linear_elasticity_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// Lesser General Public License for more details.
// LIC//
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// LIC//====================================================================
// Header file for general linear elasticity elements
 
// Include guards to prevent multiple inclusion of the header
#ifndef OOMPH_LINEAR_ELASTICITY_ELEMENTS_HEADER
#define OOMPH_LINEAR_ELASTICITY_ELEMENTS_HEADER
 
// Config header generated by autoconfig
#ifdef HAVE_CONFIG_H
#include <oomph-lib-config.h>
#endif
 
 
// OOMPH-LIB headers
#include "../generic/Qelements.h"
#include "../generic/mesh.h"
#include "../generic/hermite_elements.h"
#include "./elasticity_tensor.h"
#include "../generic/projection.h"
 
namespace oomph
{
  //=======================================================================
  /// A base class for elements that solve the equations of linear
  /// elasticity in Cartesian coordinates.
  /// Combines a few generic functions that are shared by
  /// LinearElasticityEquations
  /// and LinearElasticityEquationsWithPressure (hierher: The latter
  /// don't exist yet but will be written as soon as somebody needs them...)
  //=======================================================================
  template<unsigned DIM>
  class LinearElasticityEquationsBase : public virtual FiniteElement
  {
  public:
    /// Return the index at which the i-th unknown displacement
    /// component is stored. The default value, i, is appropriate for
    /// single-physics problems.
    virtual inline unsigned u_index_linear_elasticity(const unsigned i) const
    {
      return i;
    }
 
    /// d^2u/dt^2 at local node n
    double d2u_dt2_linear_elasticity(const unsigned& n, const unsigned& i) const
    {
      // Get the timestepper
      TimeStepper* time_stepper_pt = node_pt(n)->time_stepper_pt();
 
      // Storage for the derivative - initialise to 0
      double d2u_dt2 = 0.0;
 
      // If we are doing an unsteady solve then calculate the derivative
      if (!time_stepper_pt->is_steady())
      {
        // Get the nodal index
        const unsigned u_nodal_index = u_index_linear_elasticity(i);
 
        // Get the number of values required to represent history
        const unsigned n_time = time_stepper_pt->ntstorage();
 
        // Loop over history values
        for (unsigned t = 0; t < n_time; t++)
        {
          // Add the contribution to the derivative
          d2u_dt2 +=
            time_stepper_pt->weight(2, t) * nodal_value(t, n, u_nodal_index);
        }
      }
 
      return d2u_dt2;
    }
 
    /// Compute vector of FE interpolated displacement u at local coordinate s
    void interpolated_u_linear_elasticity(const Vector<double>& s,
                                          Vector<double>& disp) const
    {
      // Find number of nodes
      unsigned n_node = nnode();
 
      // Local shape function
      Shape psi(n_node);
 
      // Find values of shape function
      shape(s, psi);
 
      for (unsigned i = 0; i < DIM; i++)
      {
        // Index at which the nodal value is stored
        unsigned u_nodal_index = u_index_linear_elasticity(i);
 
        // Initialise value of u
        disp[i] = 0.0;
 
        // Loop over the local nodes and sum
        for (unsigned l = 0; l < n_node; l++)
        {
          disp[i] += nodal_value(l, u_nodal_index) * psi[l];
        }
      }
    }
 
    /// Return FE interpolated displacement u[i] at local coordinate s
    double interpolated_u_linear_elasticity(const Vector<double>& s,
                                            const unsigned& i) const
    {
      // Find number of nodes
      unsigned n_node = nnode();
 
      // Local shape function
      Shape psi(n_node);
 
      // Find values of shape function
      shape(s, psi);
 
      // Get nodal index at which i-th velocity is stored
      unsigned u_nodal_index = u_index_linear_elasticity(i);
 
      // 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);
    }
 
 
    /// Function pointer to function that specifies the body force
    /// as a function of the Cartesian coordinates and time FCT(t,x,b) --
    /// x and b are  Vectors!
    typedef void (*BodyForceFctPt)(const double& t,
                                   const Vector<double>& x,
                                   Vector<double>& b);
 
    /// Constructor: Set null pointers for constitutive law and for
    /// isotropic growth function. Set physical parameter values to
    /// default values, switch on inertia and set body force to zero.
    LinearElasticityEquationsBase()
      : Elasticity_tensor_pt(0),
        Lambda_sq_pt(&Default_lambda_sq_value),
        Unsteady(true),
        Body_force_fct_pt(0)
    {
    }
 
    /// Return the pointer to the elasticity_tensor
    ElasticityTensor*& elasticity_tensor_pt()
    {
      return Elasticity_tensor_pt;
    }
 
    /// Access function to the entries in the elasticity tensor
    inline double E(const unsigned& i,
                    const unsigned& j,
                    const unsigned& k,
                    const unsigned& l) const
    {
      return (*Elasticity_tensor_pt)(i, j, k, l);
    }
 
    /// Access function for timescale ratio (nondim density)
    const double& lambda_sq() const
    {
      return *Lambda_sq_pt;
    }
 
    /// Access function for pointer to timescale ratio (nondim density)
    double*& lambda_sq_pt()
    {
      return Lambda_sq_pt;
    }
 
    /// Access function: Pointer to body force function
    BodyForceFctPt& body_force_fct_pt()
    {
      return Body_force_fct_pt;
    }
 
    /// Access function: Pointer to body force function (const version)
    BodyForceFctPt body_force_fct_pt() const
    {
      return Body_force_fct_pt;
    }
 
 
    /// Switch on solid inertia
    void enable_inertia()
    {
      Unsteady = true;
    }
 
    /// Switch off solid inertia
    void disable_inertia()
    {
      Unsteady = false;
    }
 
    /// Access function to flag that switches inertia on/off (const version)
    bool is_inertia_enabled() const
    {
      return Unsteady;
    }
 
    /// Pin the element's redundant solid pressures (needed for refinement)
    virtual void pin_elemental_redundant_nodal_solid_pressures() {}
 
    ///  Loop over all elements in Vector (which typically contains
    /// all the elements in a refineable solid mesh) and pin the nodal solid
    /// pressure  degrees of freedom that are not being used. Function uses
    /// the member function
    /// - \c LinearElasticityEquationsBase<DIM>::
    ///      pin_elemental_redundant_nodal_pressure_dofs()
    /// .
    /// which is empty by default and should be implemented for
    /// elements with nodal solid pressure degrees of freedom
    /// (e.g. linear elasticity elements with continuous pressure
    /// interpolation.)
    static void pin_redundant_nodal_solid_pressures(
      const Vector<GeneralisedElement*>& element_pt)
    {
      // Loop over all elements and call the function that pins their
      // unused nodal solid pressure data
      unsigned n_element = element_pt.size();
      for (unsigned e = 0; e < n_element; e++)
      {
        dynamic_cast<LinearElasticityEquationsBase<DIM>*>(element_pt[e])
          ->pin_elemental_redundant_nodal_solid_pressures();
      }
    }
 
    /// Return the Cauchy stress tensor, as calculated
    /// from the elasticity tensor at specified local coordinate
    /// Virtual so separaete versions can (and must!) be provided
    /// for displacement and pressure-displacement formulations.
    virtual void get_stress(const Vector<double>& s,
                            DenseMatrix<double>& sigma) const = 0;
 
    /// Return the strain tensor
    void get_strain(const Vector<double>& s, DenseMatrix<double>& strain) const;
 
    /// Evaluate body force at Eulerian coordinate x at present time
    /// (returns zero vector if no body force function pointer has been set)
    inline void body_force(const Vector<double>& x, Vector<double>& b) const
    {
      // If no function has been set, return zero vector
      if (Body_force_fct_pt == 0)
      {
        // Get spatial dimension of element
        unsigned n = dim();
        for (unsigned i = 0; i < n; i++)
        {
          b[i] = 0.0;
        }
      }
      else
      {
        // Get time from timestepper of first node (note that this must
        // work -- body force only makes sense for elements that can be
        // deformed and given that the deformation of solid finite elements
        // is controlled by their nodes, nodes must exist!)
        double time = node_pt(0)->time_stepper_pt()->time_pt()->time();
 
        // Now evaluate the body force
        (*Body_force_fct_pt)(time, x, b);
      }
    }
 
 
    /// The number of "DOF types" that degrees of freedom in this element
    /// are sub-divided into: for now lump them all into one DOF type.
    /// Can be adjusted later
    unsigned ndof_types() const
    {
      return 2;
      // return 1;
    }
 
    /// Create a list of pairs for all unknowns in this element,
    /// so that the first entry in each pair contains the global equation
    /// number of the unknown, while the second one contains the number
    /// of the "DOF types" that this unknown is associated with.
    /// (Function can obviously only be called if the equation numbering
    /// scheme has been set up.)
    void get_dof_numbers_for_unknowns(
      std::list<std::pair<unsigned long, unsigned>>& dof_lookup_list) const
    {
      // temporary pair (used to store dof lookup prior to being added
      // to list)
      std::pair<unsigned long, unsigned> dof_lookup;
 
      // number of nodes
      const unsigned n_node = this->nnode();
 
      // Integer storage for local unknown
      int local_unknown = 0;
 
      // Loop over the nodes
      for (unsigned n = 0; n < n_node; n++)
      {
        // Loop over dimension
        for (unsigned i = 0; i < DIM; i++)
        {
          // If the variable is free
          local_unknown = nodal_local_eqn(n, i);
 
          // ignore pinned values
          if (local_unknown >= 0)
          {
            // store dof lookup in temporary pair: First entry in pair
            // is global equation number; second entry is dof type
            dof_lookup.first = this->eqn_number(local_unknown);
            dof_lookup.second = i;
            // dof_lookup.second = DIM;
 
            // add to list
            dof_lookup_list.push_front(dof_lookup);
          }
        }
      }
    }
 
 
  protected:
    /// Pointer to the elasticity tensor
    ElasticityTensor* Elasticity_tensor_pt;
 
    /// Timescale ratio (non-dim. density)
    double* Lambda_sq_pt;
 
    /// Flag that switches inertia on/off
    bool Unsteady;
 
    /// Pointer to body force function
    BodyForceFctPt Body_force_fct_pt;
 
    /// Static default value for timescale ratio (1.0 -- for natural scaling)
    static double Default_lambda_sq_value;
  };
 
 
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
 
 
  //=======================================================================
  /// A class for elements that solve the equations of linear elasticity
  /// in cartesian coordinates.
  //=======================================================================
  template<unsigned DIM>
  class LinearElasticityEquations
    : public virtual LinearElasticityEquationsBase<DIM>
  {
  public:
    ///  Constructor
    LinearElasticityEquations() {}
 
    /// Number of values required at node n.
    unsigned required_nvalue(const unsigned& n) const
    {
      return DIM;
    }
 
    /// Return the residuals for the solid equations (the discretised
    /// principle of virtual displacements)
    void fill_in_contribution_to_residuals(Vector<double>& residuals)
    {
      fill_in_generic_contribution_to_residuals_linear_elasticity(
        residuals, GeneralisedElement::Dummy_matrix, 0);
    }
 
    /// The jacobian is calculated by finite differences by default,
    /// We need only to take finite differences w.r.t. positional variables
    /// For this element
    void fill_in_contribution_to_jacobian(Vector<double>& residuals,
                                          DenseMatrix<double>& jacobian)
    {
      // Add the contribution to the residuals
      this->fill_in_generic_contribution_to_residuals_linear_elasticity(
        residuals, jacobian, 1);
    }
 
    /// Return the Cauchy stress tensor, as calculated
    /// from the elasticity tensor at specified local coordinate
    void get_stress(const Vector<double>& s, DenseMatrix<double>& sigma) const;
 
 
    /// Output exact solution x,y,[z],u,v,[w]
    void output_fct(std::ostream& outfile,
                    const unsigned& nplot,
                    FiniteElement::SteadyExactSolutionFctPt exact_soln_pt);
 
    /// Output exact solution x,y,[z],u,v,[w] (unsteady version)
    void output_fct(std::ostream& outfile,
                    const unsigned& nplot,
                    const double& time,
                    FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt);
 
    /// Output: x,y,[z],u,v,[w]
    void output(std::ostream& outfile)
    {
      unsigned n_plot = 5;
      output(outfile, n_plot);
    }
 
    /// Output: x,y,[z],u,v,[w]
    void output(std::ostream& outfile, const unsigned& n_plot);
 
 
    /// C-style output: x,y,[z],u,v,[w]
    void output(FILE* file_pt)
    {
      unsigned n_plot = 5;
      output(file_pt, n_plot);
    }
 
    /// Output: x,y,[z],u,v,[w]
    void output(FILE* file_pt, const unsigned& n_plot);
 
    /// Validate against exact solution.
    /// Solution is provided via function pointer.
    /// Plot at a given number of plot points and compute L2 error
    /// and L2 norm of displacement solution over element
    void compute_error(std::ostream& outfile,
                       FiniteElement::SteadyExactSolutionFctPt exact_soln_pt,
                       double& error,
                       double& norm);
 
    /// Validate against exact solution.
    /// Solution is provided via function pointer.
    /// Plot at a given number of plot points and compute L2 error
    /// and L2 norm of displacement solution over element
    /// (unsteady version)
    void compute_error(std::ostream& outfile,
                       FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt,
                       const double& time,
                       double& error,
                       double& norm);
 
  private:
    /// Private helper function to compute residuals and (if requested
    /// via flag) also the Jacobian matrix.
    virtual void fill_in_generic_contribution_to_residuals_linear_elasticity(
      Vector<double>& residuals, DenseMatrix<double>& jacobian, unsigned flag);
  };
 
 
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
 
 
  //===========================================================================
  /// An Element that solves the equations of linear elasticity
  /// in Cartesian coordinates, using QElements for the geometry
  //============================================================================
  template<unsigned DIM, unsigned NNODE_1D>
  class QLinearElasticityElement : public virtual QElement<DIM, NNODE_1D>,
                                   public virtual LinearElasticityEquations<DIM>
  {
  public:
    /// Constructor
    QLinearElasticityElement()
      : QElement<DIM, NNODE_1D>(), LinearElasticityEquations<DIM>()
    {
    }
 
    /// Output exact solution x,y,[z],u,v,[w]
    void output_fct(std::ostream& outfile,
                    const unsigned& nplot,
                    FiniteElement::SteadyExactSolutionFctPt exact_soln_pt)
    {
      LinearElasticityEquations<DIM>::output_fct(outfile, nplot, exact_soln_pt);
    }
 
    /// Output function
    void output(std::ostream& outfile)
    {
      LinearElasticityEquations<DIM>::output(outfile);
    }
 
    /// Output function
    void output(std::ostream& outfile, const unsigned& n_plot)
    {
      LinearElasticityEquations<DIM>::output(outfile, n_plot);
    }
 
 
    /// C-style output function
    void output(FILE* file_pt)
    {
      LinearElasticityEquations<DIM>::output(file_pt);
    }
 
    /// C-style output function
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      LinearElasticityEquations<DIM>::output(file_pt, n_plot);
    }
  };
 
 
  //============================================================================
  /// FaceGeometry of a linear 2D QLinearElasticityElement element
  //============================================================================
  template<>
  class FaceGeometry<QLinearElasticityElement<2, 2>>
    : public virtual QElement<1, 2>
  {
  public:
    /// Constructor must call the constructor of the underlying solid element
    FaceGeometry() : QElement<1, 2>() {}
  };
 
 
  //============================================================================
  /// FaceGeometry of a quadratic 2D QLinearElasticityElement element
  //============================================================================
  template<>
  class FaceGeometry<QLinearElasticityElement<2, 3>>
    : public virtual QElement<1, 3>
  {
  public:
    /// Constructor must call the constructor of the underlying element
    FaceGeometry() : QElement<1, 3>() {}
  };
 
 
  //============================================================================
  /// FaceGeometry of a cubic 2D QLinearElasticityElement element
  //============================================================================
  template<>
  class FaceGeometry<QLinearElasticityElement<2, 4>>
    : public virtual QElement<1, 4>
  {
  public:
    /// Constructor must call the constructor of the underlying element
    FaceGeometry() : QElement<1, 4>() {}
  };
 
 
  //============================================================================
  /// FaceGeometry of a linear 3D QLinearElasticityElement element
  //============================================================================
  template<>
  class FaceGeometry<QLinearElasticityElement<3, 2>>
    : public virtual QElement<2, 2>
  {
  public:
    /// Constructor must call the constructor of the underlying element
    FaceGeometry() : QElement<2, 2>() {}
  };
 
  //============================================================================
  /// FaceGeometry of a quadratic 3D QLinearElasticityElement element
  //============================================================================
  template<>
  class FaceGeometry<QLinearElasticityElement<3, 3>>
    : public virtual QElement<2, 3>
  {
  public:
    /// Constructor must call the constructor of the underlying element
    FaceGeometry() : QElement<2, 3>() {}
  };
 
 
  //============================================================================
  /// FaceGeometry of a cubic 3D QLinearElasticityElement element
  //============================================================================
  template<>
  class FaceGeometry<QLinearElasticityElement<3, 4>>
    : public virtual QElement<2, 4>
  {
  public:
    /// Constructor must call the constructor of the underlying element
    FaceGeometry() : QElement<2, 4>() {}
  };
 
  /// /////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////
 
 
  //==========================================================
  /// Linear elasticity upgraded to become projectable
  //==========================================================
  template<class LINEAR_ELAST_ELEMENT>
  class ProjectableLinearElasticityElement
    : public virtual ProjectableElement<LINEAR_ELAST_ELEMENT>
  {
  public:
    /// Constructor [this was only required explicitly
    /// from gcc 4.5.2 onwards...]
    ProjectableLinearElasticityElement() {}
 
 
    /// 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.
    /// In the underlying linear elasticity elements the
    /// the displacements are stored at the nodal values
    Vector<std::pair<Data*, unsigned>> data_values_of_field(const unsigned& fld)
    {
      // Create the vector
      Vector<std::pair<Data*, unsigned>> data_values;
 
      // Loop over all nodes and extract the fld-th nodal value
      unsigned nnod = this->nnode();
      for (unsigned j = 0; j < nnod; j++)
      {
        // Add the data value associated with the displacement components
        data_values.push_back(std::make_pair(this->node_pt(j), fld));
      }
 
      // Return the vector
      return data_values;
    }
 
    /// Number of fields to be projected: dim, corresponding to
    /// the displacement components
    unsigned nfields_for_projection()
    {
      return this->dim();
    }
 
    /// Number of history values to be stored for fld-th field.
    /// (includes present value!)
    unsigned nhistory_values_for_projection(const unsigned& fld)
    {
#ifdef PARANOID
      if (fld > 1)
      {
        std::stringstream error_stream;
        error_stream << "Elements only store two fields so fld can't be"
                     << " " << 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: Read out from
    /// positional timestepper  (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)
    {
      unsigned n_dim = this->dim();
      unsigned n_node = this->nnode();
      DShape dpsidx(n_node, n_dim);
 
      // Call the derivatives of the shape functions and return
      // the Jacobian
      return this->dshape_eulerian(s, psi, dpsidx);
    }
 
 
    /// 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)
    {
      unsigned n_node = this->nnode();
 
      /* #ifdef PARANOID */
      /*     unsigned n_dim=this->node_pt(0)->ndim(); */
      /* #endif */
 
      // Local shape function
      Shape psi(n_node);
 
      // Find values of shape function
      this->shape(s, psi);
 
      // Initialise value of u
      double interpolated_u = 0.0;
 
      // Sum over the local nodes at that time
      for (unsigned l = 0; l < n_node; l++)
      {
        // over-zealous I think. This will quietly do the right thing
        // even if there are additional degrees of freedom floating around.
        /* #ifdef PARANOID */
        /*       unsigned nvalue=this->node_pt(l)->nvalue(); */
        /*       if (nvalue!=n_dim) */
        /*        {         */
        /*         std::stringstream error_stream; */
        /*         error_stream  */
        /*          << "Current implementation only works for non-resized
         * nodes\n" */
        /*          << "but nvalue= " << nvalue << "!= dim = " << n_dim <<
         * std::endl; */
        /*         throw OomphLibError( */
        /*          error_stream.str(), */
        /*          OOMPH_CURRENT_FUNCTION, */
        /*          OOMPH_EXCEPTION_LOCATION); */
        /*        } */
        /* #endif */
        interpolated_u += this->nodal_value(t, l, fld) * psi[l];
      }
      return interpolated_u;
    }
 
 
    /// Return number of values in field fld
    unsigned nvalue_of_field(const unsigned& fld)
    {
      return this->nnode();
    }
 
 
    /// Return local equation number of value j in field fld.
    int local_equation(const unsigned& fld, const unsigned& j)
    {
      // over-zealous I think. This will quietly do the right thing
      // even if there are additional degrees of freedom floating around.
      /* #ifdef PARANOID */
      /*     unsigned n_dim=this->node_pt(0)->ndim(); */
      /*     unsigned nvalue=this->node_pt(j)->nvalue(); */
      /*     if (nvalue!=n_dim) */
      /*      {         */
      /*       std::stringstream error_stream; */
      /*       error_stream  */
      /*        << "Current implementation only works for non-resized nodes\n"
       */
      /*        << "but nvalue= " << nvalue << "!= dim = " << n_dim <<
       * std::endl; */
      /*       throw OomphLibError( */
      /*          error_stream.str(), */
      /*          OOMPH_CURRENT_FUNCTION, */
      /*          OOMPH_EXCEPTION_LOCATION); */
      /*      } */
      /* #endif */
      return this->nodal_local_eqn(j, fld);
    }
  };
 
 
  //=======================================================================
  /// Face geometry for element is the same as that for the underlying
  /// wrapped element
  //=======================================================================
  template<class ELEMENT>
  class FaceGeometry<ProjectableLinearElasticityElement<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<ProjectableLinearElasticityElement<ELEMENT>>>
    : public virtual FaceGeometry<FaceGeometry<ELEMENT>>
  {
  public:
    FaceGeometry() : FaceGeometry<FaceGeometry<ELEMENT>>() {}
  };
 
 
} // namespace oomph
 
#endif