axisym_displ_based_fvk_elements.cc

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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) 2006-2025 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.
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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// Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
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// LIC//
// LIC// The authors may be contacted at oomph-lib@maths.man.ac.uk.
// LIC//
// LIC//====================================================================
// Non-inline functions for axisym FoepplvonKarman elements
 
#include "axisym_displ_based_fvk_elements.h"
 
 
namespace oomph
{
  //======================================================================
  /// Set the data for the number of Variables at each node - 3
  //======================================================================
  template<unsigned NNODE_1D>
  const unsigned AxisymFoepplvonKarmanElement<NNODE_1D>::Initial_Nvalue = 3;
 
 
  //======================================================================
  /// Compute contribution to element residual Vector
  ///
  /// Pure version without hanging nodes
  //======================================================================
  void AxisymFoepplvonKarmanEquations::fill_in_contribution_to_residuals(
    Vector<double>& residuals)
  {
    // Find out how many nodes there are
    const unsigned n_node = nnode();
 
    // Set up memory for the shape and test functions
    Shape psi(n_node), test(n_node);
    DShape dpsidr(n_node, 1), dtestdr(n_node, 1);
 
    // Set the value of n_intpt
    const unsigned n_intpt = integral_pt()->nweight();
 
    // Indices at which the unknowns are stored
    const unsigned w_nodal_index = nodal_index_fvk(0);
    const unsigned laplacian_w_nodal_index = nodal_index_fvk(1);
    const unsigned u_r_nodal_index = nodal_index_fvk(2);
 
    // Local copy of parameters
    const double nu_local = nu();
    const double eta_local = eta();
 
    // Integers to store the 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);
 
      // Call the derivatives of the shape and test functions
      double J = dshape_and_dtest_eulerian_at_knot_axisym_fvk(
        ipt, psi, dpsidr, test, dtestdr);
 
      // Allocate and initialise to zero storage for the interpolated values
      double interpolated_r = 0.0;
 
      double interpolated_w = 0.0;
      double interpolated_laplacian_w = 0.0;
      double interpolated_u_r = 0.0;
 
      double interpolated_dwdr = 0.0;
      double interpolated_dlaplacian_wdr = 0.0;
      double interpolated_du_rdr = 0.0;
 
      Vector<double> nodal_value(3, 0.0);
 
      // Calculate function values and derivatives:
      //-----------------------------------------
      // Loop over nodes
      for (unsigned l = 0; l < n_node; l++)
      {
        // Get the nodal values
        nodal_value[0] = raw_nodal_value(l, w_nodal_index);
        nodal_value[1] = raw_nodal_value(l, laplacian_w_nodal_index);
        nodal_value[2] = raw_nodal_value(l, u_r_nodal_index);
 
        // Add contributions from current node/shape function
        interpolated_w += nodal_value[0] * psi(l);
        interpolated_laplacian_w += nodal_value[1] * psi(l);
        interpolated_u_r += nodal_value[2] * psi(l);
 
        interpolated_r += raw_nodal_position(l, 0) * psi(l);
 
        interpolated_dwdr += nodal_value[0] * dpsidr(l, 0);
        interpolated_dlaplacian_wdr += nodal_value[1] * dpsidr(l, 0);
        interpolated_du_rdr += nodal_value[2] * dpsidr(l, 0);
 
      } // End of loop over the nodes
 
      // Premultiply the weights and the Jacobian
      double W = w * interpolated_r * J;
 
      // Get pressure function
      //---------------------
      double pressure = 0.0;
      get_pressure_fvk(ipt, interpolated_r, pressure);
 
      // Determine the stresses
      //-----------------------
 
      double sigma_r_r = 0.0;
      double sigma_phi_phi = 0.0;
 
      if (!Linear_bending_model)
      {
        sigma_r_r =
          1.0 / (1.0 - nu_local * nu_local) *
          (interpolated_du_rdr + 0.5 * interpolated_dwdr * interpolated_dwdr +
           nu_local * 1.0 / interpolated_r * interpolated_u_r);
 
        sigma_phi_phi =
          1.0 / (1.0 - nu_local * nu_local) *
          (1.0 / interpolated_r * interpolated_u_r +
           nu_local * (interpolated_du_rdr +
                       0.5 * interpolated_dwdr * interpolated_dwdr));
      }
      else
      {
        sigma_r_r = 1.0 / (1.0 - nu_local * nu_local) *
                    (interpolated_du_rdr +
                     nu_local * 1.0 / interpolated_r * interpolated_u_r);
 
        sigma_phi_phi = 1.0 / (1.0 - nu_local * nu_local) *
                        (1.0 / interpolated_r * interpolated_u_r +
                         nu_local * interpolated_du_rdr);
      }
 
 
      // Assemble residuals and Jacobian:
      //--------------------------------
      // Loop over the test functions
      for (unsigned l = 0; l < n_node; l++)
      {
        // Get the local equation
        local_eqn = nodal_local_eqn(l, w_nodal_index);
 
        // IF it's not a boundary condition
        if (local_eqn >= 0)
        {
          residuals[local_eqn] +=
            (pressure * test(l) +
             (dtestdr(l, 0)) * interpolated_dlaplacian_wdr) *
            W;
          if (!Linear_bending_model)
          {
            residuals[local_eqn] -=
              eta_local * sigma_r_r * (dtestdr(l, 0)) * interpolated_dwdr * W;
          }
        }
 
        // Get the local equation
        local_eqn = nodal_local_eqn(l, laplacian_w_nodal_index);
 
        // IF it's not a boundary condition
        if (local_eqn >= 0)
        {
          residuals[local_eqn] += (test(l) * interpolated_laplacian_w +
                                   (dtestdr(l, 0)) * interpolated_dwdr) *
                                  W;
        }
 
        // Get the local equation
        local_eqn = nodal_local_eqn(l, u_r_nodal_index);
 
        // IF it's not a boundary condition
        if (local_eqn >= 0)
        {
          residuals[local_eqn] +=
            (sigma_r_r * dtestdr(l, 0) +
             1.0 / interpolated_r * sigma_phi_phi * test(l)) *
            W;
        }
 
      } // End of loop over test functions
    } // End of loop over integration points
  }
 
 
  //======================================================================
  /// Self-test:  Return 0 for OK
  //======================================================================
  unsigned AxisymFoepplvonKarmanEquations::self_test()
  {
    bool passed = true;
 
    // Check lower-level stuff
    if (FiniteElement::self_test() != 0)
    {
      passed = false;
    }
 
    // Return verdict
    if (passed)
    {
      return 0;
    }
    else
    {
      return 1;
    }
  }
 
 
  //======================================================================
  /// Compute in-plane stresses. Return boolean to indicate success
  /// (false if attempt to evaluate stresses at zero radius)
  //======================================================================
  void AxisymFoepplvonKarmanEquations::interpolated_stress(
    const Vector<double>& s, double& sigma_r_r, double& sigma_phi_phi) const
  {
    // No in plane stresses if linear bending
    if (Linear_bending_model)
    {
      sigma_r_r = 0.0;
      sigma_phi_phi = 0.0;
    }
    else
    {
      // Get shape fcts and derivs
      unsigned n_dim = this->dim();
      unsigned n_node = this->nnode();
      Shape psi(n_node);
      DShape dpsidr(n_node, n_dim);
 
      // Get shape fcts and derivs
      dshape_eulerian(s, psi, dpsidr);
 
      // Allocate and initialise to zero storage for the interpolated values
      double interpolated_r = 0.0;
      double interpolated_u_r = 0.0;
 
      double interpolated_dwdr = 0.0;
      double interpolated_du_rdr = 0.0;
 
      double nu_local = nu();
 
 
      // Calculate function values and derivatives:
      //-----------------------------------------
      // Loop over nodes
      for (unsigned l = 0; l < n_node; l++)
      {
        // Add contributions from current node/shape function
        interpolated_r += raw_nodal_position(l, 0) * psi(l);
        interpolated_u_r +=
          this->raw_nodal_value(l, nodal_index_fvk(2)) * psi(l);
        interpolated_dwdr +=
          this->raw_nodal_value(l, nodal_index_fvk(0)) * dpsidr(l, 0);
        interpolated_du_rdr +=
          this->raw_nodal_value(l, nodal_index_fvk(2)) * dpsidr(l, 0);
      } // End of loop over nodes
 
      // The centre stress must use a different analytical form to avoid
      // dividing by zero.  It can be found from the form below by assuming:
      // a) u_r=0 at the origin (required under axisymmetry unless there is a
      // puncture) which leads to the term u_r/r -> du_r/dr as r -> 0 (by
      // L'Hopital's rule), and
      // b) given dw/dr=0 at the origin (required for axisymmetry as there can
      // be no kink in a plate).
      // Note that this process results in s_{rr}=s_{\phi\phi} at the origin
      // which is expected as the two coordinate directions are
      // indistinguishable at the origin under axisymmetry.
      if (interpolated_r == 0.0)
      {
        // Compute the stresses:
        sigma_r_r = interpolated_du_rdr / (1.0 - nu_local);
        sigma_phi_phi = sigma_r_r;
      }
      else
      {
        // Compute the stresses:
        //---------------------
        sigma_r_r =
          1.0 / (1.0 - nu_local * nu_local) *
          (interpolated_du_rdr + 0.5 * interpolated_dwdr * interpolated_dwdr +
           nu_local * 1.0 / interpolated_r * interpolated_u_r);
 
        sigma_phi_phi =
          1.0 / (1.0 - nu_local * nu_local) *
          (1.0 / interpolated_r * interpolated_u_r +
           nu_local * (interpolated_du_rdr +
                       0.5 * interpolated_dwdr * interpolated_dwdr));
      } // End if origin
    } // End if nonlinear bending
 
  } // End of interpolated_stress function
 
  //======================================================================
  /// Output function:
  ///   r, w, u, sigma_r_r, sigma_phi_phi
  /// nplot points
  //======================================================================
  void AxisymFoepplvonKarmanEquations::output(std::ostream& outfile,
                                              const unsigned& nplot)
  {
    // Vector of local coordinates
    Vector<double> s(1);
 
    // Tecplot header info
    outfile << "ZONE\n";
 
    // Loop over plot points
    unsigned num_plot_points = nplot_points(nplot);
    for (unsigned iplot = 0; iplot < num_plot_points; iplot++)
    {
      // Get local coordinates of plot point
      get_s_plot(iplot, nplot, s);
 
      // Get stress
      double sigma_r_r = 0.0;
      double sigma_phi_phi = 0.0;
      interpolated_stress(s, sigma_r_r, sigma_phi_phi);
 
      // Output interpolated global position, displacement and stress
      outfile << interpolated_x(s, 0) << " " << interpolated_w_fvk(s) << " "
              << interpolated_u_fvk(s) << " " << sigma_r_r << " "
              << sigma_phi_phi << std::endl;
    }
  }
 
  //======================================================================
  /// C-style output function:
  ///   r,w,u
  /// nplot points
  //======================================================================
  void AxisymFoepplvonKarmanEquations::output(FILE* file_pt,
                                              const unsigned& nplot)
  {
    // Vector of local coordinates
    Vector<double> s(1);
 
    // Tecplot header info
    fprintf(file_pt, "%s", tecplot_zone_string(nplot).c_str());
 
    // Loop over plot points
    unsigned num_plot_points = nplot_points(nplot);
    for (unsigned iplot = 0; iplot < num_plot_points; iplot++)
    {
      // Get local coordinates of plot point
      get_s_plot(iplot, nplot, s);
 
      fprintf(file_pt, "%g ", interpolated_x(s, 0));
      fprintf(file_pt, "%g \n", interpolated_w_fvk(s));
      fprintf(file_pt, "%g \n", interpolated_u_fvk(s));
    }
 
    // Write tecplot footer (e.g. FE connectivity lists)
    write_tecplot_zone_footer(file_pt, nplot);
  }
 
 
  //======================================================================
  /// Output exact solution
  ///
  /// Solution is provided via function pointer.
  /// Plot at a given number of plot points.
  ///
  ///   r,w_exact
  //======================================================================
  void AxisymFoepplvonKarmanEquations::output_fct(
    std::ostream& outfile,
    const unsigned& nplot,
    FiniteElement::SteadyExactSolutionFctPt exact_soln_pt)
  {
    // Vector of local coordinates
    Vector<double> s(1);
 
    // Vector for coordinates
    Vector<double> r(1);
 
    // Tecplot header info
    outfile << tecplot_zone_string(nplot);
 
    // Exact solution Vector (here a scalar)
    // Vector<double> exact_soln(1);
    Vector<double> exact_soln(1);
 
    // Loop over plot points
    unsigned num_plot_points = nplot_points(nplot);
    for (unsigned iplot = 0; iplot < num_plot_points; iplot++)
    {
      // Get local coordinates of plot point
      get_s_plot(iplot, nplot, s);
 
      // Get r position as Vector
      interpolated_x(s, r);
 
      // Get exact solution at this point
      (*exact_soln_pt)(r, exact_soln);
 
      // Output r,w_exact
      outfile << r[0] << " ";
      outfile << exact_soln[0] << std::endl;
    }
 
    // Write tecplot footer (e.g. FE connectivity lists)
    write_tecplot_zone_footer(outfile, nplot);
  }
 
 
  //======================================================================
  /// Validate against exact solution
  ///
  /// Solution is provided via function pointer.
  /// Plot error at a given number of plot points.
  ///
  //======================================================================
  void AxisymFoepplvonKarmanEquations::compute_error(
    std::ostream& outfile,
    FiniteElement::SteadyExactSolutionFctPt exact_soln_pt,
    double& error,
    double& norm)
  {
    // Initialise
    error = 0.0;
    norm = 0.0;
 
    // Vector of local coordinates
    Vector<double> s(1);
 
    // Vector for coordintes
    Vector<double> r(1);
 
    // Find out how many nodes there are in the element
    unsigned n_node = nnode();
 
    Shape psi(n_node);
 
    // Set the value of n_intpt
    unsigned n_intpt = integral_pt()->nweight();
 
    // Tecplot
    outfile << "ZONE" << std::endl;
 
    // Exact solution Vector (here a scalar)
    // Vector<double> exact_soln(1);
    Vector<double> exact_soln(1);
 
    // Loop over the integration points
    for (unsigned ipt = 0; ipt < n_intpt; ipt++)
    {
      // Assign values of s
      s[0] = integral_pt()->knot(ipt, 0);
 
      // Get the integral weight
      double w = integral_pt()->weight(ipt);
 
      // Get jacobian of mapping
      double J = J_eulerian(s);
 
      // Premultiply the weights and the Jacobian
      double W = w * J;
 
      // Get r position as Vector
      interpolated_x(s, r);
 
      // Get FE function value
      double w_fe = interpolated_w_fvk(s);
 
      // Get exact solution at this point
      (*exact_soln_pt)(r, exact_soln);
 
      // Output r,error
      outfile << r[0] << " ";
      outfile << exact_soln[0] << " " << exact_soln[0] - w_fe << std::endl;
 
      // Add to error and norm
      norm += exact_soln[0] * exact_soln[0] * W;
      error += (exact_soln[0] - w_fe) * (exact_soln[0] - w_fe) * W;
    }
  }
 
 
  //====================================================================
  // Force build of templates
  //====================================================================
  template class AxisymFoepplvonKarmanElement<2>;
  template class AxisymFoepplvonKarmanElement<3>;
  template class AxisymFoepplvonKarmanElement<4>;
 
} // namespace oomph