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//    Version 1.0; svn revision $LastChangedRevision$
//LIC//
//LIC// $LastChangedDate$
//LIC// 
//LIC// Copyright (C) 2006-2016 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
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//LIC// 
//LIC// The authors may be contacted at oomph-lib@maths.man.ac.uk.
//LIC// 
//LIC//====================================================================
//Header file for Helmholtz elements
#ifndef OOMPH_HELMHOLTZ_ELEMENTS_HEADER
#define OOMPH_HELMHOLTZ_ELEMENTS_HEADER

 
// Config header generated by autoconfig
#ifdef HAVE_CONFIG_H
  #include <oomph-lib-config.h>
#endif


//OOMPH-LIB headers
#include "../generic/projection.h"
#include "../generic/nodes.h"
#include "../generic/Qelements.h"
#include "../generic/oomph_utilities.h"
#include "math.h"
#include <complex>

namespace oomph
{

//=============================================================
/// A class for all isoparametric elements that solve the 
/// Helmholtz equations.
/// \f[ 
/// \frac{\partial^2 u}{\partial x_i^2} + k^2 u = f(x_j) 
/// \f] 
/// This contains the generic maths. Shape functions, geometric
/// mapping etc. must get implemented in derived class.
//=============================================================
template <unsigned DIM>
class HelmholtzEquations : public virtual FiniteElement
{

public:

 /// \short Function pointer to source function fct(x,f(x)) -- 
 /// x is a Vector! 
 typedef void (*HelmholtzSourceFctPt)(const Vector<double>& x, 
                                      std::complex<double>& f);


 /// Constructor (must initialise the Source_fct_pt to null)
 HelmholtzEquations() : Source_fct_pt(0),
  K_squared_pt(0) 
  {}
 
 /// Broken copy constructor
 HelmholtzEquations(const HelmholtzEquations& dummy) 
  { 
   BrokenCopy::broken_copy("HelmholtzEquations");
  } 
 
 /// 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 HelmholtzEquations&) 
  {
   BrokenCopy::broken_assign("HelmholtzEquations");
   }*/

 /// \short 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 square of wavenumber
 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;

   }


 /// \short Number of scalars/fields output by this element. Reimplements
 /// broken virtual function in base class.
 unsigned nscalar_paraview() const
  {
   return 2;
  }

 /// \short 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_helmholtz(s));

     // Paraview need to ouput the fileds seperatly 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:
     std::stringstream error_stream;
     error_stream 
      << "Helmholtz elements only store 2 fields so i must be 0 or 1" 
      << std::endl;
     throw OomphLibError(
      error_stream.str(),
      OOMPH_CURRENT_FUNCTION,
     OOMPH_EXCEPTION_LOCATION);
       break;
      }
    } // end of plotpoint loop
  } // end scalar_value_paraview

 /// \short 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:
     std::stringstream error_stream;
     error_stream 
      << "Helmholtz elements only store 2 fields so i must be 0 or 1"
      << std::endl;
     throw OomphLibError(
      error_stream.str(),
      OOMPH_CURRENT_FUNCTION,
     OOMPH_EXCEPTION_LOCATION);
     
     // 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);
  }

 /// \short 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);

 /// \short 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);

 /// C_style output with default number of plot points
 void output(FILE* file_pt)
  {
   const unsigned n_plot=5;
   output(file_pt,n_plot);
  }

 /// \short 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);

 /// \short 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 Helmholtz elements ",
    OOMPH_CURRENT_FUNCTION,
    OOMPH_EXCEPTION_LOCATION);
  }



 /// \short 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);
 

 /// 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 Helmholtz elements",
    OOMPH_CURRENT_FUNCTION,
    OOMPH_EXCEPTION_LOCATION);
  }

 /// Access function: Pointer to source function
 HelmholtzSourceFctPt& source_fct_pt() {return Source_fct_pt;}

 /// Access function: Pointer to source function. Const version
 HelmholtzSourceFctPt 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);
    }
  }


 /// 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);
  }

 

 /// \short 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);
  }
 


 /// \short Return FE representation of function value u_helmholtz(s) 
 /// at local coordinate s
 inline std::complex<double> interpolated_u_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;
  }

 
 /// \short Self-test: Return 0 for OK
 unsigned self_test();


protected:

 /// \short 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;
 

 /// \short 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;

 /// \short 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 source function:
 HelmholtzSourceFctPt Source_fct_pt;

 /// Pointer to square of wavenumber
 double* K_squared_pt;
  
};






///////////////////////////////////////////////////////////////////////////
///////////////////////////////////////////////////////////////////////////
///////////////////////////////////////////////////////////////////////////



//======================================================================
/// QHelmholtzElement elements are linear/quadrilateral/brick-shaped 
/// Helmholtz elements with isoparametric interpolation for the function.
//======================================================================
template <unsigned DIM, unsigned NNODE_1D>
 class QHelmholtzElement : public virtual QElement<DIM,NNODE_1D>,
 public virtual HelmholtzEquations<DIM>
{

private:

 /// \short Static int that holds the number of variables at 
 /// nodes: always the same
 static const unsigned Initial_Nvalue;
 
  public:


 ///\short  Constructor: Call constructors for QElement and 
 /// Helmholtz equations
 QHelmholtzElement() : QElement<DIM,NNODE_1D>(), HelmholtzEquations<DIM>()
  {}
 
 /// Broken copy constructor
 QHelmholtzElement(const QHelmholtzElement<DIM,NNODE_1D>& dummy) 
  { 
   BrokenCopy::broken_copy("QHelmholtzElement");
  } 
 
 /// Broken assignment operator
 /*void operator=(const QHelmholtzElement<DIM,NNODE_1D>&) 
  {
   BrokenCopy::broken_assign("QHelmholtzElement");
   }*/


 /// \short  Required  # of `values' (pinned or dofs) 
 /// at node n
 inline unsigned required_nvalue(const unsigned &n) const 
  {return Initial_Nvalue;}

 /// \short Output function:  
 ///  x,y,u   or    x,y,z,u
 void output(std::ostream &outfile)
  {HelmholtzEquations<DIM>::output(outfile);}


 ///  \short 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)
  {HelmholtzEquations<DIM>::output(outfile,n_plot);}

 /// \short 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)
 {HelmholtzEquations<DIM>::output_real(outfile,phi,n_plot);}


 /// \short C-style output function:  
 ///  x,y,u   or    x,y,z,u
 void output(FILE* file_pt)
  {HelmholtzEquations<DIM>::output(file_pt);}


 ///  \short 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)
  {HelmholtzEquations<DIM>::output(file_pt,n_plot);}


 /// \short 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)
  {HelmholtzEquations<DIM>::output_fct(outfile,n_plot,exact_soln_pt);}


 /// \short 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)
 {
  HelmholtzEquations<DIM>::output_real_fct(outfile,phi,n_plot,exact_soln_pt);
 }


 /// \short 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)
  {HelmholtzEquations<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;


 /// \short 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 QHelmholtzElement<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 QHelmholtzElement<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 QHelmholtzElement 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<QHelmholtzElement<DIM,NNODE_1D> >: 
 public virtual QElement<DIM-1,NNODE_1D>
{

  public:
 
 /// \short Constructor: Call the constructor for the
 /// appropriate lower-dimensional QElement
 FaceGeometry() : QElement<DIM-1,NNODE_1D>() {}

};

////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////


//=======================================================================
/// Face geometry for the 1D QHelmholtzElement elements: Point elements
//=======================================================================
template<unsigned NNODE_1D>
class FaceGeometry<QHelmholtzElement<1,NNODE_1D> >: 
 public virtual PointElement
{

  public:
 
 /// \short Constructor: Call the constructor for the
 /// appropriate lower-dimensional QElement
 FaceGeometry() : PointElement() {}

};



////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////



//==========================================================
/// Helmholtz upgraded to become projectable
//==========================================================
 template<class HELMHOLTZ_ELEMENT>
 class ProjectableHelmholtzElement : 
  public virtual ProjectableElement<HELMHOLTZ_ELEMENT>
 {

 public:


  /// \short Constructor [this was only required explicitly
  /// from gcc 4.5.2 onwards...]
  ProjectableHelmholtzElement(){}

  /// \short 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 
       << "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;
   }

  /// \short Number of fields to be projected: 2 (real and imag part)
  unsigned nfields_for_projection()
   {
    return 2;
   }
 
  /// \short Number of history values to be stored for fld-th field. 
  /// (includes current value!)
  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();   
  }
  
  ///\short Number of positional history values
  /// (includes current value!)
  unsigned nhistory_values_for_coordinate_projection()
   {
    return this->node_pt(0)->position_time_stepper_pt()->ntstorage();
   }
  
  /// \short 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 
       << "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_helmholtz(s,psi,dpsidx,
                                                       test,dtestdx);
    return J;
   }



  /// \short 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 
       << "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_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 
       << "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();
   }

 
  ///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 
       << "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_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);     
   }




 /// \short 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<ProjectableHelmholtzElement<ELEMENT> > 
  : public virtual FaceGeometry<ELEMENT>
 {
 public:
  FaceGeometry() : FaceGeometry<ELEMENT>() {}
 };


//=======================================================================
/// Face geometry of the Face Geometry for element is the same as 
/// that for the underlying wrapped element
//=======================================================================
 template<class ELEMENT>
 class FaceGeometry<FaceGeometry<ProjectableHelmholtzElement<ELEMENT> > >
  : public virtual FaceGeometry<FaceGeometry<ELEMENT> >
 {
 public:
  FaceGeometry() : FaceGeometry<FaceGeometry<ELEMENT> >() {}
 };




}

#endif