foeppl_von_karman_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//
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// LIC// License as published by the Free Software Foundation; either
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// LIC// Lesser General Public License for more details.
// LIC//
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// Header file for FoepplvonKarman elements
#ifndef OOMPH_FOEPPLVONKARMAN_ELEMENTS_HEADER
#define OOMPH_FOEPPLVONKARMAN_ELEMENTS_HEADER
 
 
// Config header generated by autoconfig
#ifdef HAVE_CONFIG_H
#include <oomph-lib-config.h>
#endif
 
#include <sstream>
 
// OOMPH-LIB headers
#include "../generic/projection.h"
#include "../generic/nodes.h"
#include "../generic/Qelements.h"
#include "../generic/oomph_utilities.h"
 
 
namespace oomph
{
  //=============================================================
  /// A class for all isoparametric elements that solve the
  /// Foeppl von Karman equations.
  /// \f[ \nabla^4 w - \eta Diamond^4(w,\phi) = p(x,y) \f]
  /// and
  /// \f[ \nabla^4 \phi + \frac{1}{2} Diamond^4(w,w) = 0 \f]
  /// This contains the generic maths. Shape functions, geometric
  /// mapping etc. must get implemented in derived class.
  //=============================================================
  class FoepplvonKarmanEquations : public virtual FiniteElement
  {
  public:
    /// Function pointer to pressure function fct(x,f(x)) --
    /// x is a Vector!
    typedef void (*FoepplvonKarmanPressureFctPt)(const Vector<double>& x,
                                                 double& f);
 
    /// Constructor (must initialise the Pressure_fct_pt and
    /// Airy_forcing_fct_pt to null). Also set physical parameters to their
    /// default values. No volume constraint applied by default.
    FoepplvonKarmanEquations() : Pressure_fct_pt(0), Airy_forcing_fct_pt(0)
    {
      // Set all the physical constants to the default value (zero)
      Eta_pt = &Default_Physical_Constant_Value;
      Linear_bending_model = false;
 
      // No volume constraint
      Volume_constraint_pressure_external_data_index = -1;
    }
 
    /// Broken copy constructor
    FoepplvonKarmanEquations(const FoepplvonKarmanEquations& dummy) = delete;
 
    /// Broken assignment operator
    void operator=(const FoepplvonKarmanEquations&) = delete;
 
    // Access functions for the physical constants
 
    /// Eta
    const double& eta() const
    {
      return *Eta_pt;
    }
 
    /// Pointer to eta
    double*& eta_pt()
    {
      return Eta_pt;
    }
 
 
    /// Set Data value containing a single value which represents the
    /// volume control pressure as external data for this element.
    /// Only used for volume controlled problems in conjunction with
    /// FoepplvonKarmanVolumeConstraintElement.
    void set_volume_constraint_pressure_data_as_external_data(Data* data_pt)
    {
#ifdef PARANOID
      if (data_pt->nvalue() != 1)
      {
        throw OomphLibError("Data object that contains volume control pressure "
                            "should only contain a single value. ",
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // Add as external data and remember the index in the element's storage
      // scheme
      Volume_constraint_pressure_external_data_index =
        add_external_data(data_pt);
    }
 
    /// Return the index at which the i-th unknown value
    /// is stored. The default value, i, is appropriate for single-physics
    /// problems. By default, these are:
    /// 0: w
    /// 1: laplacian w
    /// 2: phi
    /// 3: laplacian phi
    /// 4-8: smooth first derivatives
    /// 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 nodal_index_fvk(const unsigned& i = 0) const
    {
      return i;
    }
 
    /// Output with default number of plot points
    void output(std::ostream& outfile)
    {
      const unsigned n_plot = 5;
      output(outfile, n_plot);
    }
 
    /// Output FE representation of soln: x,y,w at
    /// n_plot^DIM plot points
    void output(std::ostream& outfile, const unsigned& n_plot);
 
    /// C_style output with default number of plot points
    void output(FILE* file_pt)
    {
      const unsigned n_plot = 5;
      output(file_pt, n_plot);
    }
 
    /// C-style output FE representation of soln: x,y,w at
    /// n_plot^DIM plot points
    void output(FILE* file_pt, const unsigned& n_plot);
 
    /// Output exact soln: x,y,w_exact at n_plot^DIM plot points
    void output_fct(std::ostream& outfile,
                    const unsigned& n_plot,
                    FiniteElement::SteadyExactSolutionFctPt exact_soln_pt);
 
    /// Output exact soln: x,y,w_exact at
    /// n_plot^DIM plot points (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 Foeppl von Karman"
        "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(
        "There is no time-dependent compute_error() for Foeppl von Karman"
        "elements",
        OOMPH_CURRENT_FUNCTION,
        OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Access function: Pointer to pressure function
    FoepplvonKarmanPressureFctPt& pressure_fct_pt()
    {
      return Pressure_fct_pt;
    }
 
    /// Access function: Pointer to pressure function. Const version
    FoepplvonKarmanPressureFctPt pressure_fct_pt() const
    {
      return Pressure_fct_pt;
    }
 
    /// Access function: Pointer to Airy forcing function
    FoepplvonKarmanPressureFctPt& airy_forcing_fct_pt()
    {
      return Airy_forcing_fct_pt;
    }
 
    /// Access function: Pointer to Airy forcing function. Const version
    FoepplvonKarmanPressureFctPt airy_forcing_fct_pt() const
    {
      return Airy_forcing_fct_pt;
    }
 
    /// Get pressure term at (Eulerian) position x. This function is
    /// virtual to allow overloading in multi-physics problems where
    /// the strength of the pressure function might be determined by
    /// another system of equations.
    inline virtual void get_pressure_fvk(const unsigned& ipt,
                                         const Vector<double>& x,
                                         double& pressure) const
    {
      // If no pressure function has been set, return zero
      if (Pressure_fct_pt == 0)
      {
        pressure = 0.0;
      }
      else
      {
        // Get pressure strength
        (*Pressure_fct_pt)(x, pressure);
      }
    }
 
    /// Get Airy forcing term at (Eulerian) position x. This function is
    /// virtual to allow overloading in multi-physics problems where
    /// the strength of the pressure function might be determined by
    /// another system of equations.
    inline virtual void get_airy_forcing_fvk(const unsigned& ipt,
                                             const Vector<double>& x,
                                             double& airy_forcing) const
    {
      // If no pressure function has been set, return zero
      if (Airy_forcing_fct_pt == 0)
      {
        airy_forcing = 0.0;
      }
      else
      {
        // Get pressure strength
        (*Airy_forcing_fct_pt)(x, airy_forcing);
      }
    }
 
    /// Get gradient of deflection: gradient[i] = dw/dx_i
    void get_gradient_of_deflection(const Vector<double>& s,
                                    Vector<double>& gradient) const
    {
      // Find out how many nodes there are in the element
      const unsigned n_node = nnode();
 
      // Get the index at which the unknown is stored
      const unsigned w_nodal_index = nodal_index_fvk(0);
 
      // 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++)
      {
        gradient[j] = 0.0;
      }
 
      // Loop over nodes
      for (unsigned l = 0; l < n_node; l++)
      {
        // Loop over derivative directions
        for (unsigned j = 0; j < 2; j++)
        {
          gradient[j] += this->nodal_value(l, w_nodal_index) * dpsidx(l, j);
        }
      }
    }
 
    /// Fill in the residuals with this element's contribution
    void fill_in_contribution_to_residuals(Vector<double>& residuals);
 
    // void fill_in_contribution_to_jacobian(Vector<double> &residuals,
    //                                      DenseMatrix<double> &jacobian);
 
    /// Return FE representation of function value w_fvk(s)
    /// at local coordinate s (by default - if index > 0, returns
    /// FE representation of valued stored at index^th nodal index
    inline double interpolated_w_fvk(const Vector<double>& s,
                                     unsigned index = 0) const
    {
      // Find number of nodes
      const unsigned n_node = nnode();
 
      // Get the index at which the poisson unknown is stored
      const unsigned w_nodal_index = nodal_index_fvk(index);
 
      // Local shape function
      Shape psi(n_node);
 
      // Find values of shape function
      shape(s, psi);
 
      // Initialise value of u
      double interpolated_w = 0.0;
 
      // Loop over the local nodes and sum
      for (unsigned l = 0; l < n_node; l++)
      {
        interpolated_w += this->nodal_value(l, w_nodal_index) * psi[l];
      }
 
      return (interpolated_w);
    }
 
    /// Compute in-plane stresses
    void interpolated_stress(const Vector<double>& s,
                             double& sigma_xx,
                             double& sigma_yy,
                             double& sigma_xy);
 
    /// Return the integral of the displacement over the current
    /// element, effectively calculating its contribution to the volume under
    /// the membrane. Virtual so it can be overloaded in multi-physics
    /// where the volume may incorporate an offset, say.
    virtual double get_bounded_volume() const
    {
      // Number of nodes and integration points for the current element
      const unsigned n_node = nnode();
      const unsigned n_intpt = integral_pt()->nweight();
 
      // Shape functions and their derivatives
      Shape psi(n_node);
      DShape dpsidx(n_node, 2);
 
      // The nodal index at which the displacement is stored
      const unsigned w_nodal_index = nodal_index_fvk();
 
      // Initalise the integral variable
      double integral_w = 0;
 
      // Loop over the integration points
      for (unsigned ipt = 0; ipt < n_intpt; ipt++)
      {
        // Get the integral weight
        double w = integral_pt()->weight(ipt);
 
        // Get determinant of the Jacobian of the mapping
        double J = dshape_eulerian_at_knot(ipt, psi, dpsidx);
 
        // Premultiply the weight and Jacobian
        double W = w * J;
 
        // Initialise storage for the w value and nodal value
        double interpolated_w = 0;
        double w_nodal_value;
 
        // Loop over the shape functions/nodes
        for (unsigned l = 0; l < n_node; l++)
        {
          // Get the current nodal value
          w_nodal_value = raw_nodal_value(l, w_nodal_index);
          // Add the contribution to interpolated w
          interpolated_w += w_nodal_value * psi(l);
        }
 
        // Add the contribution from the current integration point
        integral_w += interpolated_w * W;
      }
 
      // Return the calculated integral
      return integral_w;
    }
 
    /// Self-test: Return 0 for OK
    unsigned self_test();
 
    /// Sets a flag to signify that we are solving the linear, pure
    /// bending equations, and pin all the nodal values that will not be used in
    /// this case
    void use_linear_bending_model()
    {
      // Set the boolean flag
      Linear_bending_model = true;
 
      // Get the index of the first FvK nodal value
      unsigned first_fvk_nodal_index = nodal_index_fvk();
 
      // Get the total number of FvK nodal values (assuming they are stored
      // contiguously) at node 0 (it's the same at all nodes anyway)
      unsigned total_fvk_nodal_indicies = 8;
 
      // Get the number of nodes in this element
      unsigned n_node = nnode();
 
      // Loop over the appropriate nodal indices
      for (unsigned index = first_fvk_nodal_index + 2;
           index < first_fvk_nodal_index + total_fvk_nodal_indicies;
           index++)
      {
        // Loop over the nodes in the element
        for (unsigned inod = 0; inod < n_node; inod++)
        {
          // Pin the nodal value at the current index
          node_pt(inod)->pin(index);
        }
      }
    }
 
 
  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_fvk(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_fvk(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const = 0;
 
    // Pointers to global physical constants
 
    /// Pointer to global eta
    double* Eta_pt;
 
    /// Pointer to pressure function:
    FoepplvonKarmanPressureFctPt Pressure_fct_pt;
 
    // mjr Ridicu-hack for made-up second pressure, q
    FoepplvonKarmanPressureFctPt Airy_forcing_fct_pt;
 
  private:
    /// Default value for physical constants
    static double Default_Physical_Constant_Value;
 
    /// Flag which stores whether we are using a linear, pure bending
    /// model instead of the full non-linear Foeppl-von Karman
    bool Linear_bending_model;
 
    /// Index of the external Data object that represents the volume
    /// constraint pressure (initialised to -1 indicating no such constraint)
    /// Gets overwritten when calling
    /// set_volume_constraint_pressure_data_as_external_data(...)
    int Volume_constraint_pressure_external_data_index;
  };
 
 
  /// ////////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////////
 
 
  // mjr QFoepplvonKarmanElements are untested!
 
  //======================================================================
  /// QFoepplvonKarmanElement elements are linear/quadrilateral/brick-shaped
  /// Foeppl von Karman elements with isoparametric interpolation for the
  /// function.
  //======================================================================
 
  template<unsigned NNODE_1D>
  class QFoepplvonKarmanElement : public virtual QElement<2, NNODE_1D>,
                                  public virtual FoepplvonKarmanEquations
  {
  private:
    /// Static int that holds the number of variables at
    /// nodes: always the same
    static const unsigned Initial_Nvalue;
 
  public:
    /// Constructor: Call constructors for QElement and
    /// FoepplvonKarmanEquations
    QFoepplvonKarmanElement()
      : QElement<2, NNODE_1D>(), FoepplvonKarmanEquations()
    {
    }
 
    /// Broken copy constructor
    QFoepplvonKarmanElement(const QFoepplvonKarmanElement<NNODE_1D>& dummy) =
      delete;
 
    /// Broken assignment operator
    void operator=(const QFoepplvonKarmanElement<NNODE_1D>&) = delete;
 
    ///  Required  # of `values' (pinned or dofs)
    /// at node n
    inline unsigned required_nvalue(const unsigned& n) const
    {
      return Initial_Nvalue;
    }
 
    /// Output function:
    ///  x,y,w
    void output(std::ostream& outfile)
    {
      FoepplvonKarmanEquations::output(outfile);
    }
 
 
    ///  Output function:
    ///   x,y,w at n_plot^DIM plot points
    void output(std::ostream& outfile, const unsigned& n_plot)
    {
      FoepplvonKarmanEquations::output(outfile, n_plot);
    }
 
 
    /// C-style output function:
    ///  x,y,w
    void output(FILE* file_pt)
    {
      FoepplvonKarmanEquations::output(file_pt);
    }
 
 
    ///  C-style output function:
    ///   x,y,w at n_plot^DIM plot points
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      FoepplvonKarmanEquations::output(file_pt, n_plot);
    }
 
 
    /// Output function for an exact solution:
    ///  x,y,w_exact at n_plot^DIM plot points
    void output_fct(std::ostream& outfile,
                    const unsigned& n_plot,
                    FiniteElement::SteadyExactSolutionFctPt exact_soln_pt)
    {
      FoepplvonKarmanEquations::output_fct(outfile, n_plot, exact_soln_pt);
    }
 
 
    /// Output function for a time-dependent exact solution.
    ///  x,y,w_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)
    {
      FoepplvonKarmanEquations::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_fvk(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_fvk(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 QFoepplvonKarmanElement<NNODE_1D>::dshape_and_dtest_eulerian_fvk(
    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 NNODE_1D>
  double QFoepplvonKarmanElement<
    NNODE_1D>::dshape_and_dtest_eulerian_at_knot_fvk(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 QFoepplvonKarmanElement 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<QFoepplvonKarmanElement<NNODE_1D>>
    : public virtual QElement<1, NNODE_1D>
  {
  public:
    /// Constructor: Call the constructor for the
    /// appropriate lower-dimensional QElement
    FaceGeometry() : QElement<1, NNODE_1D>() {}
  };
 
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
 
 
  //==========================================================
  /// Foeppl von Karman upgraded to become projectable
  //==========================================================
  template<class FVK_ELEMENT>
  class ProjectableFoepplvonKarmanElement
    : public virtual ProjectableElement<FVK_ELEMENT>
  {
  public:
    /// 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 > 7)
      {
        std::stringstream error_stream;
        error_stream
          << "Foeppl von Karman elements only store eight fields so fld must be"
          << "0 to 7 rather than " << fld << std::endl;
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // Create the vector
      unsigned nnod = this->nnode();
      Vector<std::pair<Data*, unsigned>> data_values(nnod);
 
      // Loop over all nodes
      for (unsigned j = 0; j < nnod; j++)
      {
        // Add the data value associated field: The node itself
        data_values[j] = std::make_pair(this->node_pt(j), fld);
      }
 
      // Return the vector
      return data_values;
    }
 
    /// Number of fields to be projected: Just two
    unsigned nfields_for_projection()
    {
      return 8;
    }
 
    /// Number of history values to be stored for fld-th field.
    /// (Note: count includes current value!)
    unsigned nhistory_values_for_projection(const unsigned& fld)
    {
#ifdef PARANOID
      if (fld > 7)
      {
        std::stringstream error_stream;
        error_stream
          << "Foeppl von Karman elements only store eight fields so fld must be"
          << "0 to 7 rather than " << fld << std::endl;
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
      return this->node_pt(0)->ntstorage();
    }
 
 
    /// Number of positional history values
    /// (Note: count includes current value!)
    unsigned nhistory_values_for_coordinate_projection()
    {
      return this->node_pt(0)->position_time_stepper_pt()->ntstorage();
    }
 
    /// Return Jacobian of mapping and shape functions of field fld
    /// at local coordinate s
    double jacobian_and_shape_of_field(const unsigned& fld,
                                       const Vector<double>& s,
                                       Shape& psi)
    {
#ifdef PARANOID
      if (fld > 7)
      {
        std::stringstream error_stream;
        error_stream
          << "Foeppl von Karman elements only store eight fields so fld must be"
          << "0 to 7 rather than " << fld << std::endl;
        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_fvk(s, psi, dpsidx, test, dtestdx);
      return J;
    }
 
 
    /// Return interpolated field fld at local coordinate s, at time
    /// level t (t=0: present; t>0: history values)
    double get_field(const unsigned& t,
                     const unsigned& fld,
                     const Vector<double>& s)
    {
#ifdef PARANOID
      if (fld > 7)
      {
        std::stringstream error_stream;
        error_stream
          << "Foeppl von Karman elements only store eight fields so fld must be"
          << "0 to 7 rather than " << fld << std::endl;
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
      // Find the index at which the variable is stored
      unsigned w_nodal_index = this->nodal_index_fvk(fld);
 
      // 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_w = 0.0;
 
      // Sum over the local nodes
      for (unsigned l = 0; l < n_node; l++)
      {
        interpolated_w += this->nodal_value(t, l, w_nodal_index) * psi[l];
      }
      return interpolated_w;
    }
 
 
    /// Return number of values in field fld: One per node
    unsigned nvalue_of_field(const unsigned& fld)
    {
#ifdef PARANOID
      if (fld > 7)
      {
        std::stringstream error_stream;
        error_stream
          << "Foeppl von Karman elements only store eight fields so fld must be"
          << "0 to 7 rather than " << fld << std::endl;
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
      return this->nnode();
    }
 
 
    /// Return local equation number of field fld of node j.
    int local_equation(const unsigned& fld, const unsigned& j)
    {
#ifdef PARANOID
      if (fld > 7)
      {
        std::stringstream error_stream;
        error_stream
          << "Foeppl von Karman elements only store eight fields so fld must be"
          << "0 to 7 rather than " << fld << std::endl;
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
      const unsigned w_nodal_index = this->nodal_index_fvk(fld);
      return this->nodal_local_eqn(j, w_nodal_index);
    }
  };
 
  //=======================================================================
  /// Face geometry for element is the same as that for the underlying
  /// wrapped element
  //=======================================================================
  template<class ELEMENT>
  class FaceGeometry<ProjectableFoepplvonKarmanElement<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<ProjectableFoepplvonKarmanElement<ELEMENT>>>
    : public virtual FaceGeometry<FaceGeometry<ELEMENT>>
  {
  public:
    FaceGeometry() : FaceGeometry<FaceGeometry<ELEMENT>>() {}
  };
 
} // namespace oomph
 
#endif