shell_elements.h

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// LIC// ====================================================================
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// LIC// multi-physics finite-element library, available
// LIC// at http://www.oomph-lib.org.
// LIC//
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// LIC//
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// Header file for KL shell elements
#ifndef OOMPH_KIRCHHOFF_LOVE_SHELL_ELEMENTS_HEADER
#define OOMPH_KIRCHHOFF_LOVE_SHELL_ELEMENTS_HEADER
 
// Config header generated by autoconfig
#ifdef HAVE_CONFIG_H
#include <oomph-lib-config.h>
#endif
 
// OOMPH-LIB header
#include "../generic/hermite_elements.h"
#include "../generic/geom_objects.h"
#include "../generic/fsi.h"
#include "../generic/stored_shape_function_elements.h"
#include "../generic/matrices.h"
 
namespace oomph
{
  //========================================================================
  /// A class for elements that solves the equations of Kirchhoff Love shell
  /// thin-shell theory
  //========================================================================
  class KirchhoffLoveShellEquations : public virtual SolidFiniteElement
  {
  private:
    /// Static default value for the Poisson's ratio
    static double Default_nu_value;
 
    /// Static default value for the timescale ratio (1.0 for natural scaling)
    static double Default_lambda_sq_value;
 
    /// Static default value for the thickness ratio
    static double Default_h_value;
 
    /// Boolean flag to ignore membrane terms
    bool Ignore_membrane_terms;
 
    /// Pointer to Poisson's ratio
    double* Nu_pt;
 
    /// Pointer to dimensionless wall thickness
    double* H_pt;
 
    /// Pointer to timescale ratio (non-dimensional density)
    double* Lambda_sq_pt;
 
    /// Pointer to membrane pre-stress terms -- should
    /// probably generalise this to function pointers at some point
    DenseMatrix<double*> Prestress_pt;
 
    /// Static default for prestress (set to zero)
    static double Zero_prestress;
 
  protected:
    /// Invert a DIM by DIM matrix
    inline double calculate_contravariant(double A[2][2], double Aup[2][2]);
 
    /// Default load function (zero traction)
    static void Zero_traction_fct(const Vector<double>& xi,
                                  const Vector<double>& x,
                                  const Vector<double>& N,
                                  Vector<double>& load);
 
    /// Pointer to load vector function: Its arguments are:
    /// Lagrangian coordinate, Eulerian coordinate, normal vector and
    /// load vector itself (not all of the input arguments will be
    /// required for all specific load functions but the list should
    /// cover all cases)
    void (*Load_vector_fct_pt)(const Vector<double>& xi,
                               const Vector<double>& x,
                               const Vector<double>& N,
                               Vector<double>& load);
 
 
    /// Pointer to the GeomObject that specifies the undeformed midplane
    /// of the shell
    GeomObject* Undeformed_midplane_pt;
 
 
    /// Helper function to Return the residuals for
    /// the equations of KL shell theory. This is used to prevent
    /// a call of fill_in_contribution_to_residuals in
    /// the function fill_in_contribution_to_jacobian that can
    /// lead to virtual inheritance woes if this element is ever
    /// used as part of a multi-physics element.
    void fill_in_contribution_to_residuals_shell(Vector<double>& residuals);
 
    /// Get the load vector for the computation of the rate of work
    /// done by the load. Here we simply forward this to
    /// load_vector(...) but allow it to be overloaded in derived classes
    /// to allow the computation of the rate of work due to constituent
    /// parts of the load vector (e.g. the fluid traction in an FSI
    /// problem). Pass number of integration point (dummy),
    /// Lagr. coordinate and normal vector and return the load vector
    /// (not all of the input arguments will be
    /// required for all specific load functions but the list should
    /// cover all cases).
    virtual void load_vector_for_rate_of_work_computation(
      const unsigned& intpt,
      const Vector<double>& xi,
      const Vector<double>& x,
      const Vector<double>& N,
      Vector<double>& load)
    {
      load_vector(intpt, xi, x, N, load);
    }
 
 
  public:
    /// Constructor: Initialise physical parameter values to defaults
    KirchhoffLoveShellEquations() : Undeformed_midplane_pt(0)
    {
      // Set the default values of physical parameters
      Nu_pt = &Default_nu_value;
      Lambda_sq_pt = &Default_lambda_sq_value;
      H_pt = &Default_h_value;
 
      // Don't ignore membrane terms
      Ignore_membrane_terms = false;
 
      // Default load is zero traction
      Load_vector_fct_pt = &Zero_traction_fct;
 
      // Default prestress is zero
      Prestress_pt.resize(2, 2);
      Prestress_pt(0, 0) = &Zero_prestress;
      Prestress_pt(1, 0) = &Zero_prestress;
      Prestress_pt(0, 1) = &Zero_prestress;
      Prestress_pt(1, 1) = &Zero_prestress;
    }
 
    /// Access to the load vector function pointer
    void (*&load_vector_fct_pt())(const Vector<double>& xi,
                                  const Vector<double>& x,
                                  const Vector<double>& N,
                                  Vector<double>& load)
    {
      return Load_vector_fct_pt;
    }
 
    /// Get the load vector: Pass number of integration point (dummy),
    /// Lagr. coordinate and normal vector and return the load vector
    /// (not all of the input arguments will be
    /// required for all specific load functions but the list should
    /// cover all cases). This function is virtual so it can be
    /// overloaded for FSI.
    virtual void load_vector(const unsigned& intpt,
                             const Vector<double>& xi,
                             const Vector<double>& x,
                             const Vector<double>& N,
                             Vector<double>& load)
    {
      Load_vector_fct_pt(xi, x, N, load);
    }
 
 
    /// Set pointer to (i,j)-th component of second Piola Kirchhoff
    /// membrane prestress to specified value (automatically imposes
    /// symmetry for off-diagonal entries)
    void set_prestress_pt(const unsigned& i,
                          const unsigned& j,
                          double* value_pt)
    {
      Prestress_pt(i, j) = value_pt;
      Prestress_pt(j, i) = value_pt;
    }
 
    /// Return (i,j)-th component of second Piola Kirchhoff membrane
    /// prestress
    double prestress(const unsigned& i, const unsigned& j)
    {
      return *Prestress_pt(i, j);
    }
 
    /// Set to disable the calculation of membrane terms
    void disable_membrane_terms()
    {
      Ignore_membrane_terms = true;
    }
 
    /// Set to renable the calculation of membrane terms (default)
    void enable_membrane_terms()
    {
      Ignore_membrane_terms = false;
    }
 
    /// Return the Poisson's ratio
    const double& nu() const
    {
      return *Nu_pt;
    }
 
    /// Return the wall thickness to undeformed radius ratio
    const double& h() const
    {
      return *H_pt;
    }
 
    /// Return the timescale ratio (non-dimensional density)
    const double& lambda_sq() const
    {
      return *Lambda_sq_pt;
    }
 
    /// Return a pointer to the Poisson ratio
    double*& nu_pt()
    {
      return Nu_pt;
    }
 
    /// Return a pointer to the non-dim wall thickness
    double*& h_pt()
    {
      return H_pt;
    }
 
    /// Return a pointer to timescale ratio (nondim density)
    double*& lambda_sq_pt()
    {
      return Lambda_sq_pt;
    }
 
    /// Return a reference to geometric object that specifies the shell's
    /// undeformed geometry
    GeomObject*& undeformed_midplane_pt()
    {
      return Undeformed_midplane_pt;
    }
 
    /// Get normal vector
    void get_normal(const Vector<double>& s, Vector<double>& N);
 
    /// Overload the standard fill in residuals contribution
    void fill_in_contribution_to_residuals(Vector<double>& residuals)
    {
      // Simply call the shell residuals
      fill_in_contribution_to_residuals_shell(residuals);
    }
 
    /// Return the jacobian is calculated by finite differences by default,
    void fill_in_contribution_to_jacobian(Vector<double>& residuals,
                                          DenseMatrix<double>& jacobian);
 
    /// Get potential (strain) and kinetic energy of the element
    void get_energy(double& pot_en, double& kin_en);
 
 
    /// Get strain and bending tensors; returns pair comprising the
    /// determinant of the undeformed (*.first) and deformed (*.second)
    /// midplane metric tensor.
    std::pair<double, double> get_strain_and_bend(const Vector<double>& s,
                                                  DenseDoubleMatrix& strain,
                                                  DenseDoubleMatrix& bend);
 
 
    /// Get integral of instantaneous rate of work done on
    /// the wall due to the load returned by the virtual
    /// function load_vector_for_rate_of_work_computation(...).
    /// In the current class
    /// the latter function simply de-references the external
    /// load but this can be overloaded in derived classes
    /// (e.g. in FSI elements) to determine the rate of work done
    /// by individual constituents of this load (e.g. the fluid load
    /// in an FSI problem).
    double load_rate_of_work();
 
    /// Generic FiniteElement output function
    void output(std::ostream& outfile)
    {
      FiniteElement::output(outfile);
    }
 
    /// Generic FiniteElement output function
    void output(std::ostream& outfile, const unsigned& n_plot)
    {
      FiniteElement::output(outfile, n_plot);
    }
 
    /// Generic FiniteElement output function
    void output(FILE* file_pt)
    {
      FiniteElement::output(file_pt);
    }
 
    /// Generic FiniteElement output function
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      FiniteElement::output(file_pt, n_plot);
    }
  };
 
 
  //==================================================================
  /// Matrix inversion for 2 dimensions
  //==================================================================
  double KirchhoffLoveShellEquations::calculate_contravariant(double A[2][2],
                                                              double Aup[2][2])
  {
    // Calculate determinant
    double det = A[0][0] * A[1][1] - A[0][1] * A[1][0];
 
    // Calculate entries of the inverse
    Aup[0][0] = A[1][1] / det;
    Aup[0][1] = -A[0][1] / det;
    Aup[1][0] = -A[1][0] / det;
    Aup[1][1] = A[0][0] / det;
 
    // Return determinant
    return (det);
  }
 
 
  //=======================================================================
  /// An element that solves the Kirchhoff-Love shell theory equations
  /// using Hermite interpolation (displacements
  /// and slopes are interpolated separately. The local and global
  /// (Lagrangian) coordinates  are not assumed to be aligned.
  /// N.B. It will be DOG SLOW.
  //=======================================================================
  class HermiteShellElement : public virtual SolidQHermiteElement<2>,
                              public KirchhoffLoveShellEquations
  {
  public:
    /// Constructor, there are no internal data points
    HermiteShellElement()
      : SolidQHermiteElement<2>(), KirchhoffLoveShellEquations()
    {
      // Set the number of dimensions at each node (3D nodes on 2D surface)
      set_nodal_dimension(3);
    }
 
    /// Output position veloc and accel.
    void output_with_time_dep_quantities(std::ostream& outfile,
                                         const unsigned& n_plot);
 
    /// Overload the output function
    void output(std::ostream& outfile)
    {
      SolidQHermiteElement<2>::output(outfile);
    }
 
    /// Output function
    void output(std::ostream& outfile, const unsigned& n_plot);
 
    /// Overload the output function
    void output(FILE* file_pt)
    {
      SolidQHermiteElement<2>::output(file_pt);
    }
 
    /// Output function
    void output(FILE* file_pt, const unsigned& n_plot);
  };
 
 
  //=========================================================================
  /// An element that solves the Kirchhoff-Love shell theory equations
  /// using Hermite interpolation (displacements
  /// and slopes are interpolated separately. The local and global
  /// (Lagrangian) coordinates  are assumed to be aligned so that the
  /// Jacobian of the mapping between these coordinates is diagonal.
  /// This significantly simplifies (and speeds up) the computation of the
  /// derivatives of the shape functions.
  //=========================================================================
  class DiagHermiteShellElement : public HermiteShellElement,
                                  public SolidDiagQHermiteElement<2>
  {
  public:
    /// Constructor, there are no internal data points
    DiagHermiteShellElement()
      : HermiteShellElement(), SolidDiagQHermiteElement<2>()
    {
    }
  };
 
 
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
 
 
  //=======================================================================
  /// Face geometry for the HermiteShell elements: 1D SolidQHermiteElement
  //=======================================================================
  template<>
  class FaceGeometry<HermiteShellElement>
    : public virtual SolidQHermiteElement<1>
  {
  public:
    /// Constructor [this was only required explicitly
    /// from gcc 4.5.2 onwards...]
    FaceGeometry() : SolidQHermiteElement<1>() {}
  };
 
 
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
 
 
  //=========================================================================
  /// Diag Hermite Kirchhoff Love shell "upgraded" to a FSIWallElement
  /// (and thus, by inheritance, a GeomObject), so it can be used in FSI.
  //=========================================================================
  class FSIDiagHermiteShellElement : public virtual DiagHermiteShellElement,
                                     public virtual FSIWallElement
  {
  private:
    /// Get the load vector for the computation of the rate of work
    /// done by the load. Can switch between full load and fluid load only.
    /// Overloads the version in the shell element base class.
    /// Pass number of integration point (dummy)
    /// Lagr. coordinate and normal vector and return the load vector
    /// (not all of the input arguments will be
    /// required for all specific load functions but the list should
    /// cover all cases).
    virtual void load_vector_for_rate_of_work_computation(
      const unsigned& intpt,
      const Vector<double>& xi,
      const Vector<double>& x,
      const Vector<double>& N,
      Vector<double>& load)
    {
      /// Get fluid-only load vector
      if (Compute_rate_of_work_by_load_with_fluid_load_only)
      {
        fluid_load_vector(intpt, N, load);
      }
      // Get full load vector as per default
      else
      {
        load_vector(intpt, xi, x, N, load);
      }
    }
 
    /// Boolean flag to indicate whether the normal is directed into the fluid
    bool Normal_points_into_fluid;
 
    /// Boolean flag to indicate if rate-of-work by load is to be
    /// based on the fluid traction only
    bool Compute_rate_of_work_by_load_with_fluid_load_only;
 
  public:
    /// Constructor: Create shell element as FSIWallElement (and thus,
    /// by inheritance, a GeomObject) with two Lagrangian coordinates
    /// and 3 Eulerian coordinates. By default, we assume that the
    /// normal vector computed by KirchhoffLoveShellEquations::get_normal(...)
    /// points into the fluid.
    /// If this is not the case, call the access function
    /// FSIDiagHermiteShellElement::set_normal_pointing_out_of_fluid()
    FSIDiagHermiteShellElement()
      : DiagHermiteShellElement(),
        Normal_points_into_fluid(true),
        Compute_rate_of_work_by_load_with_fluid_load_only(false)
    {
      unsigned n_lagr = 2;
      unsigned n_dim = 3;
      setup_fsi_wall_element(n_lagr, n_dim);
    }
 
    /// Destructor: empty
    ~FSIDiagHermiteShellElement() {}
 
    /// Set the normal computed by
    /// KirchhoffLoveShellEquations::get_normal(...) to point into the fluid
    void set_normal_pointing_into_fluid()
    {
      Normal_points_into_fluid = true;
    }
 
    /// Set the normal computed by
    /// KirchhoffLoveShellEquations::get_normal(...) to point out of the fluid
    void set_normal_pointing_out_of_fluid()
    {
      Normal_points_into_fluid = false;
    }
 
 
    /// Derivative of position vector w.r.t. the SolidFiniteElement's
    /// Lagrangian coordinates; evaluated at current time.
    void dposition_dlagrangian_at_local_coordinate(
      const Vector<double>& s, DenseMatrix<double>& drdxi) const;
 
 
    /// Get integral of instantaneous rate of work done on
    /// the wall due to the fluid load returned by the function
    /// fluid_load_vector(...).
    double fluid_load_rate_of_work()
    {
      Compute_rate_of_work_by_load_with_fluid_load_only = true;
      double tmp = load_rate_of_work();
      Compute_rate_of_work_by_load_with_fluid_load_only = false;
      return tmp;
    }
 
 
    /// Get the load vector: Pass number of the integration point,
    /// Lagr. coordinate, Eulerian coordinate and normal vector
    /// and return the load vector. (Not all of the input arguments will be
    /// required for all specific load functions but the list should
    /// cover all cases). We first evaluate the load function defined via
    /// KirchhoffLoveShellEquations::load_vector_fct_pt() -- this
    /// represents the non-FSI load on the shell, e.g. an external
    /// pressure load. Then we add to this the FSI load due to
    /// the traction exerted by the adjacent FSIFluidElements, taking
    /// the sign of the normal into account.
    void load_vector(const unsigned& intpt,
                     const Vector<double>& xi,
                     const Vector<double>& x,
                     const Vector<double>& N,
                     Vector<double>& load)
    {
      // Initially call the standard Load_vector_fct_pt
      Load_vector_fct_pt(xi, x, N, load);
 
      // Memory for the FSI load
      Vector<double> fsi_load(3);
 
      // Get the fluid load on the wall stress scale
      fluid_load_vector(intpt, N, fsi_load);
 
      // If the normal is outer to the fluid switch the direction
      double sign = 1.0;
      if (!Normal_points_into_fluid)
      {
        sign = -1.0;
      }
 
      // Add the FSI load to the load vector
      for (unsigned i = 0; i < 3; i++)
      {
        load[i] += sign * fsi_load[i];
      }
    }
 
    /// Get the Jacobian and residuals. Wrapper to generic FSI version;
    /// that catches the case when we replace the Jacobian by the
    /// mass matrix (for the consistent assignment of initial conditions).
    virtual void fill_in_contribution_to_jacobian(Vector<double>& residuals,
                                                  DenseMatrix<double>& jacobian)
    {
      // Call the basic shell jacobian
      DiagHermiteShellElement::fill_in_contribution_to_jacobian(residuals,
                                                                jacobian);
      // Fill in the external interaction by finite differences
      this->fill_in_jacobian_from_external_interaction_by_fd(jacobian);
    }
 
 
    /// The number of "DOF types" that degrees of freedom in this element
    /// are sub-divided into: Just the solid degrees of freedom themselves.
    unsigned ndof_types() const
    {
      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;
  };
 
 
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
  /// /////////////////////////////////////////////////////////////////////
 
 
  //======================================================================
  /// Element that allows the imposition of boundary
  /// conditions for a shell that is clamped to a 2D plane
  /// that is specified by its normal. Constraint is applied by
  /// a Lagrange multiplier.
  /// \n
  /// \b Note \b 1: Note that the introduction of the Lagrange
  /// multiplier adds two additional values (relative to the number
  /// of values before the addition of the FaceElement) to the nodes.
  /// This ensures that nodes that are shared by adjacent FaceElements
  /// are not resized repeatedly but also means that this won't work
  /// if two "edges" of the shell (that share a node) are subject
  /// to different constraints, each applied with its own
  /// independent Lagrange multiplier. In such cases a modified
  /// version of this class must be written.
  /// \n
  ///  \b Note \b 2: The FaceGeometry for a HermiteShellElement is
  /// the 1D two-node element SolidQHermiteElement<1> which has
  /// four shape functions (two nodes, two types -- representing the
  /// shape functions that interpolate the value and the derivative).
  /// These are the "correct" shape functions for the interpolation
  /// of the Lagrange multiplier and the isoparametric representation
  /// of the geometry. However, when applying the contribution from the
  /// constraint equation to the bulk equations, we have to take
  /// all four types of dof into account so the element has to
  /// reset the number of positional dofs to four. To avoid any clashes
  /// we overload (the relevant subset of) the access functions to the
  /// shape functions and their derivatives and set the shape functions
  /// associated with the spurious positional dofs to zero. This is a bit
  /// hacky but the only way (?) this can be done...
  //======================================================================
  class ClampedHermiteShellBoundaryConditionElement
    : public virtual FaceGeometry<HermiteShellElement>,
      public virtual SolidFaceElement
  {
  public:
    /// Constructor, takes the pointer to the "bulk" element and the
    /// face index.
    ClampedHermiteShellBoundaryConditionElement(
      FiniteElement* const& bulk_el_pt, const int& face_index);
 
 
    /// Broken empty constructor
    ClampedHermiteShellBoundaryConditionElement()
    {
      throw OomphLibError("Don't call empty constructor for "
                          "ClampedHermiteShellBoundaryConditionElement",
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
    }
 
    /// Broken copy constructor
    ClampedHermiteShellBoundaryConditionElement(
      const ClampedHermiteShellBoundaryConditionElement& dummy) = delete;
 
    /// Broken assignment operator
    void operator=(const ClampedHermiteShellBoundaryConditionElement&) = delete;
 
    /// Set normal vector to clamping plane
    void set_symmetry_line(const Vector<double>& normal_to_clamping_plane)
    {
      Normal_to_clamping_plane[0] = normal_to_clamping_plane[0];
      Normal_to_clamping_plane[1] = normal_to_clamping_plane[1];
      Normal_to_clamping_plane[2] = normal_to_clamping_plane[2];
    }
 
    /// Fill in the element's contribution to its residual vector
    void fill_in_contribution_to_residuals(Vector<double>& residuals);
 
 
    /// ///////////////////////////////////////////////////////////////
    // Note: We should also overload all other versions of shape(...)
    // and dshape(...) but these are the only ones that are used.
    /// ///////////////////////////////////////////////////////////////
 
 
    /// Calculate the geometric shape functions
    /// at local coordinate s. Set any "superfluous" shape functions to zero.
    void shape(const Vector<double>& s, Shape& psi) const
    {
      // Initialise all of them to zero
      unsigned n = psi.nindex1();
      unsigned m = psi.nindex2();
      for (unsigned i = 0; i < n; i++)
      {
        for (unsigned j = 0; j < m; j++)
        {
          psi(i, j) = 0.0;
        }
      }
      FaceGeometry<HermiteShellElement>::shape(s, psi);
    }
 
 
    /// Calculate the geometric shape functions
    /// at local coordinate s. Set any "superfluous" shape functions to zero.
    void dshape_local(const Vector<double>& s, Shape& psi, DShape& dpsids) const
    {
      // Initialise all of them to zero
      unsigned n = psi.nindex1();
      unsigned m = psi.nindex2();
      for (unsigned i = 0; i < n; i++)
      {
        for (unsigned j = 0; j < m; j++)
        {
          psi(i, j) = 0.0;
        }
      }
      unsigned n1 = dpsids.nindex1();
      unsigned n2 = dpsids.nindex2();
      unsigned n3 = dpsids.nindex3();
      for (unsigned i = 0; i < n1; i++)
      {
        for (unsigned j = 0; j < n2; j++)
        {
          for (unsigned k = 0; k < n3; k++)
          {
            dpsids(i, j, k) = 0.0;
          }
        }
      }
      FaceGeometry<HermiteShellElement>::dshape_local(s, psi, dpsids);
    }
 
    /// 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
    void output(std::ostream& outfile, const unsigned& n_plot);
 
    /// C-style output function -- forward to broken version in FiniteElement
    /// until somebody decides what exactly they want to plot here...
    void output(FILE* file_pt)
    {
      FiniteElement::output(file_pt);
    }
 
    /// C-style output function -- forward to broken version in
    /// FiniteElement until somebody decides what exactly they want to plot
    /// here...
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      FiniteElement::output(file_pt, n_plot);
    }
 
    /// The number of "DOF types" that degrees of freedom in this element
    /// are sub-divided into: Just the solid degrees of freedom themselves.
    unsigned ndof_types() const
    {
      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;
 
 
  private:
    /// Normal vector to the clamping plane
    Vector<double> Normal_to_clamping_plane;
  };
 
 
} // namespace oomph
 
#endif