gen_axisym_advection_diffusion_elements.h

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// LIC// ====================================================================
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// LIC// multi-physics finite-element library, available
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// LIC//
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// Header file for Advection Diffusion elements
#ifndef OOMPH_GEN_AXISYM_ADV_DIFF_ELEMENTS_HEADER
#define OOMPH_GEN_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/oomph_utilities.h"
 
namespace oomph
{
  //=============================================================
  // DOXYERROR: the maths below is wrong somehow but I have no idea what it's
  // supposed to be doing!
  /// A class for all elements that solve the Advection
  /// Diffusion equations in conservative form using isoparametric elements
  /// in a cylindrical polar coordinate system.
  /// \mbox{\boldmath$\nabla\cdot$} \left(
  /// Pe \mbox{\boldmath$w$}(\mbox{\boldmath$x$}) u
  ///  - D(\mbox{\boldmath$x$)\mbox{\boldmath$\nabla$} u\right)
  /// = f(\mbox{\boldmath$x$})
  /// This contains the generic maths. Shape functions, geometric
  /// mapping etc. must get implemented in derived class.
  //=============================================================
  class GeneralisedAxisymAdvectionDiffusionEquations
    : public virtual FiniteElement
  {
  public:
    /// Function pointer to source function fct(x,f(x)) --
    /// x is a Vector!
    typedef void (*GeneralisedAxisymAdvectionDiffusionSourceFctPt)(
      const Vector<double>& x, double& f);
 
    /// Function pointer to wind function fct(x,w(x)) --
    /// x is a Vector!
    typedef void (*GeneralisedAxisymAdvectionDiffusionWindFctPt)(
      const Vector<double>& x, Vector<double>& wind);
 
 
    /// Function pointer to a diffusivity function
    typedef void (*GeneralisedAxisymAdvectionDiffusionDiffFctPt)(
      const Vector<double>& x, DenseMatrix<double>& D);
 
    /// Constructor: Initialise the Source_fct_pt and Wind_fct_pt
    /// to null and set (pointer to) Peclet number to default
    GeneralisedAxisymAdvectionDiffusionEquations()
      : Source_fct_pt(0),
        Wind_fct_pt(0),
        Conserved_wind_fct_pt(0),
        Diff_fct_pt(0),
        ALE_is_disabled(false)
    {
      // Set Peclet number to default
      Pe_pt = &Default_peclet_number;
      // Set Peclet Strouhal number to default
      PeSt_pt = &Default_peclet_number;
    }
 
    /// Broken copy constructor
    GeneralisedAxisymAdvectionDiffusionEquations(
      const GeneralisedAxisymAdvectionDiffusionEquations& 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 GeneralisedAxisymAdvectionDiffusionEquations&) =
     * 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 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_cons_axisym_adv_diff() const
    {
      return 0;
    }
 
    /// du/dt at local node n.
    /// Uses suitably interpolated value for hanging nodes.
    double du_dt_cons_axisym_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_cons_axisym_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;
    }
 
 
    /// 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);
 
    /// Output exact soln: r,z,,u_exact at
    /// nplot^2 plot points (dummy time-dependent version to
    /// keep intel compiler happy)
    virtual void output_fct(
      std::ostream& outfile,
      const unsigned& nplot,
      const double& time,
      FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt)
    {
      throw OomphLibError("There is no time-dependent output_fct() for "
                          "Advection Diffusion elements",
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
 
 
    /// 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(
        "No time-dependent compute_error() for Advection Diffusion elements",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Integrate the concentration over the element
    double integrate_u();
 
 
    /// Access function: Pointer to source function
    GeneralisedAxisymAdvectionDiffusionSourceFctPt& source_fct_pt()
    {
      return Source_fct_pt;
    }
 
 
    /// Access function: Pointer to source function. Const version
    GeneralisedAxisymAdvectionDiffusionSourceFctPt source_fct_pt() const
    {
      return Source_fct_pt;
    }
 
 
    /// Access function: Pointer to wind function
    GeneralisedAxisymAdvectionDiffusionWindFctPt& wind_fct_pt()
    {
      return Wind_fct_pt;
    }
 
 
    /// Access function: Pointer to wind function. Const version
    GeneralisedAxisymAdvectionDiffusionWindFctPt wind_fct_pt() const
    {
      return Wind_fct_pt;
    }
 
 
    /// Access function: Pointer to additional (conservative) wind function
    GeneralisedAxisymAdvectionDiffusionWindFctPt& conserved_wind_fct_pt()
    {
      return Conserved_wind_fct_pt;
    }
 
 
    /// Access function: Pointer to additional (conservative)
    /// wind function.
    /// Const version
    GeneralisedAxisymAdvectionDiffusionWindFctPt conserved_wind_fct_pt() const
    {
      return Conserved_wind_fct_pt;
    }
 
    /// Access function: Pointer to diffusion  function
    GeneralisedAxisymAdvectionDiffusionDiffFctPt& diff_fct_pt()
    {
      return Diff_fct_pt;
    }
 
    /// Access function: Pointer to diffusion function. Const version
    GeneralisedAxisymAdvectionDiffusionDiffFctPt diff_fct_pt() const
    {
      return Diff_fct_pt;
    }
 
    /// Peclet number
    const double& pe() const
    {
      return *Pe_pt;
    }
 
    /// Pointer to Peclet number
    double*& pe_pt()
    {
      return Pe_pt;
    }
 
    /// Peclet number multiplied by Strouhal number
    const double& pe_st() const
    {
      return *PeSt_pt;
    }
 
    /// Pointer to Peclet number multipled by Strouha number
    double*& pe_st_pt()
    {
      return PeSt_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_cons_axisym_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_cons_axisym_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
      // There are three components of the wind, but only two matter
      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 additional (conservative)
    /// 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_conserved_wind_cons_axisym_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 (Conserved_wind_fct_pt == 0)
      {
        for (unsigned i = 0; i < 3; i++)
        {
          wind[i] = 0.0;
        }
      }
      else
      {
        // Get wind
        (*Conserved_wind_fct_pt)(x, wind);
      }
    }
 
 
    /// Get diffusivity tensor 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_diff_cons_axisym_adv_diff(
      const unsigned& ipt,
      const Vector<double>& s,
      const Vector<double>& x,
      DenseMatrix<double>& D) const
    {
      // If no wind function has been set, return identity
      // Again three components, but not all of them matter
      // Those in the theta direction are ignored
      if (Diff_fct_pt == 0)
      {
        for (unsigned i = 0; i < 3; i++)
        {
          for (unsigned j = 0; j < 3; j++)
          {
            if (i == j)
            {
              D(i, j) = 1.0;
            }
            else
            {
              D(i, j) = 0.0;
            }
          }
        }
      }
      else
      {
        // Get diffusivity tensor
        (*Diff_fct_pt)(x, D);
      }
    }
 
 
    /// Get flux: \f$\mbox{flux}[i] = \mbox{d}u / \mbox{d}x_i \f$
    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 = this->nnode();
 
      // Get the nodal index at which the unknown is stored
      const unsigned u_nodal_index = this->u_index_cons_axisym_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);
      }
    }
 
    /// Get flux: \f$\mbox{flux}[i] = \mbox{d}u / \mbox{d}x_i \f$
    void get_total_flux(const Vector<double>& s,
                        Vector<double>& total_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_cons_axisym_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);
 
      // Storage for the Eulerian position
      Vector<double> interpolated_x(2, 0.0);
      // Storage for the concentration
      double interpolated_u = 0.0;
      // Storage for the derivatives of the concentration
      Vector<double> interpolated_dudx(2, 0.0);
 
      // Loop over nodes
      for (unsigned l = 0; l < n_node; l++)
      {
        // Get the value at the node
        const double u_value = this->nodal_value(l, u_nodal_index);
        interpolated_u += u_value * psi(l);
        // Loop over directions
        for (unsigned j = 0; j < 2; j++)
        {
          interpolated_x[j] += this->nodal_position(l, j) * psi(l);
          interpolated_dudx[j] += u_value * dpsidx(l, j);
        }
      }
 
      // Dummy integration point
      unsigned ipt = 0;
 
      // Get the conserved wind (non-divergence free)
      Vector<double> conserved_wind(3);
      get_conserved_wind_cons_axisym_adv_diff(
        ipt, s, interpolated_x, conserved_wind);
 
      // Get diffusivity tensor
      DenseMatrix<double> D(3, 3);
      get_diff_cons_axisym_adv_diff(ipt, s, interpolated_x, D);
 
      // Calculate the total flux made up of the diffusive flux
      // and the conserved wind only bother with the
      // first two components in each case becuase there can be no
      // variation in the azimuthal direction
      for (unsigned i = 0; i < 2; i++)
      {
        total_flux[i] = 0.0;
        for (unsigned j = 0; j < 2; j++)
        {
          total_flux[i] += D(i, j) * interpolated_dudx[j];
        }
        total_flux[i] -= conserved_wind[i] * interpolated_u;
      }
    }
 
 
    /// 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_cons_axisym_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_cons_axisym_adv_diff(
        residuals, jacobian, GeneralisedElement::Dummy_matrix, 1);
    }
 
 
    /// Add the element's contribution to its residuals vector,
    /// jacobian matrix and mass matrix
    void fill_in_contribution_to_jacobian_and_mass_matrix(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      DenseMatrix<double>& mass_matrix)
    {
      // Call the generic routine with the flag set to 2
      fill_in_generic_residual_contribution_cons_axisym_adv_diff(
        residuals, jacobian, mass_matrix, 2);
    }
 
 
    /// Return FE representation of function value u(s) at local coordinate s
    inline double interpolated_u_cons_axisym_adv_diff(
      const Vector<double>& s) const
    {
      // Find number of nodes
      unsigned n_node = nnode();
 
      // Get the nodal index at which the unknown is stored
      unsigned u_nodal_index = u_index_cons_axisym_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);
    }
 
 
    /// 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_cons_axisym_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_cons_axisym_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_cons_axisym_adv_diff(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      DenseMatrix<double>& mass_matrix,
      unsigned flag);
 
    /// Pointer to global Peclet number
    double* Pe_pt;
 
    /// Pointer to global Peclet number multiplied by Strouhal number
    double* PeSt_pt;
 
    /// Pointer to source function:
    GeneralisedAxisymAdvectionDiffusionSourceFctPt Source_fct_pt;
 
    /// Pointer to wind function:
    GeneralisedAxisymAdvectionDiffusionWindFctPt Wind_fct_pt;
 
    /// Pointer to additional (conservative) wind function:
    GeneralisedAxisymAdvectionDiffusionWindFctPt Conserved_wind_fct_pt;
 
    /// Pointer to diffusivity funciton
    GeneralisedAxisymAdvectionDiffusionDiffFctPt Diff_fct_pt;
 
    /// Boolean flag to indicate if ALE formulation is disabled when
    /// time-derivatives are computed. Only set to false if you're sure
    /// that the mesh is stationary.
    bool ALE_is_disabled;
 
  private:
    /// Static default value for the Peclet number
    static double Default_peclet_number;
  };
 
 
  /// ////////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////////
 
 
  //======================================================================
  /// QGeneralisedAxisymAdvectionDiffusionElement elements are
  /// linear/quadrilateral/brick-shaped Advection Diffusion elements with
  /// isoparametric interpolation for the function.
  //======================================================================
  template<unsigned NNODE_1D>
  class QGeneralisedAxisymAdvectionDiffusionElement
    : public virtual QElement<2, NNODE_1D>,
      public virtual GeneralisedAxisymAdvectionDiffusionEquations
  {
  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
    QGeneralisedAxisymAdvectionDiffusionElement()
      : QElement<2, NNODE_1D>(), GeneralisedAxisymAdvectionDiffusionEquations()
    {
    }
 
    /// Broken copy constructor
    QGeneralisedAxisymAdvectionDiffusionElement(
      const QGeneralisedAxisymAdvectionDiffusionElement<NNODE_1D>& dummy) =
      delete;
 
    /// Broken assignment operator
    /*void operator=(const
     QGeneralisedAxisymAdvectionDiffusionElement<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)
    {
      GeneralisedAxisymAdvectionDiffusionEquations::output(outfile);
    }
 
    /// Output function:
    /// r,z,u  at n_plot^2 plot points
    void output(std::ostream& outfile, const unsigned& n_plot)
    {
      GeneralisedAxisymAdvectionDiffusionEquations::output(outfile, n_plot);
    }
 
 
    /// C-style output function:
    ///  r,z,u
    void output(FILE* file_pt)
    {
      GeneralisedAxisymAdvectionDiffusionEquations::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)
    {
      GeneralisedAxisymAdvectionDiffusionEquations::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)
    {
      GeneralisedAxisymAdvectionDiffusionEquations ::output_fct(
        outfile, n_plot, exact_soln_pt);
    }
 
 
    /// Output function for a time-dependent exact solution.
    ///  r,z,u_exact  at n_plot^2 plot points
    /// (Calls the steady version)
    void output_fct(std::ostream& outfile,
                    const unsigned& n_plot,
                    const double& time,
                    FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt)
    {
      GeneralisedAxisymAdvectionDiffusionEquations::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_cons_axisym_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_cons_axisym_adv_diff(
      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 NNODE_1D>
  double QGeneralisedAxisymAdvectionDiffusionElement<NNODE_1D>::
    dshape_and_dtest_eulerian_cons_axisym_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 QGeneralisedAxisymAdvectionDiffusionElement<NNODE_1D>::
    dshape_and_dtest_eulerian_at_knot_cons_axisym_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;
  }
 
 
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
 
 
  //=======================================================================
  /// Face geometry for the QGeneralisedAxisymAdvectionDiffusionElement
  /// 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 NNODE_1D>
  class FaceGeometry<QGeneralisedAxisymAdvectionDiffusionElement<NNODE_1D>>
    : public virtual QElement<1, NNODE_1D>
  {
  public:
    /// Constructor: Call the constructor for the
    /// appropriate lower-dimensional QElement
    FaceGeometry() : QElement<1, NNODE_1D>() {}
  };
 
} // namespace oomph
 
#endif