pml_fourier_decomposed_helmholtz_elements.h

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// LIC// ====================================================================
// LIC// This file forms part of oomph-lib, the object-oriented,
// LIC// multi-physics finite-element library, available
// LIC// at http://www.oomph-lib.org.
// LIC//
// LIC// Copyright (C) 2006-2022 Matthias Heil and Andrew Hazel
// LIC//
// LIC// This library is free software; you can redistribute it and/or
// LIC// modify it under the terms of the GNU Lesser General Public
// LIC// License as published by the Free Software Foundation; either
// LIC// version 2.1 of the License, or (at your option) any later version.
// LIC//
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// LIC// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
// LIC// Lesser General Public License for more details.
// LIC//
// LIC// You should have received a copy of the GNU Lesser General Public
// LIC// License along with this library; if not, write to the Free Software
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// LIC//
// LIC// The authors may be contacted at oomph-lib@maths.man.ac.uk.
// LIC//
// LIC//====================================================================
// Header file for Fourier-decomposed Helmholtz elements
#ifndef OOMPH_PML_FOURIER_DECOMPOSED_HELMHOLTZ_ELEMENTS_HEADER
#define OOMPH_PML_FOURIER_DECOMPOSED_HELMHOLTZ_ELEMENTS_HEADER
 
 
// Config header generated by autoconfig
#ifdef HAVE_CONFIG_H
#include <oomph-lib-config.h>
#endif
 
#include "math.h"
#include <complex>
 
 
// OOMPH-LIB headers
#include "../generic/projection.h"
#include "../generic/nodes.h"
#include "../generic/Qelements.h"
#include "../generic/oomph_utilities.h"
#include "../generic/pml_meshes.h"
#include "../generic/projection.h"
#include "../generic/oomph_definitions.h"
 
 
namespace oomph
{
  //========================================================================
  //========================================================================
  namespace Legendre_functions_helper
  {
    extern double factorial(const unsigned& l);
 
    extern double plgndr1(const unsigned& n, const double& x);
 
    extern double plgndr2(const unsigned& l,
                          const unsigned& m,
                          const double& x);
 
  } // namespace Legendre_functions_helper
 
 
 
  //=======================================================================
  //=======================================================================
  class PMLMappingAndTransformedCoordinate
  {
  public:
    PMLMappingAndTransformedCoordinate(){};
 
    virtual std::complex<double> gamma(const double& nu_i,
                                       const double& pml_width_i,
                                       const double& k_squared) = 0;
 
    virtual std::complex<double> transformed_coordinate(
      const double& nu_i,
      const double& pml_width_i,
      const double& pml_inner_boundary,
      const double& k_squared) = 0;
  };
 
 
  //=======================================================================
  //=======================================================================
  class BermudezPMLMappingAndTransformedCoordinate
    : public PMLMappingAndTransformedCoordinate
  {
  public:
    BermudezPMLMappingAndTransformedCoordinate(){};
 
    std::complex<double> gamma(const double& nu_i,
                               const double& pml_width_i,
                               const double& k_squared)
    {
      return 1.0 +
             std::complex<double>(1.0 / sqrt(k_squared), 0) *
               std::complex<double>(0.0, 1.0 / (std::fabs(pml_width_i - nu_i)));
    }
 
    std::complex<double> transformed_coordinate(
      const double& nu_i,
      const double& pml_width_i,
      const double& pml_inner_boundary,
      const double& k_squared)
    {
      double log_arg = 1.0 - std::fabs(nu_i / pml_width_i);
      return std::complex<double>(pml_inner_boundary + nu_i,
                                  -log(log_arg) / sqrt(k_squared));
    }
  };
 
 
 
 
  //=============================================================
  //=============================================================
  class PMLFourierDecomposedHelmholtzEquations
    : public virtual PMLElementBase<2>,
      public virtual FiniteElement
  {
  public:
    typedef void (*PMLFourierDecomposedHelmholtzSourceFctPt)(
      const Vector<double>& x, std::complex<double>& f);
 
    PMLFourierDecomposedHelmholtzEquations()
      : Source_fct_pt(0), K_squared_pt(0), N_pml_fourier_pt(0)
    {
      // Provide pointer to static method (Save memory)
      Pml_mapping_and_transformed_coordinate_pt =
        &PMLFourierDecomposedHelmholtzEquations::
          Default_pml_mapping_and_transformed_coordinate;
 
      Alpha_pt = &Default_Physical_Constant_Value;
    }
 
 
    PMLFourierDecomposedHelmholtzEquations(
      const PMLFourierDecomposedHelmholtzEquations& dummy) = delete;
 
    // Commented out broken assignment operator because this can lead to a
    // conflict warning when used in the virtual inheritence hierarchy.
    // Essentially the compiler doesn't realise that two separate
    // implementations of the broken function are the same and so, quite
    // rightly, it shouts.
    /*void operator=(const PMLFourierDecomposedHelmholtzEquations&) = delete;*/
 
    virtual inline std::complex<unsigned> u_index_pml_fourier_decomposed_helmholtz()
      const
    {
      return std::complex<unsigned>(0, 1);
    }
 
 
    double*& k_squared_pt()
    {
      return K_squared_pt;
    }
 
 
    double k_squared()
    {
#ifdef PARANOID
      if (K_squared_pt == 0)
      {
        throw OomphLibError(
          "Please set pointer to k_squared using access fct to pointer!",
          OOMPH_CURRENT_FUNCTION,
          OOMPH_EXCEPTION_LOCATION);
      }
#endif
      return *K_squared_pt;
    }
 
    double*& alpha_pt()
    {
      return Alpha_pt;
    }
 
 
    double alpha()
    {
      return *Alpha_pt;
    }
 
    int*& pml_fourier_wavenumber_pt()
    {
      return N_pml_fourier_pt;
    }
 
    int pml_fourier_wavenumber()
    {
      if (N_pml_fourier_pt == 0)
      {
        return 0;
      }
      else
      {
        return *N_pml_fourier_pt;
      }
    }
 
 
    void output(std::ostream& outfile)
    {
      const unsigned n_plot = 5;
      output(outfile, n_plot);
    }
 
    void output(std::ostream& outfile, const unsigned& n_plot);
 
    void output_real(std::ostream& outfile,
                     const double& phi,
                     const unsigned& n_plot);
 
    void output(FILE* file_pt)
    {
      const unsigned n_plot = 5;
      output(file_pt, n_plot);
    }
 
    void output(FILE* file_pt, const unsigned& n_plot);
 
    void output_fct(std::ostream& outfile,
                    const unsigned& n_plot,
                    FiniteElement::SteadyExactSolutionFctPt exact_soln_pt);
 
    virtual void output_fct(
      std::ostream& outfile,
      const unsigned& n_plot,
      const double& time,
      FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt)
    {
      throw OomphLibError("There is no time-dependent output_fct() for "
                          "PMLFourierDecomposedHelmholtz elements ",
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
 
 
    void output_real_fct(std::ostream& outfile,
                         const double& phi,
                         const unsigned& n_plot,
                         FiniteElement::SteadyExactSolutionFctPt exact_soln_pt);
 
 
    void compute_error(std::ostream& outfile,
                       FiniteElement::SteadyExactSolutionFctPt exact_soln_pt,
                       double& error,
                       double& norm);
 
 
    void compute_error(std::ostream& outfile,
                       FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt,
                       const double& time,
                       double& error,
                       double& norm)
    {
      throw OomphLibError("There is no time-dependent compute_error() for "
                          "PMLFourierDecomposedHelmholtz elements",
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
 
    void compute_norm(double& norm);
 
 
    PMLFourierDecomposedHelmholtzSourceFctPt& source_fct_pt()
    {
      return Source_fct_pt;
    }
 
 
    PMLFourierDecomposedHelmholtzSourceFctPt source_fct_pt() const
    {
      return Source_fct_pt;
    }
 
 
    inline virtual void get_source_pml_fourier_decomposed_helmholtz(
      const unsigned& ipt,
      const Vector<double>& x,
      std::complex<double>& source) const
    {
      // If no source function has been set, return zero
      if (Source_fct_pt == 0)
      {
        source = std::complex<double>(0.0, 0.0);
      }
      else
      {
        // Get source strength
        (*Source_fct_pt)(x, source);
      }
    }
 
    void values_to_be_pinned_on_outer_pml_boundary(
      Vector<unsigned>& values_to_pin)
    {
      values_to_pin.resize(2);
      for (unsigned j = 0; j < 2; j++)
      {
        values_to_pin[j] = j;
      }
    }
 
 
    void get_flux(const Vector<double>& s,
                  Vector<std::complex<double>>& flux) const
    {
      // Find out how many nodes there are in the element
      const unsigned n_node = nnode();
 
      // Set up memory for the shape and test functions
      Shape psi(n_node);
      DShape dpsidx(n_node, 2);
 
      // Call the derivatives of the shape and test functions
      dshape_eulerian(s, psi, dpsidx);
 
      // Initialise to zero
      const std::complex<double> zero(0.0, 0.0);
      for (unsigned j = 0; j < 2; j++)
      {
        flux[j] = zero;
      }
 
      // Loop over nodes
      for (unsigned l = 0; l < n_node; l++)
      {
        // Cache the complex value of the unknown
        const std::complex<double> u_value(
          this->nodal_value(l,
                            u_index_pml_fourier_decomposed_helmholtz().real()),
          this->nodal_value(l,
                            u_index_pml_fourier_decomposed_helmholtz().imag()));
 
        // Loop over derivative directions
        for (unsigned j = 0; j < 2; j++)
        {
          flux[j] += u_value * dpsidx(l, j);
        }
      }
    }
 
 
    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(
        residuals, GeneralisedElement::Dummy_matrix, 0);
    }
 
 
    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(
        residuals, jacobian, 1);
    }
 
 
    inline std::complex<double> interpolated_u_pml_fourier_decomposed_helmholtz(
      const Vector<double>& s) const
    {
      // Find number of nodes
      const unsigned n_node = nnode();
 
      // Local shape function
      Shape psi(n_node);
 
      // Find values of shape function
      shape(s, psi);
 
      // Initialise value of u
      std::complex<double> interpolated_u(0.0, 0.0);
 
      // Get the index at which the helmholtz unknown is stored
      const unsigned u_nodal_index_real =
        u_index_pml_fourier_decomposed_helmholtz().real();
      const unsigned u_nodal_index_imag =
        u_index_pml_fourier_decomposed_helmholtz().imag();
 
      // Loop over the local nodes and sum
      for (unsigned l = 0; l < n_node; l++)
      {
        // Make a temporary complex number from the stored data
        const std::complex<double> u_value(
          this->nodal_value(l, u_nodal_index_real),
          this->nodal_value(l, u_nodal_index_imag));
        // Add to the interpolated value
        interpolated_u += u_value * psi[l];
      }
      return interpolated_u;
    }
 
 
    unsigned self_test();
 
 
  protected:
    void compute_pml_coefficients(
      const unsigned& ipt,
      const Vector<double>& x,
      Vector<std::complex<double>>& pml_laplace_factor,
      std::complex<double>& pml_k_squared_factor)
    {
      Vector<double> nu(2);
      for (unsigned k = 0; k < 2; k++)
      {
        nu[k] = x[k] - this->Pml_inner_boundary[k];
      }
 
      Vector<double> pml_width(2);
      for (unsigned k = 0; k < 2; k++)
      {
        pml_width[k] =
          this->Pml_outer_boundary[k] - this->Pml_inner_boundary[k];
      }
 
      // Declare gamma_i vectors of complex numbers for PML weights
      Vector<std::complex<double>> pml_gamma(2);
 
      if (this->Pml_is_enabled)
      {
        // Cache k_squared to pass into mapping function
        double k_squared_local = k_squared();
 
        for (unsigned k = 0; k < 2; k++)
        {
          // If PML is enabled in the respective direction
          if (this->Pml_direction_active[k])
          {
            pml_gamma[k] = Pml_mapping_and_transformed_coordinate_pt->gamma(
              nu[k], pml_width[k], k_squared_local);
          }
          else
          {
            pml_gamma[k] = 1.0;
          }
        }
 
        pml_laplace_factor[0] = pml_gamma[1] / pml_gamma[0];
        pml_laplace_factor[1] = pml_gamma[0] / pml_gamma[1];
        pml_k_squared_factor = pml_gamma[0] * pml_gamma[1];
      }
      else
      {
        for (unsigned k = 0; k < 2; k++)
        {
          pml_laplace_factor[k] = std::complex<double>(1.0, 0.0);
        }
 
        pml_k_squared_factor = std::complex<double>(1.0, 0.0);
      }
    }
 
 
    void compute_complex_r(const unsigned& ipt,
                           const Vector<double>& x,
                           std::complex<double>& complex_r)
    {
      // Cache current position r
      double r = x[0];
 
 
      // If the complex r variable is imaginary
      if (this->Pml_is_enabled && (this->Pml_direction_active[0]))
      {
        double nu = x[0] - Pml_inner_boundary[0];
        double pml_width = Pml_outer_boundary[0] - Pml_inner_boundary[0];
        double k_squared_local = k_squared();
 
        // Determine the complex r variable
        complex_r =
          Pml_mapping_and_transformed_coordinate_pt->transformed_coordinate(
            nu, pml_width, Pml_inner_boundary[0], k_squared_local);
      }
      else
      {
        // The complex r variable is infact purely real, and
        // is equal to x[0]
        complex_r = std::complex<double>(r, 0.0);
      }
 
    } // end of compute_complex_r
 
    PMLMappingAndTransformedCoordinate*& pml_mapping_and_transformed_coordinate_pt()
    {
      return Pml_mapping_and_transformed_coordinate_pt;
    }
 
    PMLMappingAndTransformedCoordinate* const& pml_mapping_and_transformed_coordinate_pt()
      const
    {
      return Pml_mapping_and_transformed_coordinate_pt;
    }
 
    static BermudezPMLMappingAndTransformedCoordinate
      Default_pml_mapping_and_transformed_coordinate;
 
 
    virtual double dshape_and_dtest_eulerian_pml_fourier_decomposed_helmholtz(
      const Vector<double>& s,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const = 0;
 
 
    virtual double dshape_and_dtest_eulerian_at_knot_pml_fourier_decomposed_helmholtz(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const = 0;
 
    virtual void fill_in_generic_residual_contribution_pml_fourier_decomposed_helmholtz(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      const unsigned& flag);
 
    PMLFourierDecomposedHelmholtzSourceFctPt Source_fct_pt;
 
    double* K_squared_pt;
 
    PMLMappingAndTransformedCoordinate*
      Pml_mapping_and_transformed_coordinate_pt;
 
    double* Alpha_pt;
 
    static double Default_Physical_Constant_Value;
 
    int* N_pml_fourier_pt;
  };
 
 
 
 
  //======================================================================
  //======================================================================
  template<unsigned NNODE_1D>
  class QPMLFourierDecomposedHelmholtzElement
    : public virtual QElement<2, NNODE_1D>,
      public virtual PMLFourierDecomposedHelmholtzEquations
  {
  private:
    static const unsigned Initial_Nvalue;
 
  public:
    QPMLFourierDecomposedHelmholtzElement()
      : QElement<2, NNODE_1D>(), PMLFourierDecomposedHelmholtzEquations()
    {
    }
 
    QPMLFourierDecomposedHelmholtzElement(
      const QPMLFourierDecomposedHelmholtzElement<NNODE_1D>& dummy) = delete;
 
    /*void operator=(const
                   QPMLFourierDecomposedHelmholtzElement<NNODE_1D>&) = delete;*/
 
 
    inline unsigned required_nvalue(const unsigned& n) const
    {
      return Initial_Nvalue;
    }
 
    void output(std::ostream& outfile)
    {
      PMLFourierDecomposedHelmholtzEquations::output(outfile);
    }
 
    void output(std::ostream& outfile, const unsigned& n_plot)
    {
      PMLFourierDecomposedHelmholtzEquations::output(outfile, n_plot);
    }
 
    void output_real(std::ostream& outfile,
                     const double& phi,
                     const unsigned& n_plot)
    {
      PMLFourierDecomposedHelmholtzEquations::output_real(outfile, phi, n_plot);
    }
 
    void output(FILE* file_pt)
    {
      PMLFourierDecomposedHelmholtzEquations::output(file_pt);
    }
 
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      PMLFourierDecomposedHelmholtzEquations::output(file_pt, n_plot);
    }
 
    void output_fct(std::ostream& outfile,
                    const unsigned& n_plot,
                    FiniteElement::SteadyExactSolutionFctPt exact_soln_pt)
    {
      PMLFourierDecomposedHelmholtzEquations::output_fct(
        outfile, n_plot, exact_soln_pt);
    }
 
    void output_real_fct(std::ostream& outfile,
                         const double& phi,
                         const unsigned& n_plot,
                         FiniteElement::SteadyExactSolutionFctPt exact_soln_pt)
    {
      PMLFourierDecomposedHelmholtzEquations::output_real_fct(
        outfile, phi, n_plot, exact_soln_pt);
    }
 
 
    void output_fct(std::ostream& outfile,
                    const unsigned& n_plot,
                    const double& time,
                    FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt)
    {
      PMLFourierDecomposedHelmholtzEquations::output_fct(
        outfile, n_plot, time, exact_soln_pt);
    }
 
  protected:
    inline double dshape_and_dtest_eulerian_pml_fourier_decomposed_helmholtz(
      const Vector<double>& s,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const;
 
 
    inline double dshape_and_dtest_eulerian_at_knot_pml_fourier_decomposed_helmholtz(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const;
  };
 
 
  // Inline functions:
 
 
  //======================================================================
  //======================================================================
  template<unsigned NNODE_1D>
  double QPMLFourierDecomposedHelmholtzElement<NNODE_1D>::
    dshape_and_dtest_eulerian_pml_fourier_decomposed_helmholtz(
      const Vector<double>& s,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const
  {
    // Call the geometrical shape functions and derivatives
    const double J = this->dshape_eulerian(s, psi, dpsidx);
 
    // Set the test functions equal to the shape functions
    test = psi;
    dtestdx = dpsidx;
 
    // Return the jacobian
    return J;
  }
 
 
  //======================================================================
  //======================================================================
  template<unsigned NNODE_1D>
  double QPMLFourierDecomposedHelmholtzElement<NNODE_1D>::
    dshape_and_dtest_eulerian_at_knot_pml_fourier_decomposed_helmholtz(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const
  {
    // Call the geometrical shape functions and derivatives
    const double J = this->dshape_eulerian_at_knot(ipt, psi, dpsidx);
 
    // Set the pointers of the test functions
    test = psi;
    dtestdx = dpsidx;
 
    // Return the jacobian
    return J;
  }
 
 
 
  //=======================================================================
  //=======================================================================
  template<unsigned NNODE_1D>
  class FaceGeometry<QPMLFourierDecomposedHelmholtzElement<NNODE_1D>>
    : public virtual QElement<1, NNODE_1D>
  {
  public:
    FaceGeometry() : QElement<1, NNODE_1D>() {}
  };
 
 
 
 
  //==========================================================
  //==========================================================
  template<class FOURIER_DECOMPOSED_HELMHOLTZ_ELEMENT>
  class ProjectablePMLFourierDecomposedHelmholtzElement
    : public virtual ProjectableElement<FOURIER_DECOMPOSED_HELMHOLTZ_ELEMENT>
  {
  public:
    ProjectablePMLFourierDecomposedHelmholtzElement() {}
 
    Vector<std::pair<Data*, unsigned>> data_values_of_field(const unsigned& fld)
    {
#ifdef PARANOID
      if (fld > 1)
      {
        std::stringstream error_stream;
        error_stream << "Fourier decomposed Helmholtz elements only store 2 "
                        "fields so fld = "
                     << fld << " is illegal \n";
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // Create the vector
      unsigned nnod = this->nnode();
      Vector<std::pair<Data*, unsigned>> data_values(nnod);
 
      // Loop over all nodes
      for (unsigned j = 0; j < nnod; j++)
      {
        // Add the data value associated field: The node itself
        data_values[j] = std::make_pair(this->node_pt(j), fld);
      }
 
      // Return the vector
      return data_values;
    }
 
    unsigned nfields_for_projection()
    {
      return 2;
    }
 
    unsigned nhistory_values_for_projection(const unsigned& fld)
    {
#ifdef PARANOID
      if (fld > 1)
      {
        std::stringstream error_stream;
        error_stream << "Helmholtz elements only store two fields so fld = "
                     << fld << " is illegal\n";
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
      return this->node_pt(0)->ntstorage();
    }
 
    unsigned nhistory_values_for_coordinate_projection()
    {
      return this->node_pt(0)->position_time_stepper_pt()->ntstorage();
    }
 
    double jacobian_and_shape_of_field(const unsigned& fld,
                                       const Vector<double>& s,
                                       Shape& psi)
    {
#ifdef PARANOID
      if (fld > 1)
      {
        std::stringstream error_stream;
        error_stream << "Helmholtz elements only store two fields so fld = "
                     << fld << " is illegal.\n";
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
      unsigned n_dim = this->dim();
      unsigned n_node = this->nnode();
      Shape test(n_node);
      DShape dpsidx(n_node, n_dim), dtestdx(n_node, n_dim);
      double J =
        this->dshape_and_dtest_eulerian_pml_fourier_decomposed_helmholtz(
          s, psi, dpsidx, test, dtestdx);
      return J;
    }
 
 
    double get_field(const unsigned& t,
                     const unsigned& fld,
                     const Vector<double>& s)
    {
#ifdef PARANOID
      if (fld > 1)
      {
        std::stringstream error_stream;
        error_stream << "Helmholtz elements only store two fields so fld = "
                     << fld << " is illegal\n";
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
      // Find the index at which the variable is stored
      std::complex<unsigned> complex_u_nodal_index =
        this->u_index_pml_fourier_decomposed_helmholtz();
      unsigned u_nodal_index = 0;
      if (fld == 0)
      {
        u_nodal_index = complex_u_nodal_index.real();
      }
      else
      {
        u_nodal_index = complex_u_nodal_index.imag();
      }
 
 
      // Local shape function
      unsigned n_node = this->nnode();
      Shape psi(n_node);
 
      // Find values of shape function
      this->shape(s, psi);
 
      // Initialise value of u
      double interpolated_u = 0.0;
 
      // Sum over the local nodes
      for (unsigned l = 0; l < n_node; l++)
      {
        interpolated_u += this->nodal_value(t, l, u_nodal_index) * psi[l];
      }
      return interpolated_u;
    }
 
 
    unsigned nvalue_of_field(const unsigned& fld)
    {
#ifdef PARANOID
      if (fld > 1)
      {
        std::stringstream error_stream;
        error_stream << "Helmholtz elements only store two fields so fld = "
                     << fld << " is illegal\n";
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
      return this->nnode();
    }
 
 
    int local_equation(const unsigned& fld, const unsigned& j)
    {
#ifdef PARANOID
      if (fld > 1)
      {
        std::stringstream error_stream;
        error_stream << "Helmholtz elements only store two fields so fld = "
                     << fld << " is illegal\n";
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
      std::complex<unsigned> complex_u_nodal_index =
        this->u_index_pml_fourier_decomposed_helmholtz();
      unsigned u_nodal_index = 0;
      if (fld == 0)
      {
        u_nodal_index = complex_u_nodal_index.real();
      }
      else
      {
        u_nodal_index = complex_u_nodal_index.imag();
      }
      return this->nodal_local_eqn(j, u_nodal_index);
    }
 
 
    void output(std::ostream& outfile, const unsigned& nplot)
    {
      FOURIER_DECOMPOSED_HELMHOLTZ_ELEMENT::output(outfile, nplot);
    }
  };
 
 
  //=======================================================================
  //=======================================================================
  template<class ELEMENT>
  class FaceGeometry<ProjectablePMLFourierDecomposedHelmholtzElement<ELEMENT>>
    : public virtual FaceGeometry<ELEMENT>
  {
  public:
    FaceGeometry() : FaceGeometry<ELEMENT>() {}
  };
 
 
  //=======================================================================
  //=======================================================================
  template<class ELEMENT>
  class FaceGeometry<
    FaceGeometry<ProjectablePMLFourierDecomposedHelmholtzElement<ELEMENT>>>
    : public virtual FaceGeometry<FaceGeometry<ELEMENT>>
  {
  public:
    FaceGeometry() : FaceGeometry<FaceGeometry<ELEMENT>>() {}
  };
 
 
 
  //=======================================================================
  //=======================================================================
  template<unsigned NNODE_1D>
  class PMLLayerElement<QPMLFourierDecomposedHelmholtzElement<NNODE_1D>>
    : public virtual QPMLFourierDecomposedHelmholtzElement<NNODE_1D>
  {
  public:
    PMLLayerElement() : QPMLFourierDecomposedHelmholtzElement<NNODE_1D>() {}
  };
 
} // namespace oomph
 
#endif