From: sfloresm <sflores@dim.uchile.cl>
Date: Mon, 9 Nov 2015 13:46:38 +0000 (-0300)
Subject: Solve p-laplacian elliptic problems for large p
X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=53c587d95106d9dab024ffa3cabee764fbfd4b49;p=code-gallery.git

Solve p-laplacian elliptic problems for large p
---

diff --git a/ElastoplasticTorsion/CMakeLists.txt b/ElastoplasticTorsion/CMakeLists.txt
new file mode 100644
index 0000000..82fb113
--- /dev/null
+++ b/ElastoplasticTorsion/CMakeLists.txt
@@ -0,0 +1,40 @@
+##
+#  CMake script for the Elastoplastic Torsion problem
+#   at the deal.ii code-gallery 
+##
+
+# Set the name of the project and target:
+SET(TARGET "ElastoplasticTorsion")
+
+# Declare all source files the target consists of. Here, this is only
+# the one step-X.cc file, but as you expand your project you may wish
+# to add other source files as well. If your project becomes much larger,
+# you may want to either replace the following statement by something like
+#    FILE(GLOB_RECURSE TARGET_SRC  "source/*.cc")
+#    FILE(GLOB_RECURSE TARGET_INC  "include/*.h")
+#    SET(TARGET_SRC ${TARGET_SRC}  ${TARGET_INC}) 
+# or switch altogether to the large project CMakeLists.txt file discussed
+# in the "CMake in user projects" page accessible from the "User info"
+# page of the documentation.
+SET(TARGET_SRC
+  ${TARGET}.cc
+  )
+
+# Usually, you will not need to modify anything beyond this point...
+
+CMAKE_MINIMUM_REQUIRED(VERSION 2.8.8)
+
+FIND_PACKAGE(deal.II 8.0 QUIET
+  HINTS ${deal.II_DIR} ${DEAL_II_DIR} ../ ../../ $ENV{DEAL_II_DIR}
+  )
+IF(NOT ${deal.II_FOUND})
+  MESSAGE(FATAL_ERROR "\n"
+    "*** Could not locate a (sufficiently recent) version of deal.II. ***\n\n"
+    "You may want to either pass a flag -DDEAL_II_DIR=/path/to/deal.II to cmake\n"
+    "or set an environment variable \"DEAL_II_DIR\" that contains this path."
+    )
+ENDIF()
+
+DEAL_II_INITIALIZE_CACHED_VARIABLES()
+PROJECT(${TARGET})
+DEAL_II_INVOKE_AUTOPILOT()
diff --git a/ElastoplasticTorsion/EPT.prm b/ElastoplasticTorsion/EPT.prm
new file mode 100644
index 0000000..437629b
--- /dev/null
+++ b/ElastoplasticTorsion/EPT.prm
@@ -0,0 +1,71 @@
+# Listing of Parameters For the ElastoplasticTorsion Solver
+
+
+subsection Global Parameters
+	# Penalization parameter
+	set p = 100.0
+
+	# Boolean whether exact solution to compare exists
+	set known_solution=true
+end
+
+
+subsection Mesh & Refinement Parameters
+
+	# Choose the domain: 0 (circle), 1 (rectangle) or 2 (Glowinski's)
+	set Code for the domain=0 	
+
+	# Number of global mesh refinement steps applied to initial coarse grid
+	set No of initial refinements=5
+
+	# Number of global adaptive mesh refinements
+	set No of adaptive refinements=7	
+
+	# Refinement threshold
+	set top_fraction_of_cells=0.2
+
+	# Coarsening threshold
+	set bottom_fraction_of_cells=0.05
+end
+
+subsection Algorithm Parameters
+
+	#Choose of the method 0: Preconditioned descent, 1: Newton Method
+	set Descent_direction=1
+
+	# initial p
+	set init_p=25
+	
+	# step for increasing p
+	set delta_p=25
+	
+	# Maximum Number of CG iterations
+	set Max_CG_it=2000
+	
+	# Tolerance for CG iterations
+	set CG_tol=1e-9
+	
+	# initial step-length	
+	set init_step_length=1e-1
+
+	# Maximum Number of Line-Search iterations
+	set max_LS_it=50
+
+	# Line-Search tolerance constant (c1 in Nocedal-Wright)
+	set line_search_tolerence=1e-3
+		
+	# Maximum Number of inner iterations
+	set Max_inner=1000
+		
+	# Threshold on norm of the derivative to declare optimality achieved
+	set eps=1e-6	
+		
+	# Threshold on norm of the derivative to declare optimality achieved in highly refined mesh
+	set hi_eps=1e-7
+	# Number of adaptive refinement before change convergence threshold
+	set hi_th=7
+end
+
+
+
+
diff --git a/ElastoplasticTorsion/ElastoplasticTorsion.cc b/ElastoplasticTorsion/ElastoplasticTorsion.cc
new file mode 100644
index 0000000..7ce1f82
--- /dev/null
+++ b/ElastoplasticTorsion/ElastoplasticTorsion.cc
@@ -0,0 +1,1367 @@
+/* ---------------------------------------------------------------------
+ *
+ * Copyright (C) 2010 - 2015 by the deal.II authors
+ *                and Salvador Flores.
+ *
+ *
+ *
+ * This is free software; you can use it, redistribute
+ * it, and/or modify it under the terms of the GNU Lesser General
+ * Public License as published by the Free Software Foundation; either
+ * version 2.1 of the License, or (at your option) any later version.
+ * The full text of the license can be found in the file LICENSE at
+ * the top level of the deal.II distribution.
+ *
+ * ---------------------------------------------------------------------
+ *
+ * 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
+
+      step-15 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/smartpointer.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/compressed_sparsity_pattern.h>
+#include <deal.II/lac/solver_cg.h>
+#include <deal.II/lac/precondition.h>
+#include <deal.II/lac/constraint_matrix.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/tria_boundary_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 Subscriptor
+  {
+  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.read_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;
+    virtual Tensor<1,dim> gradient (const Point<dim> &pto, const unsigned int component = 0) const;
+  };
+
+  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;
+
+    virtual
+    std::vector<DataComponentInterpretation::DataComponentInterpretation>
+    get_data_component_interpretation () const;
+    virtual UpdateFlags get_needed_update_flags () const;
+  };
+
+
+  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);
+    // 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;
+    ConstraintMatrix     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;
+  };
+
+
+  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;
+  };
+
+  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
+
+
+  /*************************************************************/
+// formating
+
+  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 QTrapez<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());
+    CompressedSparsityPattern 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 coeffcients 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,
+                                              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 ()
+  {
+    Vector<float> estimated_error_per_cell (triangulation.n_active_cells());
+    KellyErrorEstimator<dim>::estimate (dof_handler,
+                                        QGauss<dim-1>(3),
+                                        typename FunctionMap<dim>::type(),
+                                        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);
+    Vector<double> tmp(dof_handler.n_dofs());
+    solution_transfer.interpolate(present_solution, tmp);
+    present_solution = tmp;
+    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 QTrapez<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 wiht 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 HyperBallBoundary<dim> boundary;
+        triangulation.set_boundary (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 ();
+
+    // initalize 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();
+        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();
+        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;
+}
diff --git a/ElastoplasticTorsion/doc/ElastoplasticTorsion-doc.pdf b/ElastoplasticTorsion/doc/ElastoplasticTorsion-doc.pdf
new file mode 100644
index 0000000..3df5d9d
Binary files /dev/null and b/ElastoplasticTorsion/doc/ElastoplasticTorsion-doc.pdf differ
diff --git a/ElastoplasticTorsion/doc/author b/ElastoplasticTorsion/doc/author
new file mode 100644
index 0000000..edaf4ed
--- /dev/null
+++ b/ElastoplasticTorsion/doc/author
@@ -0,0 +1 @@
+Salvador Flores <sflores@dim.uchile.cl	>
diff --git a/ElastoplasticTorsion/doc/builds-on b/ElastoplasticTorsion/doc/builds-on
new file mode 100644
index 0000000..8aa2ce0
--- /dev/null
+++ b/ElastoplasticTorsion/doc/builds-on
@@ -0,0 +1 @@
+step-15 step-29