pml_helmholtz_flux_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 apply prescribed flux
// boundary conditions to the Pml Helmholtz equations
#ifndef OOMPH_PML_HELMHOLTZ_FLUX_ELEMENTS_HEADER
#define OOMPH_PML_HELMHOLTZ_FLUX_ELEMENTS_HEADER
 
 
// Config header generated by autoconfig
#ifdef HAVE_CONFIG_H
#include <oomph-lib-config.h>
#endif
 
// oomph-lib ncludes
#include "../generic/Qelements.h"
#include "math.h"
#include <complex>
 
namespace oomph
{
  //======================================================================
  /// A class for elements that allow the post-processing of
  /// radiated power and flux on the boundaries of PMLHelmholtz elements.
  /// The element geometry is obtained from the  FaceGeometry<ELEMENT>
  /// policy class.
  //======================================================================
  template<class ELEMENT>
  class PMLHelmholtzPowerElement : 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.
    PMLHelmholtzPowerElement(FiniteElement* const& bulk_el_pt,
                             const int& face_index);
 
    /// Broken empty constructor
    PMLHelmholtzPowerElement()
    {
      throw OomphLibError(
        "Don't call empty constructor for PMLHelmholtzPowerElement",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Broken copy constructor
    PMLHelmholtzPowerElement(const PMLHelmholtzPowerElement& 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 PMLHelmholtzPowerElement&) = 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);
    }
 
 
    /// Return the index at which the 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 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>(0.0, 0.0));
        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 the element's contribution to the time-averaged
    /// flux
    std::complex<double> global_flux_contribution()
    {
      // Dummy output file
      std::ofstream outfile;
      return global_flux_contribution(outfile);
    }
 
    ///  Compute the element's contribution to the integral of
    /// the flux over the artificial boundary. Also output the
    /// flux  in the specified
    /// output file if it's open.
    std::complex<double> global_flux_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);
      std::complex<double> flux(0.0, 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>(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
 
 
        // define dphi_dn
        for (unsigned i = 0; i < bulk_dim; i++)
        {
          dphi_dn += interpolated_dphidx[i] * unit_normal[i];
        }
 
        // Output?
        if (outfile.is_open())
        {
          interpolated_x(s, x);
          outfile << x[0] << " " << x[1] << " " << dphi_dn.real() << " "
                  << dphi_dn.imag() << "\n";
        }
 
        // ...add to integral
        flux += dphi_dn * W;
      }
 
      return flux;
    }
 
 
  protected:
    /// 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;
  };
 
  /// ///////////////////////////////////////////////////////////////////
  /// ///////////////////////////////////////////////////////////////////
  /// ///////////////////////////////////////////////////////////////////
 
 
  //===========================================================================
  /// Constructor, takes the pointer to the "bulk" element, the
  /// index of the fixed local coordinate and its value represented
  /// by an integer (+/- 1), indicating that the face is located
  /// at the max. or min. value of the "fixed" local coordinate
  /// in the bulk element.
  //===========================================================================
  template<class ELEMENT>
  PMLHelmholtzPowerElement<ELEMENT>::PMLHelmholtzPowerElement(
    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 = dynamic_cast<ELEMENT*>(bulk_el_pt);
      // If it's three-d
      if (elem_pt->dim() == 3)
      {
        // Is it refineable
        RefineableElement* ref_el_pt =
          dynamic_cast<RefineableElement*>(elem_pt);
        if (ref_el_pt != 0)
        {
          if (this->has_hanging_nodes())
          {
            throw OomphLibError("This flux element will not work correctly if "
                                "nodes are hanging\n",
                                OOMPH_CURRENT_FUNCTION,
                                OOMPH_EXCEPTION_LOCATION);
          }
        }
      }
    }
#endif
 
    // 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 PMLHelmholtzEquation 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:
      {
        PMLHelmholtzEquations<1>* eqn_pt =
          dynamic_cast<PMLHelmholtzEquations<1>*>(bulk_el_pt);
        // If the cast has failed die
        if (eqn_pt == 0)
        {
          std::string error_string =
            "Bulk element must inherit from PMLHelmholtzEquations.";
          error_string +=
            "Nodes are one dimensional, but cannot cast the bulk element to\n";
          error_string += "PMLHelmholtzEquations<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:
      {
        PMLHelmholtzEquations<2>* eqn_pt =
          dynamic_cast<PMLHelmholtzEquations<2>*>(bulk_el_pt);
        // If the cast has failed die
        if (eqn_pt == 0)
        {
          std::string error_string =
            "Bulk element must inherit from PMLHelmholtzEquations.";
          error_string +=
            "Nodes are two dimensional, but cannot cast the bulk element to\n";
          error_string += "PMLHelmholtzEquations<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:
      {
        PMLHelmholtzEquations<3>* eqn_pt =
          dynamic_cast<PMLHelmholtzEquations<3>*>(bulk_el_pt);
        // If the cast has failed die
        if (eqn_pt == 0)
        {
          std::string error_string =
            "Bulk element must inherit from PMLHelmholtzEquations.";
          error_string += "Nodes are three dimensional, but cannot cast the "
                          "bulk element to\n";
          error_string += "PMLHelmholtzEquations<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;
    }
  }
 
 
  /// ///////////////////////////////////////////////////////////////////////
  /// ///////////////////////////////////////////////////////////////////////
  /// ///////////////////////////////////////////////////////////////////////
 
 
  //======================================================================
  /// A class for elements that allow the imposition of an
  /// applied flux on the boundaries of PMLHelmholtz elements.
  /// The element geometry is obtained from the  FaceGeometry<ELEMENT>
  /// policy class.
  //======================================================================
  template<class ELEMENT>
  class PMLHelmholtzFluxElement : public virtual FaceGeometry<ELEMENT>,
                                  public virtual FaceElement
  {
  public:
    /// Function pointer to the prescribed-flux function fct(x,f(x)) --
    /// x is a Vector and  the flux is a complex. NOTE THAT f(x) represents
    /// c^2 du/dn!
    typedef void (*PMLHelmholtzPrescribedFluxFctPt)(const Vector<double>& x,
                                                    std::complex<double>& flux);
 
    /// Constructor, takes the pointer to the "bulk" element and the
    /// index of the face to which the element is attached.
    PMLHelmholtzFluxElement(FiniteElement* const& bulk_el_pt,
                            const int& face_index);
 
    /// Broken empty constructor
    PMLHelmholtzFluxElement()
    {
      throw OomphLibError(
        "Don't call empty constructor for PMLHelmholtzFluxElement",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Broken copy constructor
    PMLHelmholtzFluxElement(const PMLHelmholtzFluxElement& dummy) = delete;
 
    /// Broken assignment operator
    /*void operator=(const PMLHelmholtzFluxElement&) = delete;*/
 
 
    /// Access function for the prescribed-flux function pointer
    PMLHelmholtzPrescribedFluxFctPt& flux_fct_pt()
    {
      return Flux_fct_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_flux(
        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_flux(
        residuals, jacobian, 1);
    }
 
 
    /// 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 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());
    }
 
 
    /// The number of "DOF types" that degrees of freedom in this element
    /// are sub-divided into: real and imaginary part
    unsigned ndof_types() const
    {
      return 2;
    }
 
    /// Create a list of pairs for all unknowns in this element,
    /// so that the first entry in each pair contains the global equation
    /// number of the unknown, while the second one contains the number
    /// of the "DOF type" that this unknown is associated with.
    /// (Function can obviously only be called if the equation numbering
    /// scheme has been set up.) Real=0; Imag=1
    void get_dof_numbers_for_unknowns(
      std::list<std::pair<unsigned long, unsigned>>& dof_lookup_list) const
    {
      // temporary pair (used to store dof lookup prior to being added to list)
      std::pair<unsigned, unsigned> dof_lookup;
 
      // number of nodes
      unsigned n_node = this->nnode();
 
      // loop over the nodes
      for (unsigned n = 0; n < n_node; n++)
      {
        // determine local eqn number for real part
        int local_eqn_number =
          this->nodal_local_eqn(n, this->U_index_helmholtz.real());
 
        // ignore pinned values
        if (local_eqn_number >= 0)
        {
          // store dof lookup in temporary pair: First entry in pair
          // is global equation number; second entry is dof type
          dof_lookup.first = this->eqn_number(local_eqn_number);
          dof_lookup.second = 0;
 
          // add to list
          dof_lookup_list.push_front(dof_lookup);
        }
 
        // determine local eqn number for imag part
        local_eqn_number =
          this->nodal_local_eqn(n, this->U_index_helmholtz.imag());
 
        // ignore pinned values
        if (local_eqn_number >= 0)
        {
          // store dof lookup in temporary pair: First entry in pair
          // is global equation number; second entry is dof type
          dof_lookup.first = this->eqn_number(local_eqn_number);
          dof_lookup.second = 1;
 
          // add to list
          dof_lookup_list.push_front(dof_lookup);
        }
      }
    }
 
  protected:
    /// Function to compute the shape and test functions and to return
    /// the Jacobian of mapping between local and global (Eulerian)
    /// coordinates
    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);
    }
 
 
    /// Function to compute the shape and test functions and to return
    /// the Jacobian of mapping between local and global (Eulerian)
    /// coordinates
    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);
    }
 
 
    /// Function to calculate the prescribed flux at a given spatial
    /// position
    void get_flux(const Vector<double>& x, std::complex<double>& flux)
    {
      // If the function pointer is zero return zero
      if (Flux_fct_pt == 0)
      {
        flux = std::complex<double>(0.0, 0.0);
      }
      // Otherwise call the function
      else
      {
        (*Flux_fct_pt)(x, flux);
      }
    }
 
 
    /// The index at which the real and imag part of the unknown is
    /// stored at the nodes
    std::complex<unsigned> U_index_helmholtz;
 
 
    /// Add the element's contribution to its residual vector.
    /// flag=1(or 0): do (or don't) compute the contribution to the
    /// Jacobian as well.
    virtual void fill_in_generic_residual_contribution_helmholtz_flux(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      const unsigned& flag);
 
 
    /// Function pointer to the (global) prescribed-flux function
    PMLHelmholtzPrescribedFluxFctPt Flux_fct_pt;
 
    /// The spatial dimension of the problem
    unsigned Dim;
  };
 
  /// ///////////////////////////////////////////////////////////////////
  /// ///////////////////////////////////////////////////////////////////
  /// ///////////////////////////////////////////////////////////////////
 
 
  //===========================================================================
  /// Constructor, takes the pointer to the "bulk" element, the
  /// index of the fixed local coordinate and its value represented
  /// by an integer (+/- 1), indicating that the face is located
  /// at the max. or min. value of the "fixed" local coordinate
  /// in the bulk element.
  //===========================================================================
  template<class ELEMENT>
  PMLHelmholtzFluxElement<ELEMENT>::PMLHelmholtzFluxElement(
    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 = dynamic_cast<ELEMENT*>(bulk_el_pt);
      // If it's three-d
      if (elem_pt->dim() == 3)
      {
        // Is it refineable
        RefineableElement* ref_el_pt =
          dynamic_cast<RefineableElement*>(elem_pt);
        if (ref_el_pt != 0)
        {
          if (this->has_hanging_nodes())
          {
            throw OomphLibError("This flux element will not work correctly if "
                                "nodes are hanging\n",
                                OOMPH_CURRENT_FUNCTION,
                                OOMPH_EXCEPTION_LOCATION);
          }
        }
      }
    }
#endif
 
    // 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);
 
    // Initialise the prescribed-flux function pointer to zero
    Flux_fct_pt = 0;
 
    // 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 PMLHelmholtzEquation 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:
      {
        PMLHelmholtzEquations<1>* eqn_pt =
          dynamic_cast<PMLHelmholtzEquations<1>*>(bulk_el_pt);
        // If the cast has failed die
        if (eqn_pt == 0)
        {
          std::string error_string =
            "Bulk element must inherit from PMLHelmholtzEquations.";
          error_string +=
            "Nodes are one dimensional, but cannot cast the bulk element to\n";
          error_string += "PMLHelmholtzEquations<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:
      {
        PMLHelmholtzEquations<2>* eqn_pt =
          dynamic_cast<PMLHelmholtzEquations<2>*>(bulk_el_pt);
        // If the cast has failed die
        if (eqn_pt == 0)
        {
          std::string error_string =
            "Bulk element must inherit from PMLHelmholtzEquations.";
          error_string +=
            "Nodes are two dimensional, but cannot cast the bulk element to\n";
          error_string += "PMLHelmholtzEquations<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:
      {
        PMLHelmholtzEquations<3>* eqn_pt =
          dynamic_cast<PMLHelmholtzEquations<3>*>(bulk_el_pt);
        // If the cast has failed die
        if (eqn_pt == 0)
        {
          std::string error_string =
            "Bulk element must inherit from PMLHelmholtzEquations.";
          error_string += "Nodes are three dimensional, but cannot cast the "
                          "bulk element to\n";
          error_string += "PMLHelmholtzEquations<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;
    }
  }
 
 
  //===========================================================================
  /// Compute the element's residual vector and the (zero) Jacobian matrix.
  //===========================================================================
  template<class ELEMENT>
  void PMLHelmholtzFluxElement<ELEMENT>::
    fill_in_generic_residual_contribution_helmholtz_flux(
      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);
 
    // Integers to hold the local equation and unknown numbers
    int local_eqn_real = 0, local_eqn_imag = 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;
 
      // Need to find position to feed into flux function, initialise to zero
      Vector<double> interpolated_x(Dim, 0.0);
 
      // Calculate Eulerian position of integration point
      for (unsigned l = 0; l < n_node; l++)
      {
        for (unsigned i = 0; i < Dim; i++)
        {
          interpolated_x[i] += nodal_position(l, i) * psif[l];
        }
      }
 
      // Get the imposed flux
      std::complex<double> flux(0.0, 0.0);
      get_flux(interpolated_x, flux);
 
      // Now add to the appropriate equations
      // Loop over the test functions
      for (unsigned l = 0; l < n_node; l++)
      {
        local_eqn_real = nodal_local_eqn(l, U_index_helmholtz.real());
        /*IF it's not a boundary condition*/
        if (local_eqn_real >= 0)
        {
          // Add the prescribed flux terms
          residuals[local_eqn_real] -= flux.real() * testf[l] * W;
 
          // Imposed traction doesn't depend upon the solution,
          // --> the Jacobian is always zero, so no Jacobian
          // terms are required
        }
        local_eqn_imag = nodal_local_eqn(l, U_index_helmholtz.imag());
        /*IF it's not a boundary condition*/
        if (local_eqn_imag >= 0)
        {
          // Add the prescribed flux terms
          residuals[local_eqn_imag] -= flux.imag() * testf[l] * W;
 
          // Imposed traction doesn't depend upon the solution,
          // --> the Jacobian is always zero, so no Jacobian
          // terms are required
        }
      }
    }
  }
 
 
} // namespace oomph
 
#endif