pml_fourier_decomposed_helmholtz_flux_elements.h

Go to the documentation of this file.

// LIC// ====================================================================
// LIC// This file forms part of oomph-lib, the object-oriented,
// LIC// multi-physics finite-element library, available
// LIC// at http://www.oomph-lib.org.
// LIC//
// LIC// Copyright (C) 2006-2024 Matthias Heil and Andrew Hazel
// LIC//
// LIC// This library is free software; you can redistribute it and/or
// LIC// modify it under the terms of the GNU Lesser General Public
// LIC// License as published by the Free Software Foundation; either
// LIC// version 2.1 of the License, or (at your option) any later version.
// LIC//
// LIC// This library is distributed in the hope that it will be useful,
// LIC// but WITHOUT ANY WARRANTY; without even the implied warranty of
// LIC// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
// LIC// Lesser General Public License for more details.
// LIC//
// LIC// You should have received a copy of the GNU Lesser General Public
// LIC// License along with this library; if not, write to the Free Software
// LIC// Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
// LIC// 02110-1301  USA.
// LIC//
// LIC// The authors may be contacted at oomph-lib@maths.man.ac.uk.
// LIC//
// LIC//====================================================================
// Header file for elements that are used to apply prescribed flux
// boundary conditions to the Fourier decomposed Helmholtz equations
#ifndef OOMPH_PML_FOURIER_DECOMPOSED_HELMHOLTZ_FLUX_ELEMENTS_HEADER
#define OOMPH_PML_FOURIER_DECOMPOSED_HELMHOLTZ_FLUX_ELEMENTS_HEADER
 
 
// Config header generated by autoconfig
#ifdef HAVE_CONFIG_H
#include <oomph-lib-config.h>
#endif
 
#include "math.h"
#include <complex>
 
// oomph-lib includes
#include "../generic/Qelements.h"
 
 
namespace oomph
{
  //======================================================================
  /// A class for elements that allow the imposition of an
  /// applied flux on the boundaries of Fourier decomposed Helmholtz elements.
  /// The element geometry is obtained from the  FaceGeometry<ELEMENT>
  /// policy class.
  //======================================================================
  template<class ELEMENT>
  class PMLFourierDecomposedHelmholtzFluxElement
    : 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
 
    typedef void (*PMLFourierDecomposedHelmholtzPrescribedFluxFctPt)(
      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.
    PMLFourierDecomposedHelmholtzFluxElement(FiniteElement* const& bulk_el_pt,
                                             const int& face_index);
 
    /// Broken empty constructor
    PMLFourierDecomposedHelmholtzFluxElement()
    {
      throw OomphLibError("Don't call empty constructor for "
                          "PMLFourierDecomposedHelmholtzFluxElement",
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Broken copy constructor
    PMLFourierDecomposedHelmholtzFluxElement(
      const PMLFourierDecomposedHelmholtzFluxElement& 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 PMLFourierDecomposedHelmholtzFluxElement&) =
     * delete;*/
 
 
    /// Access function for the prescribed-flux function pointer
    PMLFourierDecomposedHelmholtzPrescribedFluxFctPt& 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_pml_fourier_decomposed_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_pml_fourier_decomposed_helmholtz_flux(
        residuals, jacobian, 1);
    }
 
    /// 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. (Real/imag part gives real index of real/imag part).
    virtual inline std::complex<unsigned> u_index_pml_fourier_decomposed_helmholtz()
      const
    {
      return std::complex<unsigned>(
        U_index_pml_fourier_decomposed_helmholtz.real(),
        U_index_pml_fourier_decomposed_helmholtz.imag());
    }
 
 
  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_pml_fourier_decomposed_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_pml_fourier_decomposed_helmholtz_flux(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      const unsigned& flag);
 
 
    /// Function pointer to the (global) prescribed-flux function
    PMLFourierDecomposedHelmholtzPrescribedFluxFctPt Flux_fct_pt;
  };
 
  /// ///////////////////////////////////////////////////////////////////
  /// ///////////////////////////////////////////////////////////////////
  /// ///////////////////////////////////////////////////////////////////
 
 
  //===========================================================================
  /// 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>
  PMLFourierDecomposedHelmholtzFluxElement<ELEMENT>::
    PMLFourierDecomposedHelmholtzFluxElement(FiniteElement* const& bulk_el_pt,
                                             const int& face_index)
    : FaceGeometry<ELEMENT>(), FaceElement()
  {
    // 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;
 
    // Initialise index at which real and imag unknowns are stored
    U_index_pml_fourier_decomposed_helmholtz = std::complex<unsigned>(0, 1);
 
    // Now read out indices from bulk element
    PMLFourierDecomposedHelmholtzEquations* eqn_pt =
      dynamic_cast<PMLFourierDecomposedHelmholtzEquations*>(bulk_el_pt);
    // If the cast has failed die
    if (eqn_pt == 0)
    {
      std::string error_string = "Bulk element must inherit from "
                                 "PMLFourierDecomposedHelmholtzEquations.";
      throw OomphLibError(
        error_string, OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
    }
    else
    {
      // Read the index from the (cast) bulk element
      U_index_pml_fourier_decomposed_helmholtz =
        eqn_pt->u_index_pml_fourier_decomposed_helmholtz();
    }
  }
 
 
  //===========================================================================
  /// Compute the element's residual vector and the (zero) Jacobian matrix.
  //===========================================================================
  template<class ELEMENT>
  void PMLFourierDecomposedHelmholtzFluxElement<ELEMENT>::
    fill_in_generic_residual_contribution_pml_fourier_decomposed_helmholtz_flux(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      const unsigned& flag)
  {
    // Find out how many nodes there are3
    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(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 < 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(2, 0.0);
 
      // Calculate coordinates
      for (unsigned l = 0; l < n_node; l++)
      {
        // Loop over coordinates
        for (unsigned i = 0; i < 2; i++)
        {
          interpolated_x[i] += nodal_position(l, i) * psif[l];
        }
      }
 
      // first component
      double r = interpolated_x[0];
 
      // 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_pml_fourier_decomposed_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] * r * 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_pml_fourier_decomposed_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] * r * W;
 
          // Imposed traction doesn't depend upon the solution,
          // --> the Jacobian is always zero, so no Jacobian
          // terms are required
        }
      }
    }
  }
 
 
  /// //////////////////////////////////////////////////////////////////
  /// //////////////////////////////////////////////////////////////////
  /// //////////////////////////////////////////////////////////////////
 
 
  //======================================================================
  /// A class for elements that allow postprocessing of the
  /// results -- currently computes radiated power over domain
  /// boundaries.
  /// The element geometry is obtained from the  FaceGeometry<ELEMENT>
  /// policy class.
  //======================================================================
  template<class ELEMENT>
  class PMLFourierDecomposedHelmholtzPowerMonitorElement
    : 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.
    PMLFourierDecomposedHelmholtzPowerMonitorElement(
      FiniteElement* const& bulk_el_pt, const int& face_index);
 
    /// Broken empty constructor
    PMLFourierDecomposedHelmholtzPowerMonitorElement()
    {
      throw OomphLibError("Don't call empty constructor for "
                          "PMLFourierDecomposedHelmholtzPowerMonitorElement",
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Broken copy constructor
    PMLFourierDecomposedHelmholtzPowerMonitorElement(
      const PMLFourierDecomposedHelmholtzPowerMonitorElement& dummy) = delete;
 
    /// Broken assignment operator
    /*void operator=(const PMLFourierDecomposedHelmholtzPowerMonitorElement&) =
     * 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_pml_fourier_decomposed_helmholtz()
      const
    {
      return std::complex<unsigned>(
        U_index_pml_fourier_decomposed_helmholtz.real(),
        U_index_pml_fourier_decomposed_helmholtz.imag());
    }
 
    /// Compute the element's contribution to the time-averaged
    /// radiated power over the artificial boundary.
    /// NOTE: This may give the wrong result
    /// if the constitutive parameters genuinely vary!
    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 zenith angle in the specified
    /// output file if it's open. NOTE: This may give the wrong result
    /// if the constitutive parameters genuinely vary!
    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 = this->nnode();
 
      // get the dim of the bulk and local nodes
      const unsigned bulk_dim = bulk_elem_pt->dim();
      const unsigned local_dim = this->node_pt(0)->ndim();
 
      // 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 - 1);
      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 - 1); 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_pml_fourier_decomposed_helmholtz().real()),
            bulk_elem_pt->nodal_value(
              l,
              bulk_elem_pt->u_index_pml_fourier_decomposed_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_pml_fourier_decomposed_helmholtz().real()),
            nodal_value(l, u_index_pml_fourier_decomposed_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 = (interpolated_phi.real() * dphi_dn.imag() -
                            interpolated_phi.imag() * dphi_dn.real());
 
        interpolated_x(s, x);
        double theta = atan2(x[0], x[1]);
        // Output?
        if (outfile.is_open())
        {
          outfile << x[0] << " " << x[1] << " " << theta << " " << integrand
                  << "\n";
        }
 
        // ...add to integral
        power += MathematicalConstants::Pi * x[0] * integrand * W;
      }
 
      return power;
    }
 
  protected:
    /// Function to compute the 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
    {
      // Get the shape functions
      shape(s, test);
 
      unsigned nnod = nnode();
      for (unsigned i = 0; i < nnod; i++)
      {
        psi[i] = test[i];
      }
 
      // 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 < 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_pml_fourier_decomposed_helmholtz;
  };
 
 
  /// ///////////////////////////////////////////////////////////////////
  /// ///////////////////////////////////////////////////////////////////
  /// ///////////////////////////////////////////////////////////////////
 
 
  //===========================================================================
  /// Constructor, takes the pointer to the "bulk" element and the face index.
  //===========================================================================
  template<class ELEMENT>
  PMLFourierDecomposedHelmholtzPowerMonitorElement<ELEMENT>::
    PMLFourierDecomposedHelmholtzPowerMonitorElement(
      FiniteElement* const& bulk_el_pt, const int& face_index)
    : FaceGeometry<ELEMENT>(), FaceElement()
  {
    // 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);
 
    // Set up U_index_pml_fourier_decomposedhelmholtz.
    U_index_pml_fourier_decomposed_helmholtz = std::complex<unsigned>(0, 1);
 
    // Cast to the appropriate PMLFourierDecomposedHelmholtzEquation
    // 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
    PMLFourierDecomposedHelmholtzEquations* eqn_pt =
      dynamic_cast<PMLFourierDecomposedHelmholtzEquations*>(bulk_el_pt);
    if (eqn_pt == 0)
    {
      std::string error_string = "Bulk element must inherit from "
                                 "PMLFourierDecomposedHelmholtzEquations.";
      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_pml_fourier_decomposed_helmholtz =
        eqn_pt->u_index_pml_fourier_decomposed_helmholtz();
    }
  }
 
} // namespace oomph
 
#endif