mixed_order_petrov_galerkin_space_time_fluid_traction_elements.h

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// Header file for elements that are used to integrate fluid tractions
// This includes the guts (i.e. equations) because we want to inline them
// for faster operation, although it slows down the compilation!
#ifndef OOMPH_MIXED_ORDER_PETROV_GALERKIN_SPACE_TIME_FLUID_TRACTION_ELEMENTS_HEADER
#define OOMPH_MIXED_ORDER_PETROV_GALERKIN_SPACE_TIME_FLUID_TRACTION_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/Telements.h"
 
namespace oomph
{
  //======================================================================
  /// A class for elements that allow the imposition of an applied traction
  /// to the Navier--Stokes equations
  /// The geometrical information can be read from the FaceGeometry<ELEMENT>
  /// class and and thus, we can be generic enough without the need to have
  /// a separate equations class
  //======================================================================
  template<class ELEMENT>
  class NavierStokesMixedOrderSpaceTimeTractionElement
    : public virtual FaceGeometry<ELEMENT>,
      public virtual FaceElement
  {
  private:
    /// Pointer to an imposed traction function
    void (*Traction_fct_pt)(const double& time,
                            const Vector<double>& x,
                            const Vector<double>& n,
                            Vector<double>& result);
 
  public:
    /// Constructor, which takes a "bulk" element and the value of the index
    /// and its limit
    NavierStokesMixedOrderSpaceTimeTractionElement(
      FiniteElement* const& element_pt,
      const int& face_index,
      const bool& called_from_refineable_constructor = false)
      : FaceGeometry<ELEMENT>(), FaceElement()
    {
      // Attach the geometrical information to the element. N.B. This function
      // also assigns nbulk_value from the required_nvalue of the bulk element
      element_pt->build_face_element(face_index, this);
 
#ifdef PARANOID
      {
        // Check that the element is not a refineable 3D element
        if (!called_from_refineable_constructor)
        {
          // If it's 3D
          if (element_pt->dim() == 3)
          {
            // Is it refineable
            RefineableElement* ref_el_pt =
              dynamic_cast<RefineableElement*>(element_pt);
 
            // If the upcast was successful
            if (ref_el_pt != 0)
            {
              // If there are any hanging nodes
              if (this->has_hanging_nodes())
              {
                // Create an output stream
                std::ostringstream error_message_stream;
 
                // Create an error message
                error_message_stream
                  << "This flux element will not work correctly "
                  << "if nodes are hanging" << std::endl;
 
                // Throw the error message
                throw OomphLibError(error_message_stream.str(),
                                    OOMPH_CURRENT_FUNCTION,
                                    OOMPH_EXCEPTION_LOCATION);
              }
            } // if (ref_el_pt!=0)
          } // if (element_pt->dim()==3)
        } // if (!called_from_refineable_constructor)
      }
#endif
 
      // Set the body force function pointer to zero
      Traction_fct_pt = 0;
 
      // Set the dimension from the dimension of the first node
      Dim = this->node_pt(0)->ndim();
    } // End of NavierStokesMixedOrderSpaceTimeTractionElement
 
 
    /// Destructor should not delete anything
    ~NavierStokesMixedOrderSpaceTimeTractionElement() {}
 
 
    /// Access function for the imposed traction pointer
    void (*&traction_fct_pt())(const double& t,
                               const Vector<double>& x,
                               const Vector<double>& n,
                               Vector<double>& result)
    {
      // Return the function pointer
      return Traction_fct_pt;
    } // End of traction_fct_pt
 
 
    /// This function returns just the residuals
    inline void fill_in_contribution_to_residuals(Vector<double>& residuals)
    {
      // Call the generic residuals function with flag set to 0
      // using a dummy matrix argument
      fill_in_generic_residual_contribution_fluid_traction(
        residuals, GeneralisedElement::Dummy_matrix, 0);
    }
 
 
    /// This function returns the residuals and the jacobian
    inline void fill_in_contribution_to_jacobian(Vector<double>& residuals,
                                                 DenseMatrix<double>& jacobian)
    {
      // Call the generic routine with the flag set to 1
      fill_in_generic_residual_contribution_fluid_traction(
        residuals, jacobian, 1);
    }
 
 
    /// Overload the output function
    void output(std::ostream& outfile)
    {
      FiniteElement::output(outfile);
    }
 
 
    /// Output function: x,y,[z],u,v,[w],p in tecplot format
    void output(std::ostream& outfile, const unsigned& nplot)
    {
      FiniteElement::output(outfile, nplot);
    }
 
  protected:
    /// This function returns the residuals for the traction function.
    /// flag=1(or 0): do (or don't) compute the Jacobian as well.
    void fill_in_generic_residual_contribution_fluid_traction(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      const unsigned& flag);
 
 
    /// The "global" intrinsic coordinate of the element when
    /// viewed as part of a geometric object should be given by
    /// the FaceSpaceTimeElement representation, by default
    double zeta_nodal(const unsigned& n,
                      const unsigned& k,
                      const unsigned& i) const
    {
      return FaceElement::zeta_nodal(n, k, i);
    } // End of zeta_nodal
 
 
    /// Access function that returns the local equation numbers
    /// for velocity components.
    /// u_local_eqn(n,i)=local equation number or < 0 if pinned.
    /// The default is to asssume that n is the local node number
    /// and the i-th velocity component is the i-th unknown stored at the node.
    virtual inline int u_local_eqn(const unsigned& n, const unsigned& i)
    {
      return nodal_local_eqn(n, i);
    }
 
    /// Function to compute the shape and test functions and to return
    /// the Jacobian of mapping
    inline double shape_and_test_at_knot(const unsigned& ipt,
                                         Shape& psi,
                                         Shape& test) const
    {
      // Number of nodes in each direction
      const unsigned nnode_1d = 3;
 
      // Find the dimension of the element
      const unsigned el_dim = dim();
 
      // Sanity check
      if (el_dim != 2)
      {
        // Throw an error
        throw OomphLibError(
          "Can only deal with 2D space-time traction elements!",
          OOMPH_CURRENT_FUNCTION,
          OOMPH_EXCEPTION_LOCATION);
      }
 
      // Storage for the local coordinates of the integration point
      Vector<double> s(el_dim);
 
      // Set the local coordinate
      for (unsigned i = 0; i < el_dim; i++)
      {
        // Get the i-th local coordinate associated with the ipt-th knot point
        s[i] = integral_pt()->knot(ipt, i);
      }
 
      //--------------------------
      // Call the shape functions:
      //--------------------------
      //--------start_of_shape------------------------------------------------
      // Local storage
      double psi_values[2][nnode_1d];
 
      // Call the OneDimensional shape function (for x or y) with quadratic
      // interpolation
      OneDimLagrange::shape<3>(s[0], psi_values[0]);
 
      // Call the OneDimensional shape function. NOTE: The second local
      // coordinate will always be time for a space-time element!
      OneDimDiscontinuousGalerkinMixedOrderBasis::shape<3>(s[1], psi_values[1]);
 
      // Index for the shape functions
      unsigned index = 0;
 
      // Loop over the first local coordinate direction
      for (unsigned j = 0; j < nnode_1d; j++)
      {
        // Loop over the first local coordinate direction
        for (unsigned i = 0; i < nnode_1d; i++)
        {
          // Multiply the two 1D functions together to get the 2D function
          psi[index] = psi_values[0][i] * psi_values[1][j];
 
          // Increment the index
          index++;
        }
      } // for (unsigned j=0;j<nnode_1d;j++)
      //--------end_of_shape--------------------------------------------------
 
      //-------------------------
      // Call the test functions:
      //-------------------------
      //--------start_of_shape------------------------------------------------
      // Local storage
      double test_values[2][nnode_1d];
 
      // Call the OneDimensional shape function (for x or y) with quadratic
      // interpolation
      OneDimLagrange::shape<3>(s[0], test_values[0]);
 
      // Call the OneDimensional shape function. NOTE: The second local
      // coordinate will always be time for a space-time element!
      OneDimDiscontinuousGalerkinMixedOrderTest::shape<3>(s[1], test_values[1]);
 
      // Reset the value of index to loop over the test functions
      index = 0;
 
      // Loop over the first local coordinate direction
      for (unsigned j = 0; j < nnode_1d; j++)
      {
        // Loop over the first local coordinate direction
        for (unsigned i = 0; i < nnode_1d; i++)
        {
          // Multiply the two 1D functions together to get the 2D function
          test[index] = test_values[0][i] * test_values[1][j];
 
          // Increment the index
          index++;
        }
      } // for (unsigned j=0;j<nnode_1d;j++)
      //--------end_of_shape--------------------------------------------------
 
      // Return the value of the jacobian
      return J_eulerian_at_knot(ipt);
    } // End of shape_and_test_at_knot
 
 
    /// Function to calculate the traction applied to the fluid
    void get_traction(const double& time,
                      const Vector<double>& x,
                      const Vector<double>& n,
                      Vector<double>& result)
    {
      // If the function pointer is zero return zero
      if (Traction_fct_pt == 0)
      {
        // Loop over the (spatial) dimensions
        for (unsigned i = 0; i < Dim - 1; i++)
        {
          // Set the i-th traction component to zero
          result[i] = 0.0;
        }
      }
      // If the traction function pointer has been set
      else
      {
        // Call the function pointer
        (*Traction_fct_pt)(time, x, n, result);
      } // if (Traction_fct_pt==0)
    } // End of get_traction
 
 
    /// The highest dimension of the problem
    unsigned Dim;
  };
 
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
 
  //=========================================================================
  /// Function that returns the residuals for the imposed traction
  /// Navier-Stokes equations
  //=========================================================================
  template<class ELEMENT>
  void NavierStokesMixedOrderSpaceTimeTractionElement<ELEMENT>::
    fill_in_generic_residual_contribution_fluid_traction(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      const unsigned& flag)
  {
    // Find out how many nodes there are
    unsigned n_node = nnode();
 
    // Set up memory for the shape functions
    Shape psif(n_node);
 
    // Set up memory for the test functions
    Shape testf(n_node);
 
    // Number of integration points
    unsigned n_intpt = integral_pt()->nweight();
 
    // Integers to store local equation numbers
    int local_eqn = 0;
 
    // Loop over the integration points
    for (unsigned ipt = 0; ipt < n_intpt; ipt++)
    {
      // Get the integral weight
      double w = integral_pt()->weight(ipt);
 
      // Find the shape and test functions and return the Jacobian
      // of the mapping
      double J = shape_and_test_at_knot(ipt, psif, testf);
 
      // Premultiply the weights and the Jacobian
      double W = w * J;
 
      // Get continuous time
      double interpolated_t = 0.0;
 
      // Need to find position to feed into traction function
      Vector<double> interpolated_x(Dim - 1, 0.0);
 
      // Allocate space for the outer unit normal
      Vector<double> interpolated_spacetime_n(Dim, 0.0);
 
      // Allocate space for the outer unit normal (in the given time slice)
      Vector<double> interpolated_n(Dim - 1, 0.0);
 
      // Allocate space for the traction vector
      Vector<double> traction(Dim - 1, 0.0);
 
      // Calculate velocities and derivatives
      for (unsigned l = 0; l < n_node; l++)
      {
        // Update the interpolated time value
        interpolated_t += raw_nodal_position(l, Dim - 1) * psif(l);
 
        // Loop over velocity components
        for (unsigned i = 0; i < Dim - 1; i++)
        {
          // Update the i-th interpolated coordinate value
          interpolated_x[i] += nodal_position(l, i) * psif(l);
        }
      } // for (unsigned l=0;l<n_node;l++)
 
      // Calculate the outer unit normal
      outer_unit_normal(ipt, interpolated_spacetime_n);
 
#ifdef PARANOID
      // Sanity check: Make sure the normal component in the time-direction is
      // zero. If it isn't then something has gone wrong...
      if (std::fabs(interpolated_spacetime_n[Dim - 1]) > 1.0e-15)
      {
        // Create an output stream
        std::ostringstream error_message_stream;
 
        // Create the error message
        error_message_stream
          << "The component of the outer unit normal in the "
          << "time-direction\nshould be zero but the actual "
          << "value is: " << interpolated_spacetime_n[Dim - 1] << std::endl;
 
        // Throw the error message
        throw OomphLibError(error_message_stream.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // Loop over spatial coordinates
      for (unsigned i = 0; i < Dim - 1; i++)
      {
        // Copy the i-th normal component over
        interpolated_n[i] = interpolated_spacetime_n[i];
      }
 
      // Get the user-defined traction
      get_traction(interpolated_t, interpolated_x, interpolated_n, traction);
 
      // Now add to the appropriate equations:
 
      // Loop over the test functions
      for (unsigned l = 0; l < n_node; l++)
      {
        // Loop over the velocity components
        for (unsigned i = 0; i < Dim - 1; i++)
        {
          // Get the local equation number associated with this unknown and node
          local_eqn = u_local_eqn(l, i);
 
          // If it's not a boundary condition
          if (local_eqn >= 0)
          {
            // Add the user-defined traction terms
            residuals[local_eqn] += traction[i] * testf[l] * W;
 
            // Assuming the traction DOES NOT depend upon velocities
            // or pressures, the jacobian is always zero, so no jacobian
            // terms are required
          }
        } // for (unsigned i=0;i<Dim;i++)
      } // for (unsigned l=0;l<n_node;l++)
    } // for (unsigned ipt=0;ipt<n_intpt;ipt++)
  } // End of fill_in_generic_residual_contribution_fluid_traction
 
  /// //////////////////////////////////////////////////////////////////////
  /// //////////////////////////////////////////////////////////////////////
  /// //////////////////////////////////////////////////////////////////////
 
  //======================================================================
  /// A class for elements that allow the imposition of an applied traction
  /// to the Navier--Stokes equations
  /// The geometrical information can be read from the FaceGeometry<ELEMENT>
  /// class and and thus, we can be generic enough without the need to have
  /// a separate equations class.
  ///
  /// THIS IS THE REFINEABLE VERSION.
  //======================================================================
  template<class ELEMENT>
  class RefineableNavierStokesMixedOrderSpaceTimeTractionElement
    : public virtual NavierStokesMixedOrderSpaceTimeTractionElement<ELEMENT>,
      public virtual NonRefineableElementWithHangingNodes
  {
  public:
    /// Constructor, which takes a "bulk" element and the face index
    RefineableNavierStokesMixedOrderSpaceTimeTractionElement(
      FiniteElement* const& element_pt, const int& face_index)
      : // we're calling this from the constructor of the refineable version.
        NavierStokesMixedOrderSpaceTimeTractionElement<ELEMENT>(
          element_pt, face_index, true)
    {
    }
 
 
    /// Destructor should not delete anything
    ~RefineableNavierStokesMixedOrderSpaceTimeTractionElement() {}
 
 
    /// Number of continuously interpolated values are the
    /// same as those in the bulk element.
    unsigned ncont_interpolated_values() const
    {
      return dynamic_cast<ELEMENT*>(this->bulk_element_pt())
        ->ncont_interpolated_values();
    }
 
    /// This function returns just the residuals
    inline void fill_in_contribution_to_residuals(Vector<double>& residuals)
    {
      // Call the generic residuals function using a dummy matrix argument
      refineable_fill_in_generic_residual_contribution_fluid_traction(
        residuals, GeneralisedElement::Dummy_matrix, 0);
    }
 
    /// This function returns the residuals and the Jacobian
    inline void fill_in_contribution_to_jacobian(Vector<double>& residuals,
                                                 DenseMatrix<double>& jacobian)
    {
      // Call the generic routine
      refineable_fill_in_generic_residual_contribution_fluid_traction(
        residuals, jacobian, 1);
    }
 
 
  protected:
    /// This function returns the residuals for the traction function.
    /// flag=1(or 0): do (or don't) compute the Jacobian as well.
    void refineable_fill_in_generic_residual_contribution_fluid_traction(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      const unsigned& flag);
  };
 
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
  /// ////////////////////////////////////////////////////////////////////
 
  //============================================================================
  /// Function that returns the residuals for the imposed traction Navier_Stokes
  /// equations
  //============================================================================
  template<class ELEMENT>
  void RefineableNavierStokesMixedOrderSpaceTimeTractionElement<ELEMENT>::
    refineable_fill_in_generic_residual_contribution_fluid_traction(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      const unsigned& flag)
  {
    // Get the indices at which the velocity components are stored
    unsigned u_nodal_index[this->Dim - 1];
 
    // Loop over the velocity components
    for (unsigned i = 0; i < this->Dim - 1; i++)
    {
      // Get the i-th velocity component index
      u_nodal_index[i] =
        dynamic_cast<ELEMENT*>(this->bulk_element_pt())->u_index_nst(i);
    }
 
    // Find out how many nodes there are
    unsigned n_node = nnode();
 
    // Set up memory for the shape functions
    Shape psif(n_node);
 
    // Set up memory for the test functions
    Shape testf(n_node);
 
    // Number of integration points
    unsigned n_intpt = integral_pt()->nweight();
 
    // Integers to store local equation numbers
    int local_eqn = 0;
 
    // Loop over the integration points
    for (unsigned ipt = 0; ipt < n_intpt; ipt++)
    {
      // Get the integral weight
      double w = integral_pt()->weight(ipt);
 
      // Find the shape and test functions and return the Jacobian
      // of the mapping
      double J = this->shape_and_test_at_knot(ipt, psif, testf);
 
      // Premultiply the weights and the Jacobian
      double W = w * J;
 
      // Get continuous time
      double interpolated_t = 0.0;
 
      // Need to find position to feed into traction function
      Vector<double> interpolated_x(this->Dim - 1, 0.0);
 
      // Allocate space for the outer unit normal
      Vector<double> interpolated_spacetime_n(this->Dim, 0.0);
 
      // Allocate space for the outer unit normal (in the given time slice)
      Vector<double> interpolated_n(this->Dim - 1, 0.0);
 
      // Allocate space for the traction vector
      Vector<double> traction(this->Dim - 1, 0.0);
 
      // Calculate velocities and derivatives
      for (unsigned l = 0; l < n_node; l++)
      {
        // Update the interpolated time value
        interpolated_t += raw_nodal_position(l, this->Dim - 1) * psif(l);
 
        // Loop over velocity components
        for (unsigned i = 0; i < this->Dim - 1; i++)
        {
          // Update the i-th interpolated coordinate value
          interpolated_x[i] += this->nodal_position(l, i) * psif(l);
        }
      } // for (unsigned l=0;l<n_node;l++)
 
      // Calculate the outer unit normal
      this->outer_unit_normal(ipt, interpolated_spacetime_n);
 
#ifdef PARANOID
      // Sanity check: Make sure the normal component in the time-direction is
      // zero. If it isn't then something has gone wrong...
      if (std::fabs(interpolated_n[this->Dim - 1]) > 1.0e-15)
      {
        // Create an output stream
        std::ostringstream error_message_stream;
 
        // Create the error message
        error_message_stream << "The component of the outer unit normal in the "
                             << "time-direction\nshould be zero but the actual "
                             << "value is: " << interpolated_n[this->Dim - 1]
                             << std::endl;
 
        // Throw the error message
        throw OomphLibError(error_message_stream.str(),
                            OOMPH_CURRENT_FUNCTION,
                            OOMPH_EXCEPTION_LOCATION);
      }
#endif
 
      // Loop over spatial coordinates
      for (unsigned i = 0; i < this->Dim - 1; i++)
      {
        // Copy the i-th normal component over
        interpolated_n[i] = interpolated_spacetime_n[i];
      }
 
      // Get the user-defined traction
      this->get_traction(
        interpolated_t, interpolated_x, interpolated_n, traction);
 
      // Now add to the appropriate equations:
 
      // Number of master nodes and storage for the weight of the shape function
      unsigned n_master = 1;
 
      // The weight of the shape function
      double hang_weight = 1.0;
 
      // Pointer to hang info object
      HangInfo* hang_info_pt = 0;
 
      // Loop over the nodes for the test functions/equations
      //-----------------------------------------------------
      for (unsigned l = 0; l < n_node; l++)
      {
        // Local boolean to indicate whether the node is hanging
        bool is_node_hanging = this->node_pt(l)->is_hanging();
 
        // If the node is hanging
        if (is_node_hanging)
        {
          // Get the HangInfo pointer associated with this hanging node
          hang_info_pt = this->node_pt(l)->hanging_pt();
 
          // Read out number of master nodes from hanging data
          n_master = hang_info_pt->nmaster();
        }
        // Otherwise the node is its own master
        else
        {
          // If it's not hanging then it is its (one and only) master node
          n_master = 1;
        }
 
        // Loop over the master nodes
        for (unsigned m = 0; m < n_master; m++)
        {
          // Loop over velocity components for equations
          for (unsigned i = 0; i < this->Dim - 1; i++)
          {
            // Get the equation number:
 
            // If the node is hanging
            if (is_node_hanging)
            {
              // Get the equation number from the master node
              local_eqn = this->local_hang_eqn(hang_info_pt->master_node_pt(m),
                                               u_nodal_index[i]);
 
              // Get the hang weight from the master node
              hang_weight = hang_info_pt->master_weight(m);
            }
            // If the node is not hanging
            else
            {
              // Local equation number
              local_eqn = this->nodal_local_eqn(l, u_nodal_index[i]);
 
              // Node contributes with full weight
              hang_weight = 1.0;
            }
 
            // If it's not a boundary condition...
            if (local_eqn >= 0)
            {
              // Add the user-defined traction terms
              residuals[local_eqn] += traction[i] * testf[l] * W * hang_weight;
 
              // Assuming the the traction DOES NOT depend upon velocities
              // or pressures, the jacobian is always zero, so no jacobian
              // terms are required.
            }
          } // for (unsigned i=0;i<this->Dim-1;i++)
        } // for (unsigned m=0;m<n_master;m++)
      } // for (unsigned l=0;l<n_node;l++)
    } // for (unsigned ipt=0;ipt<n_intpt;ipt++)
  } // End of refineable_fill_in_generic_residual_contribution_fluid_traction
} // End of namespace oomph
 
#endif