--- /dev/null
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2023 - 2024 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.
+//
+// ---------------------------------------------------------------------
+//
+// This convergence test verifies that hanging nodes on FE_NedelecSZ
+// elements are handled correctly. The orientation of edges and faces is
+// automatically adapted in the presence of hanging edges and hanging
+// faces. Furthermore, the function make_hanging_node_constraints()
+// considers the orientation of the underlying edges and faces to prevent
+// sign conflicts.
+//
+// This test solves the real-valued Maxwell equation in 2D and 3D:
+//
+// curl(curl(E)) - E = 0,
+//
+// where we consider Dirichlet boundary data
+// such that we know the solution E.
+// In 2D:
+// E = (sin(y), sin(x))
+//
+// In 3D:
+// E = (sin(y), sin(z), 0)
+//
+// In the first step, we solve the Maxwell equation on a coarse grid.
+// After that, we refine the center of the domain so the grid contains
+// hanging edges (and faces in 3D). We expect that the L2 difference
+// between the numerical solution and the exact solution stays the same
+// or gets smaller in each step, i.e., with a finer and finer grid, we
+// converge more and more to the exact solution.
+// The geometry is chosen in such a way that it covers the most important
+// cases and is as small as possible.
+// To cover all aspects of make_hanging_node_constraints(), we have to
+// consider at least cubic base functions (that corresponds to
+// polynomial_degree = 2).
+
+#include <deal.II/base/function.h>
+#include <deal.II/base/logstream.h>
+#include <deal.II/base/quadrature_lib.h>
+
+#include <deal.II/dofs/dof_handler.h>
+#include <deal.II/dofs/dof_tools.h>
+
+#include <deal.II/fe/fe_nedelec_sz.h>
+#include <deal.II/fe/fe_system.h>
+#include <deal.II/fe/fe_values.h>
+
+#include <deal.II/grid/grid_generator.h>
+#include <deal.II/grid/grid_refinement.h>
+#include <deal.II/grid/grid_tools.h>
+#include <deal.II/grid/manifold_lib.h>
+#include <deal.II/grid/tria.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/sparse_direct.h>
+#include <deal.II/lac/sparse_matrix.h>
+#include <deal.II/lac/vector.h>
+
+#include <deal.II/numerics/vector_tools.h>
+#include <deal.II/numerics/vector_tools_boundary.h>
+
+#include <cmath>
+#include <iostream>
+
+#include "../tests.h"
+
+namespace ConvergenceTest
+{
+
+ template <int dim>
+ class SolutionValues : public Function<dim>
+ {
+ public:
+ SolutionValues()
+ : Function<dim>(dim)
+ {}
+
+ void
+ vector_value(const Point<dim> &p, Vector<double> &values) const override;
+
+ void
+ vector_value_list(const std::vector<Point<dim>> &points,
+ std::vector<Vector<double>> &value_list) const override
+ {
+ Assert(value_list.size() == points.size(),
+ ExcDimensionMismatch(value_list.size(), points.size()));
+
+ for (unsigned int p = 0; p < points.size(); p++)
+ {
+ SolutionValues<dim>::vector_value(points[p], value_list[p]);
+ }
+ }
+ };
+
+ // here we are using artificial boundary values,
+ // the benefit is, that we know the exact solution for the
+ // electric field E, i.e. this is also the exact solution
+ // to the partial differential equation we aim to solve
+ template <>
+ void
+ SolutionValues<2>::vector_value(const Point<2> &p,
+ Vector<double> &values) const
+ {
+ values(0) = sin(p[1]);
+ values(1) = sin(p[0]);
+ }
+
+ template <>
+ void
+ SolutionValues<3>::vector_value(const Point<3> &p,
+ Vector<double> &values) const
+ {
+ values(0) = sin(p[1]);
+ values(1) = sin(p[2]);
+ values(2) = 0.0;
+ }
+
+
+
+ template <int dim>
+ class MaxwellProblem
+ {
+ public:
+ MaxwellProblem(unsigned int refinements, unsigned int poly_degree);
+
+ void
+ run();
+
+ private:
+ void
+ make_grid();
+ void
+ setup_system();
+ void
+ assemble_system();
+ void
+ solve();
+ void
+ refine_grid(double radius);
+ double
+ output_error();
+
+ Triangulation<dim> triangulation;
+ DoFHandler<dim> dof_handler;
+ FE_NedelecSZ<dim> fe;
+
+ AffineConstraints<double> constraints;
+
+ SparsityPattern sparsity_pattern;
+ SparseMatrix<double> system_matrix;
+ Vector<double> solution, system_rhs;
+
+ const unsigned int refinements;
+ const unsigned int poly_degree;
+ };
+
+ template <int dim>
+ MaxwellProblem<dim>::MaxwellProblem(const unsigned int refinements,
+ const unsigned int poly_degree)
+ : dof_handler(triangulation)
+ , fe(poly_degree)
+ , refinements(refinements)
+ , poly_degree(poly_degree)
+ {}
+
+ template <int dim>
+ void
+ MaxwellProblem<dim>::make_grid()
+ {
+ GridGenerator::hyper_ball_balanced(triangulation);
+
+ static const SphericalManifold<dim> spherical_manifold;
+ triangulation.set_manifold(0, spherical_manifold);
+
+ triangulation.refine_global(refinements);
+
+ // Refine one cell at the boundary,
+ // such that the resulting grid contains
+ // hanging nodes.
+ for (auto &cell : triangulation.cell_iterators())
+ {
+ if (!cell->is_active())
+ continue;
+
+ if (cell->at_boundary())
+ {
+ cell->set_refine_flag();
+ break;
+ }
+ }
+ triangulation.execute_coarsening_and_refinement();
+ }
+
+ template <int dim>
+ void
+ MaxwellProblem<dim>::setup_system()
+ {
+ dof_handler.distribute_dofs(fe);
+
+ DynamicSparsityPattern dsp(dof_handler.n_dofs(), dof_handler.n_dofs());
+
+ DoFTools::make_sparsity_pattern(dof_handler, dsp);
+
+ constraints.clear();
+ DoFTools::make_hanging_node_constraints(dof_handler, constraints);
+
+ VectorTools::project_boundary_values_curl_conforming_l2(
+ dof_handler,
+ 0 /* vector component*/,
+ SolutionValues<dim>(),
+ 0 /* boundary id*/,
+ constraints);
+ constraints.close();
+
+ constraints.condense(dsp);
+
+ sparsity_pattern.copy_from(dsp);
+ system_matrix.reinit(sparsity_pattern);
+ system_rhs.reinit(dof_handler.n_dofs());
+ solution.reinit(dof_handler.n_dofs());
+ }
+
+ template <int dim>
+ void
+ MaxwellProblem<dim>::assemble_system()
+ {
+ system_matrix = 0;
+ system_rhs = 0;
+
+ const unsigned int curl_dim = (dim == 2) ? 1 : 3;
+
+ // choose the quadrature formulas
+ QGauss<dim> quadrature_formula(fe.degree + 2);
+
+ // get the number of quadrature points and dofs
+ const unsigned int n_q_points = quadrature_formula.size(),
+ dofs_per_cell = fe.dofs_per_cell;
+
+ // set update flags
+ FEValues<dim> fe_values(fe,
+ quadrature_formula,
+ update_values | update_gradients |
+ update_quadrature_points | update_JxW_values);
+
+ // Extractors for the real part
+ const FEValuesExtractors::Vector E_re(0);
+
+ // create the local left hand side and right hand side
+ 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);
+
+ // loop over all cells
+ for (const auto &cell : dof_handler.active_cell_iterators())
+ {
+ if (cell->is_locally_owned() == false)
+ continue;
+
+ // initialize values:
+ cell_matrix = 0;
+ cell_rhs = 0;
+ fe_values.reinit(cell);
+
+ for (const unsigned int i : fe_values.dof_indices())
+ {
+ // only compute this once
+ std::vector<Tensor<1, dim>> phi_i(n_q_points);
+ std::vector<Tensor<1, curl_dim>> curl_phi_i(n_q_points);
+ for (unsigned int q_point = 0; q_point < n_q_points; q_point++)
+ {
+ phi_i[q_point] = fe_values[E_re].value(i, q_point);
+ curl_phi_i[q_point] = fe_values[E_re].curl(i, q_point);
+ }
+
+ // we use here, that the problem is symmetrical
+ for (unsigned int j = i; j < dofs_per_cell; j++)
+ {
+ double mass_part = 0;
+ double curl_part = 0;
+
+ for (unsigned int q_point = 0; q_point < n_q_points; q_point++)
+ {
+ Tensor<1, dim> phi_j = fe_values[E_re].value(j, q_point);
+ Tensor<1, curl_dim> curl_phi_j =
+ fe_values[E_re].curl(j, q_point);
+
+ curl_part +=
+ curl_phi_i[q_point] * curl_phi_j * fe_values.JxW(q_point);
+
+ mass_part +=
+ phi_i[q_point] * phi_j * fe_values.JxW(q_point);
+ }
+
+ double mass_term = curl_part - mass_part;
+ cell_matrix(i, j) = mass_term;
+ cell_matrix(j, i) = mass_term;
+ }
+ }
+
+ 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
+ MaxwellProblem<dim>::solve()
+ {
+ SparseDirectUMFPACK A_direct;
+ A_direct.initialize(system_matrix);
+ A_direct.vmult(solution, system_rhs);
+ constraints.distribute(solution);
+ }
+
+ template <int dim>
+ void
+ MaxwellProblem<dim>::refine_grid(const double radius)
+ {
+ for (auto &cell : triangulation.cell_iterators())
+ {
+ if (!cell->is_active())
+ continue;
+
+ double distance;
+ if (dim == 3)
+ distance = std::sqrt(std::pow(cell->center()[1], 2) +
+ std::pow(cell->center()[2], 2));
+ else if (dim == 2)
+ distance = std::sqrt(std::pow(cell->center()[0], 2) +
+ std::pow(cell->center()[1], 2));
+
+ if (distance < radius)
+ cell->set_refine_flag();
+ }
+
+ triangulation.execute_coarsening_and_refinement();
+ }
+
+ template <int dim>
+ double
+ MaxwellProblem<dim>::output_error()
+ {
+ SolutionValues<dim> exact_solution;
+ Vector<double> diff_per_cell(triangulation.n_active_cells());
+ VectorTools::integrate_difference(dof_handler,
+ solution,
+ exact_solution,
+ diff_per_cell,
+ QGauss<dim>(poly_degree + 2),
+ VectorTools::L2_norm);
+ return diff_per_cell.l2_norm();
+ }
+
+ template <int dim>
+ void
+ MaxwellProblem<dim>::run()
+ {
+ deallog << "Testing for dim = " << dim
+ << ", polynomial_degree p = " << poly_degree << std::endl;
+ make_grid();
+ setup_system();
+ assemble_system();
+ solve();
+
+ // compute the error
+ double L2_error = output_error();
+
+ // check that we are close to the analytical solution
+ bool passed = (std::fabs(L2_error) < 1e-2);
+
+ if (passed)
+ deallog << "OK" << std::endl;
+ else
+ {
+ deallog << "FAILED" << std::endl;
+ deallog << "L2 Error = " << L2_error << "(threshold = 1e-2)"
+ << std::endl;
+ }
+ }
+} // namespace ConvergenceTest
+
+int
+main()
+{
+ initlog();
+
+ using namespace ConvergenceTest;
+
+ const unsigned int refinements = 1;
+ const unsigned int poly_degree = 2;
+
+ MaxwellProblem<2> maxwell_2d(refinements, poly_degree);
+ maxwell_2d.run();
+
+ MaxwellProblem<3> maxwell_3d(refinements, poly_degree);
+ maxwell_3d.run();
+
+ return 0;
+}
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