pml_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) 2006-2023 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//
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// LIC// 02110-1301  USA.
// LIC//
// LIC// The authors may be contacted at oomph-lib@maths.man.ac.uk.
// LIC//
// LIC//====================================================================
// Header file for Helmholtz elements
#ifndef OOMPH_PML_HELMHOLTZ_ELEMENTS_HEADER
#define OOMPH_PML_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/nodes.h"
#include "../generic/Qelements.h"
#include "../generic/oomph_utilities.h"
#include "../generic/pml_meshes.h"
#include "../generic/projection.h"
#include "../generic/pml_mapping_functions.h"
 
/// /////////////////////////////////////////////////////////////////
/// /////////////////////////////////////////////////////////////////
/// /////////////////////////////////////////////////////////////////
 
namespace oomph
{
  //=============================================================
  /// A class for all isoparametric elements that solve the
  /// Helmholtz equations with pml capabilities.
  /// This contains the generic maths. Shape functions, geometric
  /// mapping etc. must get implemented in derived class.
  //=============================================================
  template<unsigned DIM>
  class PMLHelmholtzEquations : public virtual PMLElementBase<DIM>,
                                public virtual FiniteElement
  {
  public:
    /// Function pointer to source function fct(x,f(x)) --
    /// x is a Vector!
    typedef void (*PMLHelmholtzSourceFctPt)(const Vector<double>& x,
                                            std::complex<double>& f);
 
    /// Constructor
    PMLHelmholtzEquations() : Source_fct_pt(0), K_squared_pt(0)
    {
      // Provide pointer to static method (Save memory)
      Pml_mapping_pt = &PMLHelmholtzEquations::Default_pml_mapping;
      Alpha_pt = &Default_Physical_Constant_Value;
    }
 
 
    /// Broken copy constructor
    PMLHelmholtzEquations(const PMLHelmholtzEquations& 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 PMLHelmholtzEquations&) = delete;*/
 
    /// Return the index at which the unknown value
    /// is stored.
    virtual inline std::complex<unsigned> u_index_helmholtz() const
    {
      return std::complex<unsigned>(0, 1);
    }
 
    /// Get pointer to k_squared
    double*& k_squared_pt()
    {
      return K_squared_pt;
    }
 
 
    /// Get the square of wavenumber
    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;
    }
 
    /// Alpha, wavenumber complex shift
    const double& alpha() const
    {
      return *Alpha_pt;
    }
 
    /// Pointer to Alpha, wavenumber complex shift
    double*& alpha_pt()
    {
      return Alpha_pt;
    }
 
 
    /// Number of scalars/fields output by this element. Reimplements
    /// broken virtual function in base class.
    unsigned nscalar_paraview() const
    {
      return 2;
    }
 
    /// Write values of the i-th scalar field at the plot points. Needs
    /// to be implemented for each new specific element type.
    void scalar_value_paraview(std::ofstream& file_out,
                               const unsigned& i,
                               const unsigned& nplot) const
    {
      // Vector of local coordinates
      Vector<double> s(DIM);
 
      // Loop over plot points
      unsigned num_plot_points = nplot_points_paraview(nplot);
      for (unsigned iplot = 0; iplot < num_plot_points; iplot++)
      {
        // Get local coordinates of plot point
        get_s_plot(iplot, nplot, s);
        std::complex<double> u(interpolated_u_pml_helmholtz(s));
 
        // Paraview need to ouput the fields separately so it loops through all
        // the elements twice
        switch (i)
        {
            // Real part first
          case 0:
            file_out << u.real() << std::endl;
            break;
 
            // Imaginary part second
          case 1:
            file_out << u.imag() << std::endl;
            break;
 
            // Never get here
          default:
#ifdef PARANOID
            std::stringstream error_stream;
            error_stream << "PML Helmholtz elements only store 2 fields (real "
                            "and imaginary) "
                         << "so i must be 0 or 1 rather "
                         << "than " << i << std::endl;
            throw OomphLibError(error_stream.str(),
                                OOMPH_CURRENT_FUNCTION,
                                OOMPH_EXCEPTION_LOCATION);
#endif
            break;
        }
      }
    }
 
    /// Write values of the i-th scalar field at the plot points. Needs
    /// to be implemented for each new specific element type.
    void scalar_value_fct_paraview(
      std::ofstream& file_out,
      const unsigned& i,
      const unsigned& nplot,
      FiniteElement::SteadyExactSolutionFctPt exact_soln_pt) const
    {
      // Vector of local coordinates
      Vector<double> s(DIM);
 
      // Vector for coordinates
      Vector<double> x(DIM);
 
      // Exact solution Vector
      Vector<double> exact_soln(2);
 
      // Loop over plot points
      unsigned num_plot_points = nplot_points_paraview(nplot);
      for (unsigned iplot = 0; iplot < num_plot_points; iplot++)
      {
        // Get local coordinates of plot point
        get_s_plot(iplot, nplot, s);
 
        // Get x position as Vector
        interpolated_x(s, x);
 
        // Get exact solution at this point
        (*exact_soln_pt)(x, exact_soln);
 
        // Paraview need to output the fields separately so it loops through all
        // the elements twice
        switch (i)
        {
            // Real part first
          case 0:
            file_out << exact_soln[0] << std::endl;
            break;
 
            // Imaginary part second
          case 1:
            file_out << exact_soln[1] << std::endl;
            break;
 
            // Never get here
          default:
#ifdef PARANOID
            std::stringstream error_stream;
            error_stream << "PML Helmholtz elements only store 2 fields (real "
                            "and imaginary) "
                         << "so i must be 0 or 1 rather "
                         << "than " << i << std::endl;
            throw OomphLibError(error_stream.str(),
                                OOMPH_CURRENT_FUNCTION,
                                OOMPH_EXCEPTION_LOCATION);
#endif
            break;
        }
      }
    }
 
    /// Name of the i-th scalar field. Default implementation
    /// returns V1 for the first one, V2 for the second etc. Can (should!)
    /// be overloaded with more meaningful names in specific elements.
    std::string scalar_name_paraview(const unsigned& i) const
    {
      switch (i)
      {
        case 0:
          return "Real part";
          break;
 
        case 1:
          return "Imaginary part";
          break;
 
          // Never get here
        default:
#ifdef PARANOID
          std::stringstream error_stream;
          error_stream << "PML Helmholtz elements only store 2 fields (real "
                          "and imaginary) "
                       << "so i must be 0 or 1 rather "
                       << "than " << i << std::endl;
          throw OomphLibError(error_stream.str(),
                              OOMPH_CURRENT_FUNCTION,
                              OOMPH_EXCEPTION_LOCATION);
#endif
 
          // Dummy return for the default statement
          return " ";
          break;
      }
    }
 
    /// Output with default number of plot points
    void output(std::ostream& outfile)
    {
      const unsigned n_plot = 5;
      output(outfile, n_plot);
    }
 
    /// Output FE representation of soln: x,y,u_re,u_im or
    /// x,y,z,u_re,u_im at  n_plot^DIM plot points
    void output(std::ostream& outfile, const unsigned& n_plot);
 
 
    /// Output function for real part of full time-dependent solution
    /// constructed by adding the scattered field
    ///  u = Re( (u_r +i u_i) exp(-i omega t)
    /// at phase angle omega t = phi computed here, to the corresponding
    /// incoming wave specified via the function pointer.
    void output_total_real(
      std::ostream& outfile,
      FiniteElement::SteadyExactSolutionFctPt incoming_wave_fct_pt,
      const double& phi,
      const unsigned& nplot);
 
 
    /// Output function for real part of full time-dependent solution
    /// u = Re( (u_r +i u_i) exp(-i omega t))
    /// at phase angle omega t = phi.
    /// x,y,u   or    x,y,z,u at n_plot plot points in each coordinate
    /// direction
    void output_real(std::ostream& outfile,
                     const double& phi,
                     const unsigned& n_plot);
 
    /// Output function for imaginary part of full time-dependent
    /// solution u = Im( (u_r +i u_i) exp(-i omega t) ) at phase angle omega t =
    /// phi. x,y,u   or    x,y,z,u at n_plot plot points in each coordinate
    /// direction
    void output_imag(std::ostream& outfile,
                     const double& phi,
                     const unsigned& n_plot);
 
    /// C_style output with default number of plot points
    void output(FILE* file_pt)
    {
      const unsigned n_plot = 5;
      output(file_pt, n_plot);
    }
 
    /// C-style output FE representation of soln: x,y,u_re,u_im or
    /// x,y,z,u_re,u_im at  n_plot^DIM plot points
    void output(FILE* file_pt, const unsigned& n_plot);
 
    /// Output exact soln: x,y,u_re_exact,u_im_exact
    /// or x,y,z,u_re_exact,u_im_exact at n_plot^DIM plot points
    void output_fct(std::ostream& outfile,
                    const unsigned& n_plot,
                    FiniteElement::SteadyExactSolutionFctPt exact_soln_pt);
 
    /// Output exact soln: (dummy time-dependent version to
    /// keep intel compiler happy)
    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 PMLHelmholtz elements.",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
    }
 
 
    /// Output function for real part of full time-dependent fct
    /// u = Re( (u_r +i u_i) exp(-i omega t)
    /// at phase angle omega t = phi.
    /// x,y,u   or    x,y,z,u at n_plot plot points in each coordinate
    /// direction
    void output_real_fct(std::ostream& outfile,
                         const double& phi,
                         const unsigned& n_plot,
                         FiniteElement::SteadyExactSolutionFctPt exact_soln_pt);
 
    /// Output function for imaginary part of full time-dependent fct
    /// u = Im( (u_r +i u_i) exp(-i omega t))
    /// at phase angle omega t = phi.
    /// x,y,u   or    x,y,z,u at n_plot plot points in each coordinate
    /// direction
    void output_imag_fct(std::ostream& outfile,
                         const double& phi,
                         const unsigned& n_plot,
                         FiniteElement::SteadyExactSolutionFctPt exact_soln_pt);
 
 
    /// Get error against and norm of exact solution
    void compute_error(std::ostream& outfile,
                       FiniteElement::SteadyExactSolutionFctPt exact_soln_pt,
                       double& error,
                       double& norm);
 
 
    /// Dummy, time dependent error checker
    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 PMLHelmholtz elements.",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
    }
 
 
    /// Compute norm of fe solution
    void compute_norm(double& norm);
 
    /// Access function: Pointer to source function
    PMLHelmholtzSourceFctPt& source_fct_pt()
    {
      return Source_fct_pt;
    }
 
    /// Access function: Pointer to source function. Const version
    PMLHelmholtzSourceFctPt source_fct_pt() const
    {
      return Source_fct_pt;
    }
 
 
    /// Get source term at (Eulerian) position x. This function is
    /// virtual to allow overloading in multi-physics problems where
    /// the strength of the source function might be determined by
    /// another system of equations.
    inline virtual void get_source_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);
      }
    }
 
    /// Pure virtual function in which we specify the
    /// values to be pinned (and set to zero) on the outer edge of
    /// the pml layer. All of them! Vector is resized internally.
    void values_to_be_pinned_on_outer_pml_boundary(
      Vector<unsigned>& values_to_pin)
    {
      values_to_pin.resize(2);
 
      values_to_pin[0] = 0;
      values_to_pin[1] = 1;
    }
 
 
    /// Get flux: flux[i] = du/dx_i for real and imag part
    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, DIM);
 
      // 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 < DIM; 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_helmholtz().real()),
          this->nodal_value(l, u_index_helmholtz().imag()));
        // Loop over derivative directions
        for (unsigned j = 0; j < DIM; j++)
        {
          flux[j] += u_value * dpsidx(l, j);
        }
      }
    }
 
 
    /// Add the element's contribution to its residual vector (wrapper)
    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(
        residuals, GeneralisedElement::Dummy_matrix, 0);
    }
 
 
    /// Add the element's contribution to its residual vector and
    /// element Jacobian matrix (wrapper)
    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(residuals, jacobian, 1);
    }
 
 
    /// Return FE representation of function value u_helmholtz(s)
    /// at local coordinate s
    inline std::complex<double> interpolated_u_pml_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_helmholtz().real();
      const unsigned u_nodal_index_imag = u_index_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;
    }
 
 
    /// Self-test: Return 0 for OK
    unsigned self_test();
 
 
    /// Compute pml coefficients at position x and integration point ipt.
    /// pml_laplace_factor is used in the residual contribution from the laplace
    /// operator, similarly pml_k_squared_factor is used in the contribution
    /// from the k^2 of the Helmholtz operator.
    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 which points from the inner boundary to x
      Vector<double> nu(DIM);
      for (unsigned k = 0; k < DIM; k++)
      {
        nu[k] = x[k] - this->Pml_inner_boundary[k];
      }
 
      /// Vector which points from the inner boundary to the edge of the
      /// boundary
      Vector<double> pml_width(DIM);
      for (unsigned k = 0; k < DIM; 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(DIM);
 
      if (this->Pml_is_enabled)
      {
        // Cache k_squared to pass into mapping function
        double k_squared_local = k_squared();
 
        for (unsigned k = 0; k < DIM; k++)
        {
          // If PML is enabled in the respective direction
          if (this->Pml_direction_active[k])
          {
            pml_gamma[k] = Pml_mapping_pt->gamma(
              nu[k], pml_width[k], k_squared_local, alpha());
          }
          else
          {
            pml_gamma[k] = 1.0;
          }
        }
 
        ///  for 2D, in order:
        /// g_y/g_x, g_x/g_y for Laplace bit and g_x*g_y for Helmholtz bit
        /// for 3D, in order: g_y*g_x/g_x, g*x*g_z/g_y, g_x*g_y/g_z for Laplace
        /// bit and g_x*g_y*g_z for Helmholtz bit
        if (DIM == 2)
        {
          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 // if (DIM == 3)
        {
          pml_laplace_factor[0] = pml_gamma[1] * pml_gamma[2] / pml_gamma[0];
          pml_laplace_factor[1] = pml_gamma[0] * pml_gamma[2] / pml_gamma[1];
          pml_laplace_factor[2] = pml_gamma[0] * pml_gamma[1] / pml_gamma[2];
          pml_k_squared_factor = pml_gamma[0] * pml_gamma[1] * pml_gamma[2];
        }
      }
      else
      {
        /// The weights all default to 1.0 as if the propagation
        /// medium is the physical domain
        for (unsigned k = 0; k < DIM; k++)
        {
          pml_laplace_factor[k] = std::complex<double>(1.0, 0.0);
        }
 
        pml_k_squared_factor = std::complex<double>(1.0, 0.0);
      }
    }
 
    /// Return a pointer to the PML Mapping object
    PMLMapping*& pml_mapping_pt()
    {
      return Pml_mapping_pt;
    }
 
    /// Return a pointer to the PML Mapping object (const version)
    PMLMapping* const& pml_mapping_pt() const
    {
      return Pml_mapping_pt;
    }
 
    /// Static so that the class doesn't need to instantiate a new default
    /// everytime it uses it
    static BermudezPMLMapping Default_pml_mapping;
 
    /// 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, 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, 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:
    /// Shape/test functions and derivs w.r.t. to global coords at
    /// local coord. s; return  Jacobian of mapping
    virtual double dshape_and_dtest_eulerian_helmholtz(
      const Vector<double>& s,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const = 0;
 
 
    /// Shape/test functions and derivs w.r.t. to global coords at
    /// integration point ipt; return  Jacobian of mapping
    virtual double dshape_and_dtest_eulerian_at_knot_helmholtz(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const = 0;
 
    /// Compute element residual Vector only (if flag=and/or element
    /// Jacobian matrix
    virtual void fill_in_generic_residual_contribution_helmholtz(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      const unsigned& flag);
 
    /// Pointer to wavenumber complex shift
    double* Alpha_pt;
 
    /// Pointer to source function:
    PMLHelmholtzSourceFctPt Source_fct_pt;
 
    /// Pointer to wave number (must be set!)
    double* K_squared_pt;
 
    /// Pointer to class which holds the pml mapping function, also known as
    /// gamma
    PMLMapping* Pml_mapping_pt;
 
    /// Static default value for the physical constants (initialised to zero)
    static double Default_Physical_Constant_Value;
  };
 
 
  /// ////////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////////
 
 
  //======================================================================
  /// QPMLHelmholtzElement elements are linear/quadrilateral/
  /// brick-shaped PMLHelmholtz elements with isoparametric
  /// interpolation for the function.
  //======================================================================
  template<unsigned DIM, unsigned NNODE_1D>
  class QPMLHelmholtzElement : public virtual QElement<DIM, NNODE_1D>,
                               public virtual PMLHelmholtzEquations<DIM>
  {
  private:
    /// Static int that holds the number of variables at
    /// nodes: always the same
    static const unsigned Initial_Nvalue;
 
  public:
    /// Constructor: Call constructors for QElement and
    /// PMLHelmholtz equations
    QPMLHelmholtzElement()
      : QElement<DIM, NNODE_1D>(), PMLHelmholtzEquations<DIM>()
    {
    }
 
    /// Broken copy constructor
    QPMLHelmholtzElement(const QPMLHelmholtzElement<DIM, NNODE_1D>& dummy) =
      delete;
 
    /// Broken assignment operator
    /*void operator=(const QPMLHelmholtzElement<DIM,NNODE_1D>&) = delete;*/
 
    ///  Required  # of `values' (pinned or dofs)
    /// at node n
    inline unsigned required_nvalue(const unsigned& n) const
    {
      return Initial_Nvalue;
    }
 
    /// Output function:
    ///  x,y,u   or    x,y,z,u
    void output(std::ostream& outfile)
    {
      PMLHelmholtzEquations<DIM>::output(outfile);
    }
 
 
    ///  Output function:
    ///   x,y,u   or    x,y,z,u at n_plot^DIM plot points
    void output(std::ostream& outfile, const unsigned& n_plot)
    {
      PMLHelmholtzEquations<DIM>::output(outfile, n_plot);
    }
 
    /// Output function for real part of full time-dependent solution
    /// u = Re( (u_r +i u_i) exp(-i omega t)
    /// at phase angle omega t = phi.
    /// x,y,u   or    x,y,z,u at n_plot plot points in each coordinate
    /// direction
    void output_real(std::ostream& outfile,
                     const double& phi,
                     const unsigned& n_plot)
    {
      PMLHelmholtzEquations<DIM>::output_real(outfile, phi, n_plot);
    }
 
    /// Output function for imaginary part of full time-dependent
    /// solution u = Im( (u_r +i u_i) exp(-i omega t)) at phase angle omega t =
    /// phi. x,y,u   or    x,y,z,u at n_plot plot points in each coordinate
    /// direction
    void output_imag(std::ostream& outfile,
                     const double& phi,
                     const unsigned& n_plot)
    {
      PMLHelmholtzEquations<DIM>::output_imag(outfile, phi, n_plot);
    }
 
 
    /// C-style output function:
    ///  x,y,u   or    x,y,z,u
    void output(FILE* file_pt)
    {
      PMLHelmholtzEquations<DIM>::output(file_pt);
    }
 
 
    ///  C-style output function:
    ///   x,y,u   or    x,y,z,u at n_plot^DIM plot points
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      PMLHelmholtzEquations<DIM>::output(file_pt, n_plot);
    }
 
 
    /// Output function for an exact solution:
    ///  x,y,u_exact   or    x,y,z,u_exact at n_plot^DIM plot points
    void output_fct(std::ostream& outfile,
                    const unsigned& n_plot,
                    FiniteElement::SteadyExactSolutionFctPt exact_soln_pt)
    {
      PMLHelmholtzEquations<DIM>::output_fct(outfile, n_plot, exact_soln_pt);
    }
 
 
    /// Output function for real part of full time-dependent fct
    /// u = Re( (u_r +i u_i) exp(-i omega t)
    /// at phase angle omega t = phi.
    /// x,y,u   or    x,y,z,u at n_plot plot points in each coordinate
    /// direction
    void output_real_fct(std::ostream& outfile,
                         const double& phi,
                         const unsigned& n_plot,
                         FiniteElement::SteadyExactSolutionFctPt exact_soln_pt)
    {
      PMLHelmholtzEquations<DIM>::output_real_fct(
        outfile, phi, n_plot, exact_soln_pt);
    }
 
    /// Output function for imaginary part of full time-dependent fct
    /// u = Im( (u_r +i u_i) exp(-i omega t))
    /// at phase angle omega t = phi.
    /// x,y,u   or    x,y,z,u at n_plot plot points in each coordinate
    /// direction
    void output_imag_fct(std::ostream& outfile,
                         const double& phi,
                         const unsigned& n_plot,
                         FiniteElement::SteadyExactSolutionFctPt exact_soln_pt)
    {
      PMLHelmholtzEquations<DIM>::output_imag_fct(
        outfile, phi, n_plot, exact_soln_pt);
    }
 
 
    /// Output function for a time-dependent exact solution.
    ///  x,y,u_exact   or    x,y,z,u_exact at n_plot^DIM plot points
    /// (Calls the steady version)
    void output_fct(std::ostream& outfile,
                    const unsigned& n_plot,
                    const double& time,
                    FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt)
    {
      PMLHelmholtzEquations<DIM>::output_fct(
        outfile, n_plot, time, exact_soln_pt);
    }
 
  protected:
    /// Shape, test functions & derivs. w.r.t. to global coords. Return
    /// Jacobian.
    inline double dshape_and_dtest_eulerian_helmholtz(const Vector<double>& s,
                                                      Shape& psi,
                                                      DShape& dpsidx,
                                                      Shape& test,
                                                      DShape& dtestdx) const;
 
 
    /// Shape, test functions & derivs. w.r.t. to global coords. at
    /// integration point ipt. Return Jacobian.
    inline double dshape_and_dtest_eulerian_at_knot_helmholtz(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const;
  };
 
 
  // Inline functions:
 
 
  //======================================================================
  /// Define the shape functions and test functions and derivatives
  /// w.r.t. global coordinates and return Jacobian of mapping.
  ///
  /// Galerkin: Test functions = shape functions
  //======================================================================
  template<unsigned DIM, unsigned NNODE_1D>
  double QPMLHelmholtzElement<DIM, NNODE_1D>::
    dshape_and_dtest_eulerian_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;
  }
 
 
  //======================================================================
  /// Define the shape functions and test functions and derivatives
  /// w.r.t. global coordinates and return Jacobian of mapping.
  ///
  /// Galerkin: Test functions = shape functions
  //======================================================================
  template<unsigned DIM, unsigned NNODE_1D>
  double QPMLHelmholtzElement<DIM, NNODE_1D>::
    dshape_and_dtest_eulerian_at_knot_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;
  }
 
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
 
 
  //=======================================================================
  /// Face geometry for the QPMLHelmholtzElement elements: The spatial
  /// dimension of the face elements is one lower than that of the
  /// bulk element but they have the same number of points
  /// along their 1D edges.
  //=======================================================================
  template<unsigned DIM, unsigned NNODE_1D>
  class FaceGeometry<QPMLHelmholtzElement<DIM, NNODE_1D>>
    : public virtual QElement<DIM - 1, NNODE_1D>
  {
  public:
    /// Constructor: Call the constructor for the
    /// appropriate lower-dimensional QElement
    FaceGeometry() : QElement<DIM - 1, NNODE_1D>() {}
  };
 
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
 
 
  //=======================================================================
  /// Face geometry for the 1D QPMLHelmholtzElement elements:
  /// Point elements
  //=======================================================================
  template<unsigned NNODE_1D>
  class FaceGeometry<QPMLHelmholtzElement<1, NNODE_1D>>
    : public virtual PointElement
  {
  public:
    /// Constructor: Call the constructor for the
    /// appropriate lower-dimensional QElement
    FaceGeometry() : PointElement() {}
  };
 
 
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
 
 
  //==========================================================
  /// PMLHelmholtz upgraded to become projectable
  //==========================================================
  template<class HELMHOLTZ_ELEMENT>
  class ProjectablePMLHelmholtzElement
    : public virtual ProjectableElement<HELMHOLTZ_ELEMENT>
  {
  public:
    /// Constructor [this was only required explicitly
    /// from gcc 4.5.2 onwards...]
    ProjectablePMLHelmholtzElement() {}
 
    /// Specify the values associated with field fld.
    /// The information is returned in a vector of pairs which comprise
    /// the Data object and the value within it, that correspond to field fld.
    Vector<std::pair<Data*, unsigned>> data_values_of_field(const unsigned& fld)
    {
#ifdef PARANOID
      if (fld > 1)
      {
        std::stringstream error_stream;
        error_stream << "PMLHelmholtz 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;
    }
 
    /// Number of fields to be projected: 2 (real and imag part)
    unsigned nfields_for_projection()
    {
      return 2;
    }
 
    /// Number of history values to be stored for fld-th field.
    /// (Note: count includes current value!)
    unsigned nhistory_values_for_projection(const unsigned& fld)
    {
#ifdef PARANOID
      if (fld > 1)
      {
        std::stringstream error_stream;
        error_stream << "PMLHelmholtz 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();
    }
 
    /// Number of positional history values
    /// (Note: count includes current value!)
    unsigned nhistory_values_for_coordinate_projection()
    {
      return this->node_pt(0)->position_time_stepper_pt()->ntstorage();
    }
 
    /// Return Jacobian of mapping and shape functions of field fld
    /// at local coordinate s
    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 << "PMLHelmholtz 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_helmholtz(
        s, psi, dpsidx, test, dtestdx);
      return J;
    }
 
 
    /// Return interpolated field fld at local coordinate s, at time
    /// level t (t=0: present; t>0: history values)
    double get_field(const unsigned& t,
                     const unsigned& fld,
                     const Vector<double>& s)
    {
#ifdef PARANOID
      if (fld > 1)
      {
        std::stringstream error_stream;
        error_stream << "PMLHelmholtz 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_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;
    }
 
 
    /// Return number of values in field fld: One per node
    unsigned nvalue_of_field(const unsigned& fld)
    {
#ifdef PARANOID
      if (fld > 1)
      {
        std::stringstream error_stream;
        error_stream << "PMLHelmholtz 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();
    }
 
 
    /// Return local equation number of value j in field fld.
    int local_equation(const unsigned& fld, const unsigned& j)
    {
#ifdef PARANOID
      if (fld > 1)
      {
        std::stringstream error_stream;
        error_stream << "PMLHelmholtz 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_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);
    }
 
    /// Output FE representation of soln: x,y,u or x,y,z,u at
    /// n_plot^DIM plot points
    void output(std::ostream& outfile, const unsigned& nplot)
    {
      HELMHOLTZ_ELEMENT::output(outfile, nplot);
    }
  };
 
 
  //=======================================================================
  /// Face geometry for element is the same as that for the underlying
  /// wrapped element
  //=======================================================================
  template<class ELEMENT>
  class FaceGeometry<ProjectablePMLHelmholtzElement<ELEMENT>>
    : public virtual FaceGeometry<ELEMENT>
  {
  public:
    FaceGeometry() : FaceGeometry<ELEMENT>() {}
  };
 
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
 
  //=======================================================================
  /// Policy class defining the elements to be used in the actual
  /// PML layers. Same!
  //=======================================================================
  template<unsigned NNODE_1D>
  class PMLLayerElement<QPMLHelmholtzElement<2, NNODE_1D>>
    : public virtual QPMLHelmholtzElement<2, NNODE_1D>
  {
  public:
    /// Constructor: Call the constructor for the
    /// appropriate QElement
    PMLLayerElement() : QPMLHelmholtzElement<2, NNODE_1D>() {}
  };
 
} // End of Namespace oomph
 
#endif