This program was contributed by Salvador Flores <[email protected]>.
It comes without any warranty or support by its authors or the authors of deal.II.

This program is part of the deal.II code gallery and consists of the following files (click to inspect):

Annotated version of ElastoplasticTorsion.cc

  /* -----------------------------------------------------------------------------
   *
   * SPDX-License-Identifier: LGPL-2.1-or-later
   * Copyright (C) 2015 by Salvador Flores
   *
   * This file is part of the deal.II code gallery.
   *
   * -----------------------------------------------------------------------------
   *
   * Author: Salvador Flores, Center for Mathematical Modelling,
   *         Universidad de Chile, 2015.
   */
  
  
  /*
   This piece of software solves the elliptic p-laplacian
   boundary-value problems:
  
     Min {∫ 1/2 W(|Du|²)+ 1/p |Du|^p -fu : u=g on ∂S }   (1)
      u
  
   for large values of p, which approximates (see Alvarez & Flores 2015)
  
     Min {∫ 1/2 W(|Du|²) -fu : |Du|<1 a.s. on S, u=g on ∂S }
      u
  
   By default W(t)=t and S=unit disk.
  
   Large portions of this code are borrowed from the deal.ii tutorials
  
        @ref step_15 "step-15" @ref step_29 "step-29".
  
   For further details see the technical report
   "Solving variational problems with uniform gradient bounds by p-Laplacian
    approximation: Elastoplastic torsion implementation using the deal.II
    library"
   available at the documentation and at http://www.dim.uchile.cl/~sflores.
  
  */
  

Include files

  #include <deal.II/base/quadrature_lib.h>
  #include <deal.II/base/function.h>
  #include <deal.II/base/logstream.h>
  #include <deal.II/base/utilities.h>
  #include <deal.II/base/convergence_table.h>
  #include <deal.II/base/observer_pointer.h>
  #include <deal.II/base/parameter_handler.h>
  #include <deal.II/base/timer.h>
  
  #include <deal.II/lac/vector.h>
  #include <deal.II/lac/full_matrix.h>
  #include <deal.II/lac/sparse_matrix.h>
  #include <deal.II/lac/solver_cg.h>
  #include <deal.II/lac/precondition.h>
  #include <deal.II/lac/affine_constraints.h>
  #include <deal.II/lac/dynamic_sparsity_pattern.h>
  
  #include <deal.II/grid/tria.h>
  #include <deal.II/grid/grid_generator.h>
  #include <deal.II/grid/tria_accessor.h>
  #include <deal.II/grid/tria_iterator.h>
  #include <deal.II/grid/manifold_lib.h>
  #include <deal.II/grid/grid_refinement.h>
  #include <deal.II/grid/grid_in.h>
  
  #include <deal.II/dofs/dof_handler.h>
  #include <deal.II/dofs/dof_accessor.h>
  #include <deal.II/dofs/dof_tools.h>
  #include <deal.II/dofs/dof_renumbering.h>
  
  #include <deal.II/fe/fe_values.h>
  #include <deal.II/fe/fe_q.h>
  
  #include <deal.II/numerics/vector_tools.h>
  #include <deal.II/numerics/matrix_tools.h>
  #include <deal.II/numerics/data_out.h>
  #include <deal.II/numerics/error_estimator.h>
  #include <deal.II/numerics/solution_transfer.h>
  
  #include <typeinfo>
  #include <fstream>
  #include <iostream>
  
  #include <deal.II/numerics/solution_transfer.h>
  

Open a namespace for this program and import everything from the dealii namespace into it.

  namespace nsp
  {
    using namespace dealii;
  

    class ParameterReader : public EnableObserverPointer
    {
    public:
      ParameterReader(ParameterHandler &);
      void read_parameters(const std::string);
    private:
      void declare_parameters();
      ParameterHandler &prm;
    };

Constructor

    ParameterReader::ParameterReader(ParameterHandler &paramhandler):
      prm(paramhandler)
    {}
  
    void ParameterReader::declare_parameters()
    {
  
      prm.enter_subsection ("Global Parameters");
      {
        prm.declare_entry("p", "100",Patterns::Double(2.1),
                          "Penalization parameter");
        prm.declare_entry("known_solution", "true",Patterns::Bool(),
                          "Whether the exact solution is known");
      }
      prm.leave_subsection ();
  
      prm.enter_subsection ("Mesh & Refinement Parameters");
      {
        prm.declare_entry("Code for the domain", "0",Patterns::Integer(0,2),
                          "Number identifying the domain in which we solve the problem");
        prm.declare_entry("No of initial refinements", "4",Patterns::Integer(0),
                          "Number of global mesh refinement steps applied to initial coarse grid");
        prm.declare_entry("No of adaptive refinements", "8",Patterns::Integer(0),
                          "Number of global adaptive mesh refinements");
        prm.declare_entry("top_fraction_of_cells", "0.25",Patterns::Double(0),
                          "refinement threshold");
        prm.declare_entry("bottom_fraction_of_cells", "0.05",Patterns::Double(0),
                          "coarsening threshold");
      }
      prm.leave_subsection ();
  
  
      prm.enter_subsection ("Algorithm Parameters");
      {
        prm.declare_entry("Descent_direction", "0",Patterns::Integer(0,1),
                          "0: Preconditioned descent, 1: Newton Method");
        prm.declare_entry("init_p", "10",Patterns::Double(2),
                          "Initial p");
        prm.declare_entry("delta_p", "50",Patterns::Double(0),
                          "increase of p");
        prm.declare_entry("Max_CG_it", "1500",Patterns::Integer(1),
                          "Maximum Number of CG iterations");
        prm.declare_entry("CG_tol", "1e-10",Patterns::Double(0),
                          "Tolerance for CG iterations");
        prm.declare_entry("max_LS_it", "45",Patterns::Integer(1),
                          "Maximum Number of LS iterations");
        prm.declare_entry("line_search_tolerence", "1e-6",Patterns::Double(0),
                          "line search tolerance constant (c1 in Nocedal-Wright)");
        prm.declare_entry("init_step_length", "1e-2",Patterns::Double(0),
                          "initial step length in line-search");
        prm.declare_entry("Max_inner", "800",Patterns::Integer(1),
                          "Maximum Number of inner iterations");
        prm.declare_entry("eps", "1.0e-8",Patterns::Double(0),
                          "Threshold on norm of the derivative to declare optimality achieved");
        prm.declare_entry("hi_eps", "1.0e-9",Patterns::Double(0),
                          "Threshold on norm of the derivative to declare optimality achieved in highly refined mesh");
        prm.declare_entry("hi_th", "8",Patterns::Integer(0),
                          "Number of adaptive refinement before change convergence threshold");
      }
      prm.leave_subsection ();
  
    }
    void ParameterReader::read_parameters (const std::string parameter_file)
    {
      declare_parameters();
      prm.parse_input (parameter_file);
    }
  

The solution of the elastoplastic torsion problem on the unit disk with rhs=4.

    template <int dim>
    class Solution : public Function<dim>
    {
    public:
      Solution () : Function<dim>() {}
      virtual double value (const Point<dim> &pto, const unsigned int component = 0) const override;
      virtual Tensor<1,dim> gradient (const Point<dim> &pto, const unsigned int component = 0) const override;
    };
  
    template <int dim>
    double Solution<dim>::value (const Point<dim> &pto,const unsigned int) const
    {
      double r=sqrt(pto.square());
      if (r<0.5)
        return -1.0*std::pow(r,2.0)+0.75;
      else
        return 1.0-r;
    }
  
  
  
    template <int dim>
    Tensor<1,dim> Solution<dim>::gradient (const Point<dim> &pto,const unsigned int) const
    {
      double r=sqrt(pto.square());
      if (r<0.5)
        return -2.0*pto;
      else
        return  -1.0*pto/r;
    }
  
  
  
  

    /*                 Compute the Lagrange multiplier (as a derived quantity)                   */
  
  
    template <int dim>
    class ComputeMultiplier : public DataPostprocessor<dim>
    {
    private:
      double p;
    public:
      ComputeMultiplier (double pe);
  
      virtual
      void compute_derived_quantities_scalar (
        const std::vector< double > &,
        const std::vector< Tensor< 1, dim > > &,
        const std::vector< Tensor< 2, dim > > &,
        const std::vector< Point< dim > > &,
        const std::vector< Point< dim > > &,
        std::vector< Vector< double > > &
      ) const;
  
      virtual std::vector<std::string> get_names () const override;
  
      virtual
      std::vector<DataComponentInterpretation::DataComponentInterpretation>
      get_data_component_interpretation () const override;
  
      virtual UpdateFlags get_needed_update_flags () const override;
    };
  
  
    template <int dim>
    ComputeMultiplier<dim>::ComputeMultiplier (double pe): p(pe)
    {}
  
  
    template <int dim>
    void ComputeMultiplier<dim>::compute_derived_quantities_scalar(
      const std::vector< double > &   /*uh*/,
      const std::vector< Tensor< 1, dim > >   &duh,
      const std::vector< Tensor< 2, dim > > &  /*dduh*/,
      const std::vector< Point< dim > > &  /* normals*/,
      const std::vector< Point< dim > > &  /*evaluation_points*/,
      std::vector< Vector< double > >    &computed_quantities )   const
    {
      const unsigned int n_quadrature_points = duh.size();
  
      for (unsigned int q=0; q<n_quadrature_points; ++q)
        {
          long  double sqrGrad=duh[q]* duh[q]; //squared norm of the gradient
          long double exponent=(p-2.0)/2*std::log(sqrGrad);
          computed_quantities[q](0) = std::sqrt(sqrGrad); // norm of the gradient
          computed_quantities[q](1)= std::exp(exponent); // multiplier
        }
    }
  
  
  
  
  
    template <int dim>
    std::vector<std::string>
    ComputeMultiplier<dim>::get_names() const
    {
      std::vector<std::string> solution_names;
      solution_names.push_back ("Gradient_norm");
      solution_names.push_back ("Lagrange_multiplier");
      return solution_names;
    }
  
  
    template <int dim>
    UpdateFlags
    ComputeMultiplier<dim>::get_needed_update_flags () const
    {
      return update_gradients;
    }
  
  
  
    template <int dim>
    std::vector<DataComponentInterpretation::DataComponentInterpretation>
    ComputeMultiplier<dim>:: get_data_component_interpretation () const
    {
      std::vector<DataComponentInterpretation::DataComponentInterpretation>
      interpretation;

norm of the gradient

interpretation.push_back (DataComponentInterpretation::component_is_scalar);

DataComponentInterpretation::component_is_scalar

@ component_is_scalar

Definition data_component_interpretation.h:50

Lagrange multiplier

      interpretation.push_back (DataComponentInterpretation::component_is_scalar);
      return interpretation;
    }

    template <int dim>
    class ElastoplasticTorsion
    {
    public:
      ElastoplasticTorsion (ParameterHandler &);
      ~ElastoplasticTorsion ();
      void run ();
  
    private:
      void setup_system (const bool initial_step);
      void assemble_system ();
      bool solve (const int inner_it);
      void init_mesh ();
      void refine_mesh ();
      void set_boundary_values ();
      double phi (const double alpha) const;
      bool checkWolfe(double &alpha, double &phi_alpha) const;
      bool determine_step_length (const int inner_it);
      void print_it_message (const int counter, bool ks);
      void output_results (unsigned int refinement) const;
      void format_convergence_tables();
      void process_solution (const unsigned int cycle);
      void process_multiplier (const unsigned int cycle,const int iter,double time);
      double dual_error () const;
      double dual_infty_error () const;
      double W (double Du2) const;
      double Wp (double Du2) const;
      double G (double Du2) const;
  
  
  
      ParameterHandler &prm;
      Triangulation<dim> triangulation;
      DoFHandler<dim> dof_handler;
      AffineConstraints<double> hanging_node_constraints;
      SparsityPattern sparsity_pattern;
      SparseMatrix<double> system_matrix;
      ConvergenceTable convergence_table;
      ConvergenceTable dual_convergence_table;
      Vector<double> present_solution;
      Vector<double> newton_update;
      Vector<double> system_rhs;
      Vector<double> grad_norm;
      Vector<double> lambda;
  
  
      double step_length,phi_zero,phi_alpha,phip,phip_zero;
      double old_step,old_phi_zero,old_phip;
      double L2_error;
      double H1_error;
      double Linfty_error;
      double dual_L1_error;
      double dual_L_infty_error;
      FE_Q<dim> fe;
      double p;
      double line_search_tolerence; // c_1 in Nocedal & Wright
      unsigned int dir_id;
      std::string elements;
      std::string Method;
  
    };
  
    /*******************************************************************************************/

Boundary condition

    template <int dim>
    class BoundaryValues : public Function<dim>
    {
    public:
      BoundaryValues () : Function<dim>() {}
  
      virtual double value (const Point<dim>   &p,
                            const unsigned int  component = 0) const override;
    };
  
  
    template <int dim>
    double BoundaryValues<dim>::value (const Point<dim> &/*pto*/,
                                       const unsigned int /*component*/) const
    {

could be anything else (theory works provided |Dg|_infty < 1/2)

      return 0.0;
  
      /* A challenging BC leading to overdetermined problems
         it is regulated by the parameter 0<eta<1.
         eta closer to 1 leads to more difficult problems.
  
      double pii=numbers::PI;
      double theta=std::atan2(p[1],p[0])+pii;
      double eta=0.9;
  
          if (theta <= 0.5)
          return eta*(theta*theta);
        else if ((theta >0.5) & (theta<= pii-0.5))
          return eta*(theta-0.25);
        else if ((theta>pii-0.5)&(theta<= pii+0.5))
          return eta*(pii-0.75-(theta-(pii-0.5))*(theta-(pii+0.5)));
        else if ((theta>pii+0.5)&(theta<= 2*pii-0.5))
          return eta*((2*pii-theta)-0.25);
        else
          return eta*((theta-2*pii)*(theta-2*pii) );*/
    }
  
  
  
    /******************************************************************************/

Right-Hand Side

    template <int dim>
    class RightHandSide : public Function<dim>
    {
    public:
      RightHandSide () : Function<dim>() {}
      virtual double value (const Point<dim> &p,
                            const unsigned int component = 0) const override;
    };
  
    template <int dim>
    double RightHandSide<dim>::value (const Point<dim> &/*p*/,
                                      const unsigned int /*component*/) const
    {

set to constant = 4, for which explicit solution to compare exists could be anything

      double return_value = 4.0;
      return return_value;
    }
  
    /*******************************************************************/

The ElastoplasticTorsion class implementation

Constructor of the class

    template <int dim>
    ElastoplasticTorsion<dim>::ElastoplasticTorsion (ParameterHandler &param):
      prm(param),
      dof_handler (triangulation),
      L2_error(1.0),
      H1_error(1.0),
      Linfty_error(1.0),
      dual_L1_error(1.0),
      dual_L_infty_error(1.0),
      fe(2)
    {
      prm.enter_subsection ("Global Parameters");
      p=prm.get_double("p");
      prm.leave_subsection ();
      prm.enter_subsection ("Algorithm Parameters");
      line_search_tolerence=prm.get_double("line_search_tolerence");
      dir_id=prm.get_integer("Descent_direction");
      prm.leave_subsection ();
      if (fe.degree==1)
        elements="P1";
      else elements="P2";
  
      if (dir_id==0)
        Method="Precond";
      else
        Method="Newton";
    }
  
  
  
    template <int dim>
    ElastoplasticTorsion<dim>::~ElastoplasticTorsion ()
    {
      dof_handler.clear ();
    }
  
    /*****************************************************************************************/

print iteration message

    template <int dim>
    void ElastoplasticTorsion<dim>::print_it_message (const int counter, bool ks)
    {
      if (ks)
        {
          process_solution (counter);
          std::cout << "iteration="<< counter+1 << "  J(u_h)= "<< phi_zero << ", H1 error: "
                    <<  H1_error  <<", W0-1,infty error: "<< Linfty_error<< " J'(u_h)(w)= "<< phip
                    << ", |J'(u_h)|= "<< system_rhs.l2_norm()<<std::endl;
        }
      else
        {
          std::cout << "iteration= " << counter+1 << " J(u_h)= "
                    << phi_alpha << " J'(u_h)= "<< phip<<std::endl;
        }
    }
  
  
    /*****************************************************************************************/

Convergence Tables

/*************************************************************/

formatting

    template <int dim>
    void ElastoplasticTorsion<dim>::format_convergence_tables()
    {
      convergence_table.set_precision("L2", 3);
      convergence_table.set_precision("H1", 3);
      convergence_table.set_precision("Linfty", 3);
      convergence_table.set_precision("function value", 3);
      convergence_table.set_precision("derivative", 3);
      dual_convergence_table.set_precision("dual_L1", 3);
      dual_convergence_table.set_precision("dual_Linfty", 3);
      dual_convergence_table.set_precision("L2", 3);
      dual_convergence_table.set_precision("H1", 3);
      dual_convergence_table.set_precision("Linfty", 3);
      convergence_table.set_scientific("L2", true);
      convergence_table.set_scientific("H1", true);
      convergence_table.set_scientific("Linfty", true);
      convergence_table.set_scientific("function value", true);
      convergence_table.set_scientific("derivative", true);
      dual_convergence_table.set_scientific("dual_L1", true);
      dual_convergence_table.set_scientific("dual_Linfty", true);
      dual_convergence_table.set_scientific("L2", true);
      dual_convergence_table.set_scientific("H1", true);
      dual_convergence_table.set_scientific("Linfty", true);
  
    }
  
    /****************************************/

fill-in entry for the solution

    template <int dim>
    void ElastoplasticTorsion<dim>::process_solution (const unsigned int it)
    {
      Vector<float> difference_per_cell (triangulation.n_active_cells());
  

compute L2 error (save to difference_per_cell)

      VectorTools::integrate_difference (dof_handler,present_solution,
                                         Solution<dim>(),difference_per_cell,QGauss<dim>(3),VectorTools::L2_norm);
      L2_error = difference_per_cell.l2_norm();
  

compute H1 error (save to difference_per_cell)

      VectorTools::integrate_difference (dof_handler,present_solution,Solution<dim>(),
                                         difference_per_cell,QGauss<dim>(3),VectorTools::H1_seminorm);
      H1_error = difference_per_cell.l2_norm();
  

compute W1infty error (save to difference_per_cell)

      const QTrapezoid<1> q_trapez;
      const QIterated<dim> q_iterated (q_trapez, 5);
      VectorTools::integrate_difference (dof_handler,present_solution,Solution<dim>(),
                                         difference_per_cell,q_iterated,VectorTools::W1infty_seminorm);
      Linfty_error = difference_per_cell.linfty_norm();
  
  
      convergence_table.add_value("cycle", it);
      convergence_table.add_value("p", p);
      convergence_table.add_value("L2", L2_error);
      convergence_table.add_value("H1", H1_error);
      convergence_table.add_value("Linfty", Linfty_error);
      convergence_table.add_value("function value", phi_alpha);
      convergence_table.add_value("derivative", phip);
    }
  
  
    /***************************************/

fill-in entry for the multiplier

    template <int dim>
    void ElastoplasticTorsion<dim>::process_multiplier (const unsigned int cycle, const int iter,double time)
    {
      const unsigned int n_active_cells=triangulation.n_active_cells();
      const unsigned int n_dofs=dof_handler.n_dofs();
      dual_L1_error=dual_error();
      dual_L_infty_error=dual_infty_error();
  
  
      dual_convergence_table.add_value("cycle", cycle);
      dual_convergence_table.add_value("p", p);
      dual_convergence_table.add_value("iteration_number", iter);
      dual_convergence_table.add_value("cpu_time", time);
      dual_convergence_table.add_value("cells", n_active_cells);
      dual_convergence_table.add_value("dofs", n_dofs);
      dual_convergence_table.add_value("L2", L2_error);
      dual_convergence_table.add_value("H1", H1_error);
      dual_convergence_table.add_value("Linfty", Linfty_error);
      dual_convergence_table.add_value("dual_L1", dual_L1_error);
      dual_convergence_table.add_value("dual_Linfty", dual_L_infty_error);
  
    }
  
  
  
  
    /****************************************************************************************/

ElastoplasticTorsion::setup_system unchanged from step-15

    template <int dim>
    void ElastoplasticTorsion<dim>::setup_system (const bool initial_step)
    {
      if (initial_step)
        {
          dof_handler.distribute_dofs (fe);
          present_solution.reinit (dof_handler.n_dofs());
          grad_norm.reinit (dof_handler.n_dofs());
          lambda.reinit (dof_handler.n_dofs());
  
          hanging_node_constraints.clear ();
          DoFTools::make_hanging_node_constraints (dof_handler,
                                                   hanging_node_constraints);
          hanging_node_constraints.close ();
        }
  
  

The remaining parts of the function

      newton_update.reinit (dof_handler.n_dofs());
      system_rhs.reinit (dof_handler.n_dofs());
      DynamicSparsityPattern c_sparsity(dof_handler.n_dofs());
      DoFTools::make_sparsity_pattern (dof_handler, c_sparsity);
      hanging_node_constraints.condense (c_sparsity);
      sparsity_pattern.copy_from(c_sparsity);
      system_matrix.reinit (sparsity_pattern);
    }
  
    /***************************************************************************************/
    /* the coefficients W, W' and G defining the problem.
  
       Min_u \int W(|Du|^2) dx
  
    They must be consistent as G(s)=W'(s)+2s W''(s) for any s>0.
    recall that they receive the SQUARED gradient. */
  
    template <int dim>
    double ElastoplasticTorsion<dim>::W (double Du2) const
    {
      return Du2;
    }
  
    template <int dim>
    double ElastoplasticTorsion<dim>::Wp (double /*Du2*/) const
    {
      return 1.0;
    }
  
    template <int dim>
    double ElastoplasticTorsion<dim>::G (double /*Du2*/) const
    {
      return 1.0;
    }
    /***************************************************************************************/
  
    template <int dim>
    void ElastoplasticTorsion<dim>::assemble_system ()
    {
  
  
      const QGauss<dim>  quadrature_formula(3);
      const RightHandSide<dim> right_hand_side;
      system_matrix = 0;
      system_rhs = 0;
  
      FEValues<dim> fe_values (fe, quadrature_formula,
                               update_gradients         |
                               update_values           |
                               update_quadrature_points |
                               update_JxW_values);
  
      const unsigned int           dofs_per_cell = fe.dofs_per_cell;
      const unsigned int           n_q_points    = quadrature_formula.size();
  
      FullMatrix<double>           cell_matrix (dofs_per_cell, dofs_per_cell);
      Vector<double>               cell_rhs (dofs_per_cell);
  
      std::vector<Tensor<1, dim> > old_solution_gradients(n_q_points);
      std::vector<types::global_dof_index>    local_dof_indices (dofs_per_cell);
  
  
      typename DoFHandler<dim>::active_cell_iterator
      cell = dof_handler.begin_active(),
      endc = dof_handler.end();
      for (; cell!=endc; ++cell)
        {
          cell_matrix = 0;
          cell_rhs = 0;
  
          fe_values.reinit (cell);
          fe_values.get_function_gradients(present_solution,
                                           old_solution_gradients);
  
          for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
            {
              long double coeff=0.0;
              long double a=old_solution_gradients[q_point] * old_solution_gradients[q_point];
              long double exponent=(p-2.0)/2*std::log(a);
              coeff= std::exp( exponent);
              for (unsigned int i=0; i<dofs_per_cell; ++i)
                {
                  for (unsigned int j=0; j<dofs_per_cell; ++j)
                    {
                      if (dir_id==1)
                        {
                          cell_matrix(i, j) +=  fe_values.shape_grad(i, q_point) *  fe_values.shape_grad(j, q_point)
                                                * (G(a)+(p-1.0)*coeff)    * fe_values.JxW(q_point);
                        }
                      else
                        {
                          cell_matrix(i, j) +=  fe_values.shape_grad(i, q_point) *  fe_values.shape_grad(j, q_point)
                                                * (Wp(a)+coeff)
                                                * fe_values.JxW(q_point);
                        }
                    }
  
                  cell_rhs(i) -= (  fe_values.shape_grad(i, q_point)
                                    * old_solution_gradients[q_point]
                                    * (Wp(a)+coeff)
                                    -right_hand_side.value(fe_values.quadrature_point(q_point))
                                    *fe_values.shape_value(i, q_point)
                                 )
                                 * fe_values.JxW(q_point);
                }
            }
  
          cell->get_dof_indices (local_dof_indices);
          for (unsigned int i=0; i<dofs_per_cell; ++i)
            {
              for (unsigned int j=0; j<dofs_per_cell; ++j)
                system_matrix.add (local_dof_indices[i],
                                   local_dof_indices[j],
                                   cell_matrix(i,j));
  
              system_rhs(local_dof_indices[i]) += cell_rhs(i);
            }
        }
  
      hanging_node_constraints.condense (system_matrix);
      hanging_node_constraints.condense (system_rhs);
  
      std::map<types::global_dof_index,double> boundary_values;
      VectorTools::interpolate_boundary_values (dof_handler,
                                                0,
                                                Functions::ZeroFunction<dim>(),
                                                boundary_values);
      MatrixTools::apply_boundary_values (boundary_values,
                                          system_matrix,
                                          newton_update,
                                          system_rhs);
    }
  
  
  
  
    /**********************************      Refine Mesh      ****************************************/

unchanged from step-15

    template <int dim>
    void ElastoplasticTorsion<dim>::refine_mesh ()
    {
      using FunctionMap = std::map<types::boundary_id, const Function<dim> *>;
  
      Vector<float> estimated_error_per_cell (triangulation.n_active_cells());
      KellyErrorEstimator<dim>::estimate (dof_handler,
                                          QGauss<dim-1>(3),
                                          FunctionMap(),
                                          present_solution,
                                          estimated_error_per_cell);
  
      prm.enter_subsection ("Mesh & Refinement Parameters");
      const double top_fraction=prm.get_double("top_fraction_of_cells");
      const double bottom_fraction=prm.get_double("bottom_fraction_of_cells");
      prm.leave_subsection ();
      GridRefinement::refine_and_coarsen_fixed_number (triangulation,
                                                       estimated_error_per_cell,
                                                       top_fraction, bottom_fraction);
  
      triangulation.prepare_coarsening_and_refinement ();
      SolutionTransfer<dim> solution_transfer(dof_handler);
      solution_transfer.prepare_for_coarsening_and_refinement(present_solution);
      triangulation.execute_coarsening_and_refinement();
      dof_handler.distribute_dofs(fe);
      present_solution.reinit(dof_handler.n_dofs());
      solution_transfer.interpolate(present_solution);
      set_boundary_values ();
      hanging_node_constraints.clear();
  
      DoFTools::make_hanging_node_constraints(dof_handler,
                                              hanging_node_constraints);
      hanging_node_constraints.close();
      hanging_node_constraints.distribute (present_solution);
      setup_system (false);
    }
  
  
    /*******************************************************************************************/

Dump the norm of the gradient and the lagrange multiplier in vtu format for visualization

    template <int dim>
    void ElastoplasticTorsion<dim>::output_results (unsigned int counter) const
    {

multiplier object contains both |Du| and lambda.

      ComputeMultiplier<dim> multiplier(p);
      DataOut<dim> data_out;
  
      data_out.attach_dof_handler (dof_handler);
      data_out.add_data_vector (present_solution, "solution");
      data_out.add_data_vector (present_solution, multiplier);
      data_out.build_patches ();
      std::ostringstream p_str;
      p_str << p<<"-cycle-"<<counter;
      std::string str = p_str.str();
      const std::string filename = "solution-" + str+".vtu";
      std::ofstream output (filename.c_str());
      data_out.write_vtu (output);
    }
  
    /********************************************************************************************/

unchanged from step-15

    template <int dim>
    void ElastoplasticTorsion<dim>::set_boundary_values ()
    {
      std::map<types::global_dof_index, double> boundary_values;
      VectorTools::interpolate_boundary_values (dof_handler,
                                                0,
                                                BoundaryValues<dim>(),
                                                boundary_values);
      for (std::map<types::global_dof_index, double>::const_iterator
           bp = boundary_values.begin();
           bp != boundary_values.end(); ++bp)
        present_solution(bp->first) = bp->second;
    }
  
  
    /****************************************************************************************/

COMPUTE \(\phi(\alpha)=J_p(u_h+\alpha w)\)

    template <int dim>
    double ElastoplasticTorsion<dim>::phi (const double alpha) const
    {
      double obj = 0.0;
      const RightHandSide<dim> right_hand_side;
      Vector<double> evaluation_point (dof_handler.n_dofs());
      evaluation_point = present_solution;  // copy of u_h
      evaluation_point.add (alpha, newton_update); // u_{n+1}=u_n+alpha w_n
  
      const QGauss<dim>  quadrature_formula(3);
      FEValues<dim> fe_values (fe, quadrature_formula,
                               update_gradients         |
                               update_values    |
                               update_quadrature_points |
                               update_JxW_values);
  
      const unsigned int           dofs_per_cell = fe.dofs_per_cell;
      const unsigned int           n_q_points    = quadrature_formula.size();
  
      Vector<double>               cell_residual (dofs_per_cell);
      std::vector<Tensor<1, dim> > gradients(n_q_points);
      std::vector<double> values(n_q_points);
  
  
      std::vector<types::global_dof_index>    local_dof_indices (dofs_per_cell);
  
      typename DoFHandler<dim>::active_cell_iterator
      cell = dof_handler.begin_active(),
      endc = dof_handler.end();
      for (; cell!=endc; ++cell)
        {
          cell_residual = 0;
          fe_values.reinit (cell);
          fe_values.get_function_gradients (evaluation_point, gradients);
          fe_values.get_function_values (evaluation_point, values);
  
  
          for (unsigned int q_point=0; q_point<n_q_points; ++q_point)
            {
              double Du2=gradients[q_point] *  gradients[q_point]; // Du2=|Du|^2
              double penalty;
              if (Du2<1.0e-10)
                penalty=0.0;
              else
                penalty=std::pow(Du2,p/2.0); // penalty=|Du|^p
  

obj+= 1/2 W(|Du|^2)+1/p |Du|^p -fu (see (1))

              obj+=(
                     (0.5*W(Du2)+penalty/p)- right_hand_side.value(fe_values.quadrature_point(q_point))*values[q_point]
                   ) * fe_values.JxW(q_point);
            }
  
        }
  
      return obj;
    }
  
  
    /***************************************************************************************************/

Compute L^1 error norm of Lagrange Multiplier with respect to exact solution (cf. Alvarez & Flores, 2015)

    template <int dim>
    double ElastoplasticTorsion<dim>::dual_error () const
    {
      double obj = 0.0;
  
      const QGauss<dim>  quadrature_formula(3);
      FEValues<dim> fe_values (fe, quadrature_formula,
                               update_gradients         |
                               update_quadrature_points |
                               update_JxW_values);
  
      const unsigned int           dofs_per_cell = fe.dofs_per_cell;
      const unsigned int           n_q_points    = quadrature_formula.size();
  
      Vector<double>               cell_residual (dofs_per_cell);
      std::vector<Tensor<1, dim> > gradients(n_q_points);
  
      std::vector<types::global_dof_index>    local_dof_indices (dofs_per_cell);
  
      typename DoFHandler<dim>::active_cell_iterator
      cell = dof_handler.begin_active(),
      endc = dof_handler.end();
      for (; cell!=endc; ++cell)
        {
          cell_residual = 0;
          fe_values.reinit (cell);
          fe_values.get_function_gradients (present_solution, gradients);
  
          for (unsigned int q_point=0; q_point<n_q_points; ++q_point)
            {
              double coeff=gradients[q_point] *  gradients[q_point] ;
              if (coeff<1.0e-15)
                coeff=0.0;
              else
                coeff=std::pow(coeff,(p-2.0)/2.0); // |Du_p|^(p-2)
  
              double r=std::sqrt(fe_values.quadrature_point(q_point).square());
              double exact=0;
              if (r>0.5)
                exact= 2*r-1;
  
              obj+=(     std::abs(coeff-exact) ) * fe_values.JxW(q_point);
            }
  
        }
  
      return obj;
    }
  
    /*******************************************************************************************/

Compute L^infinity error norm of Lagrange Multiplier with respect to exact solution (cf. Alvarez & Flores, 2015)

    template <int dim>
    double ElastoplasticTorsion<dim>::dual_infty_error () const
    {
      double obj = 0.0;
      const QTrapezoid<1> q_trapez;
      const QIterated<dim> quadrature_formula (q_trapez, 10);
  
      FEValues<dim> fe_values (fe, quadrature_formula,
                               update_gradients         |
                               update_quadrature_points );
  
      const unsigned int           dofs_per_cell = fe.dofs_per_cell;
      const unsigned int           n_q_points    = quadrature_formula.size();
  
      Vector<double>               cell_residual (dofs_per_cell);
      std::vector<Tensor<1, dim> > gradients(n_q_points);
  
      std::vector<types::global_dof_index>    local_dof_indices (dofs_per_cell);
  
      typename DoFHandler<dim>::active_cell_iterator
      cell = dof_handler.begin_active(),
      endc = dof_handler.end();
      for (; cell!=endc; ++cell)
        {
          cell_residual = 0;
          fe_values.reinit (cell);
          fe_values.get_function_gradients (present_solution, gradients);
  
          for (unsigned int q_point=0; q_point<n_q_points; ++q_point)
            {
              long double sqdGrad=gradients[q_point] *  gradients[q_point] ;
              double r=std::sqrt(fe_values.quadrature_point(q_point).square());
              double exact=0;
              if (r>0.5)
                exact= 2*r-1.0;

compute |Du|^(p-2) as exp(p-2/2*log(Du^2))

              long double exponent=(p-2.0)/2*std::log(sqdGrad);
              long double coeff=std::exp(exponent);
  
              if (std::abs(coeff-exact)>obj )
                obj=std::abs(coeff-exact);
            }
  
        }
  
      return obj;
    }
  
    /*****************************************************************************************/

check whether putative step-length satisfies sufficient decrease conditions

    template <int dim>
    bool ElastoplasticTorsion<dim>::checkWolfe(double &alpha, double &phi_alpha) const
    {
      if (phi_alpha< phi_zero+line_search_tolerence*phip*alpha )
        return true;
      else
        return false;
    }
  
  
    /*****************************************************************************************/

Find a step-length satisfying sufficient decrease condition by line-search uses quadratic interpolation

    template <int dim>
    bool ElastoplasticTorsion<dim>::determine_step_length(const int inner_it)
    {
      unsigned int it=0;
      bool done;
      double alpha,nalpha;
      prm.enter_subsection ("Algorithm Parameters");
      const unsigned int max_LS_it=prm.get_integer("max_LS_it");
      double init_SL=prm.get_double("init_step_length");
      prm.leave_subsection ();
      if (inner_it==0)
        alpha=init_SL;
      else
        {
          alpha=std::min(1.45*old_step*old_phip/phip,1.0);
        }
      phi_alpha=phi(alpha);
      std::cerr << "Step length=" << alpha << ", Value= " << phi_alpha;

check if step-size satisfies sufficient decrease condition

      done=checkWolfe(alpha,phi_alpha);
      if (done)
        std::cerr << " accepted" << std::endl;
      else
        std::cerr << " rejected" ;
  
      while ((!done) & (it<max_LS_it))
        {

new try obtained by quadratic interpolation

          nalpha=-(phip*alpha*alpha)/(2*(phi_alpha-phi_zero-phip*alpha));
  
          if (nalpha<1e-3*alpha ||  std::abs(nalpha-alpha)/alpha<1e-8)
            nalpha=alpha/2;
          else if ( phi_alpha-phi_zero>1e3*std::abs(phi_zero) )
            nalpha=alpha/10;
          alpha=nalpha;
          phi_alpha=phi(alpha);
          done=checkWolfe(alpha,phi_alpha);
          if (done)
            std::cerr << ", finished with steplength= "<< alpha<< ", fcn value= "<< phi_alpha<<std::endl;
          it=it+1;
        }
      if (!done)
        {
          std::cerr << ", max. no. of iterations reached with steplength= "<< alpha
                    << ", fcn value= "<< phi_alpha<<std::endl;
          return false;
        }
      else
        {
          step_length=alpha;
          return true;
        }
  
    }
  
    /**************************************************************************************************/

ElastoplasticTorsion::init_mesh()

    template <int dim>
    void ElastoplasticTorsion<dim>::init_mesh ()
    {

get parameters

      prm.enter_subsection ("Mesh & Refinement Parameters");
      const int domain_id=prm.get_integer("Code for the domain");
      const int init_ref=prm.get_integer("No of initial refinements");
      prm.leave_subsection ();
  
  
      if (domain_id==0)
        {

For the unit disk around the origin

          GridGenerator::hyper_ball (triangulation);
          static const SphericalManifold<dim> boundary;
          triangulation.set_manifold (0, boundary);
        }
      else if (domain_id==1)
        {

For the unit square

          GridGenerator::hyper_cube (triangulation, 0, 1);
        }
      else if (domain_id==2)
        {
          /* For Glowinski's domain
              ___    ___   __ 1
             |   |__|   |  __ .8
           |          |
           |          |
           |__________|  __ 0
  
             |   |  |   |
             0  .4 .6   1
  
          */
          Triangulation<dim> tria1;
          Triangulation<dim> tria2;
          Triangulation<dim> tria3;
          Triangulation<dim> tria4;
          Triangulation<dim> tria5;
          Triangulation<dim> tria6;
          GridGenerator::hyper_rectangle(tria1, Point<2>(0.0,0.0), Point<2>(0.4,0.8));
          GridGenerator::hyper_rectangle(tria2, Point<2>(0.0,0.8), Point<2>(0.4,1.0));
          GridGenerator::hyper_rectangle(tria3, Point<2>(0.4,0.0), Point<2>(0.6,0.8));
          GridGenerator::hyper_rectangle(tria4, Point<2>(0.6,0.0), Point<2>(1.0,0.8));
          GridGenerator::hyper_rectangle(tria5, Point<2>(0.6,0.8), Point<2>(1.0,1.0));
          GridGenerator::merge_triangulations (tria1, tria2, tria6);
          GridGenerator::merge_triangulations (tria6, tria3, tria6);
          GridGenerator::merge_triangulations (tria6, tria4, tria6);
          GridGenerator::merge_triangulations (tria6, tria5, triangulation);
        }

perform initial refinements

      triangulation.refine_global(init_ref);
    }
  
    /**************************************************************************************************/

ElastoplasticTorsion::solve(inner_it) Performs one inner iteration

    template <int dim>
    bool ElastoplasticTorsion<dim>::solve (const int inner_it)
    {
      prm.enter_subsection ("Algorithm Parameters");
      const unsigned int max_CG_it=prm.get_integer("Max_CG_it");
      const double CG_tol=prm.get_double("CG_tol");
      prm.leave_subsection ();
  
      SolverControl solver_control (max_CG_it,CG_tol);
      SolverCG<>    solver (solver_control);
  
      PreconditionSSOR<> preconditioner;
      preconditioner.initialize(system_matrix,0.25);
  
      solver.solve (system_matrix, newton_update, system_rhs,
                    preconditioner);
      hanging_node_constraints.distribute (newton_update);
      /******  save current quantities for line-search  **** */

Recall that phi(alpha)=J(u+alpha w)

      old_step=step_length;
      old_phi_zero=phi_zero;
      phi_zero=phi(0); // phi(0)=J(u)
      old_phip=phip;
      phip=-1.0*(newton_update*system_rhs); //phi'(0)=J'(u) *w, rhs=-J'(u).
      if (inner_it==0)
        phip_zero=phip;
  
      if (phip>0)   // this should not happen, step back
        {
          std::cout << "Not a descent direction!" <<std::endl;
          present_solution.add (-1.0*step_length, newton_update);
          step_length=step_length/2;
          phip=old_phip;
          return false;
        }
      else
        {
          if (determine_step_length(inner_it))
            {

update u_{n+1}=u_n+alpha w_n

              present_solution.add (step_length, newton_update);
              return true;
            }
          else return false;
        }
    }
  
  
  
    /*************************************************************************************************************/

ElastoplasticTorsion::run

    template <int dim>
    void ElastoplasticTorsion<dim>::run ()
    {
  

get parameters

      prm.enter_subsection ("Mesh & Refinement Parameters");
      const int adapt_ref=prm.get_integer("No of adaptive refinements");
      prm.leave_subsection ();
      prm.enter_subsection ("Algorithm Parameters");
      const int max_inner=prm.get_integer("Max_inner");
      const double eps=prm.get_double("eps");
      const double hi_eps=prm.get_double("hi_eps");
      const int hi_th=prm.get_integer("hi_th");
      const double init_p=prm.get_double("init_p");
      const double delta_p=prm.get_double("delta_p");
      prm.leave_subsection ();
      prm.enter_subsection ("Global Parameters");
      bool known_solution=prm.get_bool("known_solution");
      double actual_p=prm.get_double("p");
      prm.leave_subsection ();
      /************************/
  

init Timer

      Timer timer;
      double ptime=0.0;
      timer.start ();
  

initialize mesh for the selected domain

init_mesh();

setup FE space

      setup_system (true);
      set_boundary_values ();
  

init counters

      int global_it=0;    // Total inner iterations (counting both loops)
      int cycle=0;          // Total outer iterations (counting both loops)
      int refinement = 0;    //  Refinements  performed (adaptive) = outer iterations 2nd loop
  
  

prepare to start first loop

      p=init_p;
      bool well_solved=true;
  
      /*****************************          First loop      ***********************************/
      /****************** Prepare initial condition using increasing p  *************************/
      while (p<actual_p) // outer iteration, increasing p.
        {
          std::cout <<"--Preparing initial condition with p="<<p<<" iter.= " << global_it<<  "  .-- "<< std::endl;
          timer.restart();
          for (int inner_iteration=0; inner_iteration<max_inner; ++inner_iteration,++global_it)
            {
              assemble_system ();
              well_solved=solve (inner_iteration);
              print_it_message (global_it, known_solution);
              if (
                ((system_rhs.l2_norm()/std::sqrt(system_rhs.size()) <1e-4) & (cycle<1)) |
                ((system_rhs.l2_norm()/std::sqrt(system_rhs.size()) <1e-5) & (cycle>=1)) |
                !well_solved
              )
                break;
            }
          ptime=timer.cpu_time();
          if (well_solved)
            output_results (cycle);
  
          if (known_solution)
            {
              process_multiplier(cycle,global_it,ptime);

dual_convergence_table.write_tex(dual_error_table_file);

            }
          refine_mesh();
          cycle++;
          p+=delta_p;
        }
      /***************************    first loop finished        ********************/
  
  

prepare for second loop

      p=actual_p;
      well_solved=true;
  
  
      /*****************************        Second loop         *********************************/
      /**************************** Solve problem for target p  *********************************/
  
      std::cout << "============ Solving  problem with p="   <<p << " =================="  << std::endl;
      /*****    Outer iteration - refining mesh  ******************/
      while ((cycle<adapt_ref) & well_solved)
        {
          timer.restart();

inner iteration

          for (int inner_iteration=0; inner_iteration<max_inner; ++inner_iteration,++global_it)
            {
              assemble_system ();
              well_solved=solve (inner_iteration);
              print_it_message (global_it, known_solution);
  
              if (
                ((system_rhs.l2_norm()/std::sqrt(system_rhs.size()) < eps) & (refinement<hi_th)) |
                (( system_rhs.l2_norm()/ std::sqrt (system_rhs.size()) <hi_eps)  | (!well_solved))
              )
                break;
            }

inner iterations finished

          ptime=timer.cpu_time();
          if (well_solved)
            output_results (cycle);
  

compute and display error, if the explicit solution is known

          if (known_solution)
            {
              process_multiplier(cycle,global_it,ptime);
              std::cout << "finished with H1 error: " <<  H1_error << ", dual error (L1): "
                        << dual_L1_error << "dual error (L infty): "<<dual_L_infty_error <<std::endl;
            }
  

update counters

++refinement;

++cycle;

refine mesh

          std::cout << "******** Refined mesh " << cycle    << " ********"  << std::endl;
          refine_mesh();
        }// second loop
  

write convergence tables to file

      if (known_solution)
        {
          format_convergence_tables();
          std::string error_filename = "error"+Method+elements+".tex";
          std::ofstream error_table_file(error_filename.c_str());
          std::string dual_error_filename = "dual_error"+Method+elements+".tex";
          std::ofstream dual_error_table_file(dual_error_filename.c_str());
          convergence_table.write_tex(error_table_file);
          dual_convergence_table.write_tex(dual_error_table_file);
        }
    }//run()
  
  }//namespace
  
  /**********************************************************************************************/

The main function

  int main ()
  {
    try
      {
        using namespace dealii;
        using namespace nsp;
        deallog.depth_console (0);
  
        ParameterHandler prm;
        ParameterReader param(prm);
        param.read_parameters("EPT.prm");
        ElastoplasticTorsion<2> ElastoplasticTorsionProblem(prm);
        ElastoplasticTorsionProblem .run ();
      }
    catch (std::exception &exc)
      {
        std::cerr << std::endl << std::endl
                  << "----------------------------------------------------"    << std::endl;
        std::cerr << "Exception on processing: " << std::endl
                  << exc.what() << std::endl
                  << "Aborting!" << std::endl
                  << "----------------------------------------------------"
                  << std::endl;
  
        return 1;
      }
    catch (...)
      {
        std::cerr << std::endl << std::endl
                  << "----------------------------------------------------"
                  << std::endl;
        std::cerr << "Unknown exception!" << std::endl
                  << "Aborting!" << std::endl
                  << "----------------------------------------------------"
                  << std::endl;
        return 1;
      }
    return 0;
  }