multi_domain_boussinesq_elements.h

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// LIC// ====================================================================
// LIC// This file forms part of oomph-lib, the object-oriented,
// LIC// multi-physics finite-element library, available
// LIC// at http://www.oomph-lib.org.
// LIC//
// LIC// Copyright (C) 2006-2022 Matthias Heil and Andrew Hazel
// LIC//
// LIC// This library is free software; you can redistribute it and/or
// LIC// modify it under the terms of the GNU Lesser General Public
// LIC// License as published by the Free Software Foundation; either
// LIC// version 2.1 of the License, or (at your option) any later version.
// LIC//
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// LIC// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
// LIC// Lesser General Public License for more details.
// LIC//
// LIC// You should have received a copy of the GNU Lesser General Public
// LIC// License along with this library; if not, write to the Free Software
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// LIC//
// LIC// The authors may be contacted at oomph-lib@maths.man.ac.uk.
// LIC//
// LIC//====================================================================
// Header for a multi-physics elements that couple the Navier--Stokes
// and advection diffusion elements via a multi domain approach,
// giving Boussinesq convection
 
#ifndef OOMPH_MULTI_DOMAIN_BOUSSINESQ_ELEMENTS_HEADER
#define OOMPH_MULTI_DOMAIN_BOUSSINESQ_ELEMENTS_HEADER
 
// Oomph-lib headers, we require the generic, advection-diffusion
// and navier-stokes elements.
#include "generic.h"
#include "advection_diffusion.h"
#include "navier_stokes.h"
 
 
namespace oomph
{
  //=============================================================
  /// Namespace for default parameters in multi-domain Boussinesq
  //=============================================================
  namespace MultiDomainBoussinesqHelper
  {
    /// Default value for physical constants
    double Default_Physical_Constant_Value = 0.0;
 
  } // namespace MultiDomainBoussinesqHelper
  /// //////////////////////////////////////////////////////////////////////
  /// //////////////////////////////////////////////////////////////////////
  /// //////////////////////////////////////////////////////////////////////
 
 
  //======================nst_bous_class=================================
  /// Build a refineable Navier Stokes element that inherits from
  /// ElementWithExternalElement so that it can "communicate" with
  /// an advection diffusion element that provides the temperature
  /// in the body force term.
  //=====================================================================
  template<class NST_ELEMENT, class AD_ELEMENT>
  class RefineableNavierStokesBoussinesqElement
    : public virtual NST_ELEMENT,
      public virtual ElementWithExternalElement
  {
  public:
    /// Constructor: call the underlying constructors and
    /// initialise the pointer to the Rayleigh number to point
    /// to the default value of 0.0.
    RefineableNavierStokesBoussinesqElement()
      : NST_ELEMENT(), ElementWithExternalElement()
    {
      Ra_pt = &MultiDomainBoussinesqHelper::Default_Physical_Constant_Value;
 
      // There is one interaction: The effect of the advection-diffusion
      // element onto the buoyancy term
      this->set_ninteraction(1);
    }
 
    /// Access function for the Rayleigh number (const version)
    const double& ra() const
    {
      return *Ra_pt;
    }
 
    /// Access function for the pointer to the Rayleigh number
    double*& ra_pt()
    {
      return Ra_pt;
    }
 
    /// Call the underlying single-physics element's further_build()
    /// functions and make sure that the pointer to the Rayleigh number
    /// is passed to the sons. Also make sure that if the external geometric
    /// Data was ignored in the father it's also ignored in the sons
    void further_build()
    {
      NST_ELEMENT::further_build();
 
      // Cast the pointer to the father element to the specific
      // element type
      RefineableNavierStokesBoussinesqElement<NST_ELEMENT, AD_ELEMENT>*
        cast_father_element_pt = dynamic_cast<
          RefineableNavierStokesBoussinesqElement<NST_ELEMENT, AD_ELEMENT>*>(
          this->father_element_pt());
 
      // Set the pointer to the Rayleigh number to be the same as that in
      // the father
      this->Ra_pt = cast_father_element_pt->ra_pt();
 
      // Retain ignorance about external geometric data...
      if (!cast_father_element_pt->external_geometric_data_is_included())
      {
        this->ignore_external_geometric_data();
      }
    }
 
    /// Overload get_body_force_nst() to return the temperature-dependent
    /// buoyancy force, using the temperature computed by the
    /// "external" advection diffusion element associated with
    /// integration point \c ipt.
    void get_body_force_nst(const double& time,
                            const unsigned& ipt,
                            const Vector<double>& s,
                            const Vector<double>& x,
                            Vector<double>& body_force)
    {
      // Set interaction index -- there's only one interaction...
      const unsigned interaction = 0;
 
      // Get a pointer to the external element that computes the
      // the temperature -- we know it's an advection diffusion element.
      const AD_ELEMENT* adv_diff_el_pt =
        dynamic_cast<AD_ELEMENT*>(external_element_pt(interaction, ipt));
 
      // Get the temperature interpolated from the external element
      const double interpolated_t = adv_diff_el_pt->interpolated_u_adv_diff(
        external_element_local_coord(interaction, ipt));
 
      // Get vector that indicates the direction of gravity from
      // the Navier-Stokes equations
      Vector<double> gravity(NST_ELEMENT::g());
 
      // Set the temperature-dependent body force:
      const unsigned n_dim = this->dim();
      for (unsigned i = 0; i < n_dim; i++)
      {
        body_force[i] = -gravity[i] * interpolated_t * ra();
      }
 
    } // end overloaded body force
 
 
    /// Compute the element's residual vector and the Jacobian matrix.
    void fill_in_contribution_to_jacobian(Vector<double>& residuals,
                                          DenseMatrix<double>& jacobian)
    {
      // Get the analytical contribution from the basic Navier-Stokes element
      NST_ELEMENT::fill_in_contribution_to_jacobian(residuals, jacobian);
 
#ifdef USE_FD_FOR_DERIVATIVES_WRT_EXTERNAL_DATA_IN_MULTI_DOMAIN_BOUSSINESQ
 
      // Get the off-diagonal terms by finite differencing
      this->fill_in_jacobian_from_external_interaction_by_fd(residuals,
                                                             jacobian);
 
#else
 
      // Get the off-diagonal terms analytically
      this->fill_in_off_diagonal_block_analytic(residuals, jacobian);
 
#endif
    }
 
 
    /// 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 standard (Broken) function
      // which will prevent these elements from being used
      // in eigenproblems until replaced.
      FiniteElement::fill_in_contribution_to_jacobian_and_mass_matrix(
        residuals, jacobian, mass_matrix);
    }
 
 
    /// Fill in the derivatives of the body force with respect to the
    /// external unknowns
    void get_dbody_force_nst_dexternal_element_data(
      const unsigned& ipt,
      DenseMatrix<double>& result,
      Vector<unsigned>& global_eqn_number);
 
 
    /// Compute the contribution of the external
    /// degrees of freedom (temperatures) on the Navier-Stokes equations
    void fill_in_off_diagonal_block_analytic(Vector<double>& residuals,
                                             DenseMatrix<double>& jacobian)
    {
      // Local storage for the index in the nodes at which the
      // Navier-Stokes velocities are stored (we know that this
      // should be 0,1,2)
      const unsigned n_dim = this->dim();
      unsigned u_nodal_nst[n_dim];
      for (unsigned i = 0; i < n_dim; i++)
      {
        u_nodal_nst[i] = this->u_index_nst(i);
      }
 
      // Find out how many nodes there are
      const unsigned n_node = this->nnode();
 
      // Set up memory for the shape and test functions and their derivatives
      Shape psif(n_node), testf(n_node);
      DShape dpsifdx(n_node, n_dim), dtestfdx(n_node, n_dim);
 
      // Number of integration points
      const unsigned n_intpt = this->integral_pt()->nweight();
 
      // Integers to store the local equations and unknowns
      int local_eqn = 0, local_unknown = 0;
 
      // Local storage for pointers to hang_info objects
      HangInfo* hang_info_pt = 0;
 
      // Loop over the integration points
      for (unsigned ipt = 0; ipt < n_intpt; ipt++)
      {
        // Get the integral weight
        double w = this->integral_pt()->weight(ipt);
 
        // Call the derivatives of the shape and test functions
        double J = this->dshape_and_dtest_eulerian_at_knot_nst(
          ipt, psif, dpsifdx, testf, dtestfdx);
 
        // Premultiply the weights and the Jacobian
        double W = w * J;
 
        // Assemble the jacobian terms
 
        // Get the derivatives of the body force wrt the unknowns
        // of the external element
        DenseMatrix<double> dbody_dexternal_element_data;
 
        // Vector of global equation number corresponding to the external
        // element's data
        Vector<unsigned> global_eqn_number_of_external_element_data;
 
        // Get the appropriate derivatives
        this->get_dbody_force_nst_dexternal_element_data(
          ipt,
          dbody_dexternal_element_data,
          global_eqn_number_of_external_element_data);
 
        // Find out how many external data there are
        const unsigned n_external_element_data =
          global_eqn_number_of_external_element_data.size();
 
        // Loop over the test functions
        for (unsigned l = 0; l < n_node; l++)
        {
          // Assemble the contributions of the temperature to
          // the Navier--Stokes equations (which arise through the buoyancy
          // body-force term)
          unsigned n_master = 1;
          double hang_weight = 1.0;
 
          // Local bool (is the node hanging)
          bool is_node_hanging = this->node_pt(l)->is_hanging();
 
          // If the node is hanging, get the number of master nodes
          if (is_node_hanging)
          {
            hang_info_pt = this->node_pt(l)->hanging_pt();
            n_master = hang_info_pt->nmaster();
          }
          // Otherwise there is just one master node, the node itself
          else
          {
            n_master = 1;
          }
 
          // Loop over the master nodes
          for (unsigned m = 0; m < n_master; m++)
          {
            // If the node is hanging get weight from master node
            if (is_node_hanging)
            {
              // Get the hang weight from the master node
              hang_weight = hang_info_pt->master_weight(m);
            }
            else
            {
              // Node contributes with full weight
              hang_weight = 1.0;
            }
 
 
            // Loop over the velocity components in the Navier--Stokes equtions
            for (unsigned i = 0; i < n_dim; i++)
            {
              // Get the equation number
              if (is_node_hanging)
              {
                // Get the equation number from the master node
                local_eqn = this->local_hang_eqn(
                  hang_info_pt->master_node_pt(m), u_nodal_nst[i]);
              }
              else
              {
                // Local equation number
                local_eqn = this->nodal_local_eqn(l, u_nodal_nst[i]);
              }
 
              if (local_eqn >= 0)
              {
                // Loop over the external data
                for (unsigned l2 = 0; l2 < n_external_element_data; l2++)
                {
                  // Find the local equation number corresponding to the global
                  // unknown
                  local_unknown = this->local_eqn_number(
                    global_eqn_number_of_external_element_data[l2]);
                  if (local_unknown >= 0)
                  {
                    // Add contribution to jacobian matrix
                    jacobian(local_eqn, local_unknown) +=
                      dbody_dexternal_element_data(i, l2) * testf(l) *
                      hang_weight * W;
                  }
                }
              }
            }
          }
        }
      }
    }
 
    /// Classify dof numbers as in underlying element
    void get_dof_numbers_for_unknowns(
      std::list<std::pair<unsigned long, unsigned>>& dof_lookup_list) const
    {
      // Call the underlying function
      NST_ELEMENT::get_dof_numbers_for_unknowns(dof_lookup_list);
    }
 
    /// Get number of dof types from underlying element
    unsigned ndof_types() const
    {
      return NST_ELEMENT::ndof_types();
    }
 
  private:
    /// Pointer to a private data member, the Rayleigh number
    double* Ra_pt;
  };
 
 
  /// ////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////
 
 
  //======================ad_bous_class==================================
  /// Build an AdvectionDiffusionElement that inherits from
  /// ElementWithExternalElement so that it can "communicate" with the
  /// a NavierStokesElement that provides its wind.
  //=====================================================================
  template<class AD_ELEMENT, class NST_ELEMENT>
  class RefineableAdvectionDiffusionBoussinesqElement
    : public virtual AD_ELEMENT,
      public virtual ElementWithExternalElement
  {
  public:
    /// Constructor: call the underlying constructors
    RefineableAdvectionDiffusionBoussinesqElement()
      : AD_ELEMENT(), ElementWithExternalElement()
    {
      // There is one interaction
      this->set_ninteraction(1);
    }
 
 
    //-----------------------------------------------------------
    // Note: we're overloading the output functions because the
    // =====  underlying advection diffusion elements output the
    //       wind which is only available at the integration points
    //       and even then only if the multi domain interaction
    //       has been set up.
    //-----------------------------------------------------------
 
    /// Output function:
    ///  Output x, y, theta at Nplot^DIM plot points
    // Start of output function
    void output(std::ostream& outfile, const unsigned& nplot)
    {
      // vector of local coordinates
      unsigned n_dim = this->dim();
      Vector<double> s(n_dim);
 
      // Tecplot header info
      outfile << this->tecplot_zone_string(nplot);
 
      // Loop over plot points
      unsigned num_plot_points = this->nplot_points(nplot);
      for (unsigned iplot = 0; iplot < num_plot_points; iplot++)
      {
        // Get local coordinates of plot point
        this->get_s_plot(iplot, nplot, s);
 
        // Output the position of the plot point
        for (unsigned i = 0; i < n_dim; i++)
        {
          outfile << this->interpolated_x(s, i) << " ";
        }
 
        // Output the temperature (the advected variable) at the plot point
        outfile << AD_ELEMENT::interpolated_u_adv_diff(s) << std::endl;
      }
      outfile << std::endl;
 
      // Write tecplot footer (e.g. FE connectivity lists)
      this->write_tecplot_zone_footer(outfile, nplot);
 
    } // End of output function
 
 
    ///  Overload the standard output function with the broken default
    void output(std::ostream& outfile)
    {
      FiniteElement::output(outfile);
    }
 
    /// C-style output function: Broken default
    void output(FILE* file_pt)
    {
      FiniteElement::output(file_pt);
    }
 
    ///  C-style output function: Broken default
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      FiniteElement::output(file_pt, n_plot);
    }
 
 
    /// Overload the wind function in the advection-diffusion equations.
    /// This provides the coupling from the Navier--Stokes equations to the
    /// advection-diffusion equations because the wind is the fluid velocity,
    /// obtained from the "external" element
    void get_wind_adv_diff(const unsigned& ipt,
                           const Vector<double>& s,
                           const Vector<double>& x,
                           Vector<double>& wind) const
    {
      // There is only one interaction
      unsigned interaction = 0;
 
      // Dynamic cast "external" element to Navier Stokes element
      NST_ELEMENT* nst_el_pt =
        dynamic_cast<NST_ELEMENT*>(external_element_pt(interaction, ipt));
 
      // Wind is given by the velocity in the Navier Stokes element
      nst_el_pt->interpolated_u_nst(
        external_element_local_coord(interaction, ipt), wind);
 
    } // end of get_wind_adv_diff
 
 
    /// Compute the element's residual vector and the Jacobian matrix.
    void fill_in_contribution_to_jacobian(Vector<double>& residuals,
                                          DenseMatrix<double>& jacobian)
    {
      // Get the contribution from the basic advection diffusion element
      AD_ELEMENT::fill_in_contribution_to_jacobian(residuals, jacobian);
 
#ifdef USE_FD_FOR_DERIVATIVES_WRT_EXTERNAL_DATA_IN_MULTI_DOMAIN_BOUSSINESQ
 
      // Get the off-diagonal terms by finite differencing
      this->fill_in_jacobian_from_external_interaction_by_fd(residuals,
                                                             jacobian);
 
#else
 
      // Get the off-diagonal terms analytically
      this->fill_in_off_diagonal_block_analytic(residuals, jacobian);
 
#endif
    }
 
 
    /// Overload the function that must return all field data involved
    /// in the interaction with the external (Navier Stokes) element.
    /// Only the velocity dofs in the Navier Stokes element affect the
    /// interaction with the current element.
    void identify_all_field_data_for_external_interaction(
      Vector<std::set<FiniteElement*>> const& external_elements_pt,
      std::set<std::pair<Data*, unsigned>>& paired_interaction_data);
 
    /// 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 standard (Broken) function
      // which will prevent these elements from being used
      // in eigenproblems until replaced.
      FiniteElement::fill_in_contribution_to_jacobian_and_mass_matrix(
        residuals, jacobian, mass_matrix);
    }
 
 
    /// Fill in the derivatives of the wind with respect to the
    /// external unknowns
    void get_dwind_adv_diff_dexternal_element_data(
      const unsigned& ipt,
      const unsigned& i,
      Vector<double>& result,
      Vector<unsigned>& global_eqn_number)
    {
      // The interaction index is 0 in this case
      unsigned interaction = 0;
 
      // Dynamic cast "other" element to correct type
      NST_ELEMENT* source_el_pt =
        dynamic_cast<NST_ELEMENT*>(external_element_pt(interaction, ipt));
 
      // Get the external element's derivatives of the velocity with respect
      // to the data. The wind is just the Navier--Stokes velocity, so this
      // is all that's required
      source_el_pt->dinterpolated_u_nst_ddata(
        external_element_local_coord(interaction, ipt),
        i,
        result,
        global_eqn_number);
    }
 
 
    /// Compute the contribution of the external
    /// degrees of freedom (velocities) on the advection-diffusion equations
    void fill_in_off_diagonal_block_analytic(Vector<double>& residuals,
                                             DenseMatrix<double>& jacobian)
    {
      // Local storage for the index in the nodes at which the temperature
      // is stored
      const unsigned u_nodal_adv_diff = this->u_index_adv_diff();
 
      // Find out how many nodes there are
      const unsigned n_node = this->nnode();
 
      // Spatial dimension
      const unsigned n_dim = this->dim();
 
      // Set up memory for the shape and test functions and their derivatives
      Shape psi(n_node), test(n_node);
      DShape dpsidx(n_node, n_dim), dtestdx(n_node, n_dim);
 
      // Number of integration points
      const unsigned n_intpt = this->integral_pt()->nweight();
 
      // Integers to store the local equations and unknowns
      int local_eqn = 0, local_unknown = 0;
 
      // Local storage for pointers to hang_info objects
      HangInfo* hang_info_pt = 0;
 
      // Get the peclet number
      const double peclet = this->pe();
 
      // Loop over the integration points
      for (unsigned ipt = 0; ipt < n_intpt; ipt++)
      {
        // Get the integral weight
        double w = this->integral_pt()->weight(ipt);
 
        // Call the derivatives of the shape and test functions
        double J = this->dshape_and_dtest_eulerian_at_knot_adv_diff(
          ipt, psi, dpsidx, test, dtestdx);
 
        // Premultiply the weights and the Jacobian
        double W = w * J;
 
        // Calculate local values of the derivatives of the solution
        Vector<double> interpolated_dudx(n_dim, 0.0);
        // Loop over nodes
        for (unsigned l = 0; l < n_node; l++)
        {
          // Loop over directions
          for (unsigned j = 0; j < n_dim; j++)
          {
            interpolated_dudx[j] +=
              this->nodal_value(l, u_nodal_adv_diff) * dpsidx(l, j);
          }
        }
 
        // Get the derivatives of the wind wrt the unknowns
        // of the external element
        Vector<double> dwind_dexternal_element_data;
 
        // Vector of global equation number corresponding to the external
        // element's data
        Vector<unsigned> global_eqn_number_of_external_element_data;
 
        // Loop over the wind directions
        for (unsigned i2 = 0; i2 < n_dim; i2++)
        {
          // Get the appropriate derivatives
          this->get_dwind_adv_diff_dexternal_element_data(
            ipt,
            i2,
            dwind_dexternal_element_data,
            global_eqn_number_of_external_element_data);
 
 
          // Find out how many external data there are
          const unsigned n_external_element_data =
            global_eqn_number_of_external_element_data.size();
 
          // Loop over the test functions
          for (unsigned l = 0; l < n_node; l++)
          {
            // Assemble the contributions of the velocities to
            // the advection-diffusion equations
            unsigned n_master = 1;
            double hang_weight = 1.0;
 
            // Local bool (is the node hanging)
            bool is_node_hanging = this->node_pt(l)->is_hanging();
 
            // If the node is hanging, get the number of master nodes
            if (is_node_hanging)
            {
              hang_info_pt = this->node_pt(l)->hanging_pt();
              n_master = hang_info_pt->nmaster();
            }
            // Otherwise there is just one master node, the node itself
            else
            {
              n_master = 1;
            }
 
            // Loop over the master nodes
            for (unsigned m = 0; m < n_master; m++)
            {
              // If the node is hanging get weight from master node
              if (is_node_hanging)
              {
                // Get the hang weight from the master node
                hang_weight = hang_info_pt->master_weight(m);
              }
              else
              {
                // Node contributes with full weight
                hang_weight = 1.0;
              }
 
              // Get the equation number
              if (is_node_hanging)
              {
                // Get the equation number from the master node
                local_eqn = this->local_hang_eqn(
                  hang_info_pt->master_node_pt(m), u_nodal_adv_diff);
              }
              else
              {
                // Local equation number
                local_eqn = this->nodal_local_eqn(l, u_nodal_adv_diff);
              }
 
              if (local_eqn >= 0)
              {
                // Loop over the external data
                for (unsigned l2 = 0; l2 < n_external_element_data; l2++)
                {
                  // Find the local equation number corresponding to the global
                  // unknown
                  local_unknown = this->local_eqn_number(
                    global_eqn_number_of_external_element_data[l2]);
                  if (local_unknown >= 0)
                  {
                    // Add contribution to jacobian matrix
                    jacobian(local_eqn, local_unknown) -=
                      peclet * dwind_dexternal_element_data[l2] *
                      interpolated_dudx[i2] * test(l) * hang_weight * W;
                  }
                }
              }
            }
          }
        }
      }
    }
 
    /// Classify dofs for use in block preconditioner
    void get_dof_numbers_for_unknowns(
      std::list<std::pair<unsigned long, unsigned>>& dof_lookup_list) const
    {
      // number of nodes
      unsigned n_node = this->nnode();
 
      // temporary pair (used to store dof lookup prior to being added to list)
      std::pair<unsigned, unsigned> dof_lookup;
 
      // loop over the nodes
      for (unsigned n = 0; n < n_node; n++)
      {
        // find the number of values at this node
        unsigned nv = this->node_pt(n)->nvalue();
 
        // loop over these values
        for (unsigned v = 0; v < nv; v++)
        {
          // determine local eqn number
          int local_eqn_number = this->nodal_local_eqn(n, v);
 
          // ignore pinned values - far away degrees of freedom resulting
          // from hanging nodes can be ignored since these are be dealt
          // with by the element containing their master nodes
          if (local_eqn_number >= 0)
          {
            // store dof lookup in temporary pair: Global equation number
            // is the first entry in pair
            dof_lookup.first = this->eqn_number(local_eqn_number);
 
            // set dof numbers: Dof number is the second entry in pair
            dof_lookup.second = 0;
 
            // add to list
            dof_lookup_list.push_front(dof_lookup);
          }
        }
      }
    }
 
 
    /// Specify number of dof types for use in block preconditioner
    unsigned ndof_types() const
    {
      return 1;
    }
  };
 
 
  /// ////////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////////
 
 
  //================optimised_identification_of_field_data==================
  /// Overload the function that must return all field data involved
  /// in the interaction with the external (Navier Stokes) element.
  /// Only the velocity dofs in the Navier Stokes element affect the
  /// interaction with the current element.
  //=======================================================================
  template<class AD_ELEMENT, class NST_ELEMENT>
  void RefineableAdvectionDiffusionBoussinesqElement<AD_ELEMENT, NST_ELEMENT>::
    identify_all_field_data_for_external_interaction(
      Vector<std::set<FiniteElement*>> const& external_elements_pt,
      std::set<std::pair<Data*, unsigned>>& paired_interaction_data)
  {
    // There's only one interaction
    const unsigned interaction = 0;
 
    // Loop over each Navier Stokes element in the set of external elements that
    // affect the current element
    for (std::set<FiniteElement*>::iterator it =
           external_elements_pt[interaction].begin();
         it != external_elements_pt[interaction].end();
         it++)
    {
      // Cast the external element to a fluid element
      NST_ELEMENT* external_fluid_el_pt = dynamic_cast<NST_ELEMENT*>(*it);
 
      // Loop over the nodes
      unsigned nnod = external_fluid_el_pt->nnode();
      for (unsigned j = 0; j < nnod; j++)
      {
        // Pointer to node (in its incarnation as Data)
        Data* veloc_data_pt = external_fluid_el_pt->node_pt(j);
 
        // Get all velocity dofs
        const unsigned n_dim = this->dim();
        for (unsigned i = 0; i < n_dim; i++)
        {
          // Which value corresponds to the i-th velocity?
          unsigned val = external_fluid_el_pt->u_index_nst(i);
 
          // Turn pointer to Data and index of value into pair
          // and add to the set
          paired_interaction_data.insert(std::make_pair(veloc_data_pt, val));
        }
      }
    }
  } // done
 
 
  /// //////////////////////////////////////////////////////////////////////
  /// //////////////////////////////////////////////////////////////////////
  /// //////////////////////////////////////////////////////////////////////
 
 
  //======================class definitions==============================
  /// Build NavierStokesBoussinesqElement that inherits from
  /// ElementWithExternalElement so that it can "communicate"
  /// with AdvectionDiffusionElementWithExternalElement
  //=====================================================================
  template<class NST_ELEMENT, class AD_ELEMENT>
  class NavierStokesBoussinesqElement
    : public virtual NST_ELEMENT,
      public virtual ElementWithExternalElement
  {
  private:
    /// Pointer to a private data member, the Rayleigh number
    double* Ra_pt;
 
  public:
    /// Constructor: call the underlying constructors and
    /// initialise the pointer to the Rayleigh number to point
    /// to the default value of 0.0.
    NavierStokesBoussinesqElement()
      : NST_ELEMENT(), ElementWithExternalElement()
    {
      Ra_pt = &MultiDomainBoussinesqHelper::Default_Physical_Constant_Value;
 
      // There is only one interaction
      this->set_ninteraction(1);
    }
 
    /// Access function for the Rayleigh number (const version)
    const double& ra() const
    {
      return *Ra_pt;
    }
 
    /// Access function for the pointer to the Rayleigh number
    double*& ra_pt()
    {
      return Ra_pt;
    }
 
    /// Overload get_body_force_nst to get the temperature "body force"
    /// from the "source" AdvectionDiffusion element via current integration
    /// point
    void get_body_force_nst(const double& time,
                            const unsigned& ipt,
                            const Vector<double>& s,
                            const Vector<double>& x,
                            Vector<double>& result);
 
    /// Fill in the derivatives of the body force with respect to the
    /// external unknowns
    void get_dbody_force_nst_dexternal_element_data(
      const unsigned& ipt,
      DenseMatrix<double>& result,
      Vector<unsigned>& global_eqn_number);
 
 
    /// Compute the element's residual vector and the Jacobian matrix.
    /// Jacobian is computed by finite-differencing or analytically
    void fill_in_contribution_to_jacobian(Vector<double>& residuals,
                                          DenseMatrix<double>& jacobian)
    {
#ifdef USE_FD_JACOBIAN_NST_IN_MULTI_DOMAIN_BOUSSINESQ
 
      // This function computes the Jacobian by finite-differencing
      ElementWithExternalElement::fill_in_contribution_to_jacobian(residuals,
                                                                   jacobian);
 
#else
 
      // Get the contribution from the basic Navier--Stokes element
      NST_ELEMENT::fill_in_contribution_to_jacobian(residuals, jacobian);
 
      // Get the off-diagonal terms analytically
      this->fill_in_off_diagonal_block_analytic(residuals, jacobian);
 
#endif
    }
 
    /// 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 standard (Broken) function
      // which will prevent these elements from being used
      // in eigenproblems until replaced.
      FiniteElement::fill_in_contribution_to_jacobian_and_mass_matrix(
        residuals, jacobian, mass_matrix);
    }
 
    /// Compute the contribution of the external
    /// degrees of freedom (temperatures) on the Navier-Stokes equations
    void fill_in_off_diagonal_block_analytic(Vector<double>& residuals,
                                             DenseMatrix<double>& jacobian)
    {
      // Local storage for the index in the nodes at which the
      // Navier-Stokes velocities are stored (we know that this should be 0,1,2)
      const unsigned n_dim = this->dim();
      Vector<unsigned> u_nodal_nst(n_dim);
      for (unsigned i = 0; i < n_dim; i++)
      {
        u_nodal_nst[i] = this->u_index_nst(i);
      }
 
      // Find out how many nodes there are
      const unsigned n_node = this->nnode();
 
      // Set up memory for the shape and test functions and their derivatives
      Shape psif(n_node), testf(n_node);
      DShape dpsifdx(n_node, n_dim), dtestfdx(n_node, n_dim);
 
      // Number of integration points
      const unsigned n_intpt = this->integral_pt()->nweight();
 
      // Integers to store the local equations and unknowns
      int local_eqn = 0, local_unknown = 0;
 
      // Loop over the integration points
      for (unsigned ipt = 0; ipt < n_intpt; ipt++)
      {
        // Get the integral weight
        double w = this->integral_pt()->weight(ipt);
 
        // Call the derivatives of the shape and test functions
        double J = this->dshape_and_dtest_eulerian_at_knot_nst(
          ipt, psif, dpsifdx, testf, dtestfdx);
 
        // Premultiply the weights and the Jacobian
        double W = w * J;
 
        // Assemble the jacobian terms
 
        // Get the derivatives of the body force wrt the unknowns
        // of the external element
        DenseMatrix<double> dbody_dexternal_element_data;
 
        // Vector of global equation number corresponding to the external
        // element's data
        Vector<unsigned> global_eqn_number_of_external_element_data;
 
        // Get the appropriate derivatives
        this->get_dbody_force_nst_dexternal_element_data(
          ipt,
          dbody_dexternal_element_data,
          global_eqn_number_of_external_element_data);
 
        // Find out how many external data there are
        const unsigned n_external_element_data =
          global_eqn_number_of_external_element_data.size();
 
        // Loop over the test functions
        for (unsigned l = 0; l < n_node; l++)
        {
          // Assemble the contributions of the temperature to
          // the Navier--Stokes equations (which arise through the buoyancy
          // body-force term)
 
          // Loop over the velocity components in the Navier--Stokes equtions
          for (unsigned i = 0; i < n_dim; i++)
          {
            // If it's not a boundary condition
            local_eqn = this->nodal_local_eqn(l, u_nodal_nst[i]);
            if (local_eqn >= 0)
            {
              // Loop over the external data
              for (unsigned l2 = 0; l2 < n_external_element_data; l2++)
              {
                // Find the local equation number corresponding to the global
                // unknown
                local_unknown = this->local_eqn_number(
                  global_eqn_number_of_external_element_data[l2]);
                if (local_unknown >= 0)
                {
                  // Add contribution to jacobian matrix
                  jacobian(local_eqn, local_unknown) +=
                    dbody_dexternal_element_data(i, l2) * testf(l) * W;
                }
              }
            }
          }
        }
      }
    }
  };
 
 
  //============================================================
  /// Overload get_body_force_nst to get the temperature "body force"
  /// from the "source" AdvectionDiffusion element via current integration point
  //========================================================
  template<class NST_ELEMENT, class AD_ELEMENT>
  void NavierStokesBoussinesqElement<NST_ELEMENT, AD_ELEMENT>::
    get_body_force_nst(const double& time,
                       const unsigned& ipt,
                       const Vector<double>& s,
                       const Vector<double>& x,
                       Vector<double>& result)
  {
    // The interaction index is 0 in this case
    unsigned interaction = 0;
 
    // Get the temperature interpolated from the external element
    const double interpolated_t =
      dynamic_cast<AD_ELEMENT*>(external_element_pt(interaction, ipt))
        ->interpolated_u_adv_diff(
          external_element_local_coord(interaction, ipt));
 
    // Get vector that indicates the direction of gravity from
    // the Navier-Stokes equations
    Vector<double> gravity(NST_ELEMENT::g());
 
    // Temperature-dependent body force:
    const unsigned n_dim = this->dim();
    for (unsigned i = 0; i < n_dim; i++)
    {
      result[i] = -gravity[i] * interpolated_t * ra();
    }
  }
 
 
  /// Explicit definition of the face geometry of these elements
  template<class NST_ELEMENT, class AD_ELEMENT>
  class FaceGeometry<NavierStokesBoussinesqElement<NST_ELEMENT, AD_ELEMENT>>
    : public virtual FaceGeometry<NST_ELEMENT>
  {
  public:
    /// Constructor calls the constructor of the NST_ELEMENT
    /// (Only the Intel compiler seems to need this!)
    FaceGeometry() : FaceGeometry<NST_ELEMENT>() {}
 
  protected:
  };
 
  /// Explicit definition of the face geometry of these elements
  template<class NST_ELEMENT, class AD_ELEMENT>
  class FaceGeometry<
    FaceGeometry<NavierStokesBoussinesqElement<NST_ELEMENT, AD_ELEMENT>>>
    : public virtual FaceGeometry<FaceGeometry<NST_ELEMENT>>
  {
  public:
    /// Constructor calls the constructor of the NST_ELEMENT
    /// (Only the Intel compiler seems to need this!)
    FaceGeometry() : FaceGeometry<FaceGeometry<NST_ELEMENT>>() {}
 
  protected:
  };
 
 
  /// ///////////////////////////////////////////////////////////////////////
  /// ///////////////////////////////////////////////////////////////////////
  /// ///////////////////////////////////////////////////////////////////////
 
 
  //======================class definitions==============================
  /// Build AdvectionDiffusionBoussinesqElement that inherits from
  /// ElementWithExternalElement so that it can "communicate" with the
  /// Navier Stokes element
  //=====================================================================
  template<class AD_ELEMENT, class NST_ELEMENT>
  class AdvectionDiffusionBoussinesqElement
    : public virtual AD_ELEMENT,
      public virtual ElementWithExternalElement
  {
  public:
    /// Constructor: call the underlying constructors
    AdvectionDiffusionBoussinesqElement()
      : AD_ELEMENT(), ElementWithExternalElement()
    {
      // There is only one interaction
      this->set_ninteraction(1);
    }
 
    /// Overload the wind function in the advection-diffusion equations.
    /// This provides the coupling from the Navier--Stokes equations to the
    /// advection-diffusion equations because the wind is the fluid velocity,
    /// obtained from the source element in the other mesh
    void get_wind_adv_diff(const unsigned& ipt,
                           const Vector<double>& s,
                           const Vector<double>& x,
                           Vector<double>& wind) const;
 
 
    //-----------------------------------------------------------
    // Note: we're overloading the output functions because the
    // =====  underlying advection diffusion elements output the
    //       wind which is only available at the integration points
    //       and even then only if the multi domain interaction
    //       has been set up.
    //-----------------------------------------------------------
 
    /// Output function:
    ///  Output x, y, theta at Nplot^DIM plot points
    // Start of output function
    void output(std::ostream& outfile, const unsigned& nplot)
    {
      // Element dimension
      const unsigned n_dim = this->dim();
 
      // vector of local coordinates
      Vector<double> s(n_dim);
 
      // Tecplot header info
      outfile << this->tecplot_zone_string(nplot);
 
      // Loop over plot points
      unsigned num_plot_points = this->nplot_points(nplot);
      for (unsigned iplot = 0; iplot < num_plot_points; iplot++)
      {
        // Get local coordinates of plot point
        this->get_s_plot(iplot, nplot, s);
 
        // Output the position of the plot point
        for (unsigned i = 0; i < n_dim; i++)
        {
          outfile << this->interpolated_x(s, i) << " ";
        }
 
        // Output the temperature (the advected variable) at the plot point
        outfile << AD_ELEMENT::interpolated_u_adv_diff(s) << std::endl;
      }
      outfile << std::endl;
 
      // Write tecplot footer (e.g. FE connectivity lists)
      this->write_tecplot_zone_footer(outfile, nplot);
 
    } // End of output function
 
 
    ///  Overload the standard output function with the broken default
    void output(std::ostream& outfile)
    {
      FiniteElement::output(outfile);
    }
 
    /// C-style output function: Broken default
    void output(FILE* file_pt)
    {
      FiniteElement::output(file_pt);
    }
 
    ///  C-style output function: Broken default
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      FiniteElement::output(file_pt, n_plot);
    }
 
 
    /// Fill in the derivatives of the wind with respect to the
    /// external unknowns
    void get_dwind_adv_diff_dexternal_element_data(
      const unsigned& ipt,
      const unsigned& i,
      Vector<double>& result,
      Vector<unsigned>& global_eqn_number);
 
 
    /// Compute the element's residual vector and the Jacobian matrix.
    /// Jacobian is computed by finite-differencing.
    void fill_in_contribution_to_jacobian(Vector<double>& residuals,
                                          DenseMatrix<double>& jacobian)
    {
#ifdef USE_FD_JACOBIAN_IN_MULTI_DOMAIN_BOUSSINESQ
 
      // This function computes the Jacobian by finite-differencing
      ElementWithExternalElement::fill_in_contribution_to_jacobian(residuals,
                                                                   jacobian);
 
#else
 
      // Get the contribution from the basic advection-diffusion element
      AD_ELEMENT::fill_in_contribution_to_jacobian(residuals, jacobian);
 
      // Get the off-diagonal terms analytically
      this->fill_in_off_diagonal_block_analytic(residuals, jacobian);
 
#endif
    }
 
    /// 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 standard (Broken) function
      // which will prevent these elements from being used
      // in eigenproblems until replaced.
      FiniteElement::fill_in_contribution_to_jacobian_and_mass_matrix(
        residuals, jacobian, mass_matrix);
    }
 
 
    /// Compute the contribution of the external
    /// degrees of freedom (velocities) on the AdvectionDiffusion equations
    void fill_in_off_diagonal_block_analytic(Vector<double>& residuals,
                                             DenseMatrix<double>& jacobian)
    {
      // Local storage for the  index at which the temperature is stored
      const unsigned u_nodal_adv_diff = this->u_index_adv_diff();
 
      // Find out how many nodes there are
      const unsigned n_node = this->nnode();
 
      // Element dimension
      const unsigned n_dim = this->dim();
 
      // Set up memory for the shape and test functions and their derivatives
      Shape psi(n_node), test(n_node);
      DShape dpsidx(n_node, n_dim), dtestdx(n_node, n_dim);
 
      // Number of integration points
      const unsigned n_intpt = this->integral_pt()->nweight();
 
      // Integers to store the local equations and unknowns
      int local_eqn = 0, local_unknown = 0;
 
      // Get the peclet number
      const double peclet = this->pe();
 
      // Loop over the integration points
      for (unsigned ipt = 0; ipt < n_intpt; ipt++)
      {
        // Get the integral weight
        double w = this->integral_pt()->weight(ipt);
 
        // Call the derivatives of the shape and test functions
        double J = this->dshape_and_dtest_eulerian_at_knot_adv_diff(
          ipt, psi, dpsidx, test, dtestdx);
 
        // Premultiply the weights and the Jacobian
        double W = w * J;
 
        // Calculate local values of the derivatives of the solution
        Vector<double> interpolated_dudx(n_dim, 0.0);
        // Loop over nodes
        for (unsigned l = 0; l < n_node; l++)
        {
          // Loop over directions
          for (unsigned j = 0; j < n_dim; j++)
          {
            interpolated_dudx[j] +=
              this->raw_nodal_value(l, u_nodal_adv_diff) * dpsidx(l, j);
          }
        }
 
        // Get the derivatives of the wind wrt the unknowns
        // of the external element
        Vector<double> dwind_dexternal_element_data;
 
        // Vector of global equation number corresponding to the external
        // element's data
        Vector<unsigned> global_eqn_number_of_external_element_data;
 
 
        // Loop over the wind directions
        for (unsigned i2 = 0; i2 < n_dim; i2++)
        {
          // Get the appropriate derivatives
          this->get_dwind_adv_diff_dexternal_element_data(
            ipt,
            i2,
            dwind_dexternal_element_data,
            global_eqn_number_of_external_element_data);
 
          // Find out how many external data there are
          const unsigned n_external_element_data =
            global_eqn_number_of_external_element_data.size();
 
          // Loop over the test functions
          for (unsigned l = 0; l < n_node; l++)
          {
            // Assemble the contributions of the velocities
            // the Advection-Diffusion equations
 
            // If it's not a boundary condition
            local_eqn = this->nodal_local_eqn(l, u_nodal_adv_diff);
            if (local_eqn >= 0)
            {
              // Loop over the external data
              for (unsigned l2 = 0; l2 < n_external_element_data; l2++)
              {
                // Find the local equation number corresponding to the global
                // unknown
                local_unknown = this->local_eqn_number(
                  global_eqn_number_of_external_element_data[l2]);
                if (local_unknown >= 0)
                {
                  // Add contribution to jacobian matrix
                  jacobian(local_eqn, local_unknown) -=
                    peclet * dwind_dexternal_element_data[l2] *
                    interpolated_dudx[i2] * test(l) * W;
                }
              }
            }
          }
        }
      }
    }
  };
 
 
  //==========================================================================
  /// Overload the wind function in the advection-diffusion equations.
  /// This provides the coupling from the Navier--Stokes equations to the
  /// advection-diffusion equations because the wind is the fluid velocity,
  /// obtained from the source elements in the other mesh
  //==========================================================================
  template<class AD_ELEMENT, class NST_ELEMENT>
  void AdvectionDiffusionBoussinesqElement<AD_ELEMENT, NST_ELEMENT>::
    get_wind_adv_diff(const unsigned& ipt,
                      const Vector<double>& s,
                      const Vector<double>& x,
                      Vector<double>& wind) const
  {
    // The interaction index is 0 in this case
    unsigned interaction = 0;
 
    // Dynamic cast "other" element to correct type
    NST_ELEMENT* source_el_pt =
      dynamic_cast<NST_ELEMENT*>(external_element_pt(interaction, ipt));
 
    // The wind function is simply the velocity at the points of the "other" el
    source_el_pt->interpolated_u_nst(
      external_element_local_coord(interaction, ipt), wind);
  }
 
  //=========================================================================
  /// Fill in the derivatives of the wind with respect to the external
  /// unknowns in the advection-diffusion equations
  //=========================================================================
  template<class AD_ELEMENT, class NST_ELEMENT>
  void AdvectionDiffusionBoussinesqElement<AD_ELEMENT, NST_ELEMENT>::
    get_dwind_adv_diff_dexternal_element_data(
      const unsigned& ipt,
      const unsigned& i,
      Vector<double>& result,
      Vector<unsigned>& global_eqn_number)
  {
    // The interaction index is 0 in this case
    unsigned interaction = 0;
 
    // Dynamic cast "other" element to correct type
    NST_ELEMENT* source_el_pt =
      dynamic_cast<NST_ELEMENT*>(external_element_pt(interaction, ipt));
 
    // Get the external element's derivatives of the velocity with respect
    // to the data. The wind is just the Navier--Stokes velocity, so this
    // is all that's required
    source_el_pt->dinterpolated_u_nst_ddata(
      external_element_local_coord(interaction, ipt),
      i,
      result,
      global_eqn_number);
  }
 
  /// ///////////////////////////////////////////////////////////////////
  /// ///////////////////////////////////////////////////////////////////
  /// ///////////////////////////////////////////////////////////////////
 
  //=========================================================================
  /// Fill in the derivatives of the body force with respect to the external
  /// unknowns in the Navier--Stokes equations
  //=========================================================================
  template<class NST_ELEMENT, class AD_ELEMENT>
  void NavierStokesBoussinesqElement<NST_ELEMENT, AD_ELEMENT>::
    get_dbody_force_nst_dexternal_element_data(
      const unsigned& ipt,
      DenseMatrix<double>& result,
      Vector<unsigned>& global_eqn_number)
  {
    // Get vector that indicates the direction of gravity from
    // the Navier-Stokes equations
    Vector<double> gravity(NST_ELEMENT::g());
 
    // The interaction index is 0 in this case
    unsigned interaction = 0;
 
    // Get the external element's derivatives
    Vector<double> du_adv_diff_ddata;
 
    // Dynamic cast "other" element to correct type
    AD_ELEMENT* source_el_pt =
      dynamic_cast<AD_ELEMENT*>(external_element_pt(interaction, ipt));
#ifdef PARANOID
    if (source_el_pt == 0)
    {
      throw OomphLibError("External element could not be cast to AD_ELEMENT\n",
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
#endif
 
    // Get derivatives
    source_el_pt->dinterpolated_u_adv_diff_ddata(
      external_element_local_coord(interaction, ipt),
      du_adv_diff_ddata,
      global_eqn_number);
 
    // Find the number of external data
    unsigned n_external_element_data = du_adv_diff_ddata.size();
 
    // Set the size of the matrix to be returned
    const unsigned n_dim = this->dim();
    result.resize(n_dim, n_external_element_data);
 
    // Temperature-dependent body force:
    for (unsigned i = 0; i < n_dim; i++)
    {
      // Loop over the external data
      for (unsigned n = 0; n < n_external_element_data; n++)
      {
        result(i, n) = -gravity[i] * du_adv_diff_ddata[n] * ra();
      }
    }
  }
 
 
  //==========start_of_get_dbody_force=========== ===========================
  /// Fill in the derivatives of the body force with respect to the external
  /// unknowns in the Navier--Stokes equations
  //=========================================================================
  template<class NST_ELEMENT, class AD_ELEMENT>
  void RefineableNavierStokesBoussinesqElement<NST_ELEMENT, AD_ELEMENT>::
    get_dbody_force_nst_dexternal_element_data(
      const unsigned& ipt,
      DenseMatrix<double>& result,
      Vector<unsigned>& global_eqn_number)
  {
    // Get vector that indicates the direction of gravity from
    // the Navier-Stokes equations
    Vector<double> gravity(NST_ELEMENT::g());
 
    // The interaction index is 0 in this case
    unsigned interaction = 0;
 
    // Get the external element's derivatives
    Vector<double> du_adv_diff_ddata;
 
    // Dynamic cast "other" element to correct type
    AD_ELEMENT* source_el_pt =
      dynamic_cast<AD_ELEMENT*>(external_element_pt(interaction, ipt));
#ifdef PARANOID
    if (source_el_pt == 0)
    {
      throw OomphLibError("External element could not be cast to AD_ELEMENT\n",
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
#endif
 
    // Get derivatives
    source_el_pt->dinterpolated_u_adv_diff_ddata(
      external_element_local_coord(interaction, ipt),
      du_adv_diff_ddata,
      global_eqn_number);
 
    // Find the number of external data
    unsigned n_external_element_data = du_adv_diff_ddata.size();
 
    // Set the size of the matrix to be returned
    const unsigned n_dim = this->dim();
    result.resize(n_dim, n_external_element_data);
 
    // Temperature-dependent body force:
    for (unsigned i = 0; i < n_dim; i++)
    {
      // Loop over the external data
      for (unsigned n = 0; n < n_external_element_data; n++)
      {
        result(i, n) = -gravity[i] * du_adv_diff_ddata[n] * ra();
      }
    }
 
  } // end_of_get_dbody_force
 
} // namespace oomph
 
#endif