 
    // Son midpoint is located at the following coordinates in father element:
 
    // South west son
    if (son_type == SW)
    {
      s_father[0] = -0.5;
      s_father[1] = -0.5;
    }
    // South east son
    else if (son_type == SE)
    {
      s_father[0] = 0.5;
      s_father[1] = -0.5;
    }
    // North east son
    else if (son_type == NE)
    {
      s_father[0] = 0.5;
      s_father[1] = 0.5;
    }
 
    // North west son
    else if (son_type == NW)
    {
      s_father[0] = -0.5;
      s_father[1] = 0.5;
    }
 
    // Pressure value in father element
    RefineablePolarCrouzeixRaviartElement* cast_father_element_pt =
      dynamic_cast<RefineablePolarCrouzeixRaviartElement*>(father_el_pt);
 
    double press = cast_father_element_pt->interpolated_p_pnst(s_father);
 
    // Pressure value gets copied straight into internal dof:
    internal_data_pt(this->P_pnst_internal_index)->set_value(0, press);
 
    // The slopes get copied from father
    for (unsigned i = 1; i < 3; i++)
    {
      double half_father_slope =
        0.5 *
        cast_father_element_pt->internal_data_pt(this->P_pnst_internal_index)
          ->value(i);
      // Set the value in the son
      internal_data_pt(this->P_pnst_internal_index)
        ->set_value(i, half_father_slope);
    }
  }
 
} // namespace oomph
#endif