Reference documentation for deal.II version 9.5.0
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step-45.h
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1 0,
108 /*b_id2*/ 1,
109 /*direction*/ 0,
110 matched_pairs);
111@endcode
112would yield periodicity constraints such that @f$u(0,y)=u(1,y)@f$ for all
113@f$y\in[0,1]@f$.
114
115If we instead consider the parallelogram given by the convex hull of
116@f$(0,0)@f$, @f$(1,1)@f$, @f$(1,2)@f$, @f$(0,1)@f$ we can achieve the constraints
117@f$u(0,y)=u(1,y+1)@f$ by specifying an @p offset:
118@code
120 /*b_id1*/ 0,
121 /*b_id2*/ 1,
122 /*direction*/ 0,
123 matched_pairs,
124 Tensor<1, 2>(0.,1.));
125@endcode
126or
127@code
129 /*b_id1*/ 0,
130 /*b_id2*/ 1,
131 /*arbitrary direction*/ 0,
132 matched_pairs,
133 Tensor<1, 2>(1.,1.));
134@endcode
135Here, again, the assignment of boundary indicators 0 and 1 stems from
136what GridGenerator::parallelogram() documents.
137
138The resulting @p matched_pairs can be used in
139DoFTools::make_periodicity_constraints for populating an AffineConstraints
140object with periodicity constraints:
141@code
142DoFTools::make_periodicity_constraints(matched_pairs, constraints);
143@endcode
144
145Apart from this high level interface there are also variants of
146DoFTools::make_periodicity_constraints available that combine those two
147steps (see the variants of DofTools::make_periodicity_constraints).
148
149There is also a low level interface to
150DoFTools::make_periodicity_constraints if more flexibility is needed. The
151low level variant allows to directly specify two faces that shall be
152constrained:
153@code
154using namespace DoFTools;
155make_periodicity_constraints(face_1,
156 face_2,
157 affine_constraints,
158 component_mask = <default value>;
159 face_orientation = <default value>,
160 face_flip = <default value>,
161 face_rotation = <default value>,
162 matrix = <default value>);
163@endcode
164Here, we need to specify the orientation of the two faces using
165@p face_orientation, @p face_flip and @p face_orientation. For a closer description
166have a look at the documentation of DoFTools::make_periodicity_constraints.
167The remaining parameters are the same as for the high level interface apart
168from the self-explaining @p component_mask and @p affine_constraints.
169
170
171<a name="problem"></a>
172<a name="Apracticalexample"></a><h1>A practical example</h1>
173
174
175In the following, we show how to use the above functions in a more involved
176example. The task is to enforce rotated periodicity constraints for the
177velocity component of a Stokes flow.
178
179On a quarter-circle defined by @f$\Omega=\{{\bf x}\in(0,1)^2:\|{\bf x}\|\in (0.5,1)\}@f$ we are
180going to solve the Stokes problem
181@f{eqnarray*}
182 -\Delta \; \textbf{u} + \nabla p &=& (\exp(-100\|{\bf x}-(.75,0.1)^T\|^2),0)^T, \\
183 -\textrm{div}\; \textbf{u}&=&0,\\
184 \textbf{u}|_{\Gamma_1}&=&{\bf 0},
185@f}
186where the boundary @f$\Gamma_1@f$ is defined as @f$\Gamma_1 \dealcoloneq \{x\in \partial\Omega: \|x\|\in\{0.5,1\}\}@f$.
187For the remaining parts of the boundary we are going to use periodic boundary conditions, i.e.
188@f{align*}
189 u_x(0,\nu)&=-u_y(\nu,0)&\nu&\in[0,1]\\
190 u_y(0,\nu)&=u_x(\nu,0)&\nu&\in[0,1].
191@f}
192
193The mesh will be generated by GridGenerator::quarter_hyper_shell(),
194which also documents how it assigns boundary indicators to its various
195boundaries if its `colorize` argument is set to `true`.
196 *
197 *
198 * <a name="CommProg"></a>
199 * <h1> The commented program</h1>
200 *
201 * This example program is a slight modification of @ref step_22 "step-22" running in parallel
202 * using Trilinos to demonstrate the usage of periodic boundary conditions in
203 * deal.II. We thus omit to discuss the majority of the source code and only
204 * comment on the parts that deal with periodicity constraints. For the rest
205 * have a look at @ref step_22 "step-22" and the full source code at the bottom.
206 *
207
208 *
209 * In order to implement periodic boundary conditions only two functions
210 * have to be modified:
211 * - <code>StokesProblem<dim>::setup_dofs()</code>:
212 * To populate an AffineConstraints object with periodicity constraints
213 * - <code>StokesProblem<dim>::create_mesh()</code>:
214 * To supply a distributed triangulation with periodicity information.
215 *
216
217 *
218 * The rest of the program is identical to @ref step_22 "step-22", so let us skip this part
219 * and only show these two functions in the following. (The full program can be
220 * found in the "Plain program" section below, though.)
221 *
222
223 *
224 *
225
226 *
227 *
228
229 *
230 *
231 * <a name="Settingupperiodicityconstraintsondistributedtriangulations"></a>
232 * <h3>Setting up periodicity constraints on distributed triangulations</h3>
233 *
234 * @code
235 *   template <int dim>
236 *   void StokesProblem<dim>::create_mesh()
237 *   {
238 *   Point<dim> center;
239 *   const double inner_radius = .5;
240 *   const double outer_radius = 1.;
241 *  
243 *   triangulation, center, inner_radius, outer_radius, 0, true);
244 *  
245 * @endcode
246 *
247 * Before we can prescribe periodicity constraints, we need to ensure that
248 * cells on opposite sides of the domain but connected by periodic faces are
249 * part of the ghost layer if one of them is stored on the local processor.
250 * At this point we need to think about how we want to prescribe
251 * periodicity. The vertices @f$\text{vertices}_2@f$ of a face on the left
252 * boundary should be matched to the vertices @f$\text{vertices}_1@f$ of a face
253 * on the lower boundary given by @f$\text{vertices}_2=R\cdot
254 * \text{vertices}_1+b@f$ where the rotation matrix @f$R@f$ and the offset @f$b@f$ are
255 * given by
256 * @f{align*}
257 * R=\begin{pmatrix}
258 * 0&1\\-1&0
259 * \end{pmatrix},
260 * \quad
261 * b=\begin{pmatrix}0&0\end{pmatrix}.
262 * @f}
263 * The data structure we are saving the resulting information into is here
264 * based on the Triangulation.
265 *
266 * @code
267 *   std::vector<GridTools::PeriodicFacePair<
269 *   periodicity_vector;
270 *  
271 *   FullMatrix<double> rotation_matrix(dim);
272 *   rotation_matrix[0][1] = 1.;
273 *   rotation_matrix[1][0] = -1.;
274 *  
276 *   2,
277 *   3,
278 *   1,
279 *   periodicity_vector,
280 *   Tensor<1, dim>(),
281 *   rotation_matrix);
282 *  
283 * @endcode
284 *
285 * Now telling the triangulation about the desired periodicity is
286 * particularly easy by just calling
288 *
289 * @code
290 *   triangulation.add_periodicity(periodicity_vector);
291 *  
292 *   triangulation.refine_global(4 - dim);
293 *   }
294 *  
295 *  
296 *   template <int dim>
297 *   void StokesProblem<dim>::setup_dofs()
298 *   {
299 *   dof_handler.distribute_dofs(fe);
300 *  
301 *   std::vector<unsigned int> block_component(dim + 1, 0);
302 *   block_component[dim] = 1;
303 *   DoFRenumbering::component_wise(dof_handler, block_component);
304 *  
305 *   const std::vector<types::global_dof_index> dofs_per_block =
306 *   DoFTools::count_dofs_per_fe_block(dof_handler, block_component);
307 *   const unsigned int n_u = dofs_per_block[0], n_p = dofs_per_block[1];
308 *  
309 *   {
310 *   owned_partitioning.clear();
311 *   IndexSet locally_owned_dofs = dof_handler.locally_owned_dofs();
312 *   owned_partitioning.push_back(locally_owned_dofs.get_view(0, n_u));
313 *   owned_partitioning.push_back(locally_owned_dofs.get_view(n_u, n_u + n_p));
314 *  
315 *   relevant_partitioning.clear();
316 *   const IndexSet locally_relevant_dofs =
318 *   relevant_partitioning.push_back(locally_relevant_dofs.get_view(0, n_u));
319 *   relevant_partitioning.push_back(
320 *   locally_relevant_dofs.get_view(n_u, n_u + n_p));
321 *  
322 *   constraints.clear();
323 *   constraints.reinit(locally_relevant_dofs);
324 *  
325 *   const FEValuesExtractors::Vector velocities(0);
326 *  
327 *   DoFTools::make_hanging_node_constraints(dof_handler, constraints);
329 *   dof_handler,
330 *   0,
331 *   BoundaryValues<dim>(),
332 *   constraints,
333 *   fe.component_mask(velocities));
335 *   dof_handler,
336 *   1,
337 *   BoundaryValues<dim>(),
338 *   constraints,
339 *   fe.component_mask(velocities));
340 *  
341 * @endcode
342 *
343 * After we provided the mesh with the necessary information for the
344 * periodicity constraints, we are now able to actual create them. For
345 * describing the matching we are using the same approach as before, i.e.,
346 * the @f$\text{vertices}_2@f$ of a face on the left boundary should be
347 * matched to the vertices
348 * @f$\text{vertices}_1@f$ of a face on the lower boundary given by
349 * @f$\text{vertices}_2=R\cdot \text{vertices}_1+b@f$ where the rotation
350 * matrix @f$R@f$ and the offset @f$b@f$ are given by
351 * @f{align*}
352 * R=\begin{pmatrix}
353 * 0&1\\-1&0
354 * \end{pmatrix},
355 * \quad
356 * b=\begin{pmatrix}0&0\end{pmatrix}.
357 * @f}
358 * These two objects not only describe how faces should be matched but
359 * also in which sense the solution should be transformed from
360 * @f$\text{face}_2@f$ to
361 * @f$\text{face}_1@f$.
362 *
363 * @code
364 *   FullMatrix<double> rotation_matrix(dim);
365 *   rotation_matrix[0][1] = 1.;
366 *   rotation_matrix[1][0] = -1.;
367 *  
368 *   Tensor<1, dim> offset;
369 *  
370 * @endcode
371 *
372 * For setting up the constraints, we first store the periodicity
373 * information in an auxiliary object of type
374 * <code>std::vector@<GridTools::PeriodicFacePair<typename
375 * DoFHandler@<dim@>::%cell_iterator@> </code>. The periodic boundaries
376 * have the boundary indicators 2 (x=0) and 3 (y=0). All the other
377 * parameters we have set up before. In this case the direction does not
378 * matter. Due to @f$\text{vertices}_2=R\cdot \text{vertices}_1+b@f$ this is
379 * exactly what we want.
380 *
381 * @code
382 *   std::vector<
384 *   periodicity_vector;
385 *  
386 *   const unsigned int direction = 1;
387 *  
388 *   GridTools::collect_periodic_faces(dof_handler,
389 *   2,
390 *   3,
391 *   direction,
392 *   periodicity_vector,
393 *   offset,
394 *   rotation_matrix);
395 *  
396 * @endcode
397 *
398 * Next, we need to provide information on which vector valued components
399 * of the solution should be rotated. Since we choose here to just
400 * constraint the velocity and this starts at the first component of the
401 * solution vector, we simply insert a 0:
402 *
403 * @code
404 *   std::vector<unsigned int> first_vector_components;
405 *   first_vector_components.push_back(0);
406 *  
407 * @endcode
408 *
409 * After setting up all the information in periodicity_vector all we have
410 * to do is to tell make_periodicity_constraints to create the desired
411 * constraints.
412 *
413 * @code
414 *   DoFTools::make_periodicity_constraints<dim, dim>(periodicity_vector,
415 *   constraints,
416 *   fe.component_mask(
417 *   velocities),
418 *   first_vector_components);
419 *   }
420 *  
421 *   constraints.close();
422 *  
423 *   {
424 *   TrilinosWrappers::BlockSparsityPattern bsp(owned_partitioning,
425 *   owned_partitioning,
426 *   relevant_partitioning,
427 *   mpi_communicator);
428 *  
429 *   Table<2, DoFTools::Coupling> coupling(dim + 1, dim + 1);
430 *   for (unsigned int c = 0; c < dim + 1; ++c)
431 *   for (unsigned int d = 0; d < dim + 1; ++d)
432 *   if (!((c == dim) && (d == dim)))
433 *   coupling[c][d] = DoFTools::always;
434 *   else
435 *   coupling[c][d] = DoFTools::none;
436 *  
437 *   DoFTools::make_sparsity_pattern(dof_handler,
438 *   coupling,
439 *   bsp,
440 *   constraints,
441 *   false,
443 *   mpi_communicator));
444 *  
445 *   bsp.compress();
446 *  
447 *   system_matrix.reinit(bsp);
448 *   }
449 *  
450 *   {
451 *   TrilinosWrappers::BlockSparsityPattern preconditioner_bsp(
452 *   owned_partitioning,
453 *   owned_partitioning,
454 *   relevant_partitioning,
455 *   mpi_communicator);
456 *  
457 *   Table<2, DoFTools::Coupling> preconditioner_coupling(dim + 1, dim + 1);
458 *   for (unsigned int c = 0; c < dim + 1; ++c)
459 *   for (unsigned int d = 0; d < dim + 1; ++d)
460 *   if ((c == dim) && (d == dim))
461 *   preconditioner_coupling[c][d] = DoFTools::always;
462 *   else
463 *   preconditioner_coupling[c][d] = DoFTools::none;
464 *  
465 *   DoFTools::make_sparsity_pattern(dof_handler,
466 *   preconditioner_coupling,
467 *   preconditioner_bsp,
468 *   constraints,
469 *   false,
471 *   mpi_communicator));
472 *  
473 *   preconditioner_bsp.compress();
474 *  
475 *   preconditioner_matrix.reinit(preconditioner_bsp);
476 *   }
477 *  
478 *   system_rhs.reinit(owned_partitioning, mpi_communicator);
479 *   solution.reinit(owned_partitioning,
480 *   relevant_partitioning,
481 *   mpi_communicator);
482 *   }
483 *  
484 * @endcode
485 *
486 * The rest of the program is then again identical to @ref step_22 "step-22". We will omit
487 * it here now, but as before, you can find these parts in the "Plain program"
488 * section below.
489 *
490
491 *
492<a name="Results"></a><h1>Results</h1>
493
494
495The created output is not very surprising. We simply see that the solution is
496periodic with respect to the left and lower boundary:
497
498<img src="https://www.dealii.org/images/steps/developer/step-45.periodic.png" alt="">
499
500Without the periodicity constraints we would have ended up with the following solution:
501
502<img src="https://www.dealii.org/images/steps/developer/step-45.non_periodic.png" alt="">
503 *
504 *
505<a name="PlainProg"></a>
506<h1> The plain program</h1>
507@include "step-45.cc"
508*/
void clear()
Definition index_set.h:1637
Definition point.h:112
virtual void add_periodicity(const std::vector<::GridTools::PeriodicFacePair< cell_iterator > > &) override
Definition tria.cc:3720
Point< 3 > center
Point< 3 > vertices[4]
bool colorize
Definition grid_out.cc:4617
Point< 2 > first
Definition grid_out.cc:4615
unsigned int level
Definition grid_out.cc:4618
__global__ void set(Number *val, const Number s, const size_type N)
typename ::Triangulation< dim, spacedim >::cell_iterator cell_iterator
Definition tria.h:270
void make_hanging_node_constraints(const DoFHandler< dim, spacedim > &dof_handler, AffineConstraints< number > &constraints)
void make_sparsity_pattern(const DoFHandler< dim, spacedim > &dof_handler, SparsityPatternBase &sparsity_pattern, const AffineConstraints< number > &constraints=AffineConstraints< number >(), const bool keep_constrained_dofs=true, const types::subdomain_id subdomain_id=numbers::invalid_subdomain_id)
void component_wise(DoFHandler< dim, spacedim > &dof_handler, const std::vector< unsigned int > &target_component=std::vector< unsigned int >())
IndexSet extract_locally_relevant_dofs(const DoFHandler< dim, spacedim > &dof_handler)
std::vector< types::global_dof_index > count_dofs_per_fe_block(const DoFHandler< dim, spacedim > &dof, const std::vector< unsigned int > &target_block=std::vector< unsigned int >())
void make_periodicity_constraints(const FaceIterator &face_1, const std_cxx20::type_identity_t< FaceIterator > &face_2, AffineConstraints< number > &constraints, const ComponentMask &component_mask=ComponentMask(), const bool face_orientation=true, const bool face_flip=false, const bool face_rotation=false, const FullMatrix< double > &matrix=FullMatrix< double >(), const std::vector< unsigned int > &first_vector_components=std::vector< unsigned int >(), const number periodicity_factor=1.)
void parallelogram(Triangulation< dim > &tria, const Point< dim >(&corners)[dim], const bool colorize=false)
void quarter_hyper_shell(Triangulation< dim > &tria, const Point< dim > &center, const double inner_radius, const double outer_radius, const unsigned int n_cells=0, const bool colorize=false)
void collect_periodic_faces(const MeshType &mesh, const types::boundary_id b_id1, const types::boundary_id b_id2, const unsigned int direction, std::vector< PeriodicFacePair< typename MeshType::cell_iterator > > &matched_pairs, const Tensor< 1, MeshType::space_dimension > &offset=::Tensor< 1, MeshType::space_dimension >(), const FullMatrix< double > &matrix=FullMatrix< double >())
@ matrix
Contents is actually a matrix.
Point< spacedim > point(const gp_Pnt &p, const double tolerance=1e-10)
Definition utilities.cc:189
SymmetricTensor< 2, dim, Number > b(const Tensor< 2, dim, Number > &F)
SymmetricTensor< 2, dim, Number > d(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
VectorType::value_type * end(VectorType &V)
VectorType::value_type * begin(VectorType &V)
unsigned int this_mpi_process(const MPI_Comm mpi_communicator)
Definition mpi.cc:161
void interpolate_boundary_values(const Mapping< dim, spacedim > &mapping, const DoFHandler< dim, spacedim > &dof, const std::map< types::boundary_id, const Function< spacedim, number > * > &function_map, std::map< types::global_dof_index, number > &boundary_values, const ComponentMask &component_mask=ComponentMask())
int(&) functions(const void *v1, const void *v2)
const ::parallel::distributed::Triangulation< dim, spacedim > * triangulation