axisym_fluid_traction_elements.h

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// LIC// ====================================================================
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// LIC// multi-physics finite-element library, available
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// LIC//
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// Header file for elements that are used to integrate fluid tractions
// This includes the guts (i.e. equations) because we want to inline them
// for faster operation, although it slows down the compilation!
#ifndef OOMPH_AXISYM_FLUID_TRACTION_ELEMENTS_HEADER
#define OOMPH_AXISYM_FLUID_TRACTION_ELEMENTS_HEADER
 
// Config header generated by autoconfig
#ifdef HAVE_CONFIG_H
#include <oomph-lib-config.h>
#endif
 
 
// OOMPH-LIB headers
#include "../generic/shape.h"
#include "../generic/elements.h"
#include "../generic/element_with_external_element.h"
 
 
namespace oomph
{
  //=======================================================================
  //=======================================================================
  namespace AxisymmetricNavierStokesTractionElementHelper
  {
    //=======================================================================
    //=======================================================================
    extern void Zero_traction_fct(const double& time,
                                  const Vector<double>& x,
                                  const Vector<double>& N,
                                  Vector<double>& load);
 
  } // namespace AxisymmetricNavierStokesTractionElementHelper
 
 
  //======================================================================
  //======================================================================
  template<class ELEMENT>
  class AxisymmetricNavierStokesTractionElement
    : public virtual FaceGeometry<ELEMENT>,
      public virtual FaceElement
  {
  protected:
    Vector<unsigned> U_index_axisymmetric_nst_traction;
 
    void (*Traction_fct_pt)(const double& time,
                            const Vector<double>& x,
                            const Vector<double>& n,
                            Vector<double>& result);
 
 
    virtual void get_traction(const double& time,
                              const unsigned& intpt,
                              const Vector<double>& x,
                              const Vector<double>& n,
                              Vector<double>& traction) const
    {
      Traction_fct_pt(time, x, n, traction);
    }
 
 
    // This small level of indirection is required to avoid calling
    // fill_in_contribution_to_residuals in fill_in_contribution_to_jacobian
    // which causes all kinds of pain if overloading later on
    void fill_in_contribution_to_residuals_axisymmetric_nst_traction(
      Vector<double>& residuals);
 
 
  public:
    AxisymmetricNavierStokesTractionElement(FiniteElement* const& element_pt,
                                            const int& face_index)
      : FaceGeometry<ELEMENT>(), FaceElement()
    {
      // Attach the geometrical information to the element. N.B. This function
      // also assigns nbulk_value from the required_nvalue of the bulk element
      element_pt->build_face_element(face_index, this);
 
      // Find the dimension of the problem
      unsigned n_dim = element_pt->nodal_dimension();
 
      // Find the index at which the velocity unknowns are stored
      ELEMENT* cast_element_pt = dynamic_cast<ELEMENT*>(element_pt);
      this->U_index_axisymmetric_nst_traction.resize(n_dim + 1);
      for (unsigned i = 0; i < n_dim + 1; i++)
      {
        this->U_index_axisymmetric_nst_traction[i] =
          cast_element_pt->u_index_axi_nst(i);
      }
 
      // Zero traction
      Traction_fct_pt =
        &AxisymmetricNavierStokesTractionElementHelper::Zero_traction_fct;
    }
 
 
    void (*&traction_fct_pt())(const double& time,
                               const Vector<double>& x,
                               const Vector<double>& n,
                               Vector<double>& traction)
    {
      return Traction_fct_pt;
    }
 
 
    void fill_in_contribution_to_residuals(Vector<double>& residuals)
    {
      fill_in_contribution_to_residuals_axisymmetric_nst_traction(residuals);
    }
 
 
    void fill_in_contribution_to_jacobian(Vector<double>& residuals,
                                          DenseMatrix<double>& jacobian)
    {
      // Call the residuals
      fill_in_contribution_to_residuals_axisymmetric_nst_traction(residuals);
    }
 
    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)
    {
      unsigned nplot = 5;
      output(outfile, nplot);
    }
 
    unsigned nscalar_paraview() const
    {
      // Number of dimensions
      unsigned n_dim = this->nodal_dimension();
 
      return 2 * (n_dim + 1);
    }
 
    void scalar_value_paraview(std::ofstream& file_out,
                               const unsigned& k,
                               const unsigned& nplot) const
    {
      // Number of dimensions
      unsigned n_dim = this->nodal_dimension();
 
      // Find out how many nodes there are
      const unsigned n_node = nnode();
 
      // Get continuous time from timestepper of first node
      double time = node_pt(0)->time_stepper_pt()->time_pt()->time();
 
      // Set up memory for the shape functions
      Shape psi(n_node);
 
      // Local and global coordinates
      Vector<double> s(n_dim - 1);
      Vector<double> interpolated_x(n_dim);
 
      // Loop over plot points
      unsigned num_plot_points = this->nplot_points_paraview(nplot);
      for (unsigned iplot = 0; iplot < num_plot_points; iplot++)
      {
        // Get local coordinates of plot point
        this->get_s_plot(iplot, nplot, s);
 
        // Outer unit normal
        Vector<double> unit_normal(n_dim);
        outer_unit_normal(s, unit_normal);
 
        // Find the shape functions
        shape(s, psi);
 
        // Initialise to zero
        for (unsigned i = 0; i < n_dim; i++)
        {
          interpolated_x[i] = 0.0;
        }
 
        // Calculate stuff
        for (unsigned l = 0; l < n_node; l++)
        {
          // Loop over directions
          for (unsigned i = 0; i < n_dim; i++)
          {
            interpolated_x[i] += this->nodal_position(l, i) * psi[l];
          }
        }
 
        // Get the imposed traction
        Vector<double> traction(3);
 
        // Dummy integration point
        unsigned ipt = 0;
        get_traction(time, ipt, interpolated_x, unit_normal, traction);
 
        // Traction components
        if (k < n_dim + 1)
        {
          file_out << traction[k] << std::endl;
        }
        // Advection Diffusion
        else if (k < 2 * n_dim + 1 && k >= n_dim + 1)
        {
          file_out << unit_normal[k] << std::endl;
        }
        // Never get here
        else
        {
          std::stringstream error_stream;
          error_stream
            << "Axisymmetric Fluid Traction Navier-Stokes Elements only store "
            << 2 * (n_dim + 1) << " fields " << std::endl;
          throw OomphLibError(error_stream.str(),
                              OOMPH_CURRENT_FUNCTION,
                              OOMPH_EXCEPTION_LOCATION);
        }
      }
    }
 
    std::string scalar_name_paraview(const unsigned& i) const
    {
      // Number of dimensions
      unsigned n_dim = this->nodal_dimension();
 
      // Traction components
      if (i < n_dim + 1)
      {
        return "Traction component " + StringConversion::to_string(i);
      }
      // Normals
      else if (i < 2 * n_dim + 1 && i >= n_dim + 1)
      {
        return "Normal " + StringConversion::to_string(i % (n_dim + 1));
      }
      // Never get here
      else
      {
        std::stringstream error_stream;
        error_stream
          << "Axisymmetric Fluid Traction Navier-Stokes Elements only store "
          << 2 * (n_dim + 1) << " fields " << std::endl;
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
    }
 
    void output(std::ostream& outfile, const unsigned& n_plot)
    {
      // Number of dimensions
      unsigned n_dim = this->nodal_dimension();
 
      // Find out how many nodes there are
      const unsigned n_node = nnode();
 
      // Get continuous time from timestepper of first node
      double time = node_pt(0)->time_stepper_pt()->time_pt()->time();
 
      // Set up memory for the shape functions
      Shape psi(n_node);
 
      // Local and global coordinates
      Vector<double> s(n_dim - 1);
      Vector<double> interpolated_x(n_dim);
 
      // Tecplot header info
      outfile << this->tecplot_zone_string(n_plot);
 
      // Loop over plot points
      unsigned num_plot_points = this->nplot_points(n_plot);
      for (unsigned iplot = 0; iplot < num_plot_points; iplot++)
      {
        // Get local coordinates of plot point
        this->get_s_plot(iplot, n_plot, s);
 
        // Outer unit normal
        Vector<double> unit_normal(n_dim);
        outer_unit_normal(s, unit_normal);
 
        // Find the shape functions
        shape(s, psi);
 
        // Initialise to zero
        for (unsigned i = 0; i < n_dim; i++)
        {
          interpolated_x[i] = 0.0;
        }
 
        // Calculate stuff
        for (unsigned l = 0; l < n_node; l++)
        {
          // Loop over directions
          for (unsigned i = 0; i < n_dim; i++)
          {
            interpolated_x[i] += this->nodal_position(l, i) * psi[l];
          }
        }
 
        // Get the imposed traction
        Vector<double> traction(3);
 
        // Dummy integration point
        unsigned ipt = 0;
        get_traction(time, ipt, interpolated_x, unit_normal, traction);
 
        // Output the x,y,..
        for (unsigned i = 0; i < n_dim; i++)
        {
          outfile << interpolated_x[i] << " ";
        }
 
        // Output the traction components
        for (unsigned i = 0; i < n_dim + 1; i++)
        {
          outfile << traction[i] << " ";
        }
 
        // Output normal
        for (unsigned i = 0; i < n_dim; i++)
        {
          outfile << unit_normal[i] << " ";
        }
        outfile << std::endl;
      }
    }
 
    void output(FILE* file_pt)
    {
      FiniteElement::output(file_pt);
    }
 
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      FiniteElement::output(file_pt, n_plot);
    }
 
 
    void traction(const double& time,
                  const Vector<double>& s,
                  Vector<double>& traction);
  };
 
 
  //=====================================================================
  //=====================================================================
  template<class ELEMENT>
  void AxisymmetricNavierStokesTractionElement<ELEMENT>::traction(
    const double& time, const Vector<double>& s, Vector<double>& traction)
  {
    unsigned n_dim = this->nodal_dimension();
 
    // Position vector
    Vector<double> x(n_dim);
    interpolated_x(s, x);
 
    // Outer unit normal (only in r and z direction!)
    Vector<double> unit_normal(n_dim);
    outer_unit_normal(s, unit_normal);
 
    // Dummy
    unsigned ipt = 0;
 
    // Traction vector
    get_traction(time, ipt, x, unit_normal, traction);
  }
 
 
  //=====================================================================
  //=====================================================================
  template<class ELEMENT>
  void AxisymmetricNavierStokesTractionElement<ELEMENT>::
    fill_in_contribution_to_residuals_axisymmetric_nst_traction(
      Vector<double>& residuals)
  {
    // Find out how many nodes there are
    unsigned n_node = nnode();
 
    // Get continuous time from timestepper of first node
    double time = node_pt(0)->time_stepper_pt()->time_pt()->time();
 
#ifdef PARANOID
    // Find out how many positional dofs there are
    unsigned n_position_type = this->nnodal_position_type();
    if (n_position_type != 1)
    {
      throw OomphLibError("AxisymmetricNavierStokes is not yet implemented for "
                          "more than one position type",
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
#endif
 
    // Find out the dimension of the node
    unsigned n_dim = this->nodal_dimension();
 
    // Cache the nodal indices at which the velocity components are stored
    unsigned u_nodal_index[n_dim + 1];
    for (unsigned i = 0; i < n_dim + 1; i++)
    {
      u_nodal_index[i] = this->U_index_axisymmetric_nst_traction[i];
    }
 
    // Integer to hold the local equation number
    int local_eqn = 0;
 
    // Set up memory for the shape functions
    // Note that in this case, the number of lagrangian coordinates is always
    // equal to the dimension of the nodes
    Shape psi(n_node);
    DShape dpsids(n_node, n_dim - 1);
 
    // Set the value of n_intpt
    unsigned n_intpt = integral_pt()->nweight();
 
    // Loop over the integration points
    for (unsigned ipt = 0; ipt < n_intpt; ipt++)
    {
      // Get the integral weight
      double w = integral_pt()->weight(ipt);
 
      // Only need to call the local derivatives
      dshape_local_at_knot(ipt, psi, dpsids);
 
      // Calculate the Eulerian and Lagrangian coordinates
      Vector<double> interpolated_x(n_dim, 0.0);
 
      // Also calculate the surface Vectors (derivatives wrt local coordinates)
      DenseMatrix<double> interpolated_A(n_dim - 1, n_dim, 0.0);
 
      // Calculate positions and derivatives
      for (unsigned l = 0; l < n_node; l++)
      {
        // Loop over directions
        for (unsigned i = 0; i < n_dim; i++)
        {
          // Calculate the Eulerian coords
          const double x_local = nodal_position(l, i);
          interpolated_x[i] += x_local * psi(l);
 
          // Loop over LOCAL derivative directions, to calculate the tangent(s)
          for (unsigned j = 0; j < n_dim - 1; j++)
          {
            interpolated_A(j, i) += x_local * dpsids(l, j);
          }
        }
      }
 
      // Now find the local metric tensor from the tangent Vectors
      DenseMatrix<double> A(n_dim - 1);
      for (unsigned i = 0; i < n_dim - 1; i++)
      {
        for (unsigned j = 0; j < n_dim - 1; j++)
        {
          // Initialise surface metric tensor to zero
          A(i, j) = 0.0;
 
          // Take the dot product
          for (unsigned k = 0; k < n_dim; k++)
          {
            A(i, j) += interpolated_A(i, k) * interpolated_A(j, k);
          }
        }
      }
 
      // Get the outer unit normal
      Vector<double> interpolated_normal(n_dim);
      outer_unit_normal(ipt, interpolated_normal);
 
      // Find the determinant of the metric tensor
      double Adet = 0.0;
      switch (n_dim)
      {
        case 2:
          Adet = A(0, 0);
          break;
        case 3:
          Adet = A(0, 0) * A(1, 1) - A(0, 1) * A(1, 0);
          break;
        default:
          throw OomphLibError(
            "Wrong dimension in AxisymmetricNavierStokesTractionElement",
            "AxisymmetricNavierStokesTractionElement::fill_in_contribution_to_"
            "residuals()",
            OOMPH_EXCEPTION_LOCATION);
      }
 
      // Premultiply the weights and the square-root of the determinant of
      // the metric tensor
      double W = w * sqrt(Adet);
 
      // Now calculate the load
      Vector<double> traction(n_dim + 1);
      get_traction(time, ipt, interpolated_x, interpolated_normal, traction);
 
      // Loop over the test functions, nodes of the element
      for (unsigned l = 0; l < n_node; l++)
      {
        // Loop over the velocity components
        for (unsigned i = 0; i < n_dim + 1; i++)
        {
          // Equation number
          local_eqn = this->nodal_local_eqn(l, u_nodal_index[i]);
          /*IF it's not a boundary condition*/
          if (local_eqn >= 0)
          {
            // Add the loading terms to the residuals
            residuals[local_eqn] -=
              traction[i] * psi(l) * interpolated_x[0] * W;
          }
        }
      } // End of loop over shape functions
    } // End of loop over integration points
  }
 
 
 
 
  //=======================================================================
  //=======================================================================
  namespace LinearisedFSIAxisymmetricNStNoSlipBCHelper
  {
    extern double Default_strouhal_number;
 
  } // namespace LinearisedFSIAxisymmetricNStNoSlipBCHelper
 
 
  //======================================================================
  //======================================================================
  template<class FLUID_BULK_ELEMENT, class SOLID_BULK_ELEMENT>
  class LinearisedFSIAxisymmetricNStNoSlipBCElementElement
    : public virtual FaceGeometry<FLUID_BULK_ELEMENT>,
      public virtual FaceElement,
      public virtual ElementWithExternalElement
  {
  public:
    LinearisedFSIAxisymmetricNStNoSlipBCElementElement(
      FiniteElement* const& bulk_el_pt,
      const int& face_index,
      const unsigned& id = 0);
 
    LinearisedFSIAxisymmetricNStNoSlipBCElementElement(
      const LinearisedFSIAxisymmetricNStNoSlipBCElementElement& dummy) = delete;
 
    // Commented out broken assignment operator because this can lead to a
    // conflict warning when used in the virtual inheritence hierarchy.
    // Essentially the compiler doesn't realise that two separate
    // implementations of the broken function are the same and so, quite
    // rightly, it shouts.
    /*void operator=(const LinearisedFSIAxisymmetricNStNoSlipBCElementElement&)
      = delete;*/
 
 
    double*& st_pt()
    {
      return St_pt;
    }
 
    double st() const
    {
      return *St_pt;
    }
 
    inline void fill_in_contribution_to_residuals(Vector<double>& residuals)
    {
      // Call the generic residuals function with flag set to 0
      // using a dummy matrix argument
      fill_in_generic_residual_contribution_fsi_no_slip_axisym(
        residuals, GeneralisedElement::Dummy_matrix, 0);
    }
 
 
    inline 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_fsi_no_slip_axisym(
        residuals, jacobian, 1);
 
      // Derivatives w.r.t. external data
      fill_in_jacobian_from_external_interaction_by_fd(residuals, jacobian);
    }
 
    void output(std::ostream& outfile)
    {
      // Dummy
      unsigned nplot = 0;
      output(outfile, nplot);
    }
 
    void output(std::ostream& outfile, const unsigned& n_plot)
    {
      outfile << "ZONE\n";
 
      // Get the value of Nintpt
      const unsigned n_intpt = integral_pt()->nweight();
 
      // Set the Vector to hold local coordinates
      Vector<double> s_int(Dim - 1);
 
      // Loop over the integration points
      for (unsigned ipt = 0; ipt < n_intpt; ipt++)
      {
        // Assign values of s
        for (unsigned i = 0; i < (Dim - 1); i++)
        {
          s_int[i] = integral_pt()->knot(ipt, i);
        }
 
        // Get boundary coordinate
        Vector<double> zeta(1);
        interpolated_zeta(s_int, zeta);
 
        // Get velocity from adjacent solid
        SOLID_BULK_ELEMENT* ext_el_pt =
          dynamic_cast<SOLID_BULK_ELEMENT*>(external_element_pt(0, ipt));
        Vector<double> s_ext(external_element_local_coord(0, ipt));
        Vector<double> dudt(3);
        ext_el_pt->interpolated_du_dt_axisymmetric_linear_elasticity(s_ext,
                                                                     dudt);
 
        // Output
        outfile << ext_el_pt->interpolated_x(s_ext, 0) << " "
                << ext_el_pt->interpolated_x(s_ext, 1) << " " << dudt[0] << " "
                << dudt[1] << " " << dudt[2] << " " << zeta[0] << std::endl;
      }
    }
 
 
    void output(FILE* file_pt)
    {
      FaceGeometry<FLUID_BULK_ELEMENT>::output(file_pt);
    }
 
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      FaceGeometry<FLUID_BULK_ELEMENT>::output(file_pt, n_plot);
    }
 
 
  protected:
    inline double shape_and_test(const Vector<double>& s,
                                 Shape& psi,
                                 Shape& test) const
    {
      // Find number of nodes
      unsigned n_node = nnode();
 
      // Get the shape functions
      shape(s, psi);
 
      // Set the test functions to be the same as the shape functions
      for (unsigned i = 0; i < n_node; i++)
      {
        test[i] = psi[i];
      }
 
      // Return the value of the jacobian
      return J_eulerian(s);
    }
 
 
    inline double shape_and_test_at_knot(const unsigned& ipt,
                                         Shape& psi,
                                         Shape& test) const
    {
      // Find number of nodes
      unsigned n_node = nnode();
 
      // Get the shape functions
      shape_at_knot(ipt, psi);
 
      // Set the test functions to be the same as the shape functions
      for (unsigned i = 0; i < n_node; i++)
      {
        test[i] = psi[i];
      }
 
      // Return the value of the jacobian
      return J_eulerian_at_knot(ipt);
    }
 
 
  private:
    void fill_in_generic_residual_contribution_fsi_no_slip_axisym(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      const unsigned& flag);
 
    unsigned Dim;
 
    Vector<unsigned> U_index_fsi_no_slip_axisym;
 
    unsigned Id;
 
    double* St_pt;
  };
 
 
 
  //===========================================================================
  //===========================================================================
  template<class FLUID_BULK_ELEMENT, class SOLID_BULK_ELEMENT>
  LinearisedFSIAxisymmetricNStNoSlipBCElementElement<FLUID_BULK_ELEMENT,
                                                     SOLID_BULK_ELEMENT>::
    LinearisedFSIAxisymmetricNStNoSlipBCElementElement(
      FiniteElement* const& bulk_el_pt,
      const int& face_index,
      const unsigned& id)
    : FaceGeometry<FLUID_BULK_ELEMENT>(), FaceElement()
  {
    // Set source element storage: one interaction with an external element
    // that provides the velocity of the adjacent linear elasticity
    // element
    this->set_ninteraction(1);
 
    //  Store the ID of the FaceElement -- this is used to distinguish
    // it from any others
    Id = id;
 
    // Initialise pointer to fluid Strouhal number. Defaults to 1
    St_pt =
      &LinearisedFSIAxisymmetricNStNoSlipBCHelper::Default_strouhal_number;
 
    // Let the bulk element build the FaceElement, i.e. setup the pointers
    // to its nodes (by referring to the appropriate nodes in the bulk
    // element), etc.
    bulk_el_pt->build_face_element(face_index, this);
 
    // Extract the dimension of the problem from the dimension of
    // the first node
    Dim = this->node_pt(0)->ndim();
 
    // Read the index from the (cast) bulk element.
    U_index_fsi_no_slip_axisym.resize(3);
    for (unsigned i = 0; i < 3; i++)
    {
      U_index_fsi_no_slip_axisym[i] =
        dynamic_cast<FLUID_BULK_ELEMENT*>(bulk_el_pt)->u_index_axi_nst(i);
    }
 
    // We need Dim+1 additional values for each FaceElement node
    // to store the Lagrange multipliers.
    Vector<unsigned> n_additional_values(nnode(), Dim + 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.
    add_additional_values(n_additional_values, id);
  }
 
 
  //===========================================================================
  //===========================================================================
  template<class FLUID_BULK_ELEMENT, class SOLID_BULK_ELEMENT>
  void LinearisedFSIAxisymmetricNStNoSlipBCElementElement<FLUID_BULK_ELEMENT,
                                                          SOLID_BULK_ELEMENT>::
    fill_in_generic_residual_contribution_fsi_no_slip_axisym(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      const unsigned& flag)
  {
    // Find out how many nodes there are
    const unsigned n_node = nnode();
 
    // Set up memory for the shape and test functions
    Shape psif(n_node), testf(n_node);
 
    // Set the value of Nintpt
    const unsigned n_intpt = integral_pt()->nweight();
 
    // Set the Vector to hold local coordinates
    Vector<double> s(Dim - 1);
 
    // Cache the Strouhal number
    const double local_st = st();
 
    // Integers to hold the local equation and unknown numbers
    int local_eqn = 0;
 
    // Loop over the integration points
    //--------------------------------
    for (unsigned ipt = 0; ipt < n_intpt; ipt++)
    {
      // Assign values of s
      for (unsigned i = 0; i < (Dim - 1); i++)
      {
        s[i] = integral_pt()->knot(ipt, i);
      }
 
      // Get the integral weight
      double w = integral_pt()->weight(ipt);
 
      // Find the shape and test functions and return the Jacobian
      // of the mapping
      double J = shape_and_test(s, psif, testf);
 
      // Premultiply the weights and the Jacobian
      double W = w * J;
 
      // Calculate the Lagrange multiplier and the fluid veloc
      Vector<double> lambda(Dim + 1, 0.0);
      Vector<double> fluid_veloc(Dim + 1, 0.0);
 
      // Loop over nodes
      for (unsigned j = 0; j < n_node; j++)
      {
        Node* nod_pt = node_pt(j);
 
        // Cast to a boundary node
        BoundaryNodeBase* bnod_pt = dynamic_cast<BoundaryNodeBase*>(node_pt(j));
 
        // Get the index of the first nodal value associated with
        // this FaceElement
        unsigned first_index =
          bnod_pt->index_of_first_value_assigned_by_face_element(Id);
 
        // Assemble
        for (unsigned i = 0; i < Dim + 1; i++)
        {
          lambda[i] += nod_pt->value(first_index + i) * psif(j);
          fluid_veloc[i] +=
            nod_pt->value(U_index_fsi_no_slip_axisym[i]) * psif(j);
        }
      }
 
      // Get velocity from adjacent solid
      SOLID_BULK_ELEMENT* ext_el_pt =
        dynamic_cast<SOLID_BULK_ELEMENT*>(external_element_pt(0, ipt));
      Vector<double> s_ext(external_element_local_coord(0, ipt));
      Vector<double> dudt(3);
      ext_el_pt->interpolated_du_dt_axisymmetric_linear_elasticity(s_ext, dudt);
 
 
      // Now add to the appropriate equations
 
      // Loop over the test functions
      for (unsigned l = 0; l < n_node; l++)
      {
        // Loop over directions
        for (unsigned i = 0; i < Dim + 1; i++)
        {
          // Add contribution to bulk Navier Stokes equations where
          //-------------------------------------------------------
          // the Lagrange multiplier acts as a traction
          //-------------------------------------------
          local_eqn = nodal_local_eqn(l, U_index_fsi_no_slip_axisym[i]);
 
          /*IF it's not a boundary condition*/
          if (local_eqn >= 0)
          {
            // Add the Lagrange multiplier "traction" to the bulk
            residuals[local_eqn] -= lambda[i] * testf[l] * W;
 
 
            // Jacobian entries
            if (flag)
            {
              // Loop over the lagrange multiplier unknowns
              for (unsigned l2 = 0; l2 < n_node; l2++)
              {
                // Cast to a boundary node
                BoundaryNodeBase* bnod_pt =
                  dynamic_cast<BoundaryNodeBase*>(node_pt(l2));
 
                // Local unknown
                int local_unknown = nodal_local_eqn(
                  l2,
                  bnod_pt->index_of_first_value_assigned_by_face_element(Id) +
                    i);
 
                // If it's not pinned
                if (local_unknown >= 0)
                {
                  jacobian(local_eqn, local_unknown) -= psif[l2] * testf[l] * W;
                }
              }
            }
          }
 
          // Now do the Lagrange multiplier equations
          //-----------------------------------------
          // Cast to a boundary node
          BoundaryNodeBase* bnod_pt =
            dynamic_cast<BoundaryNodeBase*>(node_pt(l));
 
          // Local eqn number:
          int local_eqn = nodal_local_eqn(
            l, bnod_pt->index_of_first_value_assigned_by_face_element(Id) + i);
 
          // If it's not pinned
          if (local_eqn >= 0)
          {
            residuals[local_eqn] +=
              (local_st * dudt[i] - fluid_veloc[i]) * testf(l) * W;
 
            // Jacobian entries
            if (flag)
            {
              // Loop over the velocity unknowns [derivs w.r.t. to
              // wall velocity taken care of by fd-ing
              for (unsigned l2 = 0; l2 < n_node; l2++)
              {
                int local_unknown =
                  nodal_local_eqn(l2, U_index_fsi_no_slip_axisym[i]);
 
                /*IF it's not a boundary condition*/
                if (local_unknown >= 0)
                {
                  jacobian(local_eqn, local_unknown) -= psif[l2] * testf[l] * W;
                }
              }
            }
          }
        }
      }
    }
  }
 
} // namespace oomph
 
 
#endif