axisym_advection_diffusion_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
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// LIC// License as published by the Free Software Foundation; either
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// LIC//====================================================================
// Header file for advection diffusion elements in a spherical polar coordinate
// system
#ifndef OOMPH_AXISYM_ADV_DIFF_ELEMENTS_HEADER
#define OOMPH_AXISYM_ADV_DIFF_ELEMENTS_HEADER
 
 
// Config header generated by autoconfig
#ifdef HAVE_CONFIG_H
#include <oomph-lib-config.h>
#endif
 
// OOMPH-LIB headers
#include "../generic/nodes.h"
#include "../generic/Qelements.h"
#include "../generic/refineable_elements.h"
#include "../generic/oomph_utilities.h"
 
namespace oomph
{
  //=============================================================
  /// A class for all elements that solve the
  /// Advection Diffusion equations in a cylindrical polar coordinate system
  /// using isoparametric elements.
  /// \f[ Pe \mathbf{w}\cdot(\mathbf{x}) \nabla u = \nabla \cdot \left( \nabla u \right) + f(\mathbf{x}) \f]
  /// This contains the generic maths. Shape functions, geometric
  /// mapping etc. must get implemented in derived class.
  //=============================================================
  class AxisymAdvectionDiffusionEquations : public virtual FiniteElement
  {
  public:
    /// Function pointer to source function fct(x,f(x)) --
    /// x is a Vector!
    typedef void (*AxisymAdvectionDiffusionSourceFctPt)(const Vector<double>& x,
                                                        double& f);
 
 
    /// Function pointer to wind function fct(x,w(x)) --
    /// x is a Vector!
    typedef void (*AxisymAdvectionDiffusionWindFctPt)(const Vector<double>& x,
                                                      Vector<double>& wind);
 
 
    /// Constructor: Initialise the Source_fct_pt and Wind_fct_pt
    /// to null and set (pointer to) Peclet number to default
    AxisymAdvectionDiffusionEquations()
      : Source_fct_pt(0), Wind_fct_pt(0), ALE_is_disabled(false)
    {
      // Set pointer to Peclet number to the default value zero
      Pe_pt = &Default_peclet_number;
      PeSt_pt = &Default_peclet_number;
      D_pt = &Default_diffusion_parameter;
    }
 
    /// Broken copy constructor
    AxisymAdvectionDiffusionEquations(
      const AxisymAdvectionDiffusionEquations& 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 AxisymAdvectionDiffusionEquations&) = delete;*/
 
    /// Return the index at which the unknown value
    /// is stored. The default value, 0, is appropriate for single-physics
    /// problems, when there is only one variable, the value that satisfies
    /// the spherical advection-diffusion equation.
    /// In derived multi-physics elements, this function should be overloaded
    /// to reflect the chosen storage scheme. Note that these equations require
    /// that the unknown is always stored at the same index at each node.
    virtual inline unsigned u_index_axi_adv_diff() const
    {
      return 0;
    }
 
 
    /// du/dt at local node n.
    /// Uses suitably interpolated value for hanging nodes.
    double du_dt_axi_adv_diff(const unsigned& n) const
    {
      // Get the data's timestepper
      TimeStepper* time_stepper_pt = this->node_pt(n)->time_stepper_pt();
 
      // Initialise dudt
      double dudt = 0.0;
      // Loop over the timesteps, if there is a non Steady timestepper
      if (!time_stepper_pt->is_steady())
      {
        // Find the index at which the variable is stored
        const unsigned u_nodal_index = u_index_axi_adv_diff();
 
        // Number of timsteps (past & present)
        const unsigned n_time = time_stepper_pt->ntstorage();
 
        for (unsigned t = 0; t < n_time; t++)
        {
          dudt +=
            time_stepper_pt->weight(1, t) * nodal_value(t, n, u_nodal_index);
        }
      }
      return dudt;
    }
 
    /// Disable ALE, i.e. assert the mesh is not moving -- you do this
    /// at your own risk!
    void disable_ALE()
    {
      ALE_is_disabled = true;
    }
 
 
    /// (Re-)enable ALE, i.e. take possible mesh motion into account
    /// when evaluating the time-derivative. Note: By default, ALE is
    /// enabled, at the expense of possibly creating unnecessary work
    /// in problems where the mesh is, in fact, stationary.
    void enable_ALE()
    {
      ALE_is_disabled = false;
    }
 
 
    /// Number of scalars/fields output by this element. Reimplements
    /// broken virtual function in base class.
    unsigned nscalar_paraview() const
    {
      return 4;
    }
 
    /// 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(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 Eulerian coordinate of plot point
        Vector<double> x(2);
        interpolated_x(s, x);
 
        // Winds
        if (i < 3)
        {
          // Get the wind
          Vector<double> wind(3);
 
          // Dummy ipt argument needed... ?
          unsigned ipt = 0;
          get_wind_axi_adv_diff(ipt, s, x, wind);
 
          file_out << wind[i] << std::endl;
        }
        // Advection Diffusion
        else if (i == 3)
        {
          file_out << this->interpolated_u_axi_adv_diff(s) << std::endl;
        }
        // Never get here
        else
        {
          std::stringstream error_stream;
          error_stream << "Advection Diffusion Elements only store 4 fields "
                       << std::endl;
          throw OomphLibError(error_stream.str(),
                              OOMPH_CURRENT_FUNCTION,
                              OOMPH_EXCEPTION_LOCATION);
        }
      }
    }
 
    /// 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
    {
      // Winds
      if (i < 3)
      {
        return "Wind " + StringConversion::to_string(i);
      }
      // Advection Diffusion field
      else if (i == 3)
      {
        return "Advection Diffusion";
      }
      // Never get here
      else
      {
        std::stringstream error_stream;
        error_stream << "Advection Diffusion Elements only store 4 fields "
                     << std::endl;
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
        // Dummy return
        return " ";
      }
    }
 
    /// Output with default number of plot points
    void output(std::ostream& outfile)
    {
      unsigned nplot = 5;
      output(outfile, nplot);
    }
 
    /// Output FE representation of soln: r,z,u  at
    /// nplot^2 plot points
    void output(std::ostream& outfile, const unsigned& nplot);
 
 
    /// C_style output with default number of plot points
    void output(FILE* file_pt)
    {
      unsigned n_plot = 5;
      output(file_pt, n_plot);
    }
 
    /// C-style output FE representation of soln: r,z,u  at
    /// n_plot^2 plot points
    void output(FILE* file_pt, const unsigned& n_plot);
 
 
    /// Output exact soln: r,z,u_exact at nplot^2 plot points
    void output_fct(std::ostream& outfile,
                    const unsigned& nplot,
                    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);
 
    /// Access function: Pointer to source function
    inline AxisymAdvectionDiffusionSourceFctPt& source_fct_pt()
    {
      return Source_fct_pt;
    }
 
 
    /// Access function: Pointer to source function. Const version
    inline AxisymAdvectionDiffusionSourceFctPt source_fct_pt() const
    {
      return Source_fct_pt;
    }
 
 
    /// Access function: Pointer to wind function
    inline AxisymAdvectionDiffusionWindFctPt& wind_fct_pt()
    {
      return Wind_fct_pt;
    }
 
 
    /// Access function: Pointer to wind function. Const version
    inline AxisymAdvectionDiffusionWindFctPt wind_fct_pt() const
    {
      return Wind_fct_pt;
    }
 
    // Access functions for the physical constants
 
    /// Peclet number
    inline const double& pe() const
    {
      return *Pe_pt;
    }
 
    /// Pointer to Peclet number
    inline double*& pe_pt()
    {
      return Pe_pt;
    }
 
    /// Peclet number multiplied by Strouhal number
    inline const double& pe_st() const
    {
      return *PeSt_pt;
    }
 
    /// Pointer to Peclet number multipled by Strouha number
    inline double*& pe_st_pt()
    {
      return PeSt_pt;
    }
 
    /// Peclet number multiplied by Strouhal number
    inline const double& d() const
    {
      return *D_pt;
    }
 
    /// Pointer to Peclet number multipled by Strouha number
    inline double*& d_pt()
    {
      return D_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_axi_adv_diff(const unsigned& ipt,
                                                const Vector<double>& x,
                                                double& source) const
    {
      // If no source function has been set, return zero
      if (Source_fct_pt == 0)
      {
        source = 0.0;
      }
      else
      {
        // Get source strength
        (*Source_fct_pt)(x, source);
      }
    }
 
    /// Get wind at (Eulerian) position x and/or local coordinate s.
    /// This function is
    /// virtual to allow overloading in multi-physics problems where
    /// the wind function might be determined by
    /// another system of equations
    inline virtual void get_wind_axi_adv_diff(const unsigned& ipt,
                                              const Vector<double>& s,
                                              const Vector<double>& x,
                                              Vector<double>& wind) const
    {
      // If no wind function has been set, return zero
      if (Wind_fct_pt == 0)
      {
        for (unsigned i = 0; i < 3; i++)
        {
          wind[i] = 0.0;
        }
      }
      else
      {
        // Get wind
        (*Wind_fct_pt)(x, wind);
      }
    }
 
    /// Get flux: [du/dr,du/dz]
    void get_flux(const Vector<double>& s, Vector<double>& flux) const
    {
      // Find out how many nodes there are in the element
      const unsigned n_node = nnode();
 
      // Get the nodal index at which the unknown is stored
      const unsigned u_nodal_index = u_index_axi_adv_diff();
 
      // 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
      for (unsigned j = 0; j < 2; j++)
      {
        flux[j] = 0.0;
      }
 
      // Loop over nodes
      for (unsigned l = 0; l < n_node; l++)
      {
        const double u_value = this->nodal_value(l, u_nodal_index);
        // Add in the derivative directions
        flux[0] += u_value * dpsidx(l, 0);
        flux[1] += u_value * dpsidx(l, 1);
      }
    }
 
 
    /// 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 and using
      // a dummy matrix
      fill_in_generic_residual_contribution_axi_adv_diff(
        residuals,
        GeneralisedElement::Dummy_matrix,
        GeneralisedElement::Dummy_matrix,
        0);
    }
 
 
    /// Add the element's contribution to its residual vector and
    /// the 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_axi_adv_diff(
        residuals, jacobian, GeneralisedElement::Dummy_matrix, 1);
    }
 
 
    /// Return FE representation of function value u(s) at local coordinate s
    inline double interpolated_u_axi_adv_diff(const Vector<double>& s) const
    {
      // Find number of nodes
      const unsigned n_node = nnode();
 
      // Get the nodal index at which the unknown is stored
      const unsigned u_nodal_index = u_index_axi_adv_diff();
 
      // Local shape function
      Shape psi(n_node);
 
      // Find values of shape function
      shape(s, psi);
 
      // Initialise value of u
      double interpolated_u = 0.0;
 
      // Loop over the local nodes and sum
      for (unsigned l = 0; l < n_node; l++)
      {
        interpolated_u += nodal_value(l, u_nodal_index) * psi[l];
      }
 
      return (interpolated_u);
    }
 
 
    /// Return derivative of u at point s with respect to all data
    /// that can affect its value.
    /// In addition, return the global equation numbers corresponding to the
    /// data. This is virtual so that it can be overloaded in the
    /// refineable version
    virtual void dinterpolated_u_axi_adv_diff_ddata(
      const Vector<double>& s,
      Vector<double>& du_ddata,
      Vector<unsigned>& global_eqn_number)
    {
      // Find number of nodes
      const unsigned n_node = nnode();
 
      // Get the nodal index at which the unknown is stored
      const unsigned u_nodal_index = u_index_axi_adv_diff();
 
      // Local shape function
      Shape psi(n_node);
 
      // Find values of shape function
      shape(s, psi);
 
      // Find the number of dofs associated with interpolated u
      unsigned n_u_dof = 0;
      for (unsigned l = 0; l < n_node; l++)
      {
        int global_eqn = this->node_pt(l)->eqn_number(u_nodal_index);
        // If it's positive add to the count
        if (global_eqn >= 0)
        {
          ++n_u_dof;
        }
      }
 
      // Now resize the storage schemes
      du_ddata.resize(n_u_dof, 0.0);
      global_eqn_number.resize(n_u_dof, 0);
 
      // Loop over the nodes again and set the derivatives
      unsigned count = 0;
      for (unsigned l = 0; l < n_node; l++)
      {
        // Get the global equation number
        int global_eqn = this->node_pt(l)->eqn_number(u_nodal_index);
        // If it's positive
        if (global_eqn >= 0)
        {
          // Set the global equation number
          global_eqn_number[count] = global_eqn;
          // Set the derivative with respect to the unknown
          du_ddata[count] = psi[l];
          // Increase the counter
          ++count;
        }
      }
    }
 
 
    /// Self-test: Return 0 for OK
    unsigned self_test();
 
  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_axi_adv_diff(
      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_axi_adv_diff(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const = 0;
 
    /// Add the element's contribution to its residual vector only
    /// (if flag=and/or element  Jacobian matrix
    virtual void fill_in_generic_residual_contribution_axi_adv_diff(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      DenseMatrix<double>& mass_matrix,
      unsigned flag);
 
    // Physical constants
 
    /// Pointer to global Peclet number
    double* Pe_pt;
 
    /// Pointer to global Peclet number multiplied by Strouhal number
    double* PeSt_pt;
 
    /// Pointer to global Diffusion parameter
    double* D_pt;
 
    /// Pointer to source function:
    AxisymAdvectionDiffusionSourceFctPt Source_fct_pt;
 
    /// Pointer to wind function:
    AxisymAdvectionDiffusionWindFctPt Wind_fct_pt;
 
    /// Boolean flag to indicate whether AlE formulation is disable
    bool ALE_is_disabled;
 
  private:
    /// Static default value for the Peclet number
    static double Default_peclet_number;
 
    /// Static default value for the Peclet number
    static double Default_diffusion_parameter;
    ;
 
 
  }; // End class AxisymAdvectionDiffusionEquations
 
 
  /// ////////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////////
 
 
  //======================================================================
  /// QAxisymAdvectionDiffusionElement elements are
  /// linear/quadrilateral/brick-shaped Axisymmetric Advection Diffusion
  /// elements with isoparametric interpolation for the function.
  //======================================================================
  template<unsigned NNODE_1D>
  class QAxisymAdvectionDiffusionElement
    : public virtual QElement<2, NNODE_1D>,
      public virtual AxisymAdvectionDiffusionEquations
  {
  private:
    /// Static array of ints to hold number of variables at
    /// nodes: Initial_Nvalue[n]
    static const unsigned Initial_Nvalue;
 
  public:
    /// Constructor: Call constructors for QElement and
    /// Advection Diffusion equations
    QAxisymAdvectionDiffusionElement()
      : QElement<2, NNODE_1D>(), AxisymAdvectionDiffusionEquations()
    {
    }
 
    /// Broken copy constructor
    QAxisymAdvectionDiffusionElement(
      const QAxisymAdvectionDiffusionElement<NNODE_1D>& dummy) = delete;
 
    /// Broken assignment operator
    /*void operator=(const QAxisymAdvectionDiffusionElement<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:
    ///  r,z,u
    void output(std::ostream& outfile)
    {
      AxisymAdvectionDiffusionEquations::output(outfile);
    }
 
    /// Output function:
    ///  r,z,u  at n_plot^2 plot points
    void output(std::ostream& outfile, const unsigned& n_plot)
    {
      AxisymAdvectionDiffusionEquations::output(outfile, n_plot);
    }
 
 
    /// C-style output function:
    ///  r,z,u
    void output(FILE* file_pt)
    {
      AxisymAdvectionDiffusionEquations::output(file_pt);
    }
 
    ///  C-style output function:
    ///   r,z,u at n_plot^2 plot points
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      AxisymAdvectionDiffusionEquations::output(file_pt, n_plot);
    }
 
    /// Output function for an exact solution:
    ///  r,z,u_exact at n_plot^2 plot points
    void output_fct(std::ostream& outfile,
                    const unsigned& n_plot,
                    FiniteElement::SteadyExactSolutionFctPt exact_soln_pt)
    {
      AxisymAdvectionDiffusionEquations::output_fct(
        outfile, n_plot, exact_soln_pt);
    }
 
 
  protected:
    /// Shape, test functions & derivs. w.r.t. to global coords. Return
    /// Jacobian.
    inline double dshape_and_dtest_eulerian_axi_adv_diff(
      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_axi_adv_diff(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const;
 
  }; // End class QAxisymAdvectionDiffusionElement
 
  // 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 NNODE_1D>
  double QAxisymAdvectionDiffusionElement<
    NNODE_1D>::dshape_and_dtest_eulerian_axi_adv_diff(const Vector<double>& s,
                                                      Shape& psi,
                                                      DShape& dpsidx,
                                                      Shape& test,
                                                      DShape& dtestdx) const
  {
    // Call the geometrical shape functions and derivatives
    double J = this->dshape_eulerian(s, psi, dpsidx);
 
    // Loop over the test functions and derivatives and set them equal to the
    // shape functions
    for (unsigned i = 0; i < NNODE_1D; i++)
    {
      test[i] = psi[i];
      for (unsigned j = 0; j < 2; j++)
      {
        dtestdx(i, j) = dpsidx(i, j);
      }
    }
 
    // 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 NNODE_1D>
  double QAxisymAdvectionDiffusionElement<NNODE_1D>::
    dshape_and_dtest_eulerian_at_knot_axi_adv_diff(const unsigned& ipt,
                                                   Shape& psi,
                                                   DShape& dpsidx,
                                                   Shape& test,
                                                   DShape& dtestdx) const
  {
    // Call the geometrical shape functions and derivatives
    double J = this->dshape_eulerian_at_knot(ipt, psi, dpsidx);
 
    // Set the test functions equal to the shape functions (pointer copy)
    test = psi;
    dtestdx = dpsidx;
 
    // Return the jacobian
    return J;
  }
 
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
 
  template<unsigned NNODE_1D>
  class FaceGeometry<QAxisymAdvectionDiffusionElement<NNODE_1D>>
    : public virtual QElement<1, NNODE_1D>
  {
  public:
    /// Constructor: Call the constructor for the
    /// appropriate lower-dimensional QElement
    FaceGeometry() : QElement<1, NNODE_1D>() {}
  };
 
 
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
 
  //======================================================================
  /// A class for elements that allow the imposition of an
  /// applied Robin boundary condition on the boundaries of Steady
  /// Axisymmnetric Advection Diffusion Flux elements.
  /// \f[ -\Delta u \cdot \mathbf{n} + \alpha(r,z) u = \beta(r,z) \f]
  /// The element geometry is obtained from the FaceGeometry<ELEMENT>
  /// policy class.
  //======================================================================
  /// AT THE MOMENT THESE ARE IN SPHERICAL POLARS --- NOT CONSISTENT
 
  // template <class ELEMENT>
  // class AxisymAdvectionDiffusionFluxElement :
  // public virtual FaceGeometry<ELEMENT>,
  //  public virtual FaceElement
  // {
 
  // public:
 
 
  //  /// Function pointer to the prescribed-beta function fct(x,beta(x))
  //  --
  //  /// x is a Vector!
  //  typedef void (*AxisymAdvectionDiffusionPrescribedBetaFctPt)(
  //   const Vector<double>& x,
  //   double& beta);
 
  //  /// Function pointer to the prescribed-alpha function
  //  fct(x,alpha(x)) --
  //  /// x is a Vector!
  //  typedef void (*AxisymAdvectionDiffusionPrescribedAlphaFctPt)(
  //   const Vector<double>& x,
  //   double& alpha);
 
 
  //  /// Constructor, takes the pointer to the "bulk" element
  //  /// and the index of the face to be created
  //  AxisymAdvectionDiffusionFluxElement(FiniteElement* const &bulk_el_pt,
  //                                         const int &face_index);
 
 
  //  /// Broken empty constructor
  //  AxisymAdvectionDiffusionFluxElement()
  //   {
  //    throw OomphLibError(
  //     "Don't call empty constructor for AxisymAdvectionDiffusionFluxElement",
  //     OOMPH_CURRENT_FUNCTION,
  //     OOMPH_EXCEPTION_LOCATION);
  //   }
 
  //  /// Broken copy constructor
  //  AxisymAdvectionDiffusionFluxElement(
  //   const AxisymAdvectionDiffusionFluxElement& dummy) = delete;
 
  //  /// Broken assignment operator
  //  /*void operator=(const AxisymAdvectionDiffusionFluxElement&) = delete;*/
 
  //  /// Access function for the prescribed-beta function pointer
  //  AxisymAdvectionDiffusionPrescribedBetaFctPt& beta_fct_pt()
  //   {
  //    return Beta_fct_pt;
  //   }
 
  //  /// Access function for the prescribed-alpha function pointer
  //  AxisymAdvectionDiffusionPrescribedAlphaFctPt& alpha_fct_pt()
  //   {
  //    return Alpha_fct_pt;
  //   }
 
 
  //  /// Add the element's contribution to its residual vector
  //  inline void fill_in_contribution_to_residuals(Vector<double> &residuals)
  //   {
  //    //Call the generic residuals function with flag set to 0
  //    //using a dummy matrix
  //    fill_in_generic_residual_contribution_axi_adv_diff_flux(
  //     residuals,
  //     GeneralisedElement::Dummy_matrix,0);
  //   }
 
 
  //  /// Add the element's contribution to its residual vector and
  //  /// its Jacobian matrix
  //  inline void fill_in_contribution_to_jacobian(Vector<double> &residuals,
  //                                               DenseMatrix<double>
  //                                               &jacobian)
  //   {
  //    //Call the generic routine with the flag set to 1
  //    fill_in_generic_residual_contribution_axi_adv_diff_flux(residuals,
  //                                                        jacobian,
  //                                                        1);
  //   }
 
  //  /// Specify the value of nodal zeta from the face geometry
  //  /// The "global" intrinsic coordinate of the element when
  //  /// viewed as part of a geometric object should be given by
  //  /// the FaceElement representation, by default (needed to break
  //  /// indeterminacy if bulk element is SolidElement)
  //  double zeta_nodal(const unsigned &n, const unsigned &k,
  //                    const unsigned &i) const
  //  {return FaceElement::zeta_nodal(n,k,i);}
 
 
  //  /// Output function -- forward to broken version in FiniteElement
  //  /// until somebody decides what exactly they want to plot here...
  //  void output(std::ostream &outfile)
  //   {
  //    FiniteElement::output(outfile);
  //   }
 
  //  /// Output function -- forward to broken version in FiniteElement
  //  /// until somebody decides what exactly they want to plot here...
  //  void output(std::ostream &outfile, const unsigned &nplot)
  //   {
  //    FiniteElement::output(outfile,nplot);
  //   }
 
 
  // protected:
 
  //  /// Function to compute the shape and test functions and to return
  //  /// the Jacobian of mapping between local and global (Eulerian)
  //  /// coordinates
  //  inline double shape_and_test(const Vector<double> &s,
  //                               Shape &psi,
  //                               Shape &test) const
  //   {
  //    //Find number of nodes
  //    unsigned n_node = nnode();
 
  //    //Get the shape functions
  //    shape(s,psi);
 
  //    //Set the test functions to be the same as the shape functions
  //    for(unsigned i=0;i<n_node;i++)
  //     {
  //      test[i] = psi[i];
  //     }
 
  //    //Return the value of the jacobian
  //    return J_eulerian(s);
  //   }
 
 
  //  /// Function to compute the shape and test functions and to return
  //  /// the Jacobian of mapping between local and global (Eulerian)
  //  /// coordinates
  //  inline double shape_and_test_at_knot(const unsigned &ipt,
  //                                       Shape &psi,
  //                                       Shape &test) const
  //   {
  //    //Find number of nodes
  //    unsigned n_node = nnode();
 
  //    //Get the shape functions
  //    shape_at_knot(ipt,psi);
 
  //    //Set the test functions to be the same as the shape functions
  //    for(unsigned i=0;i<n_node;i++)
  //     {
  //      test[i] = psi[i];
  //     }
 
  //    //Return the value of the jacobian
  //    return J_eulerian_at_knot(ipt);
  //   }
 
  //  /// Function to calculate the prescribed beta at a given spatial
  //  /// position
  //  void get_beta(const Vector<double>& x, double& beta)
  //   {
  //    //If the function pointer is zero return zero
  //    if(Beta_fct_pt == 0)
  //     {
  //      beta = 0.0;
  //     }
  //    //Otherwise call the function
  //    else
  //     {
  //      (*Beta_fct_pt)(x,beta);
  //     }
  //   }
 
  //  /// Function to calculate the prescribed alpha at a given spatial
  //  /// position
  //  void get_alpha(const Vector<double>& x, double& alpha)
  //   {
  //    //If the function pointer is zero return zero
  //    if(Alpha_fct_pt == 0)
  //     {
  //      alpha = 0.0;
  //     }
  //    //Otherwise call the function
  //    else
  //     {
  //      (*Alpha_fct_pt)(x,alpha);
  //     }
  //   }
 
  // private:
 
 
  //  /// Add the element's contribution to its residual vector.
  //  /// flag=1(or 0): do (or don't) compute the Jacobian as well.
  //  void fill_in_generic_residual_contribution_axi_adv_diff_flux(
  //   Vector<double> &residuals, DenseMatrix<double> &jacobian,
  //   unsigned flag);
 
 
  //  /// Function pointer to the (global) prescribed-beta function
  //  AxisymAdvectionDiffusionPrescribedBetaFctPt Beta_fct_pt;
 
  //  /// Function pointer to the (global) prescribed-alpha function
  //  AxisymAdvectionDiffusionPrescribedAlphaFctPt Alpha_fct_pt;
 
  //  /// The index at which the unknown is stored at the nodes
  //  unsigned U_index_adv_diff;
 
 
  // };//End class AxisymAdvectionDiffusionFluxElement
 
 
  // ///////////////////////////////////////////////////////////////////////
  // ///////////////////////////////////////////////////////////////////////
  // ///////////////////////////////////////////////////////////////////////
 
 
  // //===========================================================================
  // /// Constructor, takes the pointer to the "bulk" element and the
  // index
  // /// of the face to be created
  // //===========================================================================
  // template<class ELEMENT>
  // AxisymAdvectionDiffusionFluxElement<ELEMENT>::
  // AxisymAdvectionDiffusionFluxElement(FiniteElement* const &bulk_el_pt,
  //                                           const int &face_index) :
  // FaceGeometry<ELEMENT>(), FaceElement()
 
  // {
  //  //Let the bulk element build the FaceElement, i.e. setup the pointers
  //  //to its nodes (by referring to the appropriate nodes in the bulk
  //  //element), etc.
  //  bulk_el_pt->build_face_element(face_index,this);
 
  // #ifdef PARANOID
  //  {
  //   //Check that the element is not a refineable 3d element
  //   ELEMENT* elem_pt = dynamic_cast<ELEMENT*>(bulk_el_pt);
  //   //If it's three-d
  //   if(elem_pt->dim()==3)
  //    {
  //     //Is it refineable
  //     RefineableElement* ref_el_pt=dynamic_cast<RefineableElement*>(elem_pt);
  //     if(ref_el_pt!=0)
  //      {
  //       if (this->has_hanging_nodes())
  //        {
  //         throw OomphLibError(
  //          "This flux element will not work correctly if nodes are
  //          hanging\n", OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
  //        }
  //      }
  //    }
  //  }
  // #endif
 
  //  //Initialise the prescribed-beta function pointer to zero
  //  Beta_fct_pt = 0;
 
  //  //Set up U_index_adv_diff. Initialise to zero, which probably won't change
  //  //in most cases, oh well, the price we pay for generality
  //  U_index_adv_diff = 0;
 
  //  //Cast to the appropriate AdvectionDiffusionEquation so that we can
  //  //find the index at which the variable is stored
  //  //We assume that the dimension of the full problem is the same
  //  //as the dimension of the node, if this is not the case you will have
  //  //to write custom elements, sorry
  //  AxisymAdvectionDiffusionEquations* eqn_pt =
  //  dynamic_cast<AxisymAdvectionDiffusionEquations*>(bulk_el_pt);
 
  //  //If the cast has failed die
  //  if(eqn_pt==0)
  //   {
  //      std::string error_string =
  //       "Bulk element must inherit from AxisymAdvectionDiffusionEquations.";
  //      error_string +=
  //       "Nodes are two dimensional, but cannot cast the bulk element to\n";
  //      error_string += "AxisymAdvectionDiffusionEquations<2>\n.";
  //      error_string +=
  //       "If you desire this functionality, you must implement it yourself\n";
 
  //      throw OomphLibError(
  //       error_string,
  //       OOMPH_CURRENT_FUNCTION,
  //       OOMPH_EXCEPTION_LOCATION);
  //   }
  //  else
  //   {
  //    //Read the index from the (cast) bulk element.
  //    U_index_adv_diff = eqn_pt->u_index_axi_adv_diff();
  //   }
 
 
  // }
 
 
  // //===========================================================================
  // /// Compute the element's residual vector and the (zero) Jacobian
  // /// matrix for the Robin boundary condition:
  // /// \f[ \Delta u \cdot \mathbf{n} + \alpha (\mathbf{x}) = \beta (\mathbf{x}) \f]
  // //===========================================================================
  // template<class ELEMENT>
  // void AxisymAdvectionDiffusionFluxElement<ELEMENT>::
  // fill_in_generic_residual_contribution_axi_adv_diff_flux(
  //  Vector<double> &residuals,
  //  DenseMatrix<double> &jacobian,
  //  unsigned flag)
  // {
  //  //Find out how many nodes there are
  //  const unsigned n_node = nnode();
 
  //  // Locally cache the index at which the variable is stored
  //  const unsigned u_index_axi_adv_diff = U_index_adv_diff;
 
  //  //Set up memory for the shape and test functions
  //  Shape psif(n_node), testf(n_node);
 
  //  //Set the value of n_intpt
  //  const unsigned n_intpt = integral_pt()->nweight();
 
  //  //Set the Vector to hold local coordinates
  //  Vector<double> s(1);
 
  //  //Integers used to store the local equation number and local unknown
  //  //indices for the residuals and jacobians
  //  int local_eqn=0, local_unknown=0;
 
  //  //Loop over the integration points
  //  //--------------------------------
  //  for(unsigned ipt=0;ipt<n_intpt;ipt++)
  //   {
 
  //    //Assign values of s
  //    for(unsigned i=0;i<1;i++)
  //     {
  //      s[i] = integral_pt()->knot(ipt,i);
  //     }
 
  //    //Get the integral weight
  //    double w = integral_pt()->weight(ipt);
 
  //    //Find the shape and test functions and return the Jacobian
  //    //of the mapping
  //    double J = shape_and_test(s,psif,testf);
 
  //    //Premultiply the weights and the Jacobian
  //    double W = w*J;
 
  //    //Calculate local values of the solution and its derivatives
  //    //Allocate
  //    double interpolated_u=0.0;
  //    Vector<double> interpolated_x(2,0.0);
 
  //    //Calculate position
  //    for(unsigned l=0;l<n_node;l++)
  //     {
  //      //Get the value at the node
  //      double u_value = raw_nodal_value(l,u_index_axi_adv_diff);
  //      interpolated_u += u_value*psif(l);
  //      //Loop over coordinate direction
  //      for(unsigned i=0;i<2;i++)
  //       {
  //        interpolated_x[i] += nodal_position(l,i)*psif(l);
  //       }
  //     }
 
  //    //Get the imposed beta (beta=flux when alpha=0.0)
  //    double beta;
  //    get_beta(interpolated_x,beta);
 
  //    //Get the imposed alpha
  //    double alpha;
  //    get_alpha(interpolated_x,alpha);
 
  //    //calculate the area weighting dS = r^{2} sin theta dr dtheta
  //    // r = x[0] and theta = x[1]
  //    double dS = interpolated_x[0]*interpolated_x[0]*sin(interpolated_x[1]);
 
  //    //Now add to the appropriate equations
 
  //    //Loop over the test functions
  //    for(unsigned l=0;l<n_node;l++)
  //     {
  //      //Set the local equation number
  //      local_eqn = nodal_local_eqn(l,u_index_axi_adv_diff);
  //      /*IF it's not a boundary condition*/
  //      if(local_eqn >= 0)
  //       {
  //        //Add the prescribed beta terms
  //        residuals[local_eqn] -= dS*(beta - alpha*interpolated_u)*testf(l)*W;
 
  //        //Calculate the Jacobian
  //        //----------------------
  //        if ( (flag) && (alpha!=0.0) )
  //         {
  //          //Loop over the velocity shape functions again
  //          for(unsigned l2=0;l2<n_node;l2++)
  //           {
  //            //Set the number of the unknown
  //            local_unknown = nodal_local_eqn(l2,u_index_axi_adv_diff);
 
  //            //If at a non-zero degree of freedom add in the entry
  //            if(local_unknown >= 0)
  //             {
  //              jacobian(local_eqn,local_unknown) +=
  //               dS*alpha*psif[l2]*testf[l]*W;
  //             }
  //           }
  //         }
 
  //       }
  //     } //end loop over test functions
 
  //   } //end loop over integration points
 
  // }//end fill_in_generic_residual_contribution_adv_diff_flux
 
  // } //End AxisymAdvectionDiffusionFluxElement class
} // namespace oomph
#endif