interface_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
// LIC// version 2.1 of the License, or (at your option) any later version.
// LIC//
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// LIC// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
// LIC// Lesser General Public License for more details.
// LIC//
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// LIC//
// LIC// The authors may be contacted at oomph-lib@maths.man.ac.uk.
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// LIC//====================================================================
// Header file for (one-dimensional) free surface elements
// Include guards, to prevent multiple includes
#ifndef OOMPH_INTERFACE_ELEMENTS_HEADER
#define OOMPH_INTERFACE_ELEMENTS_HEADER
 
// Config header generated by autoconfig
#ifdef HAVE_CONFIG_H
#include <oomph-lib-config.h>
#endif
 
#include "../generic/elements.h"
#include "../generic/spines.h"
#include "../generic/shape.h"
#include "../generic/hijacked_elements.h"
 
namespace oomph
{
  //========================================================================
  /// Base class for elements at the boundary of free surfaces or interfaces,
  /// used typically to impose contact angle boundary conditions.
  /// The elemental dimensions are one less than those of the
  /// surface elements, or two less than those of the original bulk elements.
  /// Thus in two-dimensional and axi-symmetric problems, are points,
  /// but in three-dimensional problems, they are lines.
  /// These boundaries may be in contact with a solid surface, in which case
  /// the normal to that surface must be provided.
  //=========================================================================
  class FluidInterfaceBoundingElement : public virtual FaceElement
  {
  private:
    /// Function pointer to a wall unit normal function. Returns the
    /// unit normal on the wall, at the specified Eulerian coordinate.
    typedef void (*WallUnitNormalFctPt)(const Vector<double>& x,
                                        Vector<double>& unit_normal);
 
    /// Pointer to a wall normal function that returns
    /// the wall unit normal as a function of position in global
    /// Eulerian coordinates.
    WallUnitNormalFctPt Wall_unit_normal_fct_pt;
 
    /// Pointer to the desired value of the contact angle (if any)
    double* Contact_angle_pt;
 
    /// Pointer to the desired value of the capillary number
    double* Ca_pt;
 
  protected:
    /// Flag used to determine whether the contact angle is to be
    /// used (0 if not), and whether it will be applied weakly as a force term
    /// in the momentum equations (1) or by hijacking the kinematic
    /// condition (2).
    unsigned Contact_angle_flag;
 
    /// Index at which the i-th velocity component is stored in the
    /// element's nodes
    Vector<unsigned> U_index_interface_boundary;
 
    /// Function that is used to determine the local equation number of
    /// the kinematic equation associated with the nodes of the element
    /// This must be overloaded depending on the node update scheme
    virtual int kinematic_local_eqn(const unsigned& n) = 0;
 
    /// Function that returns the unit normal of the bounding wall
    /// directed out of the fluid
    void wall_unit_normal(const Vector<double>& x, Vector<double>& normal)
    {
#ifdef PARANOID
      if (Wall_unit_normal_fct_pt)
      {
#endif
        (*Wall_unit_normal_fct_pt)(x, normal);
#ifdef PARANOID
      }
      else
      {
        throw OomphLibError("Wall unit normal fct has not been set",
                            "FluidInterfaceBoundingElement::wall_unit_normal()",
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
    }
 
    /// The geometric data of the parent element is included as
    /// external data and so a (bulk) node update must take place after
    /// the variation of any of this external data
    inline void update_in_external_fd(const unsigned& i)
    {
      // Update the bulk element
      bulk_element_pt()->node_update();
    }
 
    /// The only external data are these geometric data so
    /// We can omit the reset function (relying on the next update
    // function to take care of the remesh)
    inline void reset_in_external_fd(const unsigned& i) {}
 
    /// We require a final node update in the bulk element
    /// after all finite differencing
    inline void reset_after_external_fd()
    {
      // Update the bulk element
      bulk_element_pt()->node_update();
    }
 
  public:
    /// Constructor
    FluidInterfaceBoundingElement()
      : Wall_unit_normal_fct_pt(0),
        Contact_angle_pt(0),
        Ca_pt(0),
        Contact_angle_flag(0)
    {
    }
 
    /// Access function: Pointer to wall unit normal function
    WallUnitNormalFctPt& wall_unit_normal_fct_pt()
    {
      return Wall_unit_normal_fct_pt;
    }
 
    /// Access function: Pointer to wall unit normal function. Const version
    WallUnitNormalFctPt wall_unit_normal_fct_pt() const
    {
      return Wall_unit_normal_fct_pt;
    }
 
    /// Access for nodal index at which the velocity components are stored
    Vector<unsigned>& u_index_interface_boundary()
    {
      return U_index_interface_boundary;
    }
 
    /// Set a pointer to the desired contact angle. Optional boolean
    /// (defaults to true)
    /// chooses strong imposition via hijacking (true) or weak imposition
    /// via addition to momentum equation (false). The default strong imposition
    /// is appropriate for static contact angle problems.
    void set_contact_angle(double* const& angle_pt, const bool& strong = true);
 
    /// Access function to the pointer specifying the prescribed contact angle
    double*& contact_angle_pt()
    {
      return Contact_angle_pt;
    }
 
    /// Access function to the pointer specifying the capillary number
    double*& ca_pt()
    {
      return Ca_pt;
    }
 
    /// Return the value of the capillary number
    double ca()
    {
#ifdef PARANOID
      if (Ca_pt != 0)
      {
#endif
        return *Ca_pt;
#ifdef PARANOID
      }
      else
      {
        throw OomphLibError("Capillary number has not been set",
                            "FluidInterfaceBoundingElement::ca()",
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
    }
 
 
    /// Return value of the contact angle
    double& contact_angle()
    {
#ifdef PARANOID
      if (Contact_angle_pt == 0)
      {
        std::string error_message = "Contact angle not set\n";
        error_message +=
          "Please use FluidInterfaceBoundingElement::set_contact_angle()\n";
        throw OomphLibError(
          error_message, OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
      }
#endif
      return *Contact_angle_pt;
    }
 
    /// Calculate the residuals
    void fill_in_contribution_to_residuals(Vector<double>& residuals)
    {
      // Add the residual contributions using a dummy matrix
      fill_in_generic_residual_contribution_interface_boundary(
        residuals, GeneralisedElement::Dummy_matrix, 0);
    }
 
    /// Calculate the generic residuals contribution
    virtual void fill_in_generic_residual_contribution_interface_boundary(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      unsigned flag) = 0;
 
 
    /// Empty helper function to calculate the additional contributions
    /// arising from the node update strategy to the Jacobian within the
    /// integration loop. This will be overloaded by elements that require
    /// contributions to their underlying equations from boundary integrals.
    /// The shape functions, their derivatives w.r.t. to the local coordinates,
    /// the unit normal and integral weight are passed in so that they do not
    /// have to be recalculated.
    virtual void add_additional_residual_contributions_interface_boundary(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      const unsigned& flag,
      const Shape& psif,
      const DShape& dpsifds,
      const Vector<double>& interpolated_n,
      const double& W)
    {
    }
 
    /// Overload the output function
    void output(std::ostream& outfile)
    {
      FiniteElement::output(outfile);
    }
 
    /// Output function
    void output(std::ostream& outfile, const unsigned& n_plot)
    {
      FiniteElement::output(outfile, n_plot);
    }
 
    /// Overload the C-style output function
    void output(FILE* file_pt)
    {
      FiniteElement::output(file_pt);
    }
 
    /// C-style Output function
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      FiniteElement::output(file_pt, n_plot);
    }
  };
 
 
  //==========================================================================
  /// Specialisation of the interface boundary constraint to a point
  //==========================================================================
  class PointFluidInterfaceBoundingElement
    : public FluidInterfaceBoundingElement
  {
  protected:
    /// Overload the helper function to calculate the residuals and
    /// (if flag==1) the Jacobian -- this function only deals with
    /// the part of the Jacobian that can be handled generically.
    /// Specific additional contributions may be provided in
    /// add_additional_residual_contributions_interface_boundary(...)
    void fill_in_generic_residual_contribution_interface_boundary(
      Vector<double>& residuals, DenseMatrix<double>& jacobian, unsigned flag);
 
  public:
    /// Constructor
    PointFluidInterfaceBoundingElement() : FluidInterfaceBoundingElement() {}
  };
 
 
  //==========================================================================
  /// Specialisation of the interface boundary constraint to a line
  //==========================================================================
  class LineFluidInterfaceBoundingElement : public FluidInterfaceBoundingElement
  {
  protected:
    /// Overload the helper function to calculate the residuals and
    /// (if flag==true) the Jacobian -- this function only deals with
    /// the part of the Jacobian that can be handled generically.
    /// Specific additional contributions may be provided in
    /// add_additional_residual_contributions_interface_boundary()
    void fill_in_generic_residual_contribution_interface_boundary(
      Vector<double>& residuals, DenseMatrix<double>& jacobian, unsigned flag);
 
  public:
    /// Constructor
    LineFluidInterfaceBoundingElement() : FluidInterfaceBoundingElement() {}
  };
 
 
  //=======================================================================
  /// Base class establishing common interfaces and functions for all
  /// Navier-Stokes-like fluid
  /// interface elements. Namely, elements that represent either a free
  /// surface or an interface between two fluids that have distinct
  /// momentum-like equation for each velocity component.
  //======================================================================
  class FluidInterfaceElement : public virtual FaceElement
  {
    // Make the bounding element class a friend
    friend class FluidInterfaceBoundingElement;
 
  private:
    /// Pointer to the Capillary number
    double* Ca_pt;
 
    /// Pointer to the Strouhal number
    double* St_pt;
 
    /// Default value for physical constants
    static double Default_Physical_Constant_Value;
 
 
  protected:
    /// Nodal index at which the i-th velocity component is stored.
    Vector<unsigned> U_index_interface;
 
    /// The Data that contains the external  pressure is stored
    /// as external Data for the element. Which external Data item is it?
    /// (int so it can be initialised to -1, indicating that external
    /// pressure hasn't been set).
    int External_data_number_of_external_pressure;
 
    /// Pointer to the Data item that stores the external pressure
    Data* Pext_data_pt;
 
    /// Which of the values in Pext_data_pt stores the external pressure
    unsigned Index_of_external_pressure_value;
 
    /// Access function that returns the local equation number
    /// for the (scalar) kinematic equation associated with the j-th local
    /// node. This must be overloaded by specific interface elements
    /// and depends on the method for handing the free-surface deformation.
    virtual int kinematic_local_eqn(const unsigned& n) = 0;
 
    /// Access function for the local equation number that
    /// corresponds to the external pressure.
    int pext_local_eqn()
    {
#ifdef PARANOID
      if (External_data_number_of_external_pressure < 0)
      {
        throw OomphLibError("No external pressure has been set\n",
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
      return external_local_eqn(External_data_number_of_external_pressure,
                                Index_of_external_pressure_value);
    }
 
    /// Helper function to calculate the residuals and
    /// (if flag==1) the Jacobian of the equations.
    /// This is implemented generically using the surface
    /// divergence information that is overloaded in each element
    /// i.e. axisymmetric, two- or three-dimensional.
    virtual void fill_in_generic_residual_contribution_interface(
      Vector<double>& residuals, DenseMatrix<double>& jacobian, unsigned flag);
 
    /// Compute the surface gradient and surface divergence
    /// operators given the shape functions, derivatives,
    /// tangent vectors and position. All derivatives and
    /// tangent vectors should be formed
    /// with respect to the local coordinates.
    ///
    /// Return the jacobian of the surface, as well
    /// as the dpsidS, and dpsidS_div objects.
    ///
    /// This is the only
    /// function that needs to be overloaded to specify
    /// different geometries.
    ///
    /// In order to compute the surface gradient of a scalar
    /// function one needs only compute the sum over the nodes
    /// of dpsidS(l,i) * nodal_value(l,scalar_index)
    /// To compute the surface divergence of a vector quantity
    /// one computes a sum over nodes and coordinate directions
    /// dpsidS_div(l,i) * nodal_value(l,vector_index[i])
    /// In Cartesian cordinates the two surface derivatives are the
    /// same, but in Axisymmetric coordinates they are not!
    virtual double compute_surface_derivatives(
      const Shape& psi,
      const DShape& dpsids,
      const DenseMatrix<double>& interpolated_t,
      const Vector<double>& interpolated_x,
      DShape& dpsidS,
      DShape& dpsidS_div) = 0;
 
    /// Helper function to calculate the additional contributions
    /// to the resisuals and Jacobian that arise from specific node update
    /// strategies. This is called within the integration loop over the
    /// element (for efficiency) and therefore requires a fairly large
    /// number of input parameters:
    /// - the velocity shape functions and their derivatives w.r.t.
    ///   the local coordinates
    /// - the surface gradient and divergence of the velocity shape
    ///   functions
    /// - The local and Eulerian coordinates,
    /// - the outer unit normal,
    /// - the integration weight from the integration scheme
    /// - the Jacobian of the mapping between the local and global coordinates
    ///   along the element. (Note that in the axisymmmetric case this
    ///   includes the r term)!
    virtual void add_additional_residual_contributions_interface(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      const unsigned& flag,
      const Shape& psif,
      const DShape& dpsifds,
      const DShape& dpsifdS,
      const DShape& dpsifdS_div,
      const Vector<double>& s,
      const Vector<double>& interpolated_x,
      const Vector<double>& interpolated_n,
      const double& W,
      const double& J)
    {
    }
 
  public:
    /// Constructor, set the default values of the booleans and pointers (null)
    FluidInterfaceElement() : Pext_data_pt(0)
    {
      // Initialise pointer to capillary number
      Ca_pt = 0;
 
      // Set the Strouhal number to the default value
      St_pt = &Default_Physical_Constant_Value;
    }
 
    /// Virtual function that specifies the non-dimensional
    ///  surface tension as a function of local position within the element.
    /// The default behaviour is a constant surface tension of value 1.0
    /// This function can be overloaded in more specialised elements to
    /// incorporate variations in surface tension.
    virtual double sigma(const Vector<double>& s_local)
    {
      return 1.0;
    }
 
    /// Calculate the residuals by calling the generic residual contribution.
    void fill_in_contribution_to_residuals(Vector<double>& residuals)
    {
      // Add the residual contributions
      fill_in_generic_residual_contribution_interface(
        residuals, GeneralisedElement::Dummy_matrix, 0);
    }
 
 
    /// The value of the Capillary number
    const double& ca() const
    {
#ifdef PARANOID
      if (Ca_pt != 0)
      {
#endif
        return *Ca_pt;
#ifdef PARANOID
      }
      else
      {
        throw OomphLibError("Capillary number has not been set",
                            "FluidInterfaceElement::ca()",
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
    }
 
    /// Pointer to the Capillary number
    double*& ca_pt()
    {
      return Ca_pt;
    }
 
    /// The value of the Strouhal number
    const double& st() const
    {
      return *St_pt;
    }
 
    /// The pointer to the Strouhal number
    double*& st_pt()
    {
      return St_pt;
    }
 
    /// Return the i-th velocity component at local node j.
    double u(const unsigned& j, const unsigned& i)
    {
      return node_pt(j)->value(U_index_interface[i]);
    }
 
    /// Calculate the i-th velocity component at the local coordinate s.
    double interpolated_u(const Vector<double>& s, const unsigned& i);
 
    /// Return the value of the external pressure
    double pext() const
    {
      // If the external pressure has not been set, then return a
      // default value of zero.
      if (Pext_data_pt == 0)
      {
        return 0.0;
      }
      // Otherwise return the appropriate value
      else
      {
        return Pext_data_pt->value(Index_of_external_pressure_value);
      }
    }
 
    /// Set the Data that contains the single pressure value
    /// that specifies the "external pressure" for the
    /// interface/free-surface. Setting this only makes sense
    /// if the interface is, in fact, a free surface (well,
    /// an interface to another inviscid fluid if you want to be picky).
    void set_external_pressure_data(Data* external_pressure_data_pt)
    {
#ifdef PARANOID
      if (external_pressure_data_pt->nvalue() != 1)
      {
        std::ostringstream error_message;
        error_message
          << "External pressure Data must only contain a single value!\n"
          << "This one contains " << external_pressure_data_pt->nvalue()
          << std::endl;
 
        throw OomphLibError(error_message.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // Store pointer explicitly
      Pext_data_pt = external_pressure_data_pt;
 
      // Add the external pressure to the element's external Data?
      // But do not finite-difference with respect to it
      this->add_external_data(Pext_data_pt, false);
 
      // The external pressure has just been promoted to become
      // external Data of this element -- what is its number?
      External_data_number_of_external_pressure = this->nexternal_data() - 1;
 
      // Index of pressure value in Data object
      Index_of_external_pressure_value = 0;
    }
 
    /// Set the Data that contains the pressure value
    /// that specifies the "external pressure" for the
    /// interface/free-surface. Setting this only makes sense
    /// if the interface is, in fact, a free surface (well,
    /// an interface to another inviscid fluid if you want to be picky).
    /// Second argument specifies the index of the pressure
    /// value within the Data object.
    void set_external_pressure_data(
      Data* external_pressure_data_pt,
      const unsigned& index_of_external_pressure_value)
    {
      // Index of pressure value in Data object
      Index_of_external_pressure_value = index_of_external_pressure_value;
 
#ifdef PARANOID
      if (index_of_external_pressure_value >=
          external_pressure_data_pt->nvalue())
      {
        std::ostringstream error_message;
        error_message << "External pressure Data only contains "
                      << external_pressure_data_pt->nvalue() << " values\n"
                      << "You have declared value "
                      << index_of_external_pressure_value
                      << " to be the value representing the pressure\n"
                      << std::endl;
        throw OomphLibError(error_message.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // Store pointer explicitly
      Pext_data_pt = external_pressure_data_pt;
 
      // Add the external pressure to the element's external Data?
      // But do not finite-difference with respect to it
      this->add_external_data(Pext_data_pt, false);
 
      // The external pressure has just been promoted to become
      // external Data of this element -- what is its number?
      External_data_number_of_external_pressure = this->nexternal_data() - 1;
    }
 
 
    /// Create a bounding element e.g. to apply a contact angle boundary
    /// condition
    virtual FluidInterfaceBoundingElement* make_bounding_element(
      const int& face_index)
    {
      throw OomphLibError("Virtual function not yet implemented",
                          OOMPH_CURRENT_FUNCTION,
                          OOMPH_EXCEPTION_LOCATION);
      return 0;
    }
 
 
    /// Hijack the kinematic condition at the node numbers passed in
    /// the vector. The node numbers correspond to the local numbers of
    /// nodes in the associated bulk element.
    /// This is required so that contact-angle conditions can be applied
    /// by the FluidInterfaceBoundingElements.
    virtual void hijack_kinematic_conditions(
      const Vector<unsigned>& bulk_node_number) = 0;
 
    /// Overload the output function
    void output(std::ostream& outfile)
    {
      FiniteElement::output(outfile);
    }
 
    /// Output function
    void output(std::ostream& outfile, const unsigned& n_plot);
 
    /// Overload the C-style output function
    void output(FILE* file_pt)
    {
      FiniteElement::output(file_pt);
    }
 
    /// C-style Output function
    void output(FILE* file_pt, const unsigned& n_plot);
  };
 
 
  //=============================================================
  /// Class that establishes the surface derivative functions for
  /// LineElements. These are defined in a separate class so that
  /// they can be used by other interface equation-type classes.
  //=============================================================
  class LineDerivatives
  {
  public:
    // Empty Constructor
    LineDerivatives() {}
 
  protected:
    /// Fill in the specific surface derivative calculations
    double compute_surface_derivatives(
      const Shape& psi,
      const DShape& dpsids,
      const DenseMatrix<double>& interpolated_t,
      const Vector<double>& interpolated_x,
      DShape& surface_gradient,
      DShape& surface_divergence);
  };
 
 
  //=============================================================
  /// Class that establishes the surface derivative functions for
  /// AxisymmetricInterfaceElements.
  /// These are defined in a separate class so that
  /// they can be used by other interface equation-type classes.
  //=============================================================
  class AxisymmetricDerivatives
  {
  public:
    // Empty Constructor
    AxisymmetricDerivatives() {}
 
  protected:
    /// Fill in the specific surface derivative calculations
    double compute_surface_derivatives(
      const Shape& psi,
      const DShape& dpsids,
      const DenseMatrix<double>& interpolated_t,
      const Vector<double>& interpolated_x,
      DShape& surface_gradient,
      DShape& surface_divergence);
  };
 
 
  //=============================================================
  /// Class that establishes the surface derivative functions for
  /// SurfaceInterfaceElements (2D surfaces in 3D space)
  /// These are defined in a separate class so that
  /// they can be used by other interface equation-type classes.
  //=============================================================
  class SurfaceDerivatives
  {
  public:
    // Empty Constructor
    SurfaceDerivatives() {}
 
  protected:
    /// Fill in the specific surface derivative calculations
    double compute_surface_derivatives(
      const Shape& psi,
      const DShape& dpsids,
      const DenseMatrix<double>& interpolated_t,
      const Vector<double>& interpolated_x,
      DShape& surface_gradient,
      DShape& surface_divergence);
  };
 
 
} // namespace oomph
 
#endif