helmholtz_bc_elements.h

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// LIC// ====================================================================
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// LIC// multi-physics finite-element library, available
// LIC// at http://www.oomph-lib.org.
// LIC//
// LIC// Copyright (C) 2006-2024 Matthias Heil and Andrew Hazel
// LIC//
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// LIC//====================================================================
// Header file for elements that are used to apply Sommerfeld
// boundary conditions to the Helmholtz equations
#ifndef OOMPH_HELMHOLTZ_BC_ELEMENTS_HEADER
#define OOMPH_HELMHOLTZ_BC_ELEMENTS_HEADER
 
// Config header generated by autoconfig
#ifdef HAVE_CONFIG_H
#include <oomph-lib-config.h>
#endif
 
#include "math.h"
#include <complex>
 
// Get the Bessel functions
#include "oomph_crbond_bessel.h"
 
 
namespace oomph
{
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
  // Collection of the  Bessel functions used in the Helmholtz problem
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
 
  //====================================================================
  /// Namespace to provide Hankel function of the first kind
  /// and various orders -- needed for Helmholtz computations.
  //====================================================================
  namespace Hankel_functions_for_helmholtz_problem
  {
    //====================================================================
    /// Compute Hankel function of the first kind of orders 0...n and
    /// its derivates  at coordinate x. The function returns the vector
    /// then its derivative.
    //====================================================================
    void Hankel_first(const unsigned& n,
                      const double& x,
                      Vector<std::complex<double>>& h,
                      Vector<std::complex<double>>& hp)
    {
      int n_actual = 0;
      Vector<double> jn(n + 1), yn(n + 1), jnp(n + 1), ynp(n + 1);
      CRBond_Bessel::bessjyna(
        int(n), x, n_actual, &jn[0], &yn[0], &jnp[0], &ynp[0]);
#ifdef PARANOID
      if (n_actual != int(n))
      {
        std::ostringstream error_stream;
        error_stream << "CRBond_Bessel::bessjyna() only computed " << n_actual
                     << " rather than " << n << " Bessel functions.\n";
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
      for (unsigned i = 0; i < n; i++)
      {
        h[i] = std::complex<double>(jn[i], yn[i]);
        hp[i] = std::complex<double>(jnp[i], ynp[i]);
      }
    } // End of Hankel_first
 
    //====================================================================
    /// Compute Hankel function of the first kind of orders 0...n and
    /// its derivates at coordinate x. The function returns the vector
    /// then its derivative (complex version). This functionality is only
    /// required in the computation of the solution for the complex-
    /// shifted Laplacian preconditioner.
    //====================================================================
    void CHankel_first(const unsigned& n,
                       const std::complex<double>& x,
                       Vector<std::complex<double>>& h,
                       Vector<std::complex<double>>& hp)
    {
      // Set the highest order actually calculated
      int n_actual = 0;
 
      // Create a vector for the Bessel function of the 1st kind
      Vector<std::complex<double>> jn(n + 1);
 
      // Create a vector for the Bessel function of the 2nd kind
      Vector<std::complex<double>> yn(n + 1);
 
      // Create a vector for the Bessel function (1st kind) derivative
      Vector<std::complex<double>> jnp(n + 1);
 
      // Create a vector for the Bessel function (2nd kind) derivative
      Vector<std::complex<double>> ynp(n + 1);
 
      // Call the (complex) Bessel function to calculate the solution
      CRBond_Bessel::cbessjyna(
        int(n), x, n_actual, &jn[0], &yn[0], &jnp[0], &ynp[0]);
 
#ifdef PARANOID
      // Tell the user if they tried to calculate higher order terms
      if (n_actual != int(n))
      {
        // Create an output stream
        std::ostringstream error_message_stream;
 
        // Create the error message
        error_message_stream << "CRBond_Bessel::cbessjyna() only computed "
                             << n_actual << " rather than " << n
                             << " Bessel functions.\n";
 
        // Throw the error message
        throw OomphLibError(error_message_stream.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // Loop over the number of terms requested (only the first entry
      // has *actually* been calculated)
      for (unsigned i = 0; i < n; i++)
      {
        // Set the entries in the Hankel function vector
        h[i] = std::complex<double>(jn[i].real() - yn[i].imag(),
                                    jn[i].imag() + yn[i].real());
 
        // Set the entries in the Hankel function derivative vector
        hp[i] = std::complex<double>(jnp[i].real() - ynp[i].imag(),
                                     jnp[i].imag() + ynp[i].real());
      }
    } // End of Hankel_first
  } // namespace Hankel_functions_for_helmholtz_problem
 
 
  /// //////////////////////////////////////////////////////////////////
  /// //////////////////////////////////////////////////////////////////
  /// //////////////////////////////////////////////////////////////////
 
 
  //======================================================================
  /// A class for elements that allow the approximation of the
  /// Sommerfeld radiation BC.
  /// The element geometry is obtained from the  FaceGeometry<ELEMENT>
  /// policy class.
  //======================================================================
  template<class ELEMENT>
  class HelmholtzBCElementBase : public virtual FaceGeometry<ELEMENT>,
                                 public virtual FaceElement
  {
  public:
    /// Constructor, takes the pointer to the "bulk" element and the
    /// index of the face to which the element is attached.
    HelmholtzBCElementBase(FiniteElement* const& bulk_el_pt,
                           const int& face_index);
 
    /// Broken empty constructor
    HelmholtzBCElementBase()
    {
      throw OomphLibError(
        "Don't call empty constructor for HelmholtzBCElementBase",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Broken copy constructor
    HelmholtzBCElementBase(const HelmholtzBCElementBase& dummy) = delete;
 
    /// Broken assignment operator
    // 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 HelmholtzBCElementBase&) = delete;*/
 
 
    /// Specify the value of nodal zeta from the face geometry
    /// The "global" intrinsic coordinate of the element when
    /// viewed as part of a geometric object should be given by
    /// the FaceElement representation, by default (needed to break
    /// indeterminacy if bulk element is SolidElement)
    double zeta_nodal(const unsigned& n,
                      const unsigned& k,
                      const unsigned& i) const
    {
      return FaceElement::zeta_nodal(n, k, i);
    }
 
 
    /// Output function -- forward to broken version in FiniteElement
    /// until somebody decides what exactly they want to plot here...
    void output(std::ostream& outfile)
    {
      FiniteElement::output(outfile);
    }
 
    /// Output function -- forward to broken version in FiniteElement
    /// until somebody decides what exactly they want to plot here...
    void output(std::ostream& outfile, const unsigned& n_plot)
    {
      FiniteElement::output(outfile, n_plot);
    }
 
    /// C-style output function -- forward to broken version in FiniteElement
    /// until somebody decides what exactly they want to plot here...
    void output(FILE* file_pt)
    {
      FiniteElement::output(file_pt);
    }
 
    /// C-style output function -- forward to broken version in
    /// FiniteElement until somebody decides what exactly they want to plot
    /// here...
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      FiniteElement::output(file_pt, n_plot);
    }
 
    /// Return the index at which the real/imag unknown value
    /// is stored.
    virtual inline std::complex<unsigned> u_index_helmholtz() const
    {
      return std::complex<unsigned>(U_index_helmholtz.real(),
                                    U_index_helmholtz.imag());
    }
 
    /// Compute the element's contribution to the time-averaged
    /// radiated power over the artificial boundary
    double global_power_contribution()
    {
      // Dummy output file
      std::ofstream outfile;
      return global_power_contribution(outfile);
    }
 
    /// Compute the element's contribution to the time-averaged
    /// radiated power over the artificial boundary. Also output the
    /// power density as a fct of the polar angle in the specified
    /// output file if it's open.
    double global_power_contribution(std::ofstream& outfile)
    {
      // pointer to the corresponding bulk element
      ELEMENT* bulk_elem_pt = dynamic_cast<ELEMENT*>(this->bulk_element_pt());
 
      // Number of nodes in bulk element
      unsigned nnode_bulk = bulk_elem_pt->nnode();
      const unsigned n_node_local = nnode();
 
      // get the dim of the bulk and local nodes
      const unsigned bulk_dim = bulk_elem_pt->dim();
      const unsigned local_dim = this->dim();
 
      // Set up memory for the shape and test functions
      Shape psi(n_node_local);
 
      // Set up memory for the shape functions
      Shape psi_bulk(nnode_bulk);
      DShape dpsi_bulk_dx(nnode_bulk, bulk_dim);
 
      // Set up memory for the outer unit normal
      Vector<double> unit_normal(bulk_dim);
 
      // Set the value of n_intpt
      const unsigned n_intpt = integral_pt()->nweight();
 
      // Set the Vector to hold local coordinates
      Vector<double> s(local_dim);
      double power = 0.0;
 
      // Output?
      if (outfile.is_open())
      {
        outfile << "ZONE\n";
      }
 
      // Loop over the integration points
      //--------------------------------
      for (unsigned ipt = 0; ipt < n_intpt; ipt++)
      {
        // Assign values of s
        for (unsigned i = 0; i < local_dim; i++)
        {
          s[i] = integral_pt()->knot(ipt, i);
        }
        // get the outer_unit_ext vector
        this->outer_unit_normal(s, unit_normal);
 
        // Get the integral weight
        double w = integral_pt()->weight(ipt);
 
        // Get jacobian of mapping
        double J = J_eulerian(s);
 
        // Premultiply the weights and the Jacobian
        double W = w * J;
 
        // Get local coordinates in bulk element by copy construction
        Vector<double> s_bulk(local_coordinate_in_bulk(s));
 
        // Call the derivatives of the shape  functions
        // in the bulk -- must do this via s because this point
        // is not an integration point the bulk element!
        (void)bulk_elem_pt->dshape_eulerian(s_bulk, psi_bulk, dpsi_bulk_dx);
        this->shape(s, psi);
 
        // Derivs of Eulerian coordinates w.r.t. local coordinates
        std::complex<double> dphi_dn(0.0, 0.0);
        Vector<std::complex<double>> interpolated_dphidx(bulk_dim);
        std::complex<double> interpolated_phi(0.0, 0.0);
        Vector<double> x(bulk_dim);
 
        // Calculate function value and derivatives:
        //-----------------------------------------
        // Loop over nodes
        for (unsigned l = 0; l < nnode_bulk; l++)
        {
          // Get the nodal value of the helmholtz unknown
          const std::complex<double> phi_value(
            bulk_elem_pt->nodal_value(l,
                                      bulk_elem_pt->u_index_helmholtz().real()),
            bulk_elem_pt->nodal_value(
              l, bulk_elem_pt->u_index_helmholtz().imag()));
 
          // Loop over directions
          for (unsigned i = 0; i < bulk_dim; i++)
          {
            interpolated_dphidx[i] += phi_value * dpsi_bulk_dx(l, i);
          }
        } // End of loop over the bulk_nodes
 
        for (unsigned l = 0; l < n_node_local; l++)
        {
          // Get the nodal value of the helmholtz unknown
          const std::complex<double> phi_value(
            nodal_value(l, u_index_helmholtz().real()),
            nodal_value(l, u_index_helmholtz().imag()));
 
          interpolated_phi += phi_value * psi(l);
        }
 
        // define dphi_dn
        for (unsigned i = 0; i < bulk_dim; i++)
        {
          dphi_dn += interpolated_dphidx[i] * unit_normal[i];
        }
 
        // Power density
        double integrand = 0.5 * (interpolated_phi.real() * dphi_dn.imag() -
                                  interpolated_phi.imag() * dphi_dn.real());
 
        // Output?
        if (outfile.is_open())
        {
          interpolated_x(s, x);
          double phi = atan2(x[1], x[0]);
          outfile << x[0] << " " << x[1] << " " << phi << " " << integrand
                  << "\n";
        }
 
        // ...add to integral
        power += integrand * W;
      }
 
      return power;
    }
 
 
    /// Compute element's contribution to Fourier components of the
    /// solution -- length of vector indicates number of terms to be computed.
    void compute_contribution_to_fourier_components(
      Vector<std::complex<double>>& a_coeff_pos,
      Vector<std::complex<double>>& a_coeff_neg)
    {
#ifdef PARANOID
      if (a_coeff_pos.size() != a_coeff_neg.size())
      {
        std::ostringstream error_stream;
        error_stream << "a_coeff_pos and a_coeff_neg must have "
                     << "the same size. \n";
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // define the imaginary number
      const std::complex<double> I(0.0, 1.0);
 
      // Find out how many nodes there are
      const unsigned n_node = this->nnode();
 
      // Set up memory for the shape  functions
      Shape psi(n_node);
      DShape dpsi(n_node, 1);
 
      // Set the value of n_intpt
      const unsigned n_intpt = this->integral_pt()->nweight();
 
      // Set the Vector to hold local coordinates
      Vector<double> s(this->Dim - 1);
 
      // Initialise
      unsigned n = a_coeff_pos.size();
      for (unsigned i = 0; i < n; i++)
      {
        a_coeff_pos[i] = std::complex<double>(0.0, 0.0);
        a_coeff_neg[i] = std::complex<double>(0.0, 0.0);
      }
 
      // Loop over the integration points
      //--------------------------------
      for (unsigned ipt = 0; ipt < n_intpt; ipt++)
      {
        // Assign values of s
        for (unsigned i = 0; i < (this->Dim - 1); i++)
        {
          s[i] = this->integral_pt()->knot(ipt, i);
        }
 
        // Get the integral weight
        double w = this->integral_pt()->weight(ipt);
 
        // Get the shape functions
        this->dshape_local(s, psi, dpsi);
 
        // Eulerian coordinates at Gauss point
        Vector<double> interpolated_x(this->Dim, 0.0);
 
        // Derivs of Eulerian coordinates w.r.t. local coordinates
        Vector<double> interpolated_dxds(this->Dim);
        std::complex<double> interpolated_u(0.0, 0.0);
 
        // Assemble x and its derivs
        for (unsigned l = 0; l < n_node; l++)
        {
          // Loop over directions
          for (unsigned i = 0; i < this->Dim; i++)
          {
            interpolated_x[i] += this->nodal_position(l, i) * psi[l];
            interpolated_dxds[i] += this->nodal_position(l, i) * dpsi(l, 0);
          }
 
          // Get the nodal value of the helmholtz unknown
          const std::complex<double> u_value(
            this->nodal_value(l, this->U_index_helmholtz.real()),
            this->nodal_value(l, this->U_index_helmholtz.imag()));
 
          // Add to the interpolated value
          interpolated_u += u_value * psi(l);
        } // End of loop over the nodes
 
        // calculate the integral
        //-----------------------
 
        // Get polar angle
        double phi = atan2(interpolated_x[1], interpolated_x[0]);
 
        // define dphi_ds=(-yx'+y'x)/(x^2+y^2)
        double denom = (interpolated_x[0] * interpolated_x[0]) +
                       (interpolated_x[1] * interpolated_x[1]);
        double nom = -interpolated_dxds[1] * interpolated_x[0] +
                     interpolated_dxds[0] * interpolated_x[1];
        double dphi_ds = std::fabs(nom / denom);
 
        // Positive coefficients
        for (unsigned i = 0; i < n; i++)
        {
          a_coeff_pos[i] +=
            interpolated_u * exp(-I * phi * double(i)) * dphi_ds * w;
        }
        // Negative coefficients
        for (unsigned i = 1; i < n; i++)
        {
          a_coeff_neg[i] +=
            interpolated_u * exp(I * phi * double(i)) * dphi_ds * w;
        }
 
      } // End of loop over integration points
    }
 
 
  protected:
    /// Function to compute the shape and test functions and to return
    /// the Jacobian of mapping between local and global (Eulerian)
    /// coordinates
    inline double test_only(const Vector<double>& s, Shape& test) const
    {
      // Get the shape functions
      shape(s, test);
 
      // Return the value of the jacobian
      return J_eulerian(s);
    }
 
    /// Function to compute the shape, test functions and derivates
    /// and to return
    /// the Jacobian of mapping between local and global (Eulerian)
    /// coordinates
    inline double d_shape_and_test_local(const Vector<double>& s,
                                         Shape& psi,
                                         Shape& test,
                                         DShape& dpsi_ds,
                                         DShape& dtest_ds) const
    {
      // Find number of nodes
      unsigned n_node = nnode();
 
      // Get the shape functions
      dshape_local(s, psi, dpsi_ds);
 
      // Set the test functions to be the same as the shape functions
      for (unsigned i = 0; i < n_node; i++)
      {
        for (unsigned j = 0; j < (Dim - 1); j++)
        {
          test[i] = psi[i];
          dtest_ds(i, j) = dpsi_ds(i, j);
        }
      }
      // Return the value of the jacobian
      return J_eulerian(s);
    }
 
    /// The index at which the real and imag part of the unknown is
    /// stored at the nodes
    std::complex<unsigned> U_index_helmholtz;
 
    /// The spatial dimension of the problem
    unsigned Dim;
  };
 
  /// ///////////////////////////////////////////////////////////////////
  /// ///////////////////////////////////////////////////////////////////
  /// ///////////////////////////////////////////////////////////////////
 
 
  /// =================================================================
  /// Mesh for DtN boundary condition elements -- provides
  /// functionality to apply Sommerfeld radiation condtion
  /// via DtN BC
  /// =================================================================
  template<class ELEMENT>
  class HelmholtzDtNMesh : public virtual Mesh
  {
  public:
    /// Constructor: Specify radius of outer boundary and number of
    /// Fourier terms used in the computation of the gamma integral
    HelmholtzDtNMesh(const double& outer_radius, const unsigned& nfourier_terms)
      : Outer_radius(outer_radius), Nfourier_terms(nfourier_terms)
    {
    }
 
    /// Compute and store the gamma integral at all integration
    /// points of the constituent elements.
    void setup_gamma();
 
    /// Compute Fourier components of the solution -- length of
    /// vector indicates number of terms to be computed.
    void compute_fourier_components(Vector<std::complex<double>>& a_coeff_pos,
                                    Vector<std::complex<double>>& a_coeff_neg);
 
    /// Gamma integral evaluated at Gauss points
    /// for specified element
    Vector<std::complex<double>>& gamma_at_gauss_point(FiniteElement* el_pt)
    {
      return Gamma_at_gauss_point[el_pt];
    }
 
    /// Derivative of Gamma integral w.r.t global unknown, evaluated
    /// at Gauss points for specified element
    Vector<std::map<unsigned, std::complex<double>>>& d_gamma_at_gauss_point(
      FiniteElement* el_pt)
    {
      return D_Gamma_at_gauss_point[el_pt];
    }
 
    /// The outer radius
    double& outer_radius()
    {
      return Outer_radius;
    }
 
    /// Number of Fourier terms used in the computation of the
    /// gamma integral
    unsigned& nfourier_terms()
    {
      return Nfourier_terms;
    }
 
  private:
    /// Outer radius
    double Outer_radius;
 
    /// Nbr of Fourier terms used in the  Gamma computation
    unsigned Nfourier_terms;
 
 
    /// Container to store the gamma integral for given Gauss point
    /// and element
    std::map<FiniteElement*, Vector<std::complex<double>>> Gamma_at_gauss_point;
 
 
    /// Container to store the derivate of Gamma integral w.r.t
    /// global unknown evaluated at Gauss points for specified element
    std::map<FiniteElement*, Vector<std::map<unsigned, std::complex<double>>>>
      D_Gamma_at_gauss_point;
  };
 
 
  /// //////////////////////////////////////////////////////////////////
  /// //////////////////////////////////////////////////////////////////
  /// //////////////////////////////////////////////////////////////////
 
 
  //=============================================================
  /// Absorbing BC element for approximation imposition of
  /// Sommerfeld radiation condition
  //==============================================================
  template<class ELEMENT>
  class HelmholtzAbsorbingBCElement : public HelmholtzBCElementBase<ELEMENT>
  {
  public:
    /// Construct element from specification of bulk element and
    /// face index.
    HelmholtzAbsorbingBCElement(FiniteElement* const& bulk_el_pt,
                                const int& face_index)
      : HelmholtzBCElementBase<ELEMENT>(bulk_el_pt, face_index)
    {
      // Initialise pointers
      Outer_radius_pt = 0;
 
      // Initialise order of absorbing boundary condition
      ABC_order_pt = 0;
    }
 
    /// Pointer to order of absorbing boundary condition
    unsigned*& abc_order_pt()
    {
      return ABC_order_pt;
    }
 
 
    /// Add the element's contribution to its residual vector
    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_helmholtz_abc(
        residuals, GeneralisedElement::Dummy_matrix, 0);
    }
 
    /// Add the element's contribution to its residual vector and its
    /// Jacobian matrix
    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_helmholtz_abc(
        residuals, jacobian, 1);
    }
 
 
    /// Get pointer to radius of outer boundary (must be a cirle)
    double*& outer_radius_pt()
    {
      return Outer_radius_pt;
    }
 
  private:
    /// Compute the element's residual vector and the (
    /// Jacobian matrix.
    /// Overloaded version, using the abc approximation
    void fill_in_generic_residual_contribution_helmholtz_abc(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      const unsigned& flag)
    {
#ifdef PARANOID
 
      if (ABC_order_pt == 0)
      {
        throw OomphLibError("Order of ABC hasn't been set!",
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
 
      if (this->Outer_radius_pt == 0)
      {
        throw OomphLibError("Pointer to outer radius hasn't been set!",
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
 
#endif
 
      // Find out how many nodes there are
      const unsigned n_node = this->nnode();
 
      // Set up memory for the shape and test functions
      Shape psi(n_node), test(n_node);
      DShape dpsi_ds(n_node, this->Dim - 1), dtest_ds(n_node, this->Dim - 1),
        dtest_dS(n_node, this->Dim - 1), dpsi_dS(n_node, this->Dim - 1);
 
      // Set the value of Nintpt
      const unsigned n_intpt = this->integral_pt()->nweight();
 
      // Set the Vector to hold local coordinates
      Vector<double> s(this->Dim - 1);
 
      // Integers to hold the local equation and unknown numbers
      int local_eqn_real = 0, local_unknown_real = 0;
      int local_eqn_imag = 0, local_unknown_imag = 0;
 
      // Define the problem parameters
      double R = *Outer_radius_pt;
      double k =
        sqrt(dynamic_cast<ELEMENT*>(this->bulk_element_pt())->k_squared());
 
      // Loop over the integration points
      //--------------------------------
      for (unsigned ipt = 0; ipt < n_intpt; ipt++)
      {
        // Assign values of s
        for (unsigned i = 0; i < (this->Dim - 1); i++)
        {
          s[i] = this->integral_pt()->knot(ipt, i);
        }
 
        // Get the integral weight
        double w = this->integral_pt()->weight(ipt);
 
        // Find the shape test functions and derivates; return the Jacobian
        // of the mapping between local and global (Eulerian)
        // coordinates
        double J =
          this->d_shape_and_test_local(s, psi, test, dpsi_ds, dtest_ds);
 
        // Premultiply the weights and the Jacobian
        double W = w * J;
        // get the inverse of jacibian
        double inv_J = 1 / J;
 
        // Need to find position to feed into flux function,
        // initialise to zero
        std::complex<double> interpolated_u(0.0, 0.0);
        std::complex<double> du_dS(0.0, 0.0);
 
        // Calculate velocities and derivatives:loop over the nodes
        for (unsigned l = 0; l < n_node; l++)
        {
          // Loop over real and imag part
          // Get the nodal value of the helmholtz unknown
          const std::complex<double> u_value(
            this->nodal_value(l, this->U_index_helmholtz.real()),
            this->nodal_value(l, this->U_index_helmholtz.imag()));
 
          interpolated_u += u_value * psi[l];
 
          du_dS += u_value * dpsi_ds(l, 0) * inv_J;
 
          // Get the value of dtest_dS
          dtest_dS(l, 0) = dtest_ds(l, 0) * inv_J;
          // Get the value of dpsif_dS
          dpsi_dS(l, 0) = dpsi_ds(l, 0) * inv_J;
        }
 
        // use ABC first order approximation
        if (*ABC_order_pt == 1)
        {
          // Now add to the appropriate equations:use second order approximation
          // Loop over the test functions
          for (unsigned l = 0; l < n_node; l++)
          {
            local_eqn_real =
              this->nodal_local_eqn(l, this->U_index_helmholtz.real());
            local_eqn_imag =
              this->nodal_local_eqn(l, this->U_index_helmholtz.imag());
 
            // first, calculate the real part contrubution
            //-----------------------
            // IF it's not a boundary condition
            if (local_eqn_real >= 0)
            {
              // Add the second order terms:compute manually real and imag part
 
              residuals[local_eqn_real] += (k * interpolated_u.imag() +
                                            (0.5 / R) * interpolated_u.real()) *
                                           test[l] * W;
 
              // Calculate the jacobian
              //-----------------------
              if (flag)
              {
                // Loop over the shape functions again
                for (unsigned l2 = 0; l2 < n_node; l2++)
                {
                  local_unknown_real =
                    this->nodal_local_eqn(l2, this->U_index_helmholtz.real());
                  local_unknown_imag =
                    this->nodal_local_eqn(l2, this->U_index_helmholtz.imag());
                  // If at a non-zero degree of freedom add in the entry
                  if (local_unknown_real >= 0)
                  {
                    jacobian(local_eqn_real, local_unknown_real) +=
                      (0.5 / R) * psi[l2] * test[l] * W;
                  }
 
                  // If at a non-zero degree of freedom add in the entry
                  if (local_unknown_imag >= 0)
                  {
                    jacobian(local_eqn_real, local_unknown_imag) +=
                      k * psi[l2] * test[l] * W;
                  }
                }
              }
            } // end of local_eqn_real
 
            // second, calculate the imag part contrubution
            //-----------------------
            // IF it's not a boundary condition
            if (local_eqn_imag >= 0)
            {
              // Add the second order terms contibution to the residual
              residuals[local_eqn_imag] += (-k * interpolated_u.real() +
                                            (0.5 / R) * interpolated_u.imag()) *
                                           test[l] * W;
 
 
              // Calculate the jacobian
              //-----------------------
              if (flag)
              {
                // Loop over the shape functions again
                for (unsigned l2 = 0; l2 < n_node; l2++)
                {
                  local_unknown_real =
                    this->nodal_local_eqn(l2, this->U_index_helmholtz.real());
                  local_unknown_imag =
                    this->nodal_local_eqn(l2, this->U_index_helmholtz.imag());
                  // If at a non-zero degree of freedom add in the entry
                  if (local_unknown_real >= 0)
                  {
                    jacobian(local_eqn_imag, local_unknown_real) +=
                      (-k) * psi[l2] * test[l] * W;
                  }
 
                  // If at a non-zero degree of freedom add in the entry
                  if (local_unknown_imag >= 0)
                  {
                    jacobian(local_eqn_imag, local_unknown_imag) +=
                      (0.5 / R) * psi[l2] * test[l] * W;
                  }
                }
              }
            }
          } // End of loop over the nodes
        }
 
        //:use second order approximation
        if (*ABC_order_pt == 2)
        {
          // Now add to the appropriate equations:use second order approximation
          // Loop over the test functions
          for (unsigned l = 0; l < n_node; l++)
          {
            local_eqn_real =
              this->nodal_local_eqn(l, this->U_index_helmholtz.real());
            local_eqn_imag =
              this->nodal_local_eqn(l, this->U_index_helmholtz.imag());
 
            // first, calculate the real part contrubution
            //-----------------------
            // IF it's not a boundary condition
            if (local_eqn_real >= 0)
            {
              // Add the second order terms:compute manually real and imag part
 
              residuals[local_eqn_real] +=
                (k * interpolated_u.imag() +
                 (0.5 / R) * interpolated_u.real()) *
                  test[l] * W +
                ((0.125 / (k * R * R)) * interpolated_u.imag()) * test[l] * W;
 
              residuals[local_eqn_real] +=
                (-0.5 / k) * du_dS.imag() * dtest_dS(l, 0) * W;
 
              // Calculate the jacobian
              //-----------------------
              if (flag)
              {
                // Loop over the shape functions again
                for (unsigned l2 = 0; l2 < n_node; l2++)
                {
                  local_unknown_real =
                    this->nodal_local_eqn(l2, this->U_index_helmholtz.real());
                  local_unknown_imag =
                    this->nodal_local_eqn(l2, this->U_index_helmholtz.imag());
                  // If at a non-zero degree of freedom add in the entry
                  if (local_unknown_real >= 0)
                  {
                    jacobian(local_eqn_real, local_unknown_real) +=
                      (0.5 / R) * psi[l2] * test[l] * W;
                  }
 
                  // If at a non-zero degree of freedom add in the entry
                  if (local_unknown_imag >= 0)
                  {
                    jacobian(local_eqn_real, local_unknown_imag) +=
                      k * psi[l2] * test[l] * W +
                      (0.125 / (k * R * R)) * psi[l2] * test[l] * W;
 
                    jacobian(local_eqn_real, local_unknown_imag) +=
                      (-0.5 / k) * dpsi_dS(l2, 0) * dtest_dS(l, 0) * W;
                  }
                }
              }
            } // end of local_eqn_real
 
            // second, calculate the imag part contrubution
            //-----------------------
            // IF it's not a boundary condition
            if (local_eqn_imag >= 0)
            {
              // Add the second order terms contibution to the residual
              residuals[local_eqn_imag] +=
                (-k * interpolated_u.real() +
                 (0.5 / R) * interpolated_u.imag()) *
                  test[l] * W +
                ((-0.125 / (k * R * R)) * interpolated_u.real()) * test[l] * W;
 
              residuals[local_eqn_imag] +=
                (0.5 / k) * du_dS.real() * dtest_dS(l, 0) * W;
 
              // Calculate the jacobian
              //-----------------------
              if (flag)
              {
                // Loop over the shape functions again
                for (unsigned l2 = 0; l2 < n_node; l2++)
                {
                  local_unknown_real =
                    this->nodal_local_eqn(l2, this->U_index_helmholtz.real());
                  local_unknown_imag =
                    this->nodal_local_eqn(l2, this->U_index_helmholtz.imag());
                  // If at a non-zero degree of freedom add in the entry
                  if (local_unknown_real >= 0)
                  {
                    jacobian(local_eqn_imag, local_unknown_real) +=
                      (-k) * psi[l2] * test[l] * W -
                      (0.125 / (k * R * R)) * psi[l2] * test[l] * W;
 
                    jacobian(local_eqn_imag, local_unknown_real) +=
                      (0.5 / k) * dpsi_dS(l2, 0) * dtest_dS(l, 0) * W;
                  }
 
                  // If at a non-zero degree of freedom add in the entry
                  if (local_unknown_imag >= 0)
                  {
                    jacobian(local_eqn_imag, local_unknown_imag) +=
                      (0.5 / R) * psi[l2] * test[l] * W;
                  }
                }
              }
            }
          } // End of loop over the nodes
        }
 
        if (*ABC_order_pt == 3)
        {
          // Now add to the appropriate equations:use second order approximation
          // Loop over the test functions
          for (unsigned l = 0; l < n_node; l++)
          {
            local_eqn_real =
              this->nodal_local_eqn(l, this->U_index_helmholtz.real());
            local_eqn_imag =
              this->nodal_local_eqn(l, this->U_index_helmholtz.imag());
 
            // first, calculate the real part contrubution
            //-----------------------
            // IF it's not a boundary condition
            if (local_eqn_real >= 0)
            {
              // Add the second order terms:compute manually real and imag part
              residuals[local_eqn_real] +=
                ((k * (1 + 0.125 / (k * k * R * R))) * interpolated_u.imag() +
                 (0.5 / R - 0.125 / (k * k * R * R * R)) *
                   interpolated_u.real()) *
                test[l] * W;
 
              residuals[local_eqn_real] +=
                ((-0.5 / k) * du_dS.imag() +
                 (0.5 / (k * k * R)) * du_dS.real()) *
                dtest_dS(l, 0) * W;
 
              // Calculate the jacobian
              //-----------------------
              if (flag)
              {
                // Loop over the shape functions again
                for (unsigned l2 = 0; l2 < n_node; l2++)
                {
                  local_unknown_real =
                    this->nodal_local_eqn(l2, this->U_index_helmholtz.real());
                  local_unknown_imag =
                    this->nodal_local_eqn(l2, this->U_index_helmholtz.imag());
                  // If at a non-zero degree of freedom add in the entry
                  if (local_unknown_real >= 0)
                  {
                    jacobian(local_eqn_real, local_unknown_real) +=
                      (0.5 / R - 0.125 / (k * k * R * R * R)) * psi[l2] *
                      test[l] * W;
 
                    jacobian(local_eqn_real, local_unknown_real) +=
                      (0.5 / (k * k * R)) * dpsi_dS(l2, 0) * dtest_dS(l, 0) * W;
                  }
 
                  // If at a non-zero degree of freedom add in the entry
                  if (local_unknown_imag >= 0)
                  {
                    jacobian(local_eqn_real, local_unknown_imag) +=
                      (k * (1 + 0.125 / (k * k * R * R))) * psi[l2] * test[l] *
                      W;
 
                    jacobian(local_eqn_real, local_unknown_imag) +=
                      (-0.5 / k) * dpsi_dS(l2, 0) * dtest_dS(l, 0) * W;
                  }
                }
              }
            } // end of local_eqn_real
 
            // second, calculate the imag part contrubution
            //-----------------------
            // IF it's not a boundary condition
            if (local_eqn_imag >= 0)
            {
              // Add the second order terms contibution to the residual
              residuals[local_eqn_imag] +=
                ((-k * (1 + 0.125 / (k * k * R * R))) * interpolated_u.real() +
                 (0.5 / R - 0.125 / (k * k * R * R * R)) *
                   interpolated_u.imag()) *
                test[l] * W;
 
              residuals[local_eqn_imag] +=
                ((0.5 / k) * du_dS.real() +
                 (0.5 / (k * k * R)) * du_dS.imag()) *
                dtest_dS(l, 0) * W;
 
              // Calculate the jacobian
              //-----------------------
              if (flag)
              {
                // Loop over the shape functions again
                for (unsigned l2 = 0; l2 < n_node; l2++)
                {
                  local_unknown_real =
                    this->nodal_local_eqn(l2, this->U_index_helmholtz.real());
                  local_unknown_imag =
                    this->nodal_local_eqn(l2, this->U_index_helmholtz.imag());
                  // If at a non-zero degree of freedom add in the entry
                  if (local_unknown_real >= 0)
                  {
                    jacobian(local_eqn_imag, local_unknown_real) +=
                      (-k * (1 + 0.125 / (k * k * R * R))) * psi[l2] * test[l] *
                      W;
 
                    jacobian(local_eqn_imag, local_unknown_real) +=
                      (0.5 / k) * dpsi_dS(l2, 0) * dtest_dS(l, 0) * W;
                  }
                  // If at a non-zero degree of freedom add in the entry
                  if (local_unknown_imag >= 0)
                  {
                    jacobian(local_eqn_imag, local_unknown_imag) +=
                      (0.5 / R - 0.125 / (k * k * R * R * R)) * psi[l2] *
                      test[l] * W;
 
                    jacobian(local_eqn_imag, local_unknown_imag) +=
                      (0.5 / (k * k * R)) * dpsi_dS(l2, 0) * dtest_dS(l, 0) * W;
                  }
                }
              }
            }
          } // End of loop over the nodes
        }
      } // End of loop over int_pt
 
    } // End of fill_in_generic_residual_contribution_helmholtz_flux
 
  private:
    /// Pointer to radius of outer boundary (must be a circle!)
    double* Outer_radius_pt;
 
    /// Pointer to order of absorbing boundary condition
    unsigned* ABC_order_pt;
  };
 
  //=============================================================
  /// FaceElement used to apply Sommerfeld radiation conditon
  /// via Dirichlet to Neumann map.
  //==============================================================
  template<class ELEMENT>
  class HelmholtzDtNBoundaryElement : public HelmholtzBCElementBase<ELEMENT>
  {
  public:
    /// Construct element from specification of bulk element and
    /// face index.
    HelmholtzDtNBoundaryElement(FiniteElement* const& bulk_el_pt,
                                const int& face_index)
      : HelmholtzBCElementBase<ELEMENT>(bulk_el_pt, face_index)
    {
    }
 
    /// Add the element's contribution to its residual vector
    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_helmholtz_DtN_bc(
        residuals, GeneralisedElement::Dummy_matrix, 0);
    }
 
    /// Add the element's contribution to its residual vector and its
    /// Jacobian matrix
    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_helmholtz_DtN_bc(
        residuals, jacobian, 1);
    }
 
    /// Compute the contribution of the element
    /// to the Gamma integral and its derivates w.r.t
    /// to global unknows; the function takes the wavenumber and the polar
    /// angle phi as input
    void compute_gamma_contribution(
      const double& phi,
      const int& n,
      std::complex<double>& gamma_con,
      std::map<unsigned, std::complex<double>>& d_gamma_con);
 
 
    /// Access function to mesh of all DtN boundary condition elements
    /// (needed to get access to gamma values)
    HelmholtzDtNMesh<ELEMENT>* outer_boundary_mesh_pt() const
    {
      return Outer_boundary_mesh_pt;
    }
 
    /// Set mesh of all DtN boundary condition elements
    void set_outer_boundary_mesh_pt(HelmholtzDtNMesh<ELEMENT>* mesh_pt)
    {
      Outer_boundary_mesh_pt = mesh_pt;
    }
 
 
    /// Complete the setup of additional dependencies arising
    /// through the far-away interaction with other nodes in
    /// Outer_boundary_mesh_pt.
    void complete_setup_of_dependencies()
    {
      // Create a set of all nodes
      std::set<Node*> node_set;
      unsigned nel = Outer_boundary_mesh_pt->nelement();
      for (unsigned e = 0; e < nel; e++)
      {
        FiniteElement* el_pt = Outer_boundary_mesh_pt->finite_element_pt(e);
        unsigned nnod = el_pt->nnode();
        for (unsigned j = 0; j < nnod; j++)
        {
          Node* nod_pt = el_pt->node_pt(j);
 
          // Don't add copied nodes
          if (!(nod_pt->is_a_copy()))
          {
            node_set.insert(nod_pt);
          }
        }
      }
      // Now erase the current element's own nodes
      unsigned nnod = this->nnode();
      for (unsigned j = 0; j < nnod; j++)
      {
        Node* nod_pt = this->node_pt(j);
        node_set.erase(nod_pt);
 
        // If the element's node is a copy then its "master" will
        // already have been added in the set above -- remove the
        // master to avoid double counting eqn numbers
        if (nod_pt->is_a_copy())
        {
          node_set.erase(nod_pt->copied_node_pt());
        }
      }
 
      // Now declare these nodes to be the element's external Data
      for (std::set<Node*>::iterator it = node_set.begin();
           it != node_set.end();
           it++)
      {
        this->add_external_data(*it);
      }
    }
 
 
  private:
    /// Compute the element's residual vector
    /// Jacobian matrix.
    /// Overloaded version, using the gamma computed in the mesh
    void fill_in_generic_residual_contribution_helmholtz_DtN_bc(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      const unsigned& flag)
    {
      // Find out how many nodes there are
      const unsigned n_node = this->nnode();
 
      // Set up memory for the shape and test functions
      Shape test(n_node);
 
      // Set the value of Nintpt
      const unsigned n_intpt = this->integral_pt()->nweight();
 
      // Set the Vector to hold local coordinates
      Vector<double> s(this->Dim - 1);
 
      // Integers to hold the local equation and unknown numbers
      int local_eqn_real = 0, local_unknown_real = 0, global_eqn_real = 0,
          local_eqn_imag = 0, local_unknown_imag = 0, global_eqn_imag = 0;
      int external_global_eqn_real = 0, external_unknown_real = 0,
          external_global_eqn_imag = 0, external_unknown_imag = 0;
 
 
      // Get the gamma value for the current integration point
      // from the mesh
      Vector<std::complex<double>> gamma(
        Outer_boundary_mesh_pt->gamma_at_gauss_point(this));
 
      Vector<std::map<unsigned, std::complex<double>>> d_gamma(
        Outer_boundary_mesh_pt->d_gamma_at_gauss_point(this));
 
      // Loop over the integration points
      //--------------------------------
      for (unsigned ipt = 0; ipt < n_intpt; ipt++)
      {
        // Assign values of s
        for (unsigned i = 0; i < (this->Dim - 1); i++)
        {
          s[i] = this->integral_pt()->knot(ipt, i);
        }
 
        // Get the integral weight
        double w = this->integral_pt()->weight(ipt);
 
        // Find the shape test functions and derivates; return the Jacobian
        // of the mapping between local and global (Eulerian)
        // coordinates
        double J = this->test_only(s, test);
 
        // Premultiply the weights and the Jacobian
        double W = w * J;
 
        // Now add to the appropriate equations
        // Loop over the test functions:loop over the nodes
        for (unsigned l = 0; l < n_node; l++)
        {
          local_eqn_real =
            this->nodal_local_eqn(l, this->U_index_helmholtz.real());
          local_eqn_imag =
            this->nodal_local_eqn(l, this->U_index_helmholtz.imag());
 
          // IF it's not a boundary condition
          if (local_eqn_real >= 0)
          {
            // Add the gamma contribution in this int_point to the res
            residuals[local_eqn_real] -= gamma[ipt].real() * test[l] * W;
 
            // Calculate the jacobian
            //-----------------------
            if (flag)
            {
              // Loop over the shape functions again
              for (unsigned l2 = 0; l2 < n_node; l2++)
              {
                // Add the contribution of the local data
                local_unknown_real =
                  this->nodal_local_eqn(l2, this->U_index_helmholtz.real());
 
                local_unknown_imag =
                  this->nodal_local_eqn(l2, this->U_index_helmholtz.imag());
 
                // If at a non-zero degree of freedom add in the entry
                if (local_unknown_real >= 0)
                {
                  global_eqn_real = this->eqn_number(local_unknown_real);
 
                  // Add the first order terms contribution
                  jacobian(local_eqn_real, local_unknown_real) -=
                    d_gamma[ipt][global_eqn_real].real() * test[l] * W;
                }
                if (local_unknown_imag >= 0)
                {
                  global_eqn_imag = this->eqn_number(local_unknown_imag);
 
                  // Add the first order terms contribution
                  jacobian(local_eqn_real, local_unknown_imag) -=
                    d_gamma[ipt][global_eqn_imag].real() * test[l] * W;
                }
              } // End of loop over nodes l2
 
              // Add the contribution of the external data
              unsigned n_ext_data = this->nexternal_data();
              // Loop over the shape functions again
              for (unsigned l2 = 0; l2 < n_ext_data; l2++)
              {
                // Add the contribution of the local data
                external_unknown_real =
                  this->external_local_eqn(l2, this->U_index_helmholtz.real());
 
                external_unknown_imag =
                  this->external_local_eqn(l2, this->U_index_helmholtz.imag());
 
                // If at a non-zero degree of freedom add in the entry
                if (external_unknown_real >= 0)
                {
                  external_global_eqn_real =
                    this->eqn_number(external_unknown_real);
 
                  // Add the first order terms contribution
                  jacobian(local_eqn_real, external_unknown_real) -=
                    d_gamma[ipt][external_global_eqn_real].real() * test[l] * W;
                }
                if (external_unknown_imag >= 0)
                {
                  external_global_eqn_imag =
                    this->eqn_number(external_unknown_imag);
 
                  // Add the first order terms contribution
                  jacobian(local_eqn_real, external_unknown_imag) -=
                    d_gamma[ipt][external_global_eqn_imag].real() * test[l] * W;
                }
              } // End of loop over external data
            } // End of flag
          } // end of local_eqn_real
 
          if (local_eqn_imag >= 0)
          {
            // Add the gamma contribution in this int_point to the res
            residuals[local_eqn_imag] -= gamma[ipt].imag() * test[l] * W;
 
            // Calculate the jacobian
            //-----------------------
            if (flag)
            {
              // Loop over the shape functions again
              for (unsigned l2 = 0; l2 < n_node; l2++)
              {
                // Add the contribution of the local data
                local_unknown_real =
                  this->nodal_local_eqn(l2, this->U_index_helmholtz.real());
 
                local_unknown_imag =
                  this->nodal_local_eqn(l2, this->U_index_helmholtz.imag());
 
                // If at a non-zero degree of freedom add in the entry
                if (local_unknown_real >= 0)
                {
                  global_eqn_real = this->eqn_number(local_unknown_real);
 
                  // Add the first order terms contribution
                  jacobian(local_eqn_imag, local_unknown_real) -=
                    d_gamma[ipt][global_eqn_real].imag() * test[l] * W;
                }
                if (local_unknown_imag >= 0)
                {
                  global_eqn_imag = this->eqn_number(local_unknown_imag);
 
                  // Add the first order terms contribution
                  jacobian(local_eqn_imag, local_unknown_imag) -=
                    d_gamma[ipt][global_eqn_imag].imag() * test[l] * W;
                }
              } // End of loop over nodes l2
 
              // Add the contribution of the external data
              unsigned n_ext_data = this->nexternal_data();
              // Loop over the shape functions again
              for (unsigned l2 = 0; l2 < n_ext_data; l2++)
              {
                // Add the contribution of the local data
                external_unknown_real =
                  this->external_local_eqn(l2, this->U_index_helmholtz.real());
 
                external_unknown_imag =
                  this->external_local_eqn(l2, this->U_index_helmholtz.imag());
 
                // If at a non-zero degree of freedom add in the entry
                if (external_unknown_real >= 0)
                {
                  external_global_eqn_real =
                    this->eqn_number(external_unknown_real);
 
                  // Add the first order terms contribution
                  jacobian(local_eqn_imag, external_unknown_real) -=
                    d_gamma[ipt][external_global_eqn_real].imag() * test[l] * W;
                }
                if (external_unknown_imag >= 0)
                {
                  external_global_eqn_imag =
                    this->eqn_number(external_unknown_imag);
 
                  // Add the first order terms contribution
                  jacobian(local_eqn_imag, external_unknown_imag) -=
                    d_gamma[ipt][external_global_eqn_imag].imag() * test[l] * W;
                }
              } // End of loop over external data
            } // End of flag
          } // end of local_eqn_imag
        } // end of llop over yhe node
      } // End of loop over int_pt
    } // End of fill_in_generic_residual_contribution_helmholtz_flux
 
 
    /// Pointer to mesh of all DtN boundary condition elements
    /// (needed to get access to gamma values)
    HelmholtzDtNMesh<ELEMENT>* Outer_boundary_mesh_pt;
  };
 
 
  /// /////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////
 
 
  //===========start_compute_gamma_contribution==================
  /// compute the contribution of the element
  /// to the Gamma integral and its derivates w.r.t
  /// to global unknows; the function takes wavenumber n
  /// and polar angle phi as input.
  //==============================================================
  template<class ELEMENT>
  void HelmholtzDtNBoundaryElement<ELEMENT>::compute_gamma_contribution(
    const double& phi,
    const int& n,
    std::complex<double>& gamma_con,
    std::map<unsigned, std::complex<double>>& d_gamma_con)
  {
    // define the imaginary number
    const std::complex<double> I(0.0, 1.0);
 
    // Find out how many nodes there are
    const unsigned n_node = this->nnode();
 
    // Set up memory for the shape  functions
    Shape psi(n_node);
    DShape dpsi(n_node, 1);
 
    // initialise the variable
    int local_unknown_real = 0, local_unknown_imag = 0;
    int global_unknown_real = 0, global_unknown_imag = 0;
 
    // Set the value of n_intpt
    const unsigned n_intpt = this->integral_pt()->nweight();
 
    // Set the Vector to hold local coordinates
    Vector<double> s(this->Dim - 1);
 
    // Initialise
    gamma_con = std::complex<double>(0.0, 0.0);
    d_gamma_con.clear();
 
    // Loop over the integration points
    //--------------------------------
    for (unsigned ipt = 0; ipt < n_intpt; ipt++)
    {
      // Assign values of s
      for (unsigned i = 0; i < (this->Dim - 1); i++)
      {
        s[i] = this->integral_pt()->knot(ipt, i);
      }
 
      // Get the integral weight
      double w = this->integral_pt()->weight(ipt);
 
      // Get the shape functions
      this->dshape_local(s, psi, dpsi);
 
      // Eulerian coordinates at Gauss point
      Vector<double> interpolated_x(this->Dim, 0.0);
 
      // Derivs of Eulerian coordinates w.r.t. local coordinates
      Vector<double> interpolated_dxds(this->Dim);
      std::complex<double> interpolated_u(0.0, 0.0);
 
      // Assemble x and its derivs
      for (unsigned l = 0; l < n_node; l++)
      {
        // Loop over directions
        for (unsigned i = 0; i < this->Dim; i++)
        {
          interpolated_x[i] += this->nodal_position(l, i) * psi[l];
          interpolated_dxds[i] += this->nodal_position(l, i) * dpsi(l, 0);
        }
 
        // Get the nodal value of the helmholtz unknown
        std::complex<double> u_value(
          this->nodal_value(l, this->U_index_helmholtz.real()),
          this->nodal_value(l, this->U_index_helmholtz.imag()));
 
        interpolated_u += u_value * psi(l);
      } // End of loop over the nodes
 
      // calculate the integral
      //-----------------------
      // define the variable phi_p
      double phi_p = atan2(interpolated_x[1], interpolated_x[0]);
 
      // define dphi_ds=(-yx'+y'x)/(x^2+y^2)
      double denom = (interpolated_x[0] * interpolated_x[0]) +
                     (interpolated_x[1] * interpolated_x[1]);
      double nom = -interpolated_dxds[1] * interpolated_x[0] +
                   interpolated_dxds[0] * interpolated_x[1];
      double dphi_ds = std::fabs(nom / denom);
 
      // compute the element contribution to gamma
      // ALH: The awkward construction with pow and the static_cast is to
      // avoid a floating point error on my machine when running unoptimised
      // (no idea why!)
      gamma_con += (dphi_ds)*w *
                   pow(exp(I * (phi - phi_p)), static_cast<double>(n)) *
                   interpolated_u;
 
      // compute the contribution to each node to the map
      for (unsigned l = 0; l < n_node; l++)
      {
        // Add the contribution of the real local data
        local_unknown_real =
          this->nodal_local_eqn(l, this->U_index_helmholtz.real());
        if (local_unknown_real >= 0)
        {
          global_unknown_real = this->eqn_number(local_unknown_real);
          d_gamma_con[global_unknown_real] +=
            (dphi_ds)*w * exp(I * (phi - phi_p) * double(n)) * psi(l);
        }
 
        // Add the contribution of the imag local data
        local_unknown_imag =
          this->nodal_local_eqn(l, this->U_index_helmholtz.imag());
        if (local_unknown_imag >= 0)
        {
          global_unknown_imag = this->eqn_number(local_unknown_imag);
          // ALH: The awkward construction with pow and the static_cast is to
          // avoid a floating point error on my machine when running unoptimised
          // (no idea why!)
          d_gamma_con[global_unknown_imag] +=
            I * (dphi_ds)*w *
            pow(exp(I * (phi - phi_p)), static_cast<double>(n)) * psi(l);
        }
      } // end of loop over the node
    } // End of loop over integration points
  }
 
 
  /// //////////////////////////////////////////////////////////////////
  /// //////////////////////////////////////////////////////////////////
  /// //////////////////////////////////////////////////////////////////
 
 
  //===========================================================================
  /// Namespace for checking radius of nodes on (assumed to be circular)
  /// DtN boundary
  //===========================================================================
  namespace ToleranceForHelmholtzOuterBoundary
  {
    /// Relative tolerance to within radius of points on DtN boundary
    /// are allowed to deviate from specified value
    extern double Tol;
 
  } // namespace ToleranceForHelmholtzOuterBoundary
 
 
  //===========================================================================
  /// Namespace for checking radius of nodes on (assumed to be circular)
  /// DtN boundary
  //===========================================================================
  namespace ToleranceForHelmholtzOuterBoundary
  {
    /// Relative tolerance to within radius of points on DtN boundary
    /// are allowed to deviate from specified value
    double Tol = 1.0e-3;
 
  } // namespace ToleranceForHelmholtzOuterBoundary
 
  /// //////////////////////////////////////////////////////////////////
  /// //////////////////////////////////////////////////////////////////
  /// //////////////////////////////////////////////////////////////////
 
 
  /// ================================================================
  /// Compute Fourier components of the solution -- length of
  /// vector indicates number of terms to be computed.
  /// ================================================================
  template<class ELEMENT>
  void HelmholtzDtNMesh<ELEMENT>::compute_fourier_components(
    Vector<std::complex<double>>& a_coeff_pos,
    Vector<std::complex<double>>& a_coeff_neg)
  {
#ifdef PARANOID
    if (a_coeff_pos.size() != a_coeff_neg.size())
    {
      std::ostringstream error_stream;
      error_stream << "a_coeff_pos and a_coeff_neg must have "
                   << "the same size. \n";
      throw OomphLibError(
        error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
#endif
 
    // Initialise
    unsigned n = a_coeff_pos.size();
    Vector<std::complex<double>> el_a_coeff_pos(n);
    Vector<std::complex<double>> el_a_coeff_neg(n);
    for (unsigned i = 0; i < n; i++)
    {
      a_coeff_pos[i] = std::complex<double>(0.0, 0.0);
      a_coeff_neg[i] = std::complex<double>(0.0, 0.0);
    }
 
    // Loop over elements e
    unsigned nel = this->nelement();
    for (unsigned e = 0; e < nel; e++)
    {
      // Get a pointer to element
      HelmholtzBCElementBase<ELEMENT>* el_pt =
        dynamic_cast<HelmholtzBCElementBase<ELEMENT>*>(this->element_pt(e));
 
      // Compute contribution
      el_pt->compute_contribution_to_fourier_components(el_a_coeff_pos,
                                                        el_a_coeff_neg);
 
      // Add to coefficients
      for (unsigned i = 0; i < n; i++)
      {
        a_coeff_pos[i] += el_a_coeff_pos[i];
        a_coeff_neg[i] += el_a_coeff_neg[i];
      }
    }
  }
 
 
  /// ================================================================
  /// Compute and store the gamma integral and derivates
  // /w.r.t global unknows at all integration  points
  /// of the mesh's constituent elements
  //================================================================
  template<class ELEMENT>
  void HelmholtzDtNMesh<ELEMENT>::setup_gamma()
  {
#ifdef PARANOID
    {
      // Loop over elements e
      unsigned nel = this->nelement();
      for (unsigned e = 0; e < nel; e++)
      {
        FiniteElement* fe_pt = finite_element_pt(e);
        unsigned nnod = fe_pt->nnode();
        for (unsigned j = 0; j < nnod; j++)
        {
          Node* nod_pt = fe_pt->node_pt(j);
 
          // Extract nodal coordinates from node:
          Vector<double> x(2);
          x[0] = nod_pt->x(0);
          x[1] = nod_pt->x(1);
 
          // Evaluate the radial distance
          double r = sqrt(x[0] * x[0] + x[1] * x[1]);
 
          // Check
          if (Outer_radius == 0.0)
          {
            throw OomphLibError("Outer radius for DtN BC must not be zero!",
                                OOMPH_CURRENT_FUNCTION,
                                OOMPH_EXCEPTION_LOCATION);
          }
 
          if (std::fabs((r - this->Outer_radius) / Outer_radius) >
              ToleranceForHelmholtzOuterBoundary::Tol)
          {
            std::ostringstream error_stream;
            error_stream << "Node at " << x[0] << " " << x[1] << " has radius "
                         << r << " which does not "
                         << " agree with \nspecified outer radius "
                         << this->Outer_radius << " within relative tolerance "
                         << ToleranceForHelmholtzOuterBoundary::Tol
                         << ".\nYou can adjust the tolerance via\n"
                         << "ToleranceForHelmholtzOuterBoundary::Tol\n"
                         << "or recompile without PARANOID.\n";
            throw OomphLibError(error_stream.str(),
                                OOMPH_CURRENT_FUNCTION,
                                OOMPH_EXCEPTION_LOCATION);
          }
        }
      }
    }
#endif
 
 
    // Get Helmholtz parameter from bulk element
    HelmholtzDtNBoundaryElement<ELEMENT>* el_pt =
      dynamic_cast<HelmholtzDtNBoundaryElement<ELEMENT>*>(this->element_pt(0));
    double k =
      sqrt(dynamic_cast<ELEMENT*>(el_pt->bulk_element_pt())->k_squared());
 
    // Precompute factors in sum
    Vector<std::complex<double>> h_a(Nfourier_terms), hp_a(Nfourier_terms),
      q(Nfourier_terms);
    Hankel_functions_for_helmholtz_problem::Hankel_first(
      Nfourier_terms, Outer_radius * k, h_a, hp_a);
    for (unsigned i = 0; i < Nfourier_terms; i++)
    {
      q[i] = (k / (2.0 * MathematicalConstants::Pi)) * (hp_a[i] / h_a[i]);
    }
 
    // first loop over elements e
    unsigned nel = this->nelement();
    for (unsigned e = 0; e < nel; e++)
    {
      // Get a pointer to element
      HelmholtzDtNBoundaryElement<ELEMENT>* el_pt =
        dynamic_cast<HelmholtzDtNBoundaryElement<ELEMENT>*>(
          this->element_pt(e));
 
      // Set the value of n_intpt
      const unsigned n_intpt = el_pt->integral_pt()->nweight();
 
      // initialise gamma integral and its derivatives
      Vector<std::complex<double>> gamma_vector(n_intpt,
                                                std::complex<double>(0.0, 0.0));
      Vector<std::map<unsigned, std::complex<double>>> d_gamma_vector(n_intpt);
 
      // Loop over the integration points
      for (unsigned ipt = 0; ipt < n_intpt; ipt++)
      {
        // Allocate and initialise coordinate
        unsigned ndim_local = el_pt->dim();
        Vector<double> x(ndim_local + 1, 0.0);
 
        // Set the Vector to hold local coordinates
        Vector<double> s(ndim_local, 0.0);
        for (unsigned i = 0; i < ndim_local; i++)
        {
          s[i] = el_pt->integral_pt()->knot(ipt, i);
        }
 
        // Get the coordinates of the integration point
        el_pt->interpolated_x(s, x);
 
        // Polar angle
        double phi = atan2(x[1], x[0]);
 
        // Elemental contribution to gamma integral and its derivative
        std::complex<double> gamma_con_p(0.0, 0.0), gamma_con_n(0.0, 0.0);
        std::map<unsigned, std::complex<double>> d_gamma_con_p, d_gamma_con_n;
 
        // loop over the Fourier terms
        for (unsigned nn = 0; nn < Nfourier_terms; nn++)
        {
          // Second loop over the element
          // to evaluate the complete integral
          for (unsigned ee = 0; ee < nel; ee++)
          {
            HelmholtzDtNBoundaryElement<ELEMENT>* eel_pt =
              dynamic_cast<HelmholtzDtNBoundaryElement<ELEMENT>*>(
                this->element_pt(ee));
 
            // contribution of the positive term in the sum
            eel_pt->compute_gamma_contribution(
              phi, int(nn), gamma_con_p, d_gamma_con_p);
 
            // contribution of the negative term in the sum
            eel_pt->compute_gamma_contribution(
              phi, -int(nn), gamma_con_n, d_gamma_con_n);
 
            unsigned n_node = eel_pt->nnode();
            if (nn == 0)
            {
              gamma_vector[ipt] += q[nn] * gamma_con_p;
              for (unsigned l = 0; l < n_node; l++)
              {
                // Add the contribution of the real local data
                int local_unknown_p_real = eel_pt->nodal_local_eqn(
                  l, eel_pt->u_index_helmholtz().real());
                if (local_unknown_p_real >= 0)
                {
                  int global_unknown_p_real =
                    eel_pt->eqn_number(local_unknown_p_real);
                  d_gamma_vector[ipt][global_unknown_p_real] +=
                    q[nn] * d_gamma_con_p[global_unknown_p_real];
                }
 
                // Add the contribution of the imag local data
                int local_unknown_p_imag = eel_pt->nodal_local_eqn(
                  l, eel_pt->u_index_helmholtz().imag());
 
                if (local_unknown_p_imag >= 0)
                {
                  int global_unknown_p_imag =
                    eel_pt->eqn_number(local_unknown_p_imag);
 
                  d_gamma_vector[ipt][global_unknown_p_imag] +=
                    q[nn] * d_gamma_con_p[global_unknown_p_imag];
                }
              } // end of loop over the node
            } // End of if
            else
            {
              gamma_vector[ipt] += q[nn] * (gamma_con_p + gamma_con_n);
              for (unsigned l = 0; l < n_node; l++)
              {
                // Add the contribution of the real local data
                int local_unknown_real = eel_pt->nodal_local_eqn(
                  l, eel_pt->u_index_helmholtz().real());
                if (local_unknown_real >= 0)
                {
                  int global_unknown_real =
                    eel_pt->eqn_number(local_unknown_real);
                  d_gamma_vector[ipt][global_unknown_real] +=
                    q[nn] * (d_gamma_con_p[global_unknown_real] +
                             d_gamma_con_n[global_unknown_real]);
                }
                // Add the contribution of the imag local data
                int local_unknown_imag = eel_pt->nodal_local_eqn(
                  l, eel_pt->u_index_helmholtz().imag());
                if (local_unknown_imag >= 0)
                {
                  int global_unknown_imag =
                    eel_pt->eqn_number(local_unknown_imag);
                  d_gamma_vector[ipt][global_unknown_imag] +=
                    q[nn] * (d_gamma_con_p[global_unknown_imag] +
                             d_gamma_con_n[global_unknown_imag]);
                }
              } // end of loop over the node
            } // End of else
          } // End of second loop over the elements
        } // End of loop over Fourier terms
      } // end of loop over integration point
 
      // Store it in map
      Gamma_at_gauss_point[el_pt] = gamma_vector;
      D_Gamma_at_gauss_point[el_pt] = d_gamma_vector;
 
    } // end of first loop over element
  }
 
  //===========================================================================
  /// Constructor, takes the pointer to the "bulk" element and the face index
  //===========================================================================
  template<class ELEMENT>
  HelmholtzBCElementBase<ELEMENT>::HelmholtzBCElementBase(
    FiniteElement* const& bulk_el_pt, const int& face_index)
    : FaceGeometry<ELEMENT>(), FaceElement()
  {
#ifdef PARANOID
    {
      // Check that the element is not a refineable 3d element
      ELEMENT* elem_pt = new ELEMENT;
      // If it's three-d
      if (elem_pt->dim() == 3)
      {
        // Is it refineable
        if (dynamic_cast<RefineableElement*>(elem_pt))
        {
          // Issue a warning
          OomphLibWarning("This flux element will not"
                          "work correctly if nodes are hanging\n",
                          "HelmholtzBCElementBase::Constructor",
                          OOMPH_EXCEPTION_LOCATION);
        }
      }
    }
#endif
 
    // 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();
 
    // Set up U_index_helmholtz. Initialise to zero, which probably won't change
    // in most cases, oh well, the price we pay for generality
    U_index_helmholtz = std::complex<unsigned>(0, 1);
 
    // Cast to the appropriate HelmholtzEquation so that we can
    // find the index at which the variable is stored
    // We assume that the dimension of the full problem is the same
    // as the dimension of the node, if this is not the case you will have
    // to write custom elements, sorry
    switch (Dim)
    {
        // One dimensional problem
      case 1:
      {
        HelmholtzEquations<1>* eqn_pt =
          dynamic_cast<HelmholtzEquations<1>*>(bulk_el_pt);
        // If the cast has failed die
        if (eqn_pt == 0)
        {
          std::string error_string =
            "Bulk element must inherit from HelmholtzEquations.";
          error_string +=
            "Nodes are one dimensional, but cannot cast the bulk element to\n";
          error_string += "HelmholtzEquations<1>\n.";
          error_string += "If you desire this functionality, you must "
                          "implement it yourself\n";
 
          throw OomphLibError(
            error_string, OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
        }
        // Otherwise read out the value
        else
        {
          // Read the index from the (cast) bulk element
          U_index_helmholtz = eqn_pt->u_index_helmholtz();
        }
      }
      break;
 
      // Two dimensional problem
      case 2:
      {
        HelmholtzEquations<2>* eqn_pt =
          dynamic_cast<HelmholtzEquations<2>*>(bulk_el_pt);
        // If the cast has failed die
        if (eqn_pt == 0)
        {
          std::string error_string =
            "Bulk element must inherit from HelmholtzEquations.";
          error_string +=
            "Nodes are two dimensional, but cannot cast the bulk element to\n";
          error_string += "HelmholtzEquations<2>\n.";
          error_string += "If you desire this functionality, you must "
                          "implement it yourself\n";
 
          throw OomphLibError(
            error_string, OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
        }
        else
        {
          // Read the index from the (cast) bulk element
          U_index_helmholtz = eqn_pt->u_index_helmholtz();
        }
      }
 
      break;
 
      // Three dimensional problem
      case 3:
      {
        HelmholtzEquations<3>* eqn_pt =
          dynamic_cast<HelmholtzEquations<3>*>(bulk_el_pt);
        // If the cast has failed die
        if (eqn_pt == 0)
        {
          std::string error_string =
            "Bulk element must inherit from HelmholtzEquations.";
          error_string += "Nodes are three dimensional, but cannot cast the "
                          "bulk element to\n";
          error_string += "HelmholtzEquations<3>\n.";
          error_string += "If you desire this functionality, you must "
                          "implement it yourself\n";
 
          throw OomphLibError(
            error_string, OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
        }
        else
        {
          // Read the index from the (cast) bulk element
          U_index_helmholtz = eqn_pt->u_index_helmholtz();
        }
      }
      break;
 
      // Any other case is an error
      default:
        std::ostringstream error_stream;
        error_stream << "Dimension of node is " << Dim
                     << ". It should be 1,2, or 3!" << std::endl;
 
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
        break;
    }
  }
 
 
} // namespace oomph
 
#endif