adaptivity_ex3.C File Reference

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Functions
void	assemble_laplace (EquationSystems &es, const std::string &system_name)
Number	exact_solution (const Point &p, const Parameters &, const std::string &, const std::string &)
Gradient	exact_derivative (const Point &p, const Parameters &, const std::string &, const std::string &)
int	main (int argc, char **argv)

Variables
unsigned int	dim = 2
unsigned int	n_vars = 1
bool	singularity = true

Function Documentation

assemble_laplace()

void assemble_laplace	(	EquationSystems &	es,
		const std::string &	system_name
	)

Definition at line 695 of file adaptivity_ex3.C.

References libMesh::SparseMatrix< T >::add_matrix(), libMesh::NumericVector< T >::add_vector(), libMesh::FEGenericBase< OutputType >::build(), libMesh::FEType::default_quadrature_rule(), libMesh::System::get_dof_map(), libMesh::EquationSystems::get_mesh(), libMesh::EquationSystems::get_system(), libMesh::ImplicitSystem::get_system_matrix(), libMesh::libmesh_ignore(), mesh, libMesh::MeshBase::mesh_dimension(), n_vars, libMesh::PerfLog::pop(), libMesh::Utility::pow(), libMesh::PerfLog::push(), libMesh::Real, libMesh::DenseSubVector< T >::reposition(), libMesh::DenseSubMatrix< T >::reposition(), libMesh::DenseVector< T >::resize(), libMesh::DenseMatrix< T >::resize(), libMesh::ExplicitSystem::rhs, and singularity.

Referenced by main().

{
  // Ignore unused parameter warnings when !LIBMESH_ENABLE_AMR.
  libmesh_ignore(es, 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, "Laplace");

  // Declare a performance log.  Give it a descriptive
  // string to identify what part of the code we are
  // logging, since there may be many PerfLogs in an
  // application.
  PerfLog perf_log ("Matrix Assembly", false);

  // Get a constant reference to the mesh object.
  const MeshBase & mesh = es.get_mesh();

  // The dimension that we are running
  const unsigned int mesh_dim = mesh.mesh_dimension();

  // Get a reference to the LinearImplicitSystem we are solving
  LinearImplicitSystem & system = es.get_system<LinearImplicitSystem>("Laplace");

  // A reference to the DofMap object for this system.  The DofMap
  // object handles the index translation from node and element numbers
  // to degree of freedom numbers.  We will talk more about the DofMap
  // in future examples.
  const DofMap & dof_map = system.get_dof_map();

  // Get a constant reference to the Finite Element type
  // for the first (and only) variable type in the system.
  FEType fe_type = dof_map.variable_type(0);

  // Build a Finite Element object of the specified type.  Since the
  // FEBase::build() member dynamically creates memory we will
  // store the object as a std::unique_ptr<FEBase>.  This can be thought
  // of as a pointer that will clean up after itself.
  std::unique_ptr<FEBase> fe      (FEBase::build(mesh_dim, fe_type));
  std::unique_ptr<FEBase> fe_face (FEBase::build(mesh_dim, fe_type));

  // Quadrature rules for numerical integration.
  std::unique_ptr<QBase> qrule(fe_type.default_quadrature_rule(mesh_dim));
  std::unique_ptr<QBase> qface(fe_type.default_quadrature_rule(mesh_dim-1));

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

  // Here we define some references to cell-specific data that
  // will be used to assemble the linear system.
  // We begin with the element Jacobian * quadrature weight at each
  // integration point.
  const std::vector<Real> & JxW      = fe->get_JxW();

  // The physical XY locations of the quadrature points on the element.
  // These might be useful for evaluating spatially varying material
  // properties or forcing functions at the quadrature points.
  const std::vector<Point> & q_point = fe->get_xyz();

  // The element shape functions evaluated at the quadrature points.
  const std::vector<std::vector<Real>> & phi = fe->get_phi();

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

  // 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". More detail is in example 3.
  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;

  // The global system matrix
  SparseMatrix<Number> & matrix = system.get_system_matrix();

  // Now we will loop over all the elements in the mesh.  We will
  // compute the element matrix and right-hand-side contribution.  See
  // example 3 for a discussion of the element iterators.  Here we use
  // the const_active_local_elem_iterator to indicate we only want
  // to loop over elements that are assigned to the local processor
  // which are "active" in the sense of AMR.  This allows each
  // processor to compute its components of the global matrix for
  // active elements while ignoring parent elements which have been
  // refined.
  for (const auto & elem : mesh.active_local_element_ptr_range())
    {
      // Start logging the shape function initialization.
      // This is done through a simple function call with
      // the name of the event to log.
      perf_log.push("elem init");

      // 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);

      const unsigned int n_dofs =
        cast_int<unsigned int>(dof_indices.size());

      // 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 (n_dofs, n_dofs);

      Fe.resize (n_dofs);

      // Stop logging the shape function initialization.
      // If you forget to stop logging an event the PerfLog
      // object will probably catch the error and abort.
      perf_log.pop("elem init");

      // Now we will build the element matrix.  This involves
      // a quadruple loop to integrate the test functions (i) against
      // the trial functions (j) for each variable (v) at each
      // quadrature point (qp).
      //
      // Now start logging the element matrix computation
      perf_log.push ("Ke");

      std::vector<dof_id_type> dof_indices_u;
      dof_map.dof_indices (elem, dof_indices_u, 0);
      const unsigned int n_u_dofs = dof_indices_u.size();
      libmesh_assert_equal_to (n_u_dofs, phi.size());
      libmesh_assert_equal_to (n_u_dofs, dphi.size());

      for (unsigned int v=0; v != n_vars; ++v)
        {
          DenseSubMatrix<Number> Kuu(Ke);
          Kuu.reposition (v*n_u_dofs, v*n_u_dofs, n_u_dofs, n_u_dofs);

          for (unsigned int qp=0; qp<qrule->n_points(); qp++)
            for (unsigned int i=0; i != n_u_dofs; i++)
              for (unsigned int j=0; j != n_u_dofs; j++)
                Kuu(i,j) += JxW[qp]*(dphi[i][qp]*dphi[j][qp]);

          // We need a forcing function to make the 1D case interesting
          if (mesh_dim == 1)
            {
              DenseSubVector<Number> Fu(Fe);
              Fu.reposition (v*n_u_dofs, n_u_dofs);

              for (unsigned int qp=0; qp<qrule->n_points(); qp++)
                {
                  Real x = q_point[qp](0);
                  Real f = singularity ? sqrt(3.)/9.*pow(-x, -4./3.) :
                    cos(x);
                  for (unsigned int i=0; i != n_u_dofs; ++i)
                    Fu(i) += JxW[qp]*phi[i][qp]*f;
                }
            }
        }

      // Stop logging the matrix computation
      perf_log.pop ("Ke");

      // 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 SparseMatrix::add_matrix()
      // and NumericVector::add_vector() members do this for us.
      // Start logging the insertion of the local (element)
      // matrix and vector into the global matrix and vector
      LOG_SCOPE_WITH("matrix insertion", "", perf_log);

      // Use heterogenously here to handle Dirichlet as well as AMR
      // constraints.
      dof_map.heterogenously_constrain_element_matrix_and_vector(Ke, Fe, dof_indices);
      matrix.add_matrix (Ke, dof_indices);
      system.rhs->add_vector    (Fe, dof_indices);
    }

  // That's it.  We don't need to do anything else to the
  // PerfLog.  When it goes out of scope (at this function return)
  // it will print its log to the screen. Pretty easy, huh?
#endif // #ifdef LIBMESH_ENABLE_AMR
}

exact_derivative()

Gradient exact_derivative	(	const Point &	p,
		const Parameters &	,
		const std::string &	,
		const std::string &
	)

Definition at line 629 of file adaptivity_ex3.C.

References dim, libMesh::pi, libMesh::Utility::pow(), libMesh::Real, and singularity.

Referenced by main().

{
  // Gradient value to be returned.
  Gradient gradu;

  // x and y coordinates in space
  const Real x = p(0);
  const Real y = dim > 1 ? p(1) : 0.;

  if (singularity)
    {
      // We can't compute the gradient at x=0, it is not defined.
      libmesh_assert_not_equal_to (x, 0.);

      // For convenience...
      const Real tt = 2./3.;
      const Real ot = 1./3.;

      // The value of the radius, squared
      const Real r2 = x*x + y*y;

      // The boundary value, given by the exact solution,
      // u_exact = r^(2/3)*sin(2*theta/3).
      Real theta = atan2(y, x);

      // Make sure 0 <= theta <= 2*pi
      if (theta < 0)
        theta += 2. * libMesh::pi;

      // du/dx
      gradu(0) = tt*x*pow(r2,-tt)*sin(tt*theta) - pow(r2,ot)*cos(tt*theta)*tt/(1.+y*y/x/x)*y/x/x;

      // du/dy
      if (dim > 1)
        gradu(1) = tt*y*pow(r2,-tt)*sin(tt*theta) + pow(r2,ot)*cos(tt*theta)*tt/(1.+y*y/x/x)*1./x;

      if (dim > 2)
        gradu(2) = 1.;
    }
  else
    {
      const Real z = (dim > 2) ? p(2) : 0;

      gradu(0) = -sin(x) * exp(y) * (1. - z);
      if (dim > 1)
        gradu(1) = cos(x) * exp(y) * (1. - z);
      if (dim > 2)
        gradu(2) = -cos(x) * exp(y);
    }

  return gradu;
}

exact_solution()

Number exact_solution	(	const Point &	p,
		const Parameters &	,
		const std::string &	,
		const std::string &
	)

Definition at line 589 of file adaptivity_ex3.C.

References dim, libMesh::pi, libMesh::Utility::pow(), libMesh::Real, and singularity.

Referenced by main().

{
  const Real x = p(0);
  const Real y = (dim > 1) ? p(1) : 0.;

  if (singularity)
    {
      // The exact solution to the singular problem,
      // u_exact = r^(2/3)*sin(2*theta/3).
      Real theta = atan2(y, x);

      // Make sure 0 <= theta <= 2*pi
      if (theta < 0)
        theta += 2. * libMesh::pi;

      // Make the 3D solution similar
      const Real z = (dim > 2) ? p(2) : 0;

      return pow(x*x + y*y, 1./3.)*sin(2./3.*theta) + z;
    }
  else
    {
      // The exact solution to a nonsingular problem,
      // good for testing ideal convergence rates
      const Real z = (dim > 2) ? p(2) : 0;

      return cos(x) * exp(y) * (1. - z);
    }
}

main()

int main	(	int	argc,
		char **	argv
	)

Definition at line 122 of file adaptivity_ex3.C.

References libMesh::DofMap::add_algebraic_ghosting_functor(), libMesh::DofMap::add_dirichlet_boundary(), libMesh::MeshRefinement::add_p_to_h_refinement(), libMesh::System::add_variable(), libMesh::MeshBase::all_complete_order(), libMesh::MeshBase::all_second_order(), libMesh::MeshTools::Modification::all_tri(), assemble_laplace(), libMesh::System::attach_assemble_function(), libMesh::ExactErrorEstimator::attach_exact_deriv(), libMesh::ExactSolution::attach_exact_deriv(), libMesh::ExactErrorEstimator::attach_exact_value(), libMesh::ExactSolution::attach_exact_value(), libMesh::MeshTools::Generation::build_extrusion(), libMesh::MeshTools::Generation::build_line(), libMesh::MeshBase::clear(), libMesh::MeshRefinement::coarsen_fraction(), libMesh::ExactSolution::compute_error(), libMesh::default_solver_package(), dim, libMesh::PatchRecoveryErrorEstimator::estimate_error(), libMesh::UniformRefinementEstimator::estimate_error(), libMesh::JumpErrorEstimator::estimate_error(), libMesh::ExactErrorEstimator::estimate_error(), exact_derivative(), exact_grad(), exact_solution(), libMesh::ExactSolution::extra_quadrature_order(), libMesh::LinearImplicitSystem::final_linear_residual(), libMesh::MeshRefinement::flag_elements_by_error_fraction(), libMesh::System::get_dof_map(), libMesh::ExactSolution::h1_error(), libMesh::TriangleWrapper::init(), libMesh::INVALID_SOLVER_PACKAGE, libMesh::ExactSolution::l2_error(), libMesh::libmesh_assert(), libMesh::libmesh_isnan(), libMesh::MeshRefinement::max_h_level(), libMesh::ErrorVector::mean(), mesh, libMesh::LinearImplicitSystem::n_linear_iterations(), n_vars, libMesh::out, libMesh::ErrorVector::plot_error(), libMesh::Utility::pow(), libMesh::MeshBase::read(), libMesh::Real, libMesh::MeshRefinement::refine_and_coarsen_elements(), libMesh::MeshRefinement::refine_fraction(), libMesh::HPSingularity::select_refinement(), libMesh::HPCoarsenTest::select_refinement(), libMesh::HPSingularity::singular_points, singularity, libMesh::LinearImplicitSystem::solve(), libMesh::MeshRefinement::switch_h_to_p_refinement(), libMesh::TOLERANCE, libMesh::MeshRefinement::uniformly_p_refine(), libMesh::MeshRefinement::uniformly_refine(), libMesh::JumpErrorEstimator::use_unweighted_quadrature_rules, and libMesh::ExodusII_IO::write_equation_systems().

{
  // Initialize libMesh.
  LibMeshInit init (argc, argv);

  // This example requires a linear solver package.
  libmesh_example_requires(libMesh::default_solver_package() != INVALID_SOLVER_PACKAGE,
                           "--enable-petsc, --enable-trilinos, or --enable-eigen");

  // Single precision is inadequate for p refinement
  libmesh_example_requires(sizeof(Real) > 4, "--disable-singleprecision");

  // Skip adaptive examples on a non-adaptive libMesh build
#ifndef LIBMESH_ENABLE_AMR
  libmesh_example_requires(false, "--enable-amr");
#else

  // Parse the input file
  GetPot input_file("adaptivity_ex3.in");

  // But allow the command line to override it.
  input_file.parse_command_line(argc, argv);

  // Read in parameters from the input file
  const unsigned int max_r_steps    = input_file("max_r_steps", 3);
  const unsigned int max_r_level    = input_file("max_r_level", 3);
  const Real refine_percentage      = input_file("refine_percentage", 0.5);
  const Real coarsen_percentage     = input_file("coarsen_percentage", 0.5);
  const unsigned int uniform_refine = input_file("uniform_refine",0);
  const std::string refine_type     = input_file("refinement_type", "h");
  const std::string approx_type     = input_file("approx_type", "LAGRANGE");
  const unsigned int approx_order   = input_file("approx_order", 1);
  n_vars                            = input_file("n_vars", n_vars);
  const std::string element_type    = input_file("element_type", "tensor");
  const int extra_error_quadrature  = input_file("extra_error_quadrature", 0);
  const int max_linear_iterations   = input_file("max_linear_iterations", 5000);

#ifdef LIBMESH_HAVE_EXODUS_API
  const bool output_intermediate    = input_file("output_intermediate", false);
#endif

  // If libmesh is configured without second derivative support, we
  // can't run this example with Hermite elements and will therefore
  // fail gracefully.
#if !defined(LIBMESH_ENABLE_SECOND_DERIVATIVES)
  libmesh_example_requires(approx_type != "HERMITE", "--enable-second");
#endif

  dim = input_file("dimension", 2);
  const std::string indicator_type = input_file("indicator_type", "kelly");
  singularity = input_file("singularity", true);
  const bool extrusion = input_file("extrusion",false);
  const bool complete = input_file("complete",false);
  libmesh_error_msg_if(extrusion && dim < 3, "Extrusion option is only for 3D meshes");

  // Skip higher-dimensional examples on a lower-dimensional libMesh build
  libmesh_example_requires(dim <= LIBMESH_DIM, "2D/3D support");

  // Output file for plotting the error as a function of
  // the number of degrees of freedom.
  std::string approx_name = "";
  if (element_type == "tensor")
    approx_name += "bi";
  if (approx_order == 1)
    approx_name += "linear";
  else if (approx_order == 2)
    approx_name += "quadratic";
  else if (approx_order == 3)
    approx_name += "cubic";
  else if (approx_order == 4)
    approx_name += "quartic";

  std::string output_file = approx_name;
  output_file += "_";
  output_file += refine_type;
  if (uniform_refine == 0)
    output_file += "_adaptive.m";
  else
    output_file += "_uniform.m";

  std::ofstream out (output_file.c_str());
  out << "% dofs     L2-error     H1-error" << std::endl;
  out << "e = [" << std::endl;

  // Create a mesh, with dimension to be overridden later, on the
  // default MPI communicator.
  Mesh mesh(init.comm());

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

  // Declare the system we will solve, named Laplace
  LinearImplicitSystem & system =
    equation_systems.add_system<LinearImplicitSystem> ("Laplace");

  // If we're doing HP refinement, then we'll need to be able to
  // evaluate data on elements' siblings in HPCoarsenTest, which means
  // we should instruct our system's DofMap to distribute that data.
  // Create and add (and keep around! the DofMap will only be holding
  // a pointer to it!) a SiblingCoupling functor to provide that
  // instruction.
  SiblingCoupling sibling_coupling;
  if (refine_type == "hp")
    system.get_dof_map().add_algebraic_ghosting_functor(sibling_coupling);

  // Read in the mesh
  if (dim == 1)
    MeshTools::Generation::build_line(mesh, 1, -1., 0.);
  else if (dim == 2 || extrusion == true)
    mesh.read("lshaped.xda");
  else
    mesh.read("lshaped3D.xda");

  // Use triangles if the config file says so
  if (element_type == "simplex")
    MeshTools::Modification::all_tri(mesh);

  if (extrusion)
    {
      Mesh flat_mesh = mesh;
      mesh.clear();
      MeshTools::Generation::build_extrusion(mesh, flat_mesh, 1, {0,0,1});
    }

  // Use highest-order elements if the config file says so.
  if (complete)
    mesh.all_complete_order();
  // Otherwise, we used first order elements to describe the geometry,
  // but we may need second order elements to hold the degrees
  // of freedom
  else if (approx_order > 1 || refine_type != "h")
    mesh.all_second_order();

  // Mesh Refinement object
  MeshRefinement mesh_refinement(mesh);
  mesh_refinement.refine_fraction() = refine_percentage;
  mesh_refinement.coarsen_fraction() = coarsen_percentage;
  mesh_refinement.max_h_level() = max_r_level;

  // Adds the variable "u" to "Laplace", using
  // the finite element type and order specified
  // in the config file
  unsigned int u_var =
    system.add_variable("u", static_cast<Order>(approx_order),
                        Utility::string_to_enum<FEFamily>(approx_type));

  std::vector<unsigned int> all_vars(1, u_var);

  // For benchmarking purposes, add more variables if requested.
  for (unsigned int var_num=1; var_num < n_vars; ++var_num)
    {
      std::ostringstream var_name;
      var_name << "u" << var_num;
      unsigned int next_var =
        system.add_variable(var_name.str(),
                            static_cast<Order>(approx_order),
                            Utility::string_to_enum<FEFamily>(approx_type));
      all_vars.push_back(next_var);
    }

  // Give the system a pointer to the matrix assembly
  // function.
  system.attach_assemble_function (assemble_laplace);

  // Add Dirichlet boundary conditions
  WrappedFunction<Number> exact_val(system, exact_solution);
  WrappedFunction<Gradient> exact_grad(system, exact_derivative);
  DirichletBoundary exact_bc({0} , all_vars, exact_val, exact_grad);
  system.get_dof_map().add_dirichlet_boundary(exact_bc);

  // Initialize the data structures for the equation system.
  equation_systems.init();

  // Set linear solver max iterations
  equation_systems.parameters.set<unsigned int>("linear solver maximum iterations")
    = max_linear_iterations;

  // Linear solver tolerance.
  equation_systems.parameters.set<Real>("linear solver tolerance") =
    std::pow(TOLERANCE, 2.5);

  // Prints information about the system to the screen.
  equation_systems.print_info();

  // Construct ExactSolution object and attach solution functions
  ExactSolution exact_sol(equation_systems);
  exact_sol.attach_exact_value(exact_solution);
  exact_sol.attach_exact_deriv(exact_derivative);

  // Use higher quadrature order for more accurate error results
  exact_sol.extra_quadrature_order(extra_error_quadrature);

  // Compute the initial error
  exact_sol.compute_error("Laplace", "u");

  // Print out the error values
  libMesh::out << "Initial L2-Error is: "
               << exact_sol.l2_error("Laplace", "u")
               << std::endl;
  libMesh::out << "Initial H1-Error is: "
               << exact_sol.h1_error("Laplace", "u")
               << std::endl;

  // A refinement loop.
  for (unsigned int r_step=0; r_step<max_r_steps; r_step++)
    {
      libMesh::out << "Beginning Solve " << r_step << std::endl;

      // Solve the system "Laplace", just like example 2.
      system.solve();

      libMesh::out << "System has: "
                   << equation_systems.n_active_dofs()
                   << " degrees of freedom."
                   << std::endl;

      libMesh::out << "Linear solver converged at step: "
                   << system.n_linear_iterations()
                   << ", final residual: "
                   << system.final_linear_residual()
                   << std::endl;

#ifdef LIBMESH_HAVE_EXODUS_API
      // After solving the system write the solution
      // to a ExodusII-formatted plot file.
      if (output_intermediate)
        {
          std::ostringstream outfile;
          outfile << "lshaped_" << r_step << ".e";
          ExodusII_IO (mesh).write_equation_systems (outfile.str(),
                                                     equation_systems);
        }
#endif // #ifdef LIBMESH_HAVE_EXODUS_API

      // Compute the error.
      exact_sol.compute_error("Laplace", "u");

      // The error should at least be sane, but it isn't if the solver
      // failed badly enough
      if (libmesh_isnan(exact_sol.l2_error("Laplace", "u")))
        libmesh_error_msg("NaN solve result");

      // Print out the error values
      libMesh::out << "L2-Error is: "
                   << exact_sol.l2_error("Laplace", "u")
                   << std::endl;
      libMesh::out << "H1-Error is: "
                   << exact_sol.h1_error("Laplace", "u")
                   << std::endl;

      // Compute any discontinuity.  There should be none.
      {
        DiscontinuityMeasure disc;

        // This is a subclass of JumpErrorEstimator, based on
        // measuring discontinuities across sides between
        // elements, and we can tell it to use a cheaper
        // "unweighted" quadrature rule when numerically
        // integrating those discontinuities.
        disc.use_unweighted_quadrature_rules = true;

        ErrorVector disc_error;
        disc.estimate_error(system, disc_error);

        Real mean_disc_error = disc_error.mean();

        libMesh::out << "Mean discontinuity error = " << mean_disc_error << std::endl;

        // FIXME - this test fails when solving with Eigen?
#ifdef LIBMESH_ENABLE_PETSC
        libmesh_assert_less (mean_disc_error, 1e-14);
#endif
      }

      // Print to output file
      out << equation_systems.n_active_dofs() << " "
          << exact_sol.l2_error("Laplace", "u") << " "
          << exact_sol.h1_error("Laplace", "u") << std::endl;

      // Possibly refine the mesh
      if (r_step+1 != max_r_steps)
        {
          libMesh::out << "  Refining the mesh..." << std::endl;

          if (uniform_refine == 0)
            {

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

              if (indicator_type == "exact")
                {
                  // The 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.
                  // For these simple test problems, we can use
                  // numerical quadrature of the exact error between
                  // the approximate and analytic solutions.
                  // However, for real problems, we would need an error
                  // indicator which only relies on the approximate
                  // solution.
                  ExactErrorEstimator error_estimator;

                  error_estimator.attach_exact_value(exact_solution);
                  error_estimator.attach_exact_deriv(exact_derivative);

                  // We optimize in H1 norm, the default
                  // error_estimator.error_norm = H1;

                  // Compute the error for each active element using
                  // the provided indicator.  Note in general you
                  // will need to provide an error estimator
                  // specifically designed for your application.
                  error_estimator.estimate_error (system, error);
                }
              else if (indicator_type == "patch")
                {
                  // The patch recovery estimator should give a
                  // good estimate of the solution interpolation
                  // error.
                  PatchRecoveryErrorEstimator error_estimator;

                  error_estimator.estimate_error (system, error);
                }
              else if (indicator_type == "uniform")
                {
                  // Error indication based on uniform refinement
                  // is reliable, but very expensive.
                  UniformRefinementEstimator error_estimator;

                  error_estimator.estimate_error (system, error);
                }
              else
                {
                  libmesh_assert_equal_to (indicator_type, "kelly");

                  // The Kelly error estimator is based on
                  // an error bound for the Poisson problem
                  // on linear elements, but is useful for
                  // driving adaptive refinement in many problems
                  KellyErrorEstimator error_estimator;

                  // This is a subclass of JumpErrorEstimator, based on
                  // measuring gradient discontinuities across sides
                  // between elements, and we can tell it to use a
                  // cheaper "unweighted" quadrature rule when
                  // numerically integrating those discontinuities.
                  error_estimator.use_unweighted_quadrature_rules = true;

                  error_estimator.estimate_error (system, error);
                }

              // Write out the error distribution
              std::ostringstream ss;
              ss << r_step;
#ifdef LIBMESH_HAVE_EXODUS_API
#  ifdef LIBMESH_HAVE_NEMESIS_API
              std::string error_output = "error_" + ss.str() + ".n";
#  else
              std::string error_output = "error_" + ss.str() + ".e";
#  endif
#else
              std::string error_output = "error_" + ss.str() + ".gmv";
#endif
              error.plot_error(error_output, mesh);

              // This takes the error in 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 10% 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.flag_elements_by_error_fraction (error);

              // If we are doing adaptive p refinement, we want
              // elements flagged for that instead.
              if (refine_type == "p")
                mesh_refinement.switch_h_to_p_refinement();
              // If we are doing "matched hp" refinement, we
              // flag elements for both h and p
              else if (refine_type == "matchedhp")
                mesh_refinement.add_p_to_h_refinement();
              // If we are doing hp refinement, we
              // try switching some elements from h to p
              else if (refine_type == "hp")
                {
                  HPCoarsenTest hpselector;
                  hpselector.select_refinement(system);
                }
              // If we are doing "singular hp" refinement, we
              // try switching most elements from h to p
              else if (refine_type == "singularhp")
                {
                  // This only differs from p refinement for
                  // the singular problem
                  libmesh_assert (singularity);
                  HPSingularity hpselector;
                  // Our only singular point is at the origin
                  hpselector.singular_points.push_back(Point());
                  hpselector.select_refinement(system);
                }
              else
                libmesh_error_msg_if(refine_type != "h",
                                     "Unknown refinement_type = " << refine_type);

              // This call actually refines and coarsens the flagged
              // elements.
              mesh_refinement.refine_and_coarsen_elements();
            }

          else if (uniform_refine == 1)
            {
              if (refine_type == "h" || refine_type == "hp" ||
                  refine_type == "matchedhp")
                mesh_refinement.uniformly_refine(1);
              if (refine_type == "p" || refine_type == "hp" ||
                  refine_type == "matchedhp")
                mesh_refinement.uniformly_p_refine(1);
            }

          // This call reinitializes the EquationSystems object for
          // the newly refined mesh.  One of the steps in the
          // reinitialization is projecting the solution,
          // old_solution, etc... vectors from the old mesh to
          // the current one.
          equation_systems.reinit ();
        }
    }

#ifdef LIBMESH_HAVE_EXODUS_API
  // Write out the solution
  // After solving the system write the solution
  // to a ExodusII-formatted plot file.
  ExodusII_IO (mesh).write_equation_systems ("lshaped.e",
                                             equation_systems);
#endif // #ifdef LIBMESH_HAVE_EXODUS_API

  // Close up the output file.
  out << "];" << std::endl;
  out << "hold on" << std::endl;
  out << "plot(e(:,1), e(:,2), 'bo-');" << std::endl;
  out << "plot(e(:,1), e(:,3), 'ro-');" << std::endl;
  //    out << "set(gca,'XScale', 'Log');" << std::endl;
  //    out << "set(gca,'YScale', 'Log');" << std::endl;
  out << "xlabel('dofs');" << std::endl;
  out << "title('" << approx_name << " elements');" << std::endl;
  out << "legend('L2-error', 'H1-error');" << std::endl;
  //     out << "disp('L2-error linear fit');" << std::endl;
  //     out << "polyfit(log10(e(:,1)), log10(e(:,2)), 1)" << std::endl;
  //     out << "disp('H1-error linear fit');" << std::endl;
  //     out << "polyfit(log10(e(:,1)), log10(e(:,3)), 1)" << std::endl;
#endif // #ifndef LIBMESH_ENABLE_AMR

  // All done.
  return 0;
}

Variable Documentation

dim

unsigned int dim = 2

Definition at line 113 of file adaptivity_ex3.C.

Referenced by libMesh::ExactSolution::_compute_error(), libMesh::FEMContext::_do_elem_position_set(), libMesh::UniformRefinementEstimator::_estimate_error(), libMesh::MeshFunction::_gradient_on_elem(), libMesh::FEInterface::all_shape_derivs(), libMesh::FEInterface::all_shapes(), alternative_fe_assembly(), assemble(), LinearElasticity::assemble(), assemble_1D(), AssembleOptimization::assemble_A_and_F(), libMesh::ClawSystem::assemble_advection_matrices(), libMesh::ClawSystem::assemble_avg_coupling_matrices(), assemble_biharmonic(), libMesh::ClawSystem::assemble_boundary_condition_matrices(), assemble_cd(), assemble_divgrad(), assemble_elasticity(), assemble_ellipticdg(), assemble_func(), assemble_graddiv(), assemble_helmholtz(), assemble_mass(), assemble_matrices(), assemble_poisson(), assemble_SchroedingerEquation(), assemble_shell(), assemble_stokes(), assemble_wave(), libMesh::FEMContext::attach_quadrature_rules(), libMesh::FEGenericBase< FEOutputType< T >::type >::build(), libMesh::FEGenericBase< FEOutputType< T >::type >::build_InfFE(), libMesh::EquationSystems::build_parallel_solution_vector(), libMesh::EquationSystems::build_variable_names(), libMesh::System::calculate_norm(), libMesh::JumpErrorEstimator::coarse_n_flux_faces_increment(), libMesh::FEGenericBase< FEOutputType< T >::type >::coarsened_dof_values(), SolutionFunction< dim >::component(), SolutionGradient< dim >::component(), libMesh::FEMap::compute_affine_map(), libMesh::FEInterface::compute_data(), libMesh::FEMap::compute_edge_map(), compute_enriched_soln(), libMesh::InfFE< Dim, T_radial, T_map >::compute_face_functions(), libMesh::FEXYZMap::compute_face_map(), libMesh::FEMap::compute_face_map(), libMesh::FEXYZ< Dim >::compute_face_values(), compute_jacobian(), libMesh::FEMap::compute_map(), libMesh::FEMap::compute_null_map(), compute_residual(), libMesh::FEGenericBase< FEOutputType< T >::type >::compute_shape_functions(), libMesh::InfFE< Dim, T_radial, T_map >::compute_shape_functions(), libMesh::FEXYZ< Dim >::compute_shape_functions(), libMesh::FEMap::compute_single_point_map(), compute_stresses(), LinearElasticityWithContact::compute_stresses(), LinearElasticity::compute_stresses(), LargeDeformationElasticity::compute_stresses(), libMesh::InfFERadial::decay(), libMesh::InfFERadial::decay_deriv(), libMesh::FEType::default_quadrature_rule(), libMesh::MeshBase::detect_interior_parents(), libMesh::MeshFunction::discontinuous_value(), libMesh::FEMContext::elem_fe_reinit(), libMesh::FEMContext::elem_position_get(), NavierSystem::element_constraint(), NavierSystem::element_time_derivative(), SolidSystem::element_time_derivative(), libMesh::VariationalSmootherSystem::element_time_derivative(), HeatSystem::element_time_derivative(), libMesh::RBEIMConstruction::enrich_eim_approximation_on_interiors(), libMesh::JumpErrorEstimator::estimate_error(), libMesh::ExactErrorEstimator::estimate_error(), libMesh::OldSolutionValue< Output, point_output >::eval_at_node(), libMesh::OldSolutionCoefs< Output, point_output >::eval_mixed_derivatives(), libMesh::OldSolutionValue< Output, point_output >::eval_mixed_derivatives(), exact_derivative(), exact_solution(), fe_assembly(), libMesh::DynaIO::find_elem_definition(), libMesh::ElemInternal::find_interior_neighbors(), libMesh::EquationSystems::find_variable_numbers(), libMesh::MeshRefinement::flag_elements_by_elem_fraction(), libMesh::MeshRefinement::flag_elements_by_nelem_target(), NavierSystem::forcing(), form_functionA(), form_functionB(), form_matrixA(), libMesh::ClawSystem::get_advection_matrix(), libMesh::ClawSystem::get_avg_matrix(), libMesh::ClawSystem::get_boundary_condition_matrix(), libMesh::FEMContext::get_element_fe(), libMesh::FEMContext::get_element_qrule(), libMesh::FEMContext::get_side_fe(), libMesh::FEMContext::get_side_qrule(), libMesh::QGrundmann_Moller::gm_rule(), libMesh::MeshFunction::hessian(), libMesh::FEInterface::ifem_compute_data(), libMesh::FEInterface::ifem_inverse_map(), libMesh::FEInterface::ifem_map(), libMesh::FEInterface::ifem_n_dofs(), libMesh::FEInterface::ifem_n_dofs_at_node(), libMesh::FEInterface::ifem_n_dofs_per_elem(), libMesh::FEInterface::ifem_n_shape_functions(), libMesh::FEInterface::ifem_nodal_soln(), libMesh::FEInterface::ifem_shape(), libMesh::FEInterface::ifem_shape_deriv(), libMesh::LaplacianErrorEstimator::init_context(), libMesh::VariationalSmootherSystem::init_context(), HilbertSystem::init_context(), libMesh::DiscontinuityMeasure::init_context(), libMesh::WrappedFunctor< Output >::init_context(), HeatSystem::init_context(), libMesh::KellyErrorEstimator::init_context(), libMesh::FEMSystem::init_context(), libMesh::RBEIMConstruction::init_context(), libMesh::OldSolutionBase< Output, point_output >::init_context(), NavierSystem::init_data(), SolidSystem::init_data(), libMesh::FEMContext::init_internal_data(), libMesh::FEMap::init_reference_to_physical_map(), libMesh::LaplacianErrorEstimator::internal_side_integration(), libMesh::InfFEMap::inverse_map(), libMesh::FEMap::inverse_map(), libMesh::FEInterface::inverse_map(), LaplaceYoung::jacobian(), LargeDeformationElasticity::jacobian(), libMesh::PointLocatorNanoflann::kdtree_get_pt(), libMesh::InverseDistanceInterpolation< KDDim >::PointListAdaptor< KDDim >::kdtree_get_pt(), libMesh::VectorOfNodesAdaptor::kdtree_get_pt(), libMesh::LIBMESH_DEFAULT_VECTORIZED_FE(), main(), libMesh::InfFEMap::map(), libMesh::FEMap::map(), libMesh::FEInterface::map(), libMesh::HCurlFETransformation< OutputShape >::map_curl(), libMesh::H1FETransformation< OutputShape >::map_curl(), libMesh::H1FETransformation< OutputShape >::map_d2phi(), libMesh::FEMap::map_deriv(), libMesh::HDivFETransformation< OutputShape >::map_div(), libMesh::H1FETransformation< OutputShape >::map_div(), libMesh::H1FETransformation< OutputShape >::map_dphi(), libMesh::H1FETransformation< OutputShape >::map_phi(), libMesh::HDivFETransformation< OutputShape >::map_phi(), libMesh::HCurlFETransformation< OutputShape >::map_phi(), NavierSystem::mass_residual(), libMesh::FEInterface::n_dofs(), libMesh::FEInterface::n_dofs_at_node(), libMesh::FEInterface::n_dofs_at_node_function(), libMesh::FEInterface::n_dofs_per_elem(), libMesh::FEInterface::n_shape_functions(), libMesh::FEInterface::nodal_soln(), OverlappingCouplingFunctor::operator()(), libMesh::OverlapCoupling::operator()(), libMesh::WeightedPatchRecoveryErrorEstimator::EstimateError::operator()(), libMesh::PatchRecoveryErrorEstimator::EstimateError::operator()(), libMesh::MeshFunction::operator()(), libMesh::BoundaryProjectSolution::operator()(), libMesh::GenericProjector< FFunctor, GFunctor, FValue, ProjectionAction >::SortAndCopy::operator()(), libMesh::GenericProjector< FFunctor, GFunctor, FValue, ProjectionAction >::ProjectVertices::operator()(), libMesh::GenericProjector< FFunctor, GFunctor, FValue, ProjectionAction >::ProjectEdges::operator()(), libMesh::GenericProjector< FFunctor, GFunctor, FValue, ProjectionAction >::ProjectSides::operator()(), libMesh::GenericProjector< FFunctor, GFunctor, FValue, ProjectionAction >::ProjectInteriors::operator()(), libMesh::System::point_gradient(), NavierSystem::postprocess(), libMesh::RBParametrizedFunction::preevaluate_parametrized_function_on_mesh(), libMesh::RBParametrizedFunction::preevaluate_parametrized_function_on_mesh_sides(), process_cmd_line(), libMesh::QClough::QClough(), libMesh::QGauss::QGauss(), libMesh::QGrid::QGrid(), libMesh::QGrundmann_Moller::QGrundmann_Moller(), libMesh::QMonomial::QMonomial(), libMesh::QNodal::QNodal(), libMesh::QSimpson::QSimpson(), libMesh::QTrap::QTrap(), libMesh::rational_fe_weighted_shapes(), libMesh::rational_fe_weighted_shapes_derivs(), libMesh::FE< Dim, LAGRANGE_VEC >::reinit(), libMesh::JumpErrorEstimator::reinit_sides(), LaplaceYoung::residual(), LargeDeformationElasticity::residual(), LinearElasticityWithContact::residual_and_jacobian(), libMesh::FEMap::resize_quadrature_map_vectors(), SolidSystem::save_initial_mesh(), libMesh::BoundingBox::scale(), libMesh::HPCoarsenTest::select_refinement(), ElasticitySystem::set_dim(), libMesh::OverlapCoupling::set_quadrature_rule(), PerElemTest< elem_type >::setUp(), setup(), libMesh::FEInterface::shape(), libMesh::FEInterface::shape_deriv(), libMesh::FEInterface::shape_deriv_function(), libMesh::FEInterface::shape_function(), libMesh::FEInterface::shape_second_deriv(), libMesh::FEInterface::shape_second_deriv_function(), libMesh::FEInterface::shapes(), NavierSystem::side_constraint(), libMesh::FEMContext::side_fe_reinit(), libMesh::FEInterface::side_nodal_soln(), libMesh::PatchRecoveryErrorEstimator::specpoly(), libMesh::GenericProjector< FFunctor, GFunctor, FValue, ProjectionAction >::SubFunctor::SubFunctor(), InfFERadialTest::testInfQuants(), InfFERadialTest::testInfQuants_numericDeriv(), PointLocatorTest::testLocator(), QuadratureTest::testPolynomials(), SlitMeshRefinedSystemTest::testRestart(), InfFERadialTest::testSides(), InfFERadialTest::testSingleOrder(), SlitMeshRefinedSystemTest::testSystem(), libMesh::FEType::unweighted_quadrature_rule(), libMesh::FEMContext::use_default_quadrature_rules(), libMesh::FEMContext::use_unweighted_quadrature_rules(), libMesh::MeshTools::volume(), libMesh::ExodusII_IO::write_as_dimension(), libMesh::ExodusII_IO_Helper::write_as_dimension(), libMesh::EnsightIO::write_scalar_ascii(), and libMesh::EnsightIO::write_vector_ascii().

n_vars

unsigned int n_vars = 1

Definition at line 116 of file adaptivity_ex3.C.

Referenced by libMesh::UniformRefinementEstimator::_estimate_error(), libMesh::DofMap::allgather_recursive_constraints(), assemble_laplace(), libMesh::WrappedFunction< Output >::component(), libMesh::DofMap::create_dof_constraints(), libMesh::RBEIMEvaluation::distribute_bfs(), libMesh::JumpErrorEstimator::estimate_error(), libMesh::AdjointResidualErrorEstimator::estimate_error(), libMesh::ExactErrorEstimator::estimate_error(), libMesh::ErrorEstimator::estimate_errors(), libMesh::RBEIMEvaluation::gather_bfs(), libMesh::RBEIMEvaluation::get_interior_basis_function_as_vec_helper(), libMesh::RBEIMEvaluation::get_interior_basis_function_sizes(), libMesh::LaplacianErrorEstimator::init_context(), libMesh::DiscontinuityMeasure::init_context(), libMesh::KellyErrorEstimator::init_context(), libMesh::PetscDMWrapper::init_petscdm(), libMesh::RadialBasisInterpolation< KDDim, RBF >::interpolate_field_data(), main(), libMesh::RBEIMEvaluation::node_distribute_bfs(), libMesh::RBEIMEvaluation::node_gather_bfs(), libMesh::WrappedFunction< Output >::operator()(), libMesh::WeightedPatchRecoveryErrorEstimator::EstimateError::operator()(), libMesh::PatchRecoveryErrorEstimator::EstimateError::operator()(), libMesh::RadialBasisInterpolation< KDDim, RBF >::prepare_for_use(), libMesh::InterMeshProjection::project_system_vectors(), libMesh::System::project_vector(), libMesh::System::projection_matrix(), libMesh::RBEIMEvaluation::read_in_interior_basis_functions(), libMesh::RBEIMEvaluation::read_in_node_basis_functions(), libMesh::RBEIMEvaluation::read_in_side_basis_functions(), libMesh::JumpErrorEstimator::reinit_sides(), libMesh::HPCoarsenTest::select_refinement(), libMesh::RBEIMEvaluation::side_distribute_bfs(), libMesh::RBEIMEvaluation::side_gather_bfs(), libMesh::MEDITIO::write_ascii(), libMesh::GMVIO::write_ascii_new_impl(), libMesh::GMVIO::write_ascii_old_impl(), libMesh::GMVIO::write_binary(), libMesh::GMVIO::write_discontinuous_gmv(), libMesh::UCDIO::write_header(), libMesh::RBEIMEvaluation::write_out_interior_basis_functions(), libMesh::RBEIMEvaluation::write_out_node_basis_functions(), libMesh::RBEIMEvaluation::write_out_side_basis_functions(), libMesh::GmshIO::write_post(), and libMesh::GnuPlotIO::write_solution().

singularity

bool singularity = true

Definition at line 119 of file adaptivity_ex3.C.

Referenced by assemble_laplace(), exact_derivative(), exact_solution(), and main().