specific_node_update_interface_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
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// LIC// 02110-1301  USA.
// LIC//
// LIC// The authors may be contacted at oomph-lib@maths.man.ac.uk.
// LIC//
// LIC//====================================================================
// Header file for specific (two-dimensional) fluid free surface elements
 
// Include guards, to prevent multiple includes
#ifndef OOMPH_SPECIFIC_NODE_UPDATE_INTERFACE_ELEMENTS_HEADER
#define OOMPH_SPECIFIC_NODE_UPDATE_INTERFACE_ELEMENTS_HEADER
 
// Config header generated by autoconfig
#ifdef HAVE_CONFIG_H
#include <oomph-lib-config.h>
#endif
 
// OOMPH-LIB headers
#include "../generic/Qelements.h"
#include "../generic/spines.h"
#include "../generic/hijacked_elements.h"
#include "interface_elements.h"
 
namespace oomph
{
  //=======================================================================
  //=======================================================================
  template<class ELEMENT>
  class BoundingElementType
  {
  };
 
  //======================================================================
  //=====================================================================
  template<class ELEMENT>
  class FluidInterfaceAdditionalValues
  {
  private:
    // Issue an error message if this base class is called
    void error_message()
    {
      std::ostringstream error_message;
      error_message
        << "The generic FluidInterfaceAdditionalValues<ELEMENT> class has "
           "been\n"
        << "called. This should never happen. If you are creating your own\n"
        << "surface equations you must overload the policy class to specify\n"
        << "how many additional values are required at the surface nodes.\n"
        << "For an example, see "
           "src/fluid_interface/surfactant_transport_elements.h\n"
        << std::endl;
      throw OomphLibError(
        error_message.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
 
  public:
    FluidInterfaceAdditionalValues<ELEMENT>() {}
 
    inline unsigned nadditional_values(const unsigned& n)
    {
      error_message();
      return 0;
    }
 
    inline void setup_equation_indices(ELEMENT* const& element_pt,
                                       const unsigned& id)
    {
      error_message();
    }
  };
 
 
  //======================================================================
  //=====================================================================
  template<>
  class FluidInterfaceAdditionalValues<FluidInterfaceElement>
  {
  public:
    FluidInterfaceAdditionalValues<FluidInterfaceElement>() {}
 
    inline unsigned nadditional_values(const unsigned& n)
    {
      return 0;
    }
 
    inline void setup_equation_indices(FluidInterfaceElement* const& element_pt,
                                       const unsigned& id)
    {
    }
  };
 
 
  //-------------SPINE NODE UPDATE CLASSES-------------------------------
  //---------------------------------------------------------------------
 
  //======================================================================
  //======================================================================
  template<class EQUATION_CLASS, class DERIVATIVE_CLASS, class ELEMENT>
  class SpineUpdateFluidInterfaceElement
    : public virtual Hijacked<SpineElement<FaceGeometry<ELEMENT>>>,
      public virtual EQUATION_CLASS,
      public DERIVATIVE_CLASS
  {
  private:
    int kinematic_local_eqn(const unsigned& n)
    {
      return this->spine_local_eqn(n);
    }
 
    void hijack_kinematic_conditions(const Vector<unsigned>& bulk_node_number)
    {
      // Loop over all the node numbers that are passed in
      for (Vector<unsigned>::const_iterator it = bulk_node_number.begin();
           it != bulk_node_number.end();
           ++it)
      {
        // Hijack the spine heights. (and delete the returned data object)
        delete this->hijack_nodal_spine_value(*it, 0);
      }
    }
 
  protected:
    double compute_surface_derivatives(
      const Shape& psi,
      const DShape& dpsids,
      const DenseMatrix<double>& interpolated_t,
      const Vector<double>& interpolated_x,
      DShape& surface_gradient,
      DShape& surface_divergence)
    {
      return DERIVATIVE_CLASS::compute_surface_derivatives(psi,
                                                           dpsids,
                                                           interpolated_t,
                                                           interpolated_x,
                                                           surface_gradient,
                                                           surface_divergence);
    }
 
 
  public:
    SpineUpdateFluidInterfaceElement(FiniteElement* const& element_pt,
                                     const int& face_index,
                                     const unsigned& id = 0)
      : Hijacked<SpineElement<FaceGeometry<ELEMENT>>>(),
        EQUATION_CLASS(),
        DERIVATIVE_CLASS()
    {
      // Attach the geometrical information to the element, by
      // making the face element from the bulk element
      element_pt->build_face_element(face_index, this);
 
#ifdef PARANOID
      // Is it refineable
      RefineableElement* ref_el_pt =
        dynamic_cast<RefineableElement*>(element_pt);
      if (ref_el_pt != 0)
      {
        if (this->has_hanging_nodes())
        {
          throw OomphLibError("This interface element will not work correctly "
                              "if nodes are hanging\n",
                              OOMPH_CURRENT_FUNCTION,
                              OOMPH_EXCEPTION_LOCATION);
        }
      }
#endif
 
      // Find the index at which the velocity unknowns are stored
      // from the bulk element and allocate the local storage
      ELEMENT* cast_element_pt = dynamic_cast<ELEMENT*>(element_pt);
      // Find number of momentum equation required
      const unsigned n_u_index = cast_element_pt->n_u_nst();
      this->U_index_interface.resize(n_u_index);
      for (unsigned i = 0; i < n_u_index; i++)
      {
        this->U_index_interface[i] = cast_element_pt->u_index_nst(i);
      }
 
      // Add any additional values required by the equations class
      unsigned n_node_face = this->nnode();
      // Create an instance of the policy class that determines
      // how many additional values are required
      FluidInterfaceAdditionalValues<EQUATION_CLASS>*
        interface_additional_values_pt =
          new FluidInterfaceAdditionalValues<EQUATION_CLASS>();
      // Do we need to add any values
      // By default the spines don't need to so we can save
      // some effort here
      bool add_values = false;
      // Storage for the number of additional values required
      // for each node
      Vector<unsigned> additional_data_values(n_node_face);
      for (unsigned i = 0; i < n_node_face; ++i)
      {
        // Read out additional values for each node
        additional_data_values[i] =
          interface_additional_values_pt->nadditional_values(i);
        // If any of these are non-zero it will set the boolean to true
        add_values |= additional_data_values[i];
      }
 
      // If storage is needed allocate it and call
      // the associated local index setup function
      if (add_values)
      {
        this->add_additional_values(additional_data_values, id);
        // Call any local setup from the interface policy class
        interface_additional_values_pt->setup_equation_indices(this, id);
      }
 
      // Delete the policy class, it's work is done.
      delete interface_additional_values_pt;
      interface_additional_values_pt = 0;
    }
 
    void fill_in_contribution_to_jacobian(Vector<double>& residuals,
                                          DenseMatrix<double>& jacobian)
    {
      // Call the generic routine with the flag set to 1
      EQUATION_CLASS::fill_in_generic_residual_contribution_interface(
        residuals, jacobian, 1);
 
      // Call the generic routine to handle the shape derivatives
      this->fill_in_jacobian_from_geometric_data(jacobian);
    }
 
    void add_additional_residual_contributions_interface(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      const unsigned& flag,
      const Shape& psif,
      const DShape& dpsifds,
      const DShape& dpsifdS,
      const DShape& dpsifdS_div,
      const Vector<double>& s,
      const Vector<double>& interpolated_x,
      const Vector<double>& interpolated_n,
      const double& W,
      const double& J)
    {
      EQUATION_CLASS::add_additional_residual_contributions_interface(
        residuals,
        jacobian,
        flag,
        psif,
        dpsifds,
        dpsifdS,
        dpsifdS_div,
        s,
        interpolated_x,
        interpolated_n,
        W,
        J);
    }
 
    void output(std::ostream& outfile)
    {
      EQUATION_CLASS::output(outfile);
    }
 
    void output(std::ostream& outfile, const unsigned& n_plot)
    {
      EQUATION_CLASS::output(outfile, n_plot);
    }
 
    void output(FILE* file_pt)
    {
      EQUATION_CLASS::output(file_pt);
    }
 
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      EQUATION_CLASS::output(file_pt, n_plot);
    }
 
 
    virtual FluidInterfaceBoundingElement* make_bounding_element(
      const int& face_index)
    {
      // Create a temporary pointer to the appropriate FaceElement read our from
      // our policy class
      BoundingElementType<SpineUpdateFluidInterfaceElement<EQUATION_CLASS,
                                                           DERIVATIVE_CLASS,
                                                           ELEMENT>>*
        face_el_pt = new BoundingElementType<
          SpineUpdateFluidInterfaceElement<EQUATION_CLASS,
                                           DERIVATIVE_CLASS,
                                           ELEMENT>>;
 
      // Attach the geometrical information to the new element
      this->build_face_element(face_index, face_el_pt);
 
      // Set the index at which the velocity nodes are stored
      face_el_pt->u_index_interface_boundary() = this->U_index_interface;
 
      // Set the value of the nbulk_value, the node is not resized
      // in this bounding element,
      // so it will just be the actual nvalue here
      // There is some ambiguity about what this means (however)
      // We are interpreting it to mean the number of
      // values in this FaceElement before creating the new
      // bounding element.
      const unsigned n_node_bounding = face_el_pt->nnode();
      for (unsigned n = 0; n < n_node_bounding; n++)
      {
        face_el_pt->nbulk_value(n) = face_el_pt->node_pt(n)->nvalue();
      }
 
      // Set of unique geometric data that is used to update the bulk,
      // but is not used to update the face
      std::set<Data*> unique_additional_geom_data;
 
      // Get all the geometric data for this (bulk) element
      this->assemble_set_of_all_geometric_data(unique_additional_geom_data);
 
      // Now assemble the set of geometric data for the face element
      std::set<Data*> unique_face_geom_data_pt;
      face_el_pt->assemble_set_of_all_geometric_data(unique_face_geom_data_pt);
 
      // Erase the face geometric data from the additional data
      for (std::set<Data*>::iterator it = unique_face_geom_data_pt.begin();
           it != unique_face_geom_data_pt.end();
           ++it)
      {
        unique_additional_geom_data.erase(*it);
      }
 
      // Finally add all unique additional data as external data
      for (std::set<Data*>::iterator it = unique_additional_geom_data.begin();
           it != unique_additional_geom_data.end();
           ++it)
      {
        face_el_pt->add_external_data(*it);
      }
 
      // Return the value of the pointer
      return face_el_pt;
    }
  };
 
 
  //=====================================================================
  //=====================================================================
  template<class ELEMENT>
  class SpinePointFluidInterfaceBoundingElement
    : public Hijacked<SpineElement<FaceGeometry<FaceGeometry<ELEMENT>>>>,
      public PointFluidInterfaceBoundingElement
 
  {
  public:
    SpinePointFluidInterfaceBoundingElement()
      : Hijacked<SpineElement<FaceGeometry<FaceGeometry<ELEMENT>>>>(),
        PointFluidInterfaceBoundingElement()
    {
    }
 
    void output(std::ostream& outfile)
    {
      FiniteElement::output(outfile);
    }
 
    void output(std::ostream& outfile, const unsigned& n_plot)
    {
      FluidInterfaceBoundingElement::output(outfile, n_plot);
    }
 
    void output(FILE* file_pt)
    {
      FiniteElement::output(file_pt);
    }
 
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      FluidInterfaceBoundingElement::output(file_pt, n_plot);
    }
 
    void fill_in_contribution_to_jacobian(Vector<double>& residuals,
                                          DenseMatrix<double>& jacobian)
    {
      // Call the generic routine with the flag set to 1
      this->fill_in_generic_residual_contribution_interface_boundary(
        residuals, jacobian, 1);
      // Call generic FD routine for the external data
      this->fill_in_jacobian_from_external_by_fd(jacobian);
 
      // Call the generic routine to handle the spine variables
      this->fill_in_jacobian_from_geometric_data(jacobian);
    }
 
    int kinematic_local_eqn(const unsigned& n)
    {
      return this->spine_local_eqn(n);
    }
  };
 
 
  //=========================================================================
  //========================================================================
  template<class ELEMENT>
  class SpineLineFluidInterfaceBoundingElement
    : public Hijacked<SpineElement<FaceGeometry<FaceGeometry<ELEMENT>>>>,
      public LineFluidInterfaceBoundingElement
 
  {
  public:
    SpineLineFluidInterfaceBoundingElement()
      : Hijacked<SpineElement<FaceGeometry<FaceGeometry<ELEMENT>>>>(),
        LineFluidInterfaceBoundingElement()
    {
    }
 
    void output(std::ostream& outfile)
    {
      FiniteElement::output(outfile);
    }
 
    void output(std::ostream& outfile, const unsigned& n_plot)
    {
      FluidInterfaceBoundingElement::output(outfile, n_plot);
    }
 
    void output(FILE* file_pt)
    {
      FiniteElement::output(file_pt);
    }
 
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      FluidInterfaceBoundingElement::output(file_pt, n_plot);
    }
 
 
    void fill_in_contribution_to_jacobian(Vector<double>& residuals,
                                          DenseMatrix<double>& jacobian)
    {
      // Call the generic routine with the flag set to 1
      this->fill_in_generic_residual_contribution_interface_boundary(
        residuals, jacobian, 1);
      // Call generic FD routine for the external data
      this->fill_in_jacobian_from_external_by_fd(jacobian);
 
      // Call the generic routine to handle the spine variables
      this->fill_in_jacobian_from_geometric_data(jacobian);
    }
 
 
    int kinematic_local_eqn(const unsigned& n)
    {
      // Kinematic bc is always associated with the n-th spine height
      return this->spine_local_eqn(n);
    }
  };
 
 
  //============GEOMETRIC SPECIALISATIONS============================
 
 
  // Specialise the spine update template class to concrete 1D case
  template<class ELEMENT>
  class SpineLineFluidInterfaceElement
    : public SpineUpdateFluidInterfaceElement<FluidInterfaceElement,
                                              LineDerivatives,
                                              ELEMENT>
  {
  public:
    SpineLineFluidInterfaceElement(FiniteElement* const& element_pt,
                                   const int& face_index)
      : SpineUpdateFluidInterfaceElement<FluidInterfaceElement,
                                         LineDerivatives,
                                         ELEMENT>(element_pt, face_index)
    {
    }
  };
 
 
  // Define the BoundingElement type associated with the 1D surface element
  template<class ELEMENT>
  class BoundingElementType<
    SpineUpdateFluidInterfaceElement<FluidInterfaceElement,
                                     LineDerivatives,
                                     ELEMENT>>
    : public SpinePointFluidInterfaceBoundingElement<ELEMENT>
  {
  public:
    BoundingElementType<SpineUpdateFluidInterfaceElement<FluidInterfaceElement,
                                                         LineDerivatives,
                                                         ELEMENT>>()
      : SpinePointFluidInterfaceBoundingElement<ELEMENT>()
    {
    }
  };
 
 
  // Specialise Spine update case to concrete axisymmetric case
  template<class ELEMENT>
  class SpineAxisymmetricFluidInterfaceElement
    : public SpineUpdateFluidInterfaceElement<FluidInterfaceElement,
                                              AxisymmetricDerivatives,
                                              ELEMENT>
  {
  public:
    SpineAxisymmetricFluidInterfaceElement(FiniteElement* const& element_pt,
                                           const int& face_index)
      : SpineUpdateFluidInterfaceElement<FluidInterfaceElement,
                                         AxisymmetricDerivatives,
                                         ELEMENT>(element_pt, face_index)
    {
    }
  };
 
  // Define the bounding element associated with the axisymmetric fluid
  // interface element
  template<class ELEMENT>
  class BoundingElementType<
    SpineUpdateFluidInterfaceElement<FluidInterfaceElement,
                                     AxisymmetricDerivatives,
                                     ELEMENT>>
    : public SpinePointFluidInterfaceBoundingElement<ELEMENT>
  {
  public:
    BoundingElementType<
      SpineUpdateFluidInterfaceElement<FluidInterfaceElement,
                                       AxisymmetricDerivatives,
                                       ELEMENT>>()
      : SpinePointFluidInterfaceBoundingElement<ELEMENT>()
    {
    }
  };
 
  // Specialise Spine update case to concrete 2D case
  template<class ELEMENT>
  class SpineSurfaceFluidInterfaceElement
    : public SpineUpdateFluidInterfaceElement<FluidInterfaceElement,
                                              SurfaceDerivatives,
                                              ELEMENT>
  {
  public:
    SpineSurfaceFluidInterfaceElement(FiniteElement* const& element_pt,
                                      const int& face_index)
      : SpineUpdateFluidInterfaceElement<FluidInterfaceElement,
                                         SurfaceDerivatives,
                                         ELEMENT>(element_pt, face_index)
    {
    }
  };
 
  // Define the bounding element type for the 2D surface
  template<class ELEMENT>
  class BoundingElementType<
    SpineUpdateFluidInterfaceElement<FluidInterfaceElement,
                                     SurfaceDerivatives,
                                     ELEMENT>>
    : public SpineLineFluidInterfaceBoundingElement<ELEMENT>
  {
  public:
    BoundingElementType<SpineUpdateFluidInterfaceElement<FluidInterfaceElement,
                                                         SurfaceDerivatives,
                                                         ELEMENT>>()
      : SpineLineFluidInterfaceBoundingElement<ELEMENT>()
    {
    }
  };
 
 
 
  //-------------ELASTIC NODE UPDATE CLASSES-------------------------------
  //---------------------------------------------------------------------
 
 
  //=======================================================================
  //======================================================================
  template<class EQUATION_CLASS, class DERIVATIVE_CLASS, class ELEMENT>
  class ElasticUpdateFluidInterfaceElement
    : public virtual Hijacked<FaceGeometry<ELEMENT>>,
      public EQUATION_CLASS,
      public DERIVATIVE_CLASS
  {
  private:
    Vector<unsigned> Lagrange_index;
 
    inline unsigned lagrange_index(const unsigned& n)
    {
      return this->Lagrange_index[n];
    }
 
    inline int kinematic_local_eqn(const unsigned& n)
    {
      // Get the index of the nodal value associated with Lagrange multiplier
      return this->nodal_local_eqn(n, this->lagrange_index(n));
    }
 
    void hijack_kinematic_conditions(const Vector<unsigned>& bulk_node_number)
    {
      // Loop over all the nodes that are passed in
      for (Vector<unsigned>::const_iterator it = bulk_node_number.begin();
           it != bulk_node_number.end();
           ++it)
      {
        // Hijack the appropriate value and delete the returned Node
        delete this->hijack_nodal_value(*it, this->lagrange_index(*it));
      }
    }
 
  protected:
    double compute_surface_derivatives(
      const Shape& psi,
      const DShape& dpsids,
      const DenseMatrix<double>& interpolated_t,
      const Vector<double>& interpolated_x,
      DShape& surface_gradient,
      DShape& surface_divergence)
    {
      return DERIVATIVE_CLASS::compute_surface_derivatives(psi,
                                                           dpsids,
                                                           interpolated_t,
                                                           interpolated_x,
                                                           surface_gradient,
                                                           surface_divergence);
    }
 
 
  public:
    ElasticUpdateFluidInterfaceElement(FiniteElement* const& element_pt,
                                       const int& face_index,
                                       const unsigned& id = 0)
      : FaceGeometry<ELEMENT>(), EQUATION_CLASS(), DERIVATIVE_CLASS()
    {
      // Attach the geometrical information to the element
      // This function also assigned nbulk_value from required_nvalue of the
      // bulk element
      element_pt->build_face_element(face_index, this);
 
#ifdef PARANOID
      // Is it refineable
      RefineableElement* ref_el_pt =
        dynamic_cast<RefineableElement*>(element_pt);
      if (ref_el_pt != 0)
      {
        if (this->has_hanging_nodes())
        {
          throw OomphLibError(
            "This flux element will not work correctly if nodes are hanging\n",
            OOMPH_CURRENT_FUNCTION,
            OOMPH_EXCEPTION_LOCATION);
        }
      }
#endif
 
      // Find the index at which the velocity unknowns are stored
      // from the bulk element and resize the local storage scheme
      ELEMENT* cast_element_pt = dynamic_cast<ELEMENT*>(element_pt);
      const unsigned n_u_index = cast_element_pt->n_u_nst();
      this->U_index_interface.resize(n_u_index);
      for (unsigned i = 0; i < n_u_index; i++)
      {
        this->U_index_interface[i] = cast_element_pt->u_index_nst(i);
      }
 
      // Read out the number of nodes on the face
      unsigned n_node_face = this->nnode();
 
      // Create an instance of the policy class that determines
      // how many additional values are required
      FluidInterfaceAdditionalValues<EQUATION_CLASS>*
        interface_additional_values_pt =
          new FluidInterfaceAdditionalValues<EQUATION_CLASS>();
 
      // Set the additional data values in the face
      // There is always also one additional values at each node --- the
      // Lagrange multiplier
      Vector<unsigned> additional_data_values(n_node_face);
      for (unsigned n = 0; n < n_node_face; n++)
      {
        // Now add one to the addtional values at every single node
        additional_data_values[n] =
          interface_additional_values_pt->nadditional_values(n) + 1;
      }
 
      // Now add storage for Lagrange multipliers and set the map containing
      // the position of the first entry of this face element's
      // additional values.
      this->add_additional_values(additional_data_values, id);
 
      // Now I can just store the lagrange index offset to give the storage
      // location of the nodes
      Lagrange_index.resize(n_node_face);
      for (unsigned n = 0; n < n_node_face; ++n)
      {
        Lagrange_index[n] =
          additional_data_values[n] - 1 +
          dynamic_cast<BoundaryNodeBase*>(this->node_pt(n))
            ->index_of_first_value_assigned_by_face_element(id);
      }
 
      // Call any local setup from the interface policy class
      interface_additional_values_pt->setup_equation_indices(this, id);
 
      // Can now delete the policy class
      delete interface_additional_values_pt;
      interface_additional_values_pt = 0;
    }
 
    double zeta_nodal(const unsigned& n,
                      const unsigned& k,
                      const unsigned& i) const
    {
      return FaceElement::zeta_nodal(n, k, i);
    }
 
    double& lagrange(const unsigned& n)
    {
      return *this->node_pt(n)->value_pt(this->lagrange_index(n));
    }
 
 
    void fill_in_contribution_to_jacobian(Vector<double>& residuals,
                                          DenseMatrix<double>& jacobian)
    {
      // Call the generic routine with the flag set to 1
      EQUATION_CLASS::fill_in_generic_residual_contribution_interface(
        residuals, jacobian, 1);
 
      // Call the generic finite difference routine for the solid variables
      this->fill_in_jacobian_from_solid_position_by_fd(jacobian);
    }
 
    void output(std::ostream& outfile)
    {
      EQUATION_CLASS::output(outfile);
    }
 
    void output(std::ostream& outfile, const unsigned& n_plot)
    {
      EQUATION_CLASS::output(outfile, n_plot);
    }
 
    void output(FILE* file_pt)
    {
      EQUATION_CLASS::output(file_pt);
    }
 
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      EQUATION_CLASS::output(file_pt, n_plot);
    }
 
 
    void add_additional_residual_contributions_interface(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      const unsigned& flag,
      const Shape& psif,
      const DShape& dpsifds,
      const DShape& dpsifdS,
      const DShape& dpsifdS_div,
      const Vector<double>& s,
      const Vector<double>& interpolated_x,
      const Vector<double>& interpolated_n,
      const double& W,
      const double& J)
    {
      EQUATION_CLASS::add_additional_residual_contributions_interface(
        residuals,
        jacobian,
        flag,
        psif,
        dpsifds,
        dpsifdS,
        dpsifdS_div,
        s,
        interpolated_x,
        interpolated_n,
        W,
        J);
 
      // Assemble Lagrange multiplier by loop over the shape functions
      const unsigned n_node = this->nnode();
      // Read out the dimension of the node
      const unsigned nodal_dimension = this->nodal_dimension();
 
      double interpolated_lagrange = 0.0;
      for (unsigned l = 0; l < n_node; l++)
      {
        // Note same shape functions used for lagrange multiplier field
        interpolated_lagrange += lagrange(l) * psif(l);
      }
 
      int local_eqn = 0, local_unknown = 0;
 
      // Loop over the shape functions to assemble contributions
      for (unsigned l = 0; l < n_node; l++)
      {
        // Loop over the directions
        for (unsigned i = 0; i < nodal_dimension; i++)
        {
          // Now using the same shape functions for the elastic equations,
          // so we can stay in the loop
          local_eqn = this->position_local_eqn(l, 0, i);
          if (local_eqn >= 0)
          {
            // Add in the Lagrange multiplier contribution
            residuals[local_eqn] -=
              interpolated_lagrange * interpolated_n[i] * psif(l) * J * W;
 
            // Do the Jacobian calculation
            if (flag)
            {
              // Loop over the nodes
              for (unsigned l2 = 0; l2 < n_node; l2++)
              {
                // Dependence on solid positions will be handled by FDs
                // That leaves the Lagrange multiplier contribution
                local_unknown = this->kinematic_local_eqn(l2);
                if (local_unknown >= 0)
                {
                  jacobian(local_eqn, local_unknown) -=
                    psif(l2) * interpolated_n[i] * psif(l) * J * W;
                }
              }
            } // End of Jacobian calculation
          }
        }
      } // End of loop over shape functions
    }
 
 
    virtual FluidInterfaceBoundingElement* make_bounding_element(
      const int& face_index)
    {
      // Create a temporary pointer to the appropriate FaceElement
      BoundingElementType<ElasticUpdateFluidInterfaceElement<EQUATION_CLASS,
                                                             DERIVATIVE_CLASS,
                                                             ELEMENT>>*
        face_el_pt = new BoundingElementType<
          ElasticUpdateFluidInterfaceElement<EQUATION_CLASS,
                                             DERIVATIVE_CLASS,
                                             ELEMENT>>;
 
      // Attach the geometrical information to the new element
      this->build_face_element(face_index, face_el_pt);
 
      // Set the index at which the velocity nodes are stored
      face_el_pt->u_index_interface_boundary() = this->U_index_interface;
 
      // Set the value of the nbulk_value, the node is not resized
      // in this bounding element,
      // so it will just be the actual nvalue here
      // There is some ambiguity about what this means (however)
      const unsigned n_node_bounding = face_el_pt->nnode();
      // Storage for lagrange multiplier index at the face nodes
      Vector<unsigned> local_lagrange_index(n_node_bounding);
      for (unsigned n = 0; n < n_node_bounding; n++)
      {
        face_el_pt->nbulk_value(n) = face_el_pt->node_pt(n)->nvalue();
        // At the moment the assumption is that it is stored at all nodes, but
        // that is consistent with the assumption in this element
        local_lagrange_index[n] =
          this->Lagrange_index[face_el_pt->bulk_node_number(n)];
      }
 
      // Pass the ID and offset down
      face_el_pt->set_lagrange_index(local_lagrange_index);
 
      // Find the nodes
      std::set<SolidNode*> set_of_solid_nodes;
      const unsigned n_node = this->nnode();
      for (unsigned n = 0; n < n_node; n++)
      {
        set_of_solid_nodes.insert(static_cast<SolidNode*>(this->node_pt(n)));
      }
 
      // Delete the nodes from the face
      // n_node = face_el_pt->nnode();
      for (unsigned n = 0; n < n_node_bounding; n++)
      {
        // Set the value of the nbulk_value, from the present element
        face_el_pt->nbulk_value(n) =
          this->nbulk_value(face_el_pt->bulk_node_number(n));
 
        // Now delete the nodes from the set
        set_of_solid_nodes.erase(
          static_cast<SolidNode*>(face_el_pt->node_pt(n)));
      }
 
      // Now add these as external data
      for (std::set<SolidNode*>::iterator it = set_of_solid_nodes.begin();
           it != set_of_solid_nodes.end();
           ++it)
      {
        face_el_pt->add_external_data((*it)->variable_position_pt());
      }
 
 
      // Return the value of the pointer
      return face_el_pt;
    }
  };
 
 
  //=========================================================================
  //========================================================================
  template<class ELEMENT>
  class ElasticPointFluidInterfaceBoundingElement
    : public FaceGeometry<FaceGeometry<ELEMENT>>,
      public PointFluidInterfaceBoundingElement,
      public virtual SolidFiniteElement
 
  {
  private:
    Vector<unsigned> Lagrange_index;
 
  public:
    void set_lagrange_index(const Vector<unsigned>& lagrange_index)
    {
      Lagrange_index = lagrange_index;
    }
 
 
    double zeta_nodal(const unsigned& n,
                      const unsigned& k,
                      const unsigned& i) const
    {
      return FaceElement::zeta_nodal(n, k, i);
    }
 
 
    ElasticPointFluidInterfaceBoundingElement()
      : FaceGeometry<FaceGeometry<ELEMENT>>(),
        PointFluidInterfaceBoundingElement()
    {
    }
 
    void output(std::ostream& outfile)
    {
      FiniteElement::output(outfile);
    }
 
    void output(std::ostream& outfile, const unsigned& n_plot)
    {
      FluidInterfaceBoundingElement::output(outfile, n_plot);
    }
 
    void output(FILE* file_pt)
    {
      FiniteElement::output(file_pt);
    }
 
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      FluidInterfaceBoundingElement::output(file_pt, n_plot);
    }
 
    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_interface_boundary(
        residuals, jacobian, 1);
      // Call the generic FD routine to get external data
      this->fill_in_jacobian_from_external_by_fd(jacobian);
 
      // Call the generic finite difference routine to handle the solid
      // variables
      this->fill_in_jacobian_from_solid_position_by_fd(jacobian);
    }
 
    inline int kinematic_local_eqn(const unsigned& n)
    {
      return this->nodal_local_eqn(n, this->Lagrange_index[n]);
    }
  };
 
 
  //=========================================================================
  //========================================================================
  template<class ELEMENT>
  class ElasticLineFluidInterfaceBoundingElement
    : public FaceGeometry<FaceGeometry<ELEMENT>>,
      public LineFluidInterfaceBoundingElement,
      public virtual SolidFiniteElement
 
  {
    Vector<unsigned> Lagrange_index;
 
  public:
    void set_lagrange_index(const Vector<unsigned>& lagrange_index)
    {
      Lagrange_index = lagrange_index;
    }
 
    ElasticLineFluidInterfaceBoundingElement()
      : FaceGeometry<FaceGeometry<ELEMENT>>(),
        LineFluidInterfaceBoundingElement()
    {
    }
 
    double zeta_nodal(const unsigned& n,
                      const unsigned& k,
                      const unsigned& i) const
    {
      return FaceElement::zeta_nodal(n, k, i);
    }
 
 
    void output(std::ostream& outfile)
    {
      FiniteElement::output(outfile);
    }
 
    void output(std::ostream& outfile, const unsigned& n_plot)
    {
      FluidInterfaceBoundingElement::output(outfile, n_plot);
    }
 
    void output(FILE* file_pt)
    {
      FiniteElement::output(file_pt);
    }
 
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      FluidInterfaceBoundingElement::output(file_pt, n_plot);
    }
 
    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_interface_boundary(
        residuals, jacobian, 1);
 
      // Call the generic FD routine to get externals
      this->fill_in_jacobian_from_external_by_fd(jacobian);
 
      // Call the generic finite difference routine to handle the solid
      // variables
      this->fill_in_jacobian_from_solid_position_by_fd(jacobian);
    }
 
    int kinematic_local_eqn(const unsigned& n)
    {
      // Read out the kinematic constraint from the Id which is passed down
      // from the constructing element
      return this->nodal_local_eqn(n, this->Lagrange_index[n]);
    }
  };
 
 
  //==================GEOMETRIC SPECIALISATIONS==========================
 
 
  template<class ELEMENT>
  class ElasticLineFluidInterfaceElement
    : public ElasticUpdateFluidInterfaceElement<FluidInterfaceElement,
                                                LineDerivatives,
                                                ELEMENT>
  {
  public:
    ElasticLineFluidInterfaceElement(FiniteElement* const& element_pt,
                                     const int& face_index,
                                     const unsigned& id = 0)
      : ElasticUpdateFluidInterfaceElement<FluidInterfaceElement,
                                           LineDerivatives,
                                           ELEMENT>(element_pt, face_index, id)
    {
    }
  };
 
  template<class ELEMENT>
  class BoundingElementType<
    ElasticUpdateFluidInterfaceElement<FluidInterfaceElement,
                                       LineDerivatives,
                                       ELEMENT>>
    : public ElasticPointFluidInterfaceBoundingElement<ELEMENT>
  {
  public:
    BoundingElementType<
      ElasticUpdateFluidInterfaceElement<FluidInterfaceElement,
                                         LineDerivatives,
                                         ELEMENT>>()
      : ElasticPointFluidInterfaceBoundingElement<ELEMENT>()
    {
    }
  };
 
 
  template<class ELEMENT>
  class ElasticAxisymmetricFluidInterfaceElement
    : public ElasticUpdateFluidInterfaceElement<FluidInterfaceElement,
                                                AxisymmetricDerivatives,
                                                ELEMENT>
  {
  public:
    ElasticAxisymmetricFluidInterfaceElement(FiniteElement* const& element_pt,
                                             const int& face_index,
                                             const unsigned& id = 0)
      : ElasticUpdateFluidInterfaceElement<FluidInterfaceElement,
                                           AxisymmetricDerivatives,
                                           ELEMENT>(element_pt, face_index, id)
    {
    }
  };
 
  // Define the bounding element associated with the axsymmetric elastic fluid
  // interface element
  template<class ELEMENT>
  class BoundingElementType<
    ElasticUpdateFluidInterfaceElement<FluidInterfaceElement,
                                       AxisymmetricDerivatives,
                                       ELEMENT>>
    : public ElasticPointFluidInterfaceBoundingElement<ELEMENT>
  {
  public:
    BoundingElementType<
      ElasticUpdateFluidInterfaceElement<FluidInterfaceElement,
                                         AxisymmetricDerivatives,
                                         ELEMENT>>()
      : ElasticPointFluidInterfaceBoundingElement<ELEMENT>()
    {
    }
  };
 
 
  template<class ELEMENT>
  class ElasticSurfaceFluidInterfaceElement
    : public ElasticUpdateFluidInterfaceElement<FluidInterfaceElement,
                                                SurfaceDerivatives,
                                                ELEMENT>
  {
  public:
    ElasticSurfaceFluidInterfaceElement(FiniteElement* const& element_pt,
                                        const int& face_index,
                                        const unsigned& id = 0)
      : ElasticUpdateFluidInterfaceElement<FluidInterfaceElement,
                                           SurfaceDerivatives,
                                           ELEMENT>(element_pt, face_index, id)
    {
    }
  };
 
 
  // Define the bounding element associated with the 2D surface elements
  template<class ELEMENT>
  class BoundingElementType<
    ElasticUpdateFluidInterfaceElement<FluidInterfaceElement,
                                       SurfaceDerivatives,
                                       ELEMENT>>
    : public ElasticLineFluidInterfaceBoundingElement<ELEMENT>
  {
  public:
    BoundingElementType<
      ElasticUpdateFluidInterfaceElement<FluidInterfaceElement,
                                         SurfaceDerivatives,
                                         ELEMENT>>()
      : ElasticLineFluidInterfaceBoundingElement<ELEMENT>()
    {
    }
  };
 
 
} // namespace oomph
 
#endif