axisym_solid_traction_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
// LIC// modify it under the terms of the GNU Lesser General Public
// 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//
// LIC// This library is distributed in the hope that it will be useful,
// LIC// but WITHOUT ANY WARRANTY; without even the implied warranty of
// LIC// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
// LIC// Lesser General Public License for more details.
// LIC//
// LIC// You should have received a copy of the GNU Lesser General Public
// LIC// License along with this library; if not, write to the Free Software
// LIC// Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
// LIC// 02110-1301  USA.
// LIC//
// LIC// The authors may be contacted at oomph-lib@maths.man.ac.uk.
// LIC//
// LIC//====================================================================
// Header file for elements that are used to apply surface loads to
// the equations of elasticity
 
#ifndef OOMPH_AXISYMM_SOLID_TRACTION_ELEMENTS_HEADER
#define OOMPH_AXISYMM_SOLID_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/hermite_elements.h"
 
namespace oomph
{
  //======================================================================
  /// A class for elements that allow the imposition of an applied traction
  /// in the principle of virtual displacements.
  /// 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 AxisymmetricSolidTractionElement : 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);
 
  protected:
    /// Return the surface traction force
    void get_traction(const double& time,
                      const Vector<double>& x,
                      const Vector<double>& n,
                      Vector<double>& result) const
    {
      // If the function pointer is zero return zero
      if (Traction_fct_pt == 0)
      {
        // Loop over dimensions and set body forces to zero
        // It's axisymmetric, so we only have "two" dimensions
        for (unsigned i = 0; i < 2; i++)
        {
          result[i] = 0.0;
        }
      }
      // Otherwise call the function
      else
      {
        (*Traction_fct_pt)(time, x, n, result);
      }
    }
 
  public:
    /// Constructor, which takes a "bulk" element and
    /// the value of the index and its limit
    AxisymmetricSolidTractionElement(FiniteElement* const& element_pt,
                                     const int& face_index)
      : 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);
 
      // Set the body force function pointer to zero
      Traction_fct_pt = 0;
    }
 
    /// Return the imposed traction pointer
    void (*&traction_fct_pt())(const double&,
                               const Vector<double>&,
                               const Vector<double>&,
                               Vector<double>&)
    {
      return Traction_fct_pt;
    }
 
    /// Return the residuals
    void fill_in_contribution_to_residuals(Vector<double>& residuals);
 
    /// Return the jacobian
    void fill_in_contribution_to_jacobian(Vector<double>& residuals,
                                          DenseMatrix<double>& jacobian)
    {
      fill_in_contribution_to_residuals(residuals);
      // Call the generic FD jacobian calculation
      FaceGeometry<ELEMENT>::fill_in_jacobian_from_solid_position_by_fd(
        jacobian);
    }
 
    /// 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& n_plot)
    {
      FiniteElement::output(outfile, n_plot);
    }
 
    /// Overload the output function
    void output(FILE* file_pt)
    {
      FiniteElement::output(file_pt);
    }
 
    /// Output function: x,y,[z],u,v,[w],p in tecplot format
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      FiniteElement::output(file_pt, n_plot);
    }
  };
 
 
  /// //////////////////////////////////////////////////////////////////////
  /// //////////////////////////////////////////////////////////////////////
  /// //////////////////////////////////////////////////////////////////////
 
 
  //=======================================================================
  /// Return the residuals for the AxisymmetricSolidTractionElements
  //=======================================================================
  template<class ELEMENT>
  void AxisymmetricSolidTractionElement<
    ELEMENT>::fill_in_contribution_to_residuals(Vector<double>& residuals)
  {
    // Find out how many nodes there are
    unsigned n_node = nnode();
    // Find out how many positional dofs there are
    unsigned n_position_type = nnodal_position_type();
 
    // Integer to hold the local equation number
    int local_eqn = 0;
 
    // Set up memory for the shape functions
    // The surface is 1D, so we only have one local derivative
    Shape psi(n_node, n_position_type);
    DShape dpsids(n_node, n_position_type, 1);
 
    // Set the value of n_intpt
    unsigned n_intpt = integral_pt()->nweight();
 
    // Loop over the integration points
    for (unsigned ipt = 0; ipt < n_intpt; ipt++)
    {
      // Get the integral weight
      double w = integral_pt()->weight(ipt);
 
      // Only need to call the local derivatives
      dshape_local_at_knot(ipt, psi, dpsids);
 
      // Calculate the global position and lagrangian coordinate
      Vector<double> interpolated_x(2, 0.0), interpolated_xi(2, 0.0);
      // Calculate the global and lagrangian derivtives wrt the local
      // coordinates
      Vector<double> interpolated_dxds(2, 0.0), interpolated_dxids(2, 0.0);
 
      // Calculate displacements and derivatives
      for (unsigned l = 0; l < n_node; l++)
      {
        // Loop over positional dofs
        for (unsigned k = 0; k < n_position_type; k++)
        {
          // Loop over the number of lagrangian coordinates (2)
          for (unsigned i = 0; i < 2; i++)
          {
            // Calculate the global position
            interpolated_x[i] +=
              nodal_position_gen(l, bulk_position_type(k), i) * psi(l, k);
            interpolated_xi[i] +=
              this->lagrangian_position_gen(l, bulk_position_type(k), i) *
              psi(l, k);
            // Calculate the derivatives of the global and lagrangian
            // coordinates
            interpolated_dxds[i] +=
              nodal_position_gen(l, bulk_position_type(k), i) * dpsids(l, k, 0);
            interpolated_dxids[i] +=
              this->lagrangian_position_gen(l, bulk_position_type(k), i) *
              dpsids(l, k, 0);
          }
        }
      }
 
      // Now calculate the entries of the deformed surface metric tensor
      // Now find the local deformed metric tensor from the tangent Vectors
      DenseMatrix<double> A(2);
      // The off-diagonal terms are Zero
      A(0, 1) = A(1, 0) = 0.0;
      // The diagonal terms are a little complicated
      A(0, 0) =
        (interpolated_dxds[0] - interpolated_x[1] * interpolated_dxids[1]) *
          (interpolated_dxds[0] - interpolated_x[1] * interpolated_dxids[1]) +
        (interpolated_dxds[1] + interpolated_x[0] * interpolated_dxids[1]) *
          (interpolated_dxds[1] + interpolated_x[0] * interpolated_dxids[1]);
 
 
      A(1, 1) = (interpolated_x[0] * sin(interpolated_xi[1]) +
                 interpolated_x[1] * cos(interpolated_xi[1])) *
                (interpolated_x[0] * sin(interpolated_xi[1]) +
                 interpolated_x[1] * cos(interpolated_xi[1]));
 
      // Premultiply the weights and the square-root of the determinant of
      // the metric tensor
      double W = w * sqrt(A(0, 0) * A(1, 1));
 
      // Also find the normal -- just the cross product of the metric tensors
      // but I want to express it in terms of e_r and e_theta components
      // N.B. There is an issue at theta = 0,pi, where the normal is e_{r},
      // but given that I never assemble it, should be OK!
      // The minus sign is chosen to ensure that the normal is really outward
      Vector<double> interpolated_normal(2);
      // Component in the e_{r} direction
      interpolated_normal[0] =
        -1.0 *
        (interpolated_x[0] * sin(interpolated_xi[1]) +
         interpolated_x[1] * cos(interpolated_xi[1])) *
        (interpolated_dxds[1] + interpolated_x[0] * interpolated_dxids[1]);
      // Component in the e_{theta} direction
      interpolated_normal[1] =
        -1.0 *
        (interpolated_x[0] * sin(interpolated_xi[1]) +
         interpolated_x[1] * cos(interpolated_xi[1])) *
        (interpolated_x[1] * interpolated_dxids[1] - interpolated_dxds[0]);
 
      // If we're on the north or south face need to flip normal
      if (s_fixed_value() == -1)
      {
        interpolated_normal[0] *= -1.0;
        interpolated_normal[1] *= -1.0;
      }
 
      // Now adjust and scale the normal
      double length = 0.0;
      for (unsigned i = 0; i < 2; i++)
      {
        interpolated_normal[i] *= normal_sign();
        length += interpolated_normal[i] * interpolated_normal[i];
      }
      for (unsigned i = 0; i < 2; i++)
      {
        interpolated_normal[i] /= sqrt(length);
      }
 
      // Now calculate the load
      Vector<double> traction(2);
 
      // Normal is outwards
      get_traction(time(), interpolated_x, interpolated_normal, traction);
 
      //=====LOAD TERMS  FROM PRINCIPLE OF VIRTUAL DISPLACEMENTS========
 
      // Loop over the test functions, nodes of the element
      for (unsigned l = 0; l < n_node; l++)
      {
        // Loop of types of dofs
        for (unsigned k = 0; k < n_position_type; k++)
        {
          // Loop over the displacement components
          for (unsigned i = 0; i < 2; i++)
          {
            local_eqn = this->position_local_eqn(l, bulk_position_type(k), i);
            /*IF it's not a boundary condition*/
            if (local_eqn >= 0)
            {
              // Add the loading terms to the residuals
              residuals[local_eqn] -= traction[i] * psi(l, k) * W;
            }
          }
        } // End of if not boundary condition
      } // End of loop over shape functions
    } // End of loop over integration points
  }
 
 
} // namespace oomph
 
#endif