poisson_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) 2006-2024 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//====================================================================
// Non-inline functions for Poisson elements
#include "poisson_elements.h"
 
 
namespace oomph
{
  //======================================================================
  /// Set the data for the number of Variables at each node, always one
  /// in every case
  //======================================================================
  template<unsigned DIM, unsigned NNODE_1D>
  const unsigned QPoissonElement<DIM, NNODE_1D>::Initial_Nvalue = 1;
 
 
  //======================================================================
  /// Compute element residual Vector and/or element Jacobian matrix
  ///
  /// flag=1: compute both
  /// flag=0: compute only residual Vector
  ///
  /// Pure version without hanging nodes
  //======================================================================
  template<unsigned DIM>
  void PoissonEquations<DIM>::fill_in_generic_residual_contribution_poisson(
    Vector<double>& residuals,
    DenseMatrix<double>& jacobian,
    const unsigned& flag)
  {
    // 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 dpsidx(n_node, DIM), dtestdx(n_node, DIM);
 
    // Index at which the poisson unknown is stored
    const unsigned u_nodal_index = u_index_poisson();
 
    // Set the value of n_intpt
    const unsigned n_intpt = integral_pt()->nweight();
 
    // Integers to store the local equation and unknown numbers
    int local_eqn = 0, local_unknown = 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_poisson(
        ipt, psi, dpsidx, test, dtestdx);
 
      // Premultiply the weights and the Jacobian
      double W = w * J;
 
      // Calculate local values of unknown
      // Allocate and initialise to zero
      double interpolated_u = 0.0;
      Vector<double> interpolated_x(DIM, 0.0);
      Vector<double> interpolated_dudx(DIM, 0.0);
 
      // Calculate function value and derivatives:
      //-----------------------------------------
      // Loop over nodes
      for (unsigned l = 0; l < n_node; l++)
      {
        // Get the nodal value of the poisson unknown
        double u_value = raw_nodal_value(l, u_nodal_index);
        interpolated_u += u_value * psi(l);
        // Loop over directions
        for (unsigned j = 0; j < DIM; j++)
        {
          interpolated_x[j] += raw_nodal_position(l, j) * psi(l);
          interpolated_dudx[j] += u_value * dpsidx(l, j);
        }
      }
 
 
      // Get source function
      //-------------------
      double source;
      get_source_poisson(ipt, interpolated_x, source);
 
      // 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, u_nodal_index);
        // IF it's not a boundary condition
        if (local_eqn >= 0)
        {
          // Add body force/source term here
          residuals[local_eqn] += source * test(l) * W;
 
          // The Poisson bit itself
          for (unsigned k = 0; k < DIM; k++)
          {
            residuals[local_eqn] += interpolated_dudx[k] * dtestdx(l, k) * W;
          }
 
          // Calculate the jacobian
          //-----------------------
          if (flag)
          {
            // Loop over the velocity shape functions again
            for (unsigned l2 = 0; l2 < n_node; l2++)
            {
              local_unknown = nodal_local_eqn(l2, u_nodal_index);
              // If at a non-zero degree of freedom add in the entry
              if (local_unknown >= 0)
              {
                // Add contribution to Elemental Matrix
                for (unsigned i = 0; i < DIM; i++)
                {
                  jacobian(local_eqn, local_unknown) +=
                    dpsidx(l2, i) * dtestdx(l, i) * W;
                }
              }
            }
          }
        }
      }
 
    } // End of loop over integration points
  }
 
 
  //======================================================================
  /// Compute derivatives of elemental residual vector with respect
  /// to nodal coordinates (fully analytical).
  /// dresidual_dnodal_coordinates(l,i,j) = d res(l) / dX_{ij}
  /// Overloads the FD-based version in the FE base class.
  //======================================================================
  template<unsigned DIM>
  void PoissonEquations<DIM>::get_dresidual_dnodal_coordinates(
    RankThreeTensor<double>& dresidual_dnodal_coordinates)
  {
    // Determine number of nodes in element
    const unsigned n_node = nnode();
 
    // Set up memory for the shape and test functions
    Shape psi(n_node), test(n_node);
    DShape dpsidx(n_node, DIM), dtestdx(n_node, DIM);
 
    // Deriatives of shape fct derivatives w.r.t. nodal coords
    RankFourTensor<double> d_dpsidx_dX(DIM, n_node, n_node, DIM);
    RankFourTensor<double> d_dtestdx_dX(DIM, n_node, n_node, DIM);
 
    // Derivative of Jacobian of mapping w.r.t. to nodal coords
    DenseMatrix<double> dJ_dX(DIM, n_node);
 
    // Derivatives of derivative of u w.r.t. nodal coords
    RankThreeTensor<double> d_dudx_dX(DIM, n_node, DIM);
 
    // Source function and its gradient
    double source;
    Vector<double> d_source_dx(DIM);
 
    // Index at which the poisson unknown is stored
    const unsigned u_nodal_index = u_index_poisson();
 
    // Determine the number of integration points
    const unsigned n_intpt = integral_pt()->nweight();
 
    // Integer to store the local equation number
    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/test functions, as well as the
      // derivatives of these w.r.t. nodal coordinates and the derivative
      // of the jacobian of the mapping w.r.t. nodal coordinates
      const double J = dshape_and_dtest_eulerian_at_knot_poisson(
        ipt, psi, dpsidx, d_dpsidx_dX, test, dtestdx, d_dtestdx_dX, dJ_dX);
 
      // Calculate local values
      // Allocate and initialise to zero
      Vector<double> interpolated_x(DIM, 0.0);
      Vector<double> interpolated_dudx(DIM, 0.0);
 
      // Calculate function value and derivatives:
      // -----------------------------------------
      // Loop over nodes
      for (unsigned l = 0; l < n_node; l++)
      {
        // Get the nodal value of the Poisson unknown
        double u_value = raw_nodal_value(l, u_nodal_index);
 
        // Loop over directions
        for (unsigned i = 0; i < DIM; i++)
        {
          interpolated_x[i] += raw_nodal_position(l, i) * psi(l);
          interpolated_dudx[i] += u_value * dpsidx(l, i);
        }
      }
 
      // Calculate derivative of du/dx_i w.r.t. nodal positions X_{pq}
      for (unsigned q = 0; q < n_node; q++)
      {
        // Loop over coordinate directions
        for (unsigned p = 0; p < DIM; p++)
        {
          for (unsigned i = 0; i < DIM; i++)
          {
            double aux = 0.0;
            for (unsigned j = 0; j < n_node; j++)
            {
              aux +=
                raw_nodal_value(j, u_nodal_index) * d_dpsidx_dX(p, q, j, i);
            }
            d_dudx_dX(p, q, i) = aux;
          }
        }
      }
 
      // Get source function
      get_source_poisson(ipt, interpolated_x, source);
 
      // Get gradient of source function
      get_source_gradient_poisson(ipt, interpolated_x, d_source_dx);
 
      // Assemble d res_{local_eqn} / d X_{pq}
      // -------------------------------------
 
      // Loop over the test functions
      for (unsigned l = 0; l < n_node; l++)
      {
        // Get the local equation
        local_eqn = nodal_local_eqn(l, u_nodal_index);
 
        // IF it's not a boundary condition
        if (local_eqn >= 0)
        {
          // Loop over coordinate directions
          for (unsigned p = 0; p < DIM; p++)
          {
            // Loop over nodes
            for (unsigned q = 0; q < n_node; q++)
            {
              double sum = source * test(l) * dJ_dX(p, q) +
                           d_source_dx[p] * test(l) * psi(q) * J;
 
              for (unsigned i = 0; i < DIM; i++)
              {
                sum += interpolated_dudx[i] * (dtestdx(l, i) * dJ_dX(p, q) +
                                               d_dtestdx_dX(p, q, l, i) * J) +
                       d_dudx_dX(p, q, i) * dtestdx(l, i) * J;
              }
 
              // Multiply through by integration weight
              dresidual_dnodal_coordinates(local_eqn, p, q) += sum * w;
            }
          }
        }
      }
    } // End of loop over integration points
  }
 
 
  //======================================================================
  /// Self-test:  Return 0 for OK
  //======================================================================
  template<unsigned DIM>
  unsigned PoissonEquations<DIM>::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;
    }
  }
 
 
  //======================================================================
  /// Output function:
  ///
  ///   x,y,u   or    x,y,z,u
  ///
  /// nplot points in each coordinate direction
  //======================================================================
  template<unsigned DIM>
  void PoissonEquations<DIM>::output(std::ostream& outfile,
                                     const unsigned& nplot)
  {
    // Vector of local coordinates
    Vector<double> s(DIM);
 
    // Tecplot header info
    outfile << tecplot_zone_string(nplot);
 
    // 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);
 
      for (unsigned i = 0; i < DIM; i++)
      {
        outfile << interpolated_x(s, i) << " ";
      }
      outfile << interpolated_u_poisson(s) << std::endl;
    }
 
    // Write tecplot footer (e.g. FE connectivity lists)
    write_tecplot_zone_footer(outfile, nplot);
  }
 
 
  //======================================================================
  /// C-style output function:
  ///
  ///   x,y,u   or    x,y,z,u
  ///
  /// nplot points in each coordinate direction
  //======================================================================
  template<unsigned DIM>
  void PoissonEquations<DIM>::output(FILE* file_pt, const unsigned& nplot)
  {
    // Vector of local coordinates
    Vector<double> s(DIM);
 
    // 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);
 
      for (unsigned i = 0; i < DIM; i++)
      {
        fprintf(file_pt, "%g ", interpolated_x(s, i));
      }
      fprintf(file_pt, "%g \n", interpolated_u_poisson(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.
  ///
  ///   x,y,u_exact    or    x,y,z,u_exact
  //======================================================================
  template<unsigned DIM>
  void PoissonEquations<DIM>::output_fct(
    std::ostream& outfile,
    const unsigned& nplot,
    FiniteElement::SteadyExactSolutionFctPt exact_soln_pt)
  {
    // Vector of local coordinates
    Vector<double> s(DIM);
 
    // Vector for coordintes
    Vector<double> x(DIM);
 
    // Tecplot header info
    outfile << tecplot_zone_string(nplot);
 
    // Exact solution Vector (here a scalar)
    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 x position as Vector
      interpolated_x(s, x);
 
      // Get exact solution at this point
      (*exact_soln_pt)(x, exact_soln);
 
      // Output x,y,...,u_exact
      for (unsigned i = 0; i < DIM; i++)
      {
        outfile << x[i] << " ";
      }
      outfile << exact_soln[0] << std::endl;
    }
 
    // Write tecplot footer (e.g. FE connectivity lists)
    write_tecplot_zone_footer(outfile, nplot);
  }
 
 
  //=======================================================================
  /// Compute norm of the solution
  //=======================================================================
  template<unsigned DIM>
  void PoissonEquations<DIM>::compute_norm(double& norm)
  {
    // Initialise
    norm = 0.0;
 
    // Vector of local coordinates
    Vector<double> s(DIM);
 
    // Solution
    double u = 0.0;
 
    // Find out how many nodes there are in the element
    unsigned n_node = this->nnode();
 
    Shape psi(n_node);
 
    // Set the value of n_intpt
    unsigned n_intpt = this->integral_pt()->nweight();
 
    // Loop over the integration points
    for (unsigned ipt = 0; ipt < n_intpt; ipt++)
    {
      // Assign values of s
      for (unsigned i = 0; i < DIM; i++)
      {
        s[i] = this->integral_pt()->knot(ipt, i);
      }
 
      // Get the integral weight
      double w = this->integral_pt()->weight(ipt);
 
      // Get jacobian of mapping
      double J = this->J_eulerian(s);
 
      // Premultiply the weights and the Jacobian
      double W = w * J;
 
      // Get FE function value
      u = this->interpolated_u_poisson(s);
 
      // Add to  norm
      norm += u * u * W;
    }
  }
 
  //======================================================================
  /// Validate against exact solution
  ///
  /// Solution is provided via function pointer.
  /// Plot error at a given number of plot points.
  ///
  //======================================================================
  template<unsigned DIM>
  void PoissonEquations<DIM>::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(DIM);
 
    // Vector for coordintes
    Vector<double> x(DIM);
 
    // 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);
 
    // Loop over the integration points
    for (unsigned ipt = 0; ipt < n_intpt; ipt++)
    {
      // Assign values of s
      for (unsigned i = 0; i < DIM; i++)
      {
        s[i] = integral_pt()->knot(ipt, i);
      }
 
      // 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 x position as Vector
      interpolated_x(s, x);
 
      // Get FE function value
      double u_fe = interpolated_u_poisson(s);
 
      // Get exact solution at this point
      (*exact_soln_pt)(x, exact_soln);
 
      // Output x,y,...,error
      for (unsigned i = 0; i < DIM; i++)
      {
        outfile << x[i] << " ";
      }
      outfile << exact_soln[0] << " " << exact_soln[0] - u_fe << std::endl;
 
      // Add to error and norm
      norm += exact_soln[0] * exact_soln[0] * W;
      error += (exact_soln[0] - u_fe) * (exact_soln[0] - u_fe) * W;
    }
  }
 
 
  //====================================================================
  // Force build of templates
  //====================================================================
  template class QPoissonElement<1, 2>;
  template class QPoissonElement<1, 3>;
  template class QPoissonElement<1, 4>;
 
  template class QPoissonElement<2, 2>;
  template class QPoissonElement<2, 3>;
  template class QPoissonElement<2, 4>;
 
  template class QPoissonElement<3, 2>;
  template class QPoissonElement<3, 3>;
  template class QPoissonElement<3, 4>;
 
} // namespace oomph