1 // The libMesh Finite Element Library.
2 // Copyright (C) 2002-2020 Benjamin S. Kirk, John W. Peterson, Roy H. Stogner
3
4 // This library is free software; you can redistribute it and/or
5 // modify it under the terms of the GNU Lesser General Public
6 // License as published by the Free Software Foundation; either
7 // version 2.1 of the License, or (at your option) any later version.
8
9 // This library is distributed in the hope that it will be useful,
10 // but WITHOUT ANY WARRANTY; without even the implied warranty of
11 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
12 // Lesser General Public License for more details.
13
14 // You should have received a copy of the GNU Lesser General Public
15 // License along with this library; if not, write to the Free Software
16 // Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
17
18
19
20 // <h1>Adaptivity Example 1 - Solving 1D PDE Using Adaptive Mesh Refinement</h1>
21 // \author Benjamin S. Kirk
22 // \date 2003
23 //
24 // This example demonstrates how to solve a simple 1D problem
25 // using adaptive mesh refinement. The PDE that is solved is:
26 // -epsilon*u''(x) + u(x) = 1, on the domain [0,1] with boundary conditions
27 // u(0) = u(1) = 0 and where epsilon << 1.
28 //
29 // The approach used to solve 1D problems in libMesh is virtually identical to
30 // solving 2D or 3D problems, so in this sense this example represents a good
31 // starting point for new users. Note that many concepts are used in this
32 // example which are explained more fully in subsequent examples.
33
34 // Libmesh includes
35 #include "libmesh/mesh.h"
36 #include "libmesh/mesh_generation.h"
37 #include "libmesh/edge_edge3.h"
38 #include "libmesh/gnuplot_io.h"
39 #include "libmesh/equation_systems.h"
40 #include "libmesh/linear_implicit_system.h"
41 #include "libmesh/fe.h"
42 #include "libmesh/getpot.h"
43 #include "libmesh/quadrature_gauss.h"
44 #include "libmesh/sparse_matrix.h"
45 #include "libmesh/dof_map.h"
46 #include "libmesh/numeric_vector.h"
47 #include "libmesh/dense_matrix.h"
48 #include "libmesh/dense_vector.h"
49 #include "libmesh/error_vector.h"
50 #include "libmesh/kelly_error_estimator.h"
51 #include "libmesh/mesh_refinement.h"
52 #include "libmesh/enum_solver_package.h"
53
54 // Bring in everything from the libMesh namespace
55 using namespace libMesh;
56
57 void assemble_1D(EquationSystems & es,
58 const std::string & system_name);
59
main(int argc,char ** argv)60 int main(int argc, char ** argv)
61 {
62 // Initialize the library. This is necessary because the library
63 // may depend on a number of other libraries (i.e. MPI and PETSc)
64 // that require initialization before use. When the LibMeshInit
65 // object goes out of scope, other libraries and resources are
66 // finalized.
67 LibMeshInit init (argc, argv);
68
69 // This example requires a linear solver package.
70 libmesh_example_requires(libMesh::default_solver_package() != INVALID_SOLVER_PACKAGE,
71 "--enable-petsc, --enable-trilinos, or --enable-eigen");
72
73 // Skip adaptive examples on a non-adaptive libMesh build
74 #ifndef LIBMESH_ENABLE_AMR
75 libmesh_example_requires(false, "--enable-amr");
76 #else
77
78 // Create a mesh, with dimension to be overridden later, on the
79 // default MPI communicator.
80 Mesh mesh(init.comm());
81
82 GetPot command_line (argc, argv);
83
84 int n = 4;
85 if (command_line.search(1, "-n"))
86 n = command_line.next(n);
87
88 // Build a 1D mesh with 4 elements from x=0 to x=1, using
89 // EDGE3 (i.e. quadratic) 1D elements. They are called EDGE3 elements
90 // because a quadratic element contains 3 nodes.
91 MeshTools::Generation::build_line(mesh, n, 0., 1., EDGE3);
92
93 // Define the equation systems object and the system we are going
94 // to solve. See Introduction Example 2 for more details.
95 EquationSystems equation_systems(mesh);
96 LinearImplicitSystem & system = equation_systems.add_system
97 <LinearImplicitSystem>("1D");
98
99 // Add a variable "u" to the system, using second-order approximation
100 system.add_variable("u", SECOND);
101
102 // Give the system a pointer to the matrix assembly function. This
103 // will be called when needed by the library.
104 system.attach_assemble_function(assemble_1D);
105
106 // Define the mesh refinement object that takes care of adaptively
107 // refining the mesh.
108 MeshRefinement mesh_refinement(mesh);
109
110 // These parameters determine the proportion of elements that will
111 // be refined and coarsened. Any element within 30% of the maximum
112 // error on any element will be refined, and any element within 30%
113 // of the minimum error on any element might be coarsened
114 mesh_refinement.refine_fraction() = 0.7;
115 mesh_refinement.coarsen_fraction() = 0.3;
116 // We won't refine any element more than 5 times in total
117 mesh_refinement.max_h_level() = 5;
118
119 // Initialize the data structures for the equation system.
120 equation_systems.init();
121
122 // Refinement parameters
123 const unsigned int max_r_steps = 5; // Refine the mesh 5 times
124
125 // Define the refinement loop
126 for (unsigned int r_step=0; r_step<=max_r_steps; r_step++)
127 {
128 // Solve the equation system
129 equation_systems.get_system("1D").solve();
130
131 // We need to ensure that the mesh is not refined on the last iteration
132 // of this loop, since we do not want to refine the mesh unless we are
133 // going to solve the equation system for that refined mesh.
134 if (r_step != max_r_steps)
135 {
136 // Error estimation objects, see Adaptivity Example 2 for details
137 ErrorVector error;
138 KellyErrorEstimator error_estimator;
139 error_estimator.use_unweighted_quadrature_rules = true;
140
141 // Compute the error for each active element
142 error_estimator.estimate_error(system, error);
143
144 // Output error estimate magnitude
145 libMesh::out << "Error estimate\nl2 norm = "
146 << error.l2_norm()
147 << "\nmaximum = "
148 << error.maximum()
149 << std::endl;
150
151 // Flag elements to be refined and coarsened
152 mesh_refinement.flag_elements_by_error_fraction (error);
153
154 // Perform refinement and coarsening
155 mesh_refinement.refine_and_coarsen_elements();
156
157 // Reinitialize the equation_systems object for the newly refined
158 // mesh. One of the steps in this is project the solution onto the
159 // new mesh
160 equation_systems.reinit();
161 }
162 }
163
164 // Construct gnuplot plotting object, pass in mesh, title of plot
165 // and boolean to indicate use of grid in plot. The grid is used to
166 // show the edges of each element in the mesh.
167 GnuPlotIO plot(mesh, "Adaptivity Example 1", GnuPlotIO::GRID_ON);
168
169 // Write out script to be called from within gnuplot:
170 // Load gnuplot, then type "call 'gnuplot_script'" from gnuplot prompt
171 plot.write_equation_systems("gnuplot_script", equation_systems);
172 #endif // #ifndef LIBMESH_ENABLE_AMR
173
174 // All done. libMesh objects are destroyed here. Because the
175 // LibMeshInit object was created first, its destruction occurs
176 // last, and it's destructor finalizes any external libraries and
177 // checks for leaked memory.
178 return 0;
179 }
180
181
182
183
184 // Define the matrix assembly function for the 1D PDE we are solving
assemble_1D(EquationSystems & es,const std::string & system_name)185 void assemble_1D(EquationSystems & es,
186 const std::string & system_name)
187 {
188 // Ignore unused parameter warnings when !LIBMESH_ENABLE_AMR.
189 libmesh_ignore(es, system_name);
190
191 #ifdef LIBMESH_ENABLE_AMR
192
193 // It is a good idea to check we are solving the correct system
194 libmesh_assert_equal_to (system_name, "1D");
195
196 // Get a reference to the mesh object
197 const MeshBase & mesh = es.get_mesh();
198
199 // The dimension we are using, i.e. dim==1
200 const unsigned int dim = mesh.mesh_dimension();
201
202 // Get a reference to the system we are solving
203 LinearImplicitSystem & system = es.get_system<LinearImplicitSystem>("1D");
204
205 // Get a reference to the DofMap object for this system. The DofMap object
206 // handles the index translation from node and element numbers to degree of
207 // freedom numbers. DofMap's are discussed in more detail in future examples.
208 const DofMap & dof_map = system.get_dof_map();
209
210 // Get a constant reference to the Finite Element type for the first
211 // (and only) variable in the system.
212 FEType fe_type = dof_map.variable_type(0);
213
214 // Build a finite element object of the specified type. The build
215 // function dynamically allocates memory so we use a std::unique_ptr in this case.
216 // A std::unique_ptr is a pointer that cleans up after itself. See examples 3 and 4
217 // for more details on std::unique_ptr.
218 std::unique_ptr<FEBase> fe(FEBase::build(dim, fe_type));
219
220 // Tell the finite element object to use fifth order Gaussian quadrature
221 QGauss qrule(dim, FIFTH);
222 fe->attach_quadrature_rule(&qrule);
223
224 // Here we define some references to cell-specific data that will be used to
225 // assemble the linear system.
226
227 // The element Jacobian * quadrature weight at each integration point.
228 const std::vector<Real> & JxW = fe->get_JxW();
229
230 // The element shape functions evaluated at the quadrature points.
231 const std::vector<std::vector<Real>> & phi = fe->get_phi();
232
233 // The element shape function gradients evaluated at the quadrature points.
234 const std::vector<std::vector<RealGradient>> & dphi = fe->get_dphi();
235
236 // Declare a dense matrix and dense vector to hold the element matrix
237 // and right-hand-side contribution
238 DenseMatrix<Number> Ke;
239 DenseVector<Number> Fe;
240
241 // This vector will hold the degree of freedom indices for the element.
242 // These define where in the global system the element degrees of freedom
243 // get mapped.
244 std::vector<dof_id_type> dof_indices;
245
246 // We now loop over all the active elements in the mesh in order to calculate
247 // the matrix and right-hand-side contribution from each element. Use a
248 // const_element_iterator to loop over the elements. We make
249 // el_end const as it is used only for the stopping condition of the loop.
250 for (const auto & elem : mesh.active_local_element_ptr_range())
251 {
252 // Get the degree of freedom indices for the current element.
253 // These define where in the global matrix and right-hand-side this
254 // element will contribute to.
255 dof_map.dof_indices(elem, dof_indices);
256
257 // Compute the element-specific data for the current element. This
258 // involves computing the location of the quadrature points (q_point)
259 // and the shape functions (phi, dphi) for the current element.
260 fe->reinit(elem);
261
262 // Store the number of local degrees of freedom contained in this element
263 const unsigned int n_dofs =
264 cast_int<unsigned int>(dof_indices.size());
265 libmesh_assert_equal_to (n_dofs, phi.size());
266
267 // We resize and zero out Ke and Fe (resize() also clears the matrix and
268 // vector). In this example, all elements in the mesh are EDGE3's, so
269 // Ke will always be 3x3, and Fe will always be 3x1. If the mesh contained
270 // different element types, then the size of Ke and Fe would change.
271 Ke.resize(n_dofs, n_dofs);
272 Fe.resize(n_dofs);
273
274
275 // Now loop over quadrature points to handle numerical integration
276 for (unsigned int qp=0; qp<qrule.n_points(); qp++)
277 {
278 // Now build the element matrix and right-hand-side using loops to
279 // integrate the test functions (i) against the trial functions (j).
280 for (unsigned int i=0; i != n_dofs; i++)
281 {
282 Fe(i) += JxW[qp]*phi[i][qp];
283
284 for (unsigned int j=0; j != n_dofs; j++)
285 {
286 Ke(i,j) += JxW[qp]*(1.e-3*dphi[i][qp]*dphi[j][qp] +
287 phi[i][qp]*phi[j][qp]);
288 }
289 }
290 }
291
292
293 // At this point we have completed the matrix and RHS summation. The
294 // final step is to apply boundary conditions, which in this case are
295 // simple Dirichlet conditions with u(0) = u(1) = 0.
296
297 // Define the penalty parameter used to enforce the BC's
298 double penalty = 1.e10;
299
300 // Loop over the sides of this element. For a 1D element, the "sides"
301 // are defined as the nodes on each edge of the element, i.e. 1D elements
302 // have 2 sides.
303 for (auto s : elem->side_index_range())
304 {
305 // If this element has a nullptr neighbor, then it is on the edge of the
306 // mesh and we need to enforce a boundary condition using the penalty
307 // method.
308 if (elem->neighbor_ptr(s) == nullptr)
309 {
310 Ke(s,s) += penalty;
311 Fe(s) += 0*penalty;
312 }
313 }
314
315 // This is a function call that is necessary when using adaptive
316 // mesh refinement. See Adaptivity Example 2 for more details.
317 dof_map.constrain_element_matrix_and_vector (Ke, Fe, dof_indices);
318
319 // Add Ke and Fe to the global matrix and right-hand-side.
320 system.matrix->add_matrix(Ke, dof_indices);
321 system.rhs->add_vector(Fe, dof_indices);
322 }
323 #endif // #ifdef LIBMESH_ENABLE_AMR
324 }
325