adaptivity_ex2.C

/* The libMesh Finite Element Library. */
/* Copyright (C) 2003  Benjamin S. Kirk */

/* This library is free software; you can 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. */

/* This library is distributed in the hope that it will be useful, */
/* but WITHOUT ANY WARRANTY; without even the implied warranty of */
/* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU */
/* Lesser General Public License for more details. */

/* You should have received a copy of the GNU Lesser General Public */
/* License along with this library; if not, write to the Free Software */
/* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA */

  // <h1>Adaptivity Example 2 - Solving a Transient System with Adaptive Mesh Refinement</h1>
  // 
  // This example shows how a simple, linear transient
  // system can be solved in parallel.  The system is simple
  // scalar convection-diffusion with a specified external
  // velocity.  The initial condition is given, and the
  // solution is advanced in time with a standard Crank-Nicolson
  // time-stepping strategy.
  //
  // Also, we use this example to demonstrate writing out and reading
  // in of solutions. We do 25 time steps, then save the solution
  // and do another 25 time steps starting from the saved solution.
 
// C++ include files that we need
#include <iostream>
#include <algorithm>
#include <cstdlib> // *must* precede <cmath> for proper std:abs() on PGI, Sun Studio CC
#include <cmath>

// Basic include file needed for the mesh functionality.
#include "libmesh/libmesh.h"
#include "libmesh/serial_mesh.h"
#include "libmesh/mesh_refinement.h"
#include "libmesh/gmv_io.h"
#include "libmesh/equation_systems.h"
#include "libmesh/fe.h"
#include "libmesh/quadrature_gauss.h"
#include "libmesh/dof_map.h"
#include "libmesh/sparse_matrix.h"
#include "libmesh/numeric_vector.h"
#include "libmesh/dense_matrix.h"
#include "libmesh/dense_vector.h"

#include "libmesh/getpot.h"

// This example will solve a linear transient system,
// so we need to include the \p TransientLinearImplicitSystem definition.
#include "libmesh/transient_system.h"
#include "libmesh/linear_implicit_system.h"
#include "libmesh/vector_value.h"

// To refine the mesh we need an \p ErrorEstimator
// object to figure out which elements to refine.
#include "libmesh/error_vector.h"
#include "libmesh/kelly_error_estimator.h"

// The definition of a geometric element
#include "libmesh/elem.h"

// Bring in everything from the libMesh namespace
using namespace libMesh;

// Function prototype.  This function will assemble the system
// matrix and right-hand-side at each time step.  Note that
// since the system is linear we technically do not need to
// assmeble the matrix at each time step, but we will anyway.
// In subsequent examples we will employ adaptive mesh refinement,
// and with a changing mesh it will be necessary to rebuild the
// system matrix.
void assemble_cd (EquationSystems& es,
                  const std::string& system_name);

// Function prototype.  This function will initialize the system.
// Initialization functions are optional for systems.  They allow
// you to specify the initial values of the solution.  If an
// initialization function is not provided then the default (0)
// solution is provided.
void init_cd (EquationSystems& es,
              const std::string& system_name);

// Exact solution function prototype.  This gives the exact
// solution as a function of space and time.  In this case the
// initial condition will be taken as the exact solution at time 0,
// as will the Dirichlet boundary conditions at time t.
Real exact_solution (const Real x,
                     const Real y,
                     const Real t);

Number exact_value (const Point& p,
                    const Parameters& parameters,
                    const std::string&,
                    const std::string&)
{
  return exact_solution(p(0), p(1), parameters.get<Real> ("time"));
}



// Begin the main program.  Note that the first
// statement in the program throws an error if
// you are in complex number mode, since this
// example is only intended to work with real
// numbers.
int main (int argc, char** argv)
{
  // Initialize libMesh.
  LibMeshInit init (argc, argv);

#ifndef LIBMESH_ENABLE_AMR
  libmesh_example_assert(false, "--enable-amr");
#else

  // Our Trilinos interface does not yet support adaptive transient
  // problems
  libmesh_example_assert(libMesh::default_solver_package() != TRILINOS_SOLVERS, "--enable-petsc");

  // Brief message to the user regarding the program name
  // and command line arguments.

  // Use commandline parameter to specify if we are to
  // read in an initial solution or generate it ourself
  std::cout << "Usage:\n"
    <<"\t " << argv[0] << " -init_timestep 0\n"
    << "OR\n"
    <<"\t " << argv[0] << " -read_solution -init_timestep 26\n"
    << std::endl;

  std::cout << "Running: " << argv[0];

  for (int i=1; i<argc; i++)
    std::cout << " " << argv[i];

  std::cout << std::endl << std::endl;

  // Create a GetPot object to parse the command line
  GetPot command_line (argc, argv);


  // This boolean value is obtained from the command line, it is true
  // if the flag "-read_solution" is present, false otherwise.
  // It indicates whether we are going to read in
  // the mesh and solution files "saved_mesh.xda" and "saved_solution.xda"
  // or whether we are going to start from scratch by just reading
  // "mesh.xda"
  const bool read_solution   = command_line.search("-read_solution");

  // This value is also obtained from the commandline and it specifies the
  // initial value for the t_step looping variable. We must
  // distinguish between the two cases here, whether we read in the 
  // solution or we started from scratch, so that we do not overwrite the
  // gmv output files.
  unsigned int init_timestep = 0;
  
  // Search the command line for the "init_timestep" flag and if it is
  // present, set init_timestep accordingly.
  if(command_line.search("-init_timestep"))
    init_timestep = command_line.next(0);
  else
    {
      if (libMesh::processor_id() == 0)
        std::cerr << "ERROR: Initial timestep not specified\n" << std::endl;

      // This handy function will print the file name, line number,
      // and then abort.  Currrently the library does not use C++
      // exception handling.
      libmesh_error();
    }

  // This value is also obtained from the command line, and specifies
  // the number of time steps to take.
  unsigned int n_timesteps = 0;

  // Again do a search on the command line for the argument
  if(command_line.search("-n_timesteps"))
    n_timesteps = command_line.next(0);
  else
    {
      std::cout << "ERROR: Number of timesteps not specified\n" << std::endl;
      libmesh_error();
    }


  // Skip this 2D example if libMesh was compiled as 1D-only.
  libmesh_example_assert(2 <= LIBMESH_DIM, "2D support");

  // Create a new mesh.
  // We still need some work on automatic parallel restarts with
  // ParallelMesh
  SerialMesh mesh;

  // Create an equation systems object.
  EquationSystems equation_systems (mesh);
  MeshRefinement mesh_refinement (mesh);

  // First we process the case where we do not read in the solution
  if(!read_solution)
    {
      // Read the mesh from file.
      mesh.read ("mesh.xda");

      // Again do a search on the command line for an argument
      unsigned int n_refinements = 5;
      if(command_line.search("-n_refinements"))
        n_refinements = command_line.next(0);

      // Uniformly refine the mesh 5 times
      if(!read_solution)
        mesh_refinement.uniformly_refine (n_refinements);

      // Print information about the mesh to the screen.
      mesh.print_info();
      
      
      // Declare the system and its variables.
      // Begin by creating a transient system
      // named "Convection-Diffusion".
      TransientLinearImplicitSystem & system = 
        equation_systems.add_system<TransientLinearImplicitSystem> 
        ("Convection-Diffusion");

      // Adds the variable "u" to "Convection-Diffusion".  "u"
      // will be approximated using first-order approximation.
      system.add_variable ("u", FIRST);

      // Give the system a pointer to the matrix assembly
      // and initialization functions.
      system.attach_assemble_function (assemble_cd);
      system.attach_init_function (init_cd);

      // Initialize the data structures for the equation system.
      equation_systems.init ();
    }
  // Otherwise we read in the solution and mesh
  else 
    {
      // Read in the mesh stored in "saved_mesh.xda"
      mesh.read("saved_mesh.xda");

      // Print information about the mesh to the screen.
      mesh.print_info();

      // Read in the solution stored in "saved_solution.xda"
      equation_systems.read("saved_solution.xda", libMeshEnums::READ);

      // Get a reference to the system so that we can call update() on it
      TransientLinearImplicitSystem & system = 
        equation_systems.get_system<TransientLinearImplicitSystem> 
        ("Convection-Diffusion");

      // We need to call update to put system in a consistent state
      // with the solution that was read in
      system.update();

      // Attach the same matrix assembly function as above. Note, we do not
      // have to attach an init() function since we are initializing the
      // system by reading in "saved_solution.xda"
      system.attach_assemble_function (assemble_cd);

      // Print out the H1 norm of the saved solution, for verification purposes:
      Real H1norm = system.calculate_norm(*system.solution, SystemNorm(H1));

      std::cout << "Initial H1 norm = " << H1norm << std::endl << std::endl;
    }

  // Prints information about the system to the screen.
  equation_systems.print_info();
    
  equation_systems.parameters.set<unsigned int>
    ("linear solver maximum iterations") = 250;
  equation_systems.parameters.set<Real>
    ("linear solver tolerance") = TOLERANCE;
    
  if(!read_solution)
    // Write out the initial condition
    GMVIO(mesh).write_equation_systems ("out.gmv.000",
                                        equation_systems);
  else
    // Write out the solution that was read in
    GMVIO(mesh).write_equation_systems ("solution_read_in.gmv",
                                        equation_systems);

  // The Convection-Diffusion system requires that we specify
  // the flow velocity.  We will specify it as a RealVectorValue
  // data type and then use the Parameters object to pass it to
  // the assemble function.
  equation_systems.parameters.set<RealVectorValue>("velocity") = 
    RealVectorValue (0.8, 0.8);

  // The Convection-Diffusion system also requires a specified
  // diffusivity.  We use an isotropic (hence Real) value.
  equation_systems.parameters.set<Real>("diffusivity") = 0.01;
    
  // Solve the system "Convection-Diffusion".  This will be done by
  // looping over the specified time interval and calling the
  // \p solve() member at each time step.  This will assemble the
  // system and call the linear solver.

  // Since only \p TransientLinearImplicitSystems (and systems
  // derived from them) contain old solutions, to use the
  // old_local_solution later we now need to specify the system
  // type when we ask for it.
  TransientLinearImplicitSystem &  system =
    equation_systems.get_system<TransientLinearImplicitSystem>("Convection-Diffusion");

  const Real dt = 0.025;
  system.time   = init_timestep*dt;
  
  // We do 25 timesteps both before and after writing out the
  // intermediate solution
  for(unsigned int t_step=init_timestep; 
                   t_step<(init_timestep+n_timesteps); 
                   t_step++)
    {
      // Increment the time counter, set the time and the
      // time step size as parameters in the EquationSystem.
      system.time += dt;

      equation_systems.parameters.set<Real> ("time") = system.time;
      equation_systems.parameters.set<Real> ("dt")   = dt;

      // A pretty update message
      std::cout << " Solving time step ";
      
      // Add a set of scope braces to enforce data locality.
      {
        std::ostringstream out;

        out << std::setw(2)
            << std::right
            << t_step
            << ", time="
            << std::fixed
            << std::setw(6)
            << std::setprecision(3)
            << std::setfill('0')
            << std::left
            << system.time
            <<  "...";

        std::cout << out.str() << std::endl;
      }
      
      // At this point we need to update the old
      // solution vector.  The old solution vector
      // will be the current solution vector from the
      // previous time step.

      *system.old_local_solution = *system.current_local_solution;
      
      // The number of refinement steps per time step.
      const unsigned int max_r_steps = 2;
      
      // A refinement loop.
      for (unsigned int r_step=0; r_step<max_r_steps; r_step++)
        {
          // Assemble & solve the linear system
          system.solve();

          // Print out the H1 norm, for verification purposes:
          Real H1norm = system.calculate_norm(*system.solution, SystemNorm(H1));

          std::cout << "H1 norm = " << H1norm << std::endl;
          
          // Possibly refine the mesh
          if (r_step+1 != max_r_steps)
            {
              std::cout << "  Refining the mesh..." << std::endl;

              // The \p ErrorVector is a particular \p StatisticsVector
              // for computing error information on a finite element mesh.
              ErrorVector error;

              // The \p ErrorEstimator class interrogates a finite element
              // solution and assigns to each element a positive error value.
              // This value is used for deciding which elements to refine
              // and which to coarsen.
              //ErrorEstimator* error_estimator = new KellyErrorEstimator;
              KellyErrorEstimator error_estimator;
              
              // Compute the error for each active element using the provided
              // \p flux_jump indicator.  Note in general you will need to
              // provide an error estimator specifically designed for your
              // application.
              error_estimator.estimate_error (system,
                                              error);
              
              // This takes the error in \p error and decides which elements
              // will be coarsened or refined.  Any element within 20% of the
              // maximum error on any element will be refined, and any
              // element within 7% of the minimum error on any element might
              // be coarsened. Note that the elements flagged for refinement
              // will be refined, but those flagged for coarsening _might_ be
              // coarsened.
              mesh_refinement.refine_fraction() = 0.80;
              mesh_refinement.coarsen_fraction() = 0.07;
              mesh_refinement.max_h_level() = 5;
              mesh_refinement.flag_elements_by_error_fraction (error);
              
              // This call actually refines and coarsens the flagged
              // elements.
              mesh_refinement.refine_and_coarsen_elements();
              
              // This call reinitializes the \p EquationSystems object for
              // the newly refined mesh.  One of the steps in the
              // reinitialization is projecting the \p solution,
              // \p old_solution, etc... vectors from the old mesh to
              // the current one.
              equation_systems.reinit ();
            }            
        }
        
      // Again do a search on the command line for an argument
      unsigned int output_freq = 10;
      if(command_line.search("-output_freq"))
        output_freq = command_line.next(0);

      // Output every 10 timesteps to file.
      if ( (t_step+1)%output_freq == 0)
        {
          std::ostringstream file_name;

          file_name << "out.gmv."
                    << std::setw(3)
                    << std::setfill('0')
                    << std::right
                    << t_step+1;

          GMVIO(mesh).write_equation_systems (file_name.str(),
                                              equation_systems);
        }
    }

  if(!read_solution)
    {
      // Print out the H1 norm of the saved solution, for verification purposes:
      Real H1norm = system.calculate_norm(*system.solution, SystemNorm(H1));

      std::cout << "Final H1 norm = " << H1norm << std::endl << std::endl;

      mesh.write("saved_mesh.xda");
      equation_systems.write("saved_solution.xda", libMeshEnums::WRITE);
      GMVIO(mesh).write_equation_systems ("saved_solution.gmv",
                                          equation_systems);
    }
#endif // #ifndef LIBMESH_ENABLE_AMR
  
  return 0;
}

// Here we define the initialization routine for the
// Convection-Diffusion system.  This routine is
// responsible for applying the initial conditions to
// the system.
void init_cd (EquationSystems& es,
              const std::string& system_name)
{
  // It is a good idea to make sure we are initializing
  // the proper system.
  libmesh_assert_equal_to (system_name, "Convection-Diffusion");

  // Get a reference to the Convection-Diffusion system object.
  TransientLinearImplicitSystem & system =
    es.get_system<TransientLinearImplicitSystem>("Convection-Diffusion");

  // Project initial conditions at time 0
  es.parameters.set<Real> ("time") = system.time = 0;
  
  system.project_solution(exact_value, NULL, es.parameters);
}



// This function defines the assembly routine which
// will be called at each time step.  It is responsible
// for computing the proper matrix entries for the
// element stiffness matrices and right-hand sides.
void assemble_cd (EquationSystems& es,
                  const std::string& system_name)
{
#ifdef LIBMESH_ENABLE_AMR
  // It is a good idea to make sure we are assembling
  // the proper system.
  libmesh_assert_equal_to (system_name, "Convection-Diffusion");
  
  // Get a constant reference to the mesh object.
  const MeshBase& mesh = es.get_mesh();
  
  // The dimension that we are running
  const unsigned int dim = mesh.mesh_dimension();
  
  // Get a reference to the Convection-Diffusion system object.
  TransientLinearImplicitSystem & system =
    es.get_system<TransientLinearImplicitSystem> ("Convection-Diffusion");
  
  // Get the Finite Element type for the first (and only) 
  // variable in the system.
  FEType fe_type = system.variable_type(0);
  
  // Build a Finite Element object of the specified type.  Since the
  // \p FEBase::build() member dynamically creates memory we will
  // store the object as an \p AutoPtr<FEBase>.  This can be thought
  // of as a pointer that will clean up after itself.
  AutoPtr<FEBase> fe      (FEBase::build(dim, fe_type));
  AutoPtr<FEBase> fe_face (FEBase::build(dim, fe_type));
  
  // A Gauss quadrature rule for numerical integration.
  // Let the \p FEType object decide what order rule is appropriate.
  QGauss qrule (dim,   fe_type.default_quadrature_order());
  QGauss qface (dim-1, fe_type.default_quadrature_order());

  // Tell the finite element object to use our quadrature rule.
  fe->attach_quadrature_rule      (&qrule);
  fe_face->attach_quadrature_rule (&qface);

  // Here we define some references to cell-specific data that
  // will be used to assemble the linear system.  We will start
  // with the element Jacobian * quadrature weight at each integration point.
  const std::vector<Real>& JxW      = fe->get_JxW();
  const std::vector<Real>& JxW_face = fe_face->get_JxW();
  
  // The element shape functions evaluated at the quadrature points.
  const std::vector<std::vector<Real> >& phi = fe->get_phi();
  const std::vector<std::vector<Real> >& psi = fe_face->get_phi();

  // The element shape function gradients evaluated at the quadrature
  // points.
  const std::vector<std::vector<RealGradient> >& dphi = fe->get_dphi();

  // The XY locations of the quadrature points used for face integration
  const std::vector<Point>& qface_points = fe_face->get_xyz();
    
  // A reference to the \p DofMap object for this system.  The \p DofMap
  // object handles the index translation from node and element numbers
  // to degree of freedom numbers.  We will talk more about the \p DofMap
  // in future examples.
  const DofMap& dof_map = system.get_dof_map();
  
  // Define data structures to contain the element matrix
  // and right-hand-side vector contribution.  Following
  // basic finite element terminology we will denote these
  // "Ke" and "Fe".
  DenseMatrix<Number> Ke;
  DenseVector<Number> Fe;
  
  // This vector will hold the degree of freedom indices for
  // the element.  These define where in the global system
  // the element degrees of freedom get mapped.
  std::vector<dof_id_type> dof_indices;

  // Here we extract the velocity & parameters that we put in the
  // EquationSystems object.
  const RealVectorValue velocity =
    es.parameters.get<RealVectorValue> ("velocity");

  const Real diffusivity =
    es.parameters.get<Real> ("diffusivity");

  const Real dt = es.parameters.get<Real>   ("dt");

  // Now we will loop over all the elements in the mesh that
  // live on the local processor. We will compute the element
  // matrix and right-hand-side contribution.  Since the mesh
  // will be refined we want to only consider the ACTIVE elements,
  // hence we use a variant of the \p active_elem_iterator.
  MeshBase::const_element_iterator       el     = mesh.active_local_elements_begin();
  const MeshBase::const_element_iterator end_el = mesh.active_local_elements_end(); 
  
  for ( ; el != end_el; ++el)
    {    
      // Store a pointer to the element we are currently
      // working on.  This allows for nicer syntax later.
      const Elem* elem = *el;
      
      // Get the degree of freedom indices for the
      // current element.  These define where in the global
      // matrix and right-hand-side this element will
      // contribute to.
      dof_map.dof_indices (elem, dof_indices);

      // Compute the element-specific data for the current
      // element.  This involves computing the location of the
      // quadrature points (q_point) and the shape functions
      // (phi, dphi) for the current element.
      fe->reinit (elem);
      
      // Zero the element matrix and right-hand side before
      // summing them.  We use the resize member here because
      // the number of degrees of freedom might have changed from
      // the last element.  Note that this will be the case if the
      // element type is different (i.e. the last element was a
      // triangle, now we are on a quadrilateral).
      Ke.resize (dof_indices.size(),
                 dof_indices.size());

      Fe.resize (dof_indices.size());
      
      // Now we will build the element matrix and right-hand-side.
      // Constructing the RHS requires the solution and its
      // gradient from the previous timestep.  This myst be
      // calculated at each quadrature point by summing the
      // solution degree-of-freedom values by the appropriate
      // weight functions.
      for (unsigned int qp=0; qp<qrule.n_points(); qp++)
        {
          // Values to hold the old solution & its gradient.
          Number   u_old = 0.;
          Gradient grad_u_old;
          
          // Compute the old solution & its gradient.
          for (unsigned int l=0; l<phi.size(); l++)
            {
              u_old      += phi[l][qp]*system.old_solution  (dof_indices[l]);
              
              // This will work,
              // grad_u_old += dphi[l][qp]*system.old_solution (dof_indices[l]);
              // but we can do it without creating a temporary like this:
              grad_u_old.add_scaled (dphi[l][qp],system.old_solution (dof_indices[l]));
            }
          
          // Now compute the element matrix and RHS contributions.
          for (unsigned int i=0; i<phi.size(); i++)
            {
              // The RHS contribution
              Fe(i) += JxW[qp]*(
                                // Mass matrix term
                                u_old*phi[i][qp] + 
                                -.5*dt*(
                                        // Convection term
                                        // (grad_u_old may be complex, so the
                                        // order here is important!)
                                        (grad_u_old*velocity)*phi[i][qp] +
                                        
                                        // Diffusion term
                                        diffusivity*(grad_u_old*dphi[i][qp]))     
                                );
              
              for (unsigned int j=0; j<phi.size(); j++)
                {
                  // The matrix contribution
                  Ke(i,j) += JxW[qp]*(
                                      // Mass-matrix
                                      phi[i][qp]*phi[j][qp] + 
                                      .5*dt*(
                                             // Convection term
                                             (velocity*dphi[j][qp])*phi[i][qp] +
                                             // Diffusion term
                                             diffusivity*(dphi[i][qp]*dphi[j][qp]))      
                                      );
                }
            } 
        } 

      // At this point the interior element integration has
      // been completed.  However, we have not yet addressed
      // boundary conditions.  For this example we will only
      // consider simple Dirichlet boundary conditions imposed
      // via the penalty method. 
      //        
      // The following loops over the sides of the element.
      // If the element has no neighbor on a side then that
      // side MUST live on a boundary of the domain.
      {
        // The penalty value.  
        const Real penalty = 1.e10;

        // The following loops over the sides of the element.
        // If the element has no neighbor on a side then that
        // side MUST live on a boundary of the domain.
        for (unsigned int s=0; s<elem->n_sides(); s++)
          if (elem->neighbor(s) == NULL)
            {
              fe_face->reinit(elem,s);
              
              for (unsigned int qp=0; qp<qface.n_points(); qp++)
                {
                  const Number value = exact_solution (qface_points[qp](0),
                                                       qface_points[qp](1),
                                                       system.time);
                                                       
                  // RHS contribution
                  for (unsigned int i=0; i<psi.size(); i++)
                    Fe(i) += penalty*JxW_face[qp]*value*psi[i][qp];

                  // Matrix contribution
                  for (unsigned int i=0; i<psi.size(); i++)
                    for (unsigned int j=0; j<psi.size(); j++)
                      Ke(i,j) += penalty*JxW_face[qp]*psi[i][qp]*psi[j][qp];
                }
            } 
      } 

      
      // We have now built the element matrix and RHS vector in terms
      // of the element degrees of freedom.  However, it is possible
      // that some of the element DOFs are constrained to enforce
      // solution continuity, i.e. they are not really "free".  We need
      // to constrain those DOFs in terms of non-constrained DOFs to
      // ensure a continuous solution.  The
      // \p DofMap::constrain_element_matrix_and_vector() method does
      // just that.
      dof_map.constrain_element_matrix_and_vector (Ke, Fe, dof_indices);
      
      // The element matrix and right-hand-side are now built
      // for this element.  Add them to the global matrix and
      // right-hand-side vector.  The \p SparseMatrix::add_matrix()
      // and \p NumericVector::add_vector() members do this for us.
      system.matrix->add_matrix (Ke, dof_indices);
      system.rhs->add_vector    (Fe, dof_indices);
      
    }
  // Finished computing the sytem matrix and right-hand side.
#endif // #ifdef LIBMESH_ENABLE_AMR
}