--- /dev/null
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2021 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library 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.md at
+// the top level directory of deal.II.
+//
+// ---------------------------------------------------------------------
+
+
+// Step-08 on a simplex mesh. Following incompatible modifications had to be
+// made:
+// - Change the FE_Q to Simplex::FE_P.
+// - Change QGauss to Simplex::QGauss.
+// - Use MappingFE (Do not use default mapping).
+// - Convert triangulation to a triangulation based on simplices.
+// - Use refine_global() instead of execute_coarsening_and_refinement().
+// - Reduce number of cycles from 7 to 4 to reduce runtime.
+
+#include <deal.II/base/function.h>
+#include <deal.II/base/quadrature_lib.h>
+#include <deal.II/base/tensor.h>
+
+#include <deal.II/dofs/dof_accessor.h>
+#include <deal.II/dofs/dof_handler.h>
+#include <deal.II/dofs/dof_tools.h>
+
+#include <deal.II/fe/fe_q.h>
+#include <deal.II/fe/fe_system.h>
+#include <deal.II/fe/fe_values.h>
+#include <deal.II/fe/mapping_fe.h>
+
+#include <deal.II/grid/grid_generator.h>
+#include <deal.II/grid/grid_refinement.h>
+#include <deal.II/grid/tria.h>
+#include <deal.II/grid/tria_accessor.h>
+#include <deal.II/grid/tria_iterator.h>
+
+#include <deal.II/lac/affine_constraints.h>
+#include <deal.II/lac/dynamic_sparsity_pattern.h>
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/precondition.h>
+#include <deal.II/lac/solver_cg.h>
+#include <deal.II/lac/sparse_matrix.h>
+#include <deal.II/lac/vector.h>
+
+#include <deal.II/numerics/data_out.h>
+#include <deal.II/numerics/error_estimator.h>
+#include <deal.II/numerics/matrix_tools.h>
+#include <deal.II/numerics/vector_tools.h>
+
+#include <deal.II/simplex/fe_lib.h>
+#include <deal.II/simplex/grid_generator.h>
+#include <deal.II/simplex/quadrature_lib.h>
+
+#include <fstream>
+#include <iostream>
+
+#include "../tests.h"
+
+namespace Step8
+{
+ using namespace dealii;
+
+ template <int dim>
+ class ElasticProblem
+ {
+ public:
+ ElasticProblem();
+ void
+ run();
+
+ private:
+ void
+ setup_system();
+ void
+ assemble_system();
+ void
+ solve();
+ void
+ refine_grid();
+ void
+ output_results(const unsigned int cycle) const;
+
+ Triangulation<dim> triangulation;
+ DoFHandler<dim> dof_handler;
+ MappingFE<dim> mapping;
+
+ FESystem<dim> fe;
+
+ AffineConstraints<double> constraints;
+
+ SparsityPattern sparsity_pattern;
+ SparseMatrix<double> system_matrix;
+
+ Vector<double> solution;
+ Vector<double> system_rhs;
+ };
+
+ template <int dim>
+ void
+ right_hand_side(const std::vector<Point<dim>> &points,
+ std::vector<Tensor<1, dim>> & values)
+ {
+ Assert(values.size() == points.size(),
+ ExcDimensionMismatch(values.size(), points.size()));
+ Assert(dim >= 2, ExcNotImplemented());
+
+ Point<dim> point_1, point_2;
+ point_1(0) = 0.5;
+ point_2(0) = -0.5;
+
+ for (unsigned int point_n = 0; point_n < points.size(); ++point_n)
+ {
+ if (((points[point_n] - point_1).norm_square() < 0.2 * 0.2) ||
+ ((points[point_n] - point_2).norm_square() < 0.2 * 0.2))
+ values[point_n][0] = 1.0;
+ else
+ values[point_n][0] = 0.0;
+
+ if (points[point_n].norm_square() < 0.2 * 0.2)
+ values[point_n][1] = 1.0;
+ else
+ values[point_n][1] = 0.0;
+ }
+ }
+
+
+ template <int dim>
+ ElasticProblem<dim>::ElasticProblem()
+ : dof_handler(triangulation)
+ , mapping(Simplex::FE_P<dim>(1))
+ , fe(Simplex::FE_P<dim>(1), dim)
+ {}
+
+ template <int dim>
+ void
+ ElasticProblem<dim>::setup_system()
+ {
+ dof_handler.distribute_dofs(fe);
+ solution.reinit(dof_handler.n_dofs());
+ system_rhs.reinit(dof_handler.n_dofs());
+
+ constraints.clear();
+ DoFTools::make_hanging_node_constraints(dof_handler, constraints);
+ VectorTools::interpolate_boundary_values(
+ mapping, dof_handler, 0, Functions::ZeroFunction<dim>(dim), constraints);
+ constraints.close();
+
+ DynamicSparsityPattern dsp(dof_handler.n_dofs(), dof_handler.n_dofs());
+ DoFTools::make_sparsity_pattern(dof_handler, dsp, constraints, false);
+ sparsity_pattern.copy_from(dsp);
+
+ system_matrix.reinit(sparsity_pattern);
+ }
+
+ template <int dim>
+ void
+ ElasticProblem<dim>::assemble_system()
+ {
+ Simplex::QGauss<dim> quadrature_formula(fe.degree + 1);
+ FEValues<dim> fe_values(mapping,
+ fe,
+ quadrature_formula,
+ update_values | update_gradients |
+ update_quadrature_points | update_JxW_values);
+
+ const unsigned int dofs_per_cell = fe.n_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<types::global_dof_index> local_dof_indices(dofs_per_cell);
+
+ std::vector<double> lambda_values(n_q_points);
+ std::vector<double> mu_values(n_q_points);
+
+ Functions::ConstantFunction<dim> lambda(1.), mu(1.);
+
+ std::vector<Tensor<1, dim>> rhs_values(n_q_points);
+
+ for (const auto &cell : dof_handler.active_cell_iterators())
+ {
+ cell_matrix = 0;
+ cell_rhs = 0;
+
+ fe_values.reinit(cell);
+
+ lambda.value_list(fe_values.get_quadrature_points(), lambda_values);
+ mu.value_list(fe_values.get_quadrature_points(), mu_values);
+ right_hand_side(fe_values.get_quadrature_points(), rhs_values);
+
+ for (const unsigned int i : fe_values.dof_indices())
+ {
+ const unsigned int component_i =
+ fe.system_to_component_index(i).first;
+
+ for (const unsigned int j : fe_values.dof_indices())
+ {
+ const unsigned int component_j =
+ fe.system_to_component_index(j).first;
+
+ for (const unsigned int q_point :
+ fe_values.quadrature_point_indices())
+ {
+ cell_matrix(i, j) +=
+ ((fe_values.shape_grad(i, q_point)[component_i] *
+ fe_values.shape_grad(j, q_point)[component_j] *
+ lambda_values[q_point]) +
+ (fe_values.shape_grad(i, q_point)[component_j] *
+ fe_values.shape_grad(j, q_point)[component_i] *
+ mu_values[q_point]) +
+ ((component_i == component_j) ? //
+ (fe_values.shape_grad(i, q_point) * //
+ fe_values.shape_grad(j, q_point) * //
+ mu_values[q_point]) : //
+ 0) //
+ ) * //
+ fe_values.JxW(q_point); //
+ }
+ }
+ }
+
+ for (const unsigned int i : fe_values.dof_indices())
+ {
+ const unsigned int component_i =
+ fe.system_to_component_index(i).first;
+
+ for (const unsigned int q_point :
+ fe_values.quadrature_point_indices())
+ cell_rhs(i) += fe_values.shape_value(i, q_point) *
+ rhs_values[q_point][component_i] *
+ fe_values.JxW(q_point);
+ }
+
+ cell->get_dof_indices(local_dof_indices);
+ constraints.distribute_local_to_global(
+ cell_matrix, cell_rhs, local_dof_indices, system_matrix, system_rhs);
+ }
+ }
+
+ template <int dim>
+ void
+ ElasticProblem<dim>::solve()
+ {
+ SolverControl solver_control(1000, 1e-12);
+ SolverCG<Vector<double>> cg(solver_control);
+
+ PreconditionSSOR<SparseMatrix<double>> preconditioner;
+ preconditioner.initialize(system_matrix, 1.2);
+
+ cg.solve(system_matrix, solution, system_rhs, preconditioner);
+
+ constraints.distribute(solution);
+ }
+
+ template <int dim>
+ void
+ ElasticProblem<dim>::refine_grid()
+ {
+ Vector<float> estimated_error_per_cell(triangulation.n_active_cells());
+
+ Simplex::QGauss<dim - 1> quadrature(fe.degree + 1);
+ KellyErrorEstimator<dim>::estimate(
+ mapping, dof_handler, quadrature, {}, solution, estimated_error_per_cell);
+
+ GridRefinement::refine_and_coarsen_fixed_number(triangulation,
+ estimated_error_per_cell,
+ 0.3,
+ 0.03);
+
+ triangulation.refine_global();
+ }
+
+ template <int dim>
+ void
+ ElasticProblem<dim>::output_results(const unsigned int cycle) const
+ {
+ DataOut<dim> data_out;
+ data_out.attach_dof_handler(dof_handler);
+
+ std::vector<std::string> solution_names;
+ switch (dim)
+ {
+ case 1:
+ solution_names.emplace_back("displacement");
+ break;
+ case 2:
+ solution_names.emplace_back("x_displacement");
+ solution_names.emplace_back("y_displacement");
+ break;
+ case 3:
+ solution_names.emplace_back("x_displacement");
+ solution_names.emplace_back("y_displacement");
+ solution_names.emplace_back("z_displacement");
+ break;
+ default:
+ Assert(false, ExcNotImplemented());
+ }
+
+ data_out.add_data_vector(solution, solution_names);
+ data_out.build_patches(mapping);
+
+ std::ofstream output("solution-" + std::to_string(cycle) + ".vtk");
+ data_out.write_vtk(output);
+ }
+
+ template <int dim>
+ void
+ ElasticProblem<dim>::run()
+ {
+ for (unsigned int cycle = 0; cycle < 5; ++cycle)
+ {
+ std::cout << "Cycle " << cycle << ':' << std::endl;
+
+ if (cycle == 0)
+ {
+ Triangulation<dim> temp;
+ GridGenerator::hyper_cube(temp, -1, 1);
+ GridGenerator::convert_hypercube_to_simplex_mesh(temp,
+ triangulation);
+ }
+ else
+ refine_grid();
+
+ std::cout << " Number of active cells: "
+ << triangulation.n_active_cells() << std::endl;
+
+ setup_system();
+
+ std::cout << " Number of degrees of freedom: " << dof_handler.n_dofs()
+ << std::endl;
+
+ assemble_system();
+ solve();
+ output_results(cycle);
+ }
+ }
+} // namespace Step8
+
+int
+main()
+{
+ try
+ {
+ Step8::ElasticProblem<2> elastic_problem_2d;
+ elastic_problem_2d.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;
+}
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