110would yield periodicity constraints such that @f$u(0,y)=u(1,y)@f$
for all
113If we instead consider the
parallelogram given by the convex hull of
114@f$(0,0)@f$, @f$(1,1)@f$, @f$(1,2)@f$, @f$(0,1)@f$ we can achieve the constraints
115@f$u(0,y)=u(1,y+1)@f$ by specifying an @p offset:
133Here, again, the assignment of boundary indicators 0 and 1 stems from
136The resulting @p matched_pairs can be used in
138object with periodicity constraints:
140DoFTools::make_periodicity_constraints(matched_pairs, constraints);
143Apart from this high
level interface there are also variants of
144DoFTools::make_periodicity_constraints available that combine those two
145steps (see the variants of DofTools::make_periodicity_constraints).
147There is also a low
level interface to
148DoFTools::make_periodicity_constraints if more flexibility is needed. The
149low
level variant allows to directly specify two faces that shall be
153make_periodicity_constraints(face_1,
156 component_mask = <default value>;
157 face_orientation = <default value>,
158 face_flip = <default value>,
159 face_rotation = <default value>,
160 matrix = <default value>);
162Here, we need to specify the orientation of the two faces using
163@p face_orientation, @p face_flip and @p face_orientation. For a closer description
164have a look at the documentation of
DoFTools::make_periodicity_constraints.
165The remaining parameters are the same as for the high
level interface apart
166from the self-explaining @p component_mask and @p affine_constraints.
169<a name="step-45-problem"></a>
170<a name="step_45-Apracticalexample"></a><h1>A practical example</h1>
173In the following, we show how to use the above functions in a more involved
174example. The task is to enforce rotated periodicity constraints for the
175velocity component of a Stokes flow.
177On a quarter-circle defined by @f$\Omega=\{{\bf x}\in(0,1)^2:\|{\bf x}\|\in (0.5,1)\}@f$ we are
178going to solve the Stokes problem
180 -\Delta \; \textbf{u} + \nabla p &=& (\exp(-100\|{\bf x}-(.75,0.1)^T\|^2),0)^T, \\
181 -\textrm{div}\; \textbf{u}&=&0,\\
182 \textbf{u}|_{\Gamma_1}&=&{\bf 0},
184where the boundary @f$\Gamma_1@f$ is defined as @f$\Gamma_1 \dealcoloneq \{x\in \partial\Omega: \|x\|\in\{0.5,1\}\}@f$.
185For the remaining parts of the boundary we are going to use periodic boundary conditions, i.e.
187 u_x(0,\nu)&=-u_y(\nu,0)&\nu&\in[0,1]\\
188 u_y(0,\nu)&=u_x(\nu,0)&\nu&\in[0,1].
192which also documents how it assigns boundary indicators to its various
193boundaries
if its `
colorize` argument is set to `
true`.
196 * <a name=
"step_45-CommProg"></a>
197 * <h1> The commented program</h1>
199 * This example program is a slight modification of @ref step_22
"step-22" running in
parallel
200 *
using Trilinos to demonstrate the usage of periodic boundary conditions in
201 * deal.II. We thus omit to discuss the majority of the source code and only
202 * comment on the parts that deal with periodicity constraints. For the rest
203 * have a look at @ref step_22
"step-22" and the full source code at the bottom.
207 * In order to implement periodic boundary conditions only two
functions
208 * have to be modified:
209 * - <code>StokesProblem<dim>::setup_dofs()</code>:
211 * - <code>StokesProblem<dim>::create_mesh()</code>:
212 * To supply a distributed
triangulation with periodicity information.
216 * The rest of the program is identical to @ref step_22
"step-22", so let us skip
this part
217 * and only show these two
functions in the following. (The full program can be
218 * found in the
"Plain program" section below, though.)
229 * <a name=
"step_45-Settingupperiodicityconstraintsondistributedtriangulations"></a>
230 * <h3>Setting up periodicity constraints on distributed triangulations</h3>
234 *
void StokesProblem<dim>::create_mesh()
237 *
const double inner_radius = .5;
238 *
const double outer_radius = 1.;
245 * Before we can prescribe periodicity constraints, we need to ensure that
246 * cells on opposite sides of the domain but connected by periodic faces are
247 * part of the ghost layer
if one of them is stored on the local processor.
248 * At
this point we need to think about how we want to prescribe
250 * boundary should be matched to the
vertices @f$\text{
vertices}_1@f$ of a face
251 * on the lower boundary given by @f$\text{
vertices}_2=R\cdot
252 * \text{
vertices}_1+
b@f$ where the rotation
matrix @f$R@f$ and the offset @f$b@f$ are
261 * The data structure we are saving the resulting information into is here
267 * periodicity_vector;
270 * rotation_matrix[0][1] = 1.;
271 * rotation_matrix[1][0] = -1.;
277 * periodicity_vector,
283 * Now telling the
triangulation about the desired periodicity is
284 * particularly easy by just calling
295 *
void StokesProblem<dim>::setup_dofs()
297 * dof_handler.distribute_dofs(fe);
299 * std::vector<unsigned int> block_component(dim + 1, 0);
300 * block_component[dim] = 1;
303 *
const std::vector<types::global_dof_index> dofs_per_block =
305 *
const unsigned int n_u = dofs_per_block[0], n_p = dofs_per_block[1];
308 * owned_partitioning.clear();
309 *
const IndexSet &locally_owned_dofs = dof_handler.locally_owned_dofs();
310 * owned_partitioning.push_back(locally_owned_dofs.get_view(0, n_u));
311 * owned_partitioning.push_back(locally_owned_dofs.get_view(n_u, n_u + n_p));
313 * relevant_partitioning.
clear();
314 *
const IndexSet locally_relevant_dofs =
316 * relevant_partitioning.push_back(locally_relevant_dofs.get_view(0, n_u));
317 * relevant_partitioning.push_back(
318 * locally_relevant_dofs.get_view(n_u, n_u + n_p));
320 * constraints.clear();
321 * constraints.reinit(locally_owned_dofs, locally_relevant_dofs);
329 * BoundaryValues<dim>(),
331 * fe.component_mask(velocities));
335 * BoundaryValues<dim>(),
337 * fe.component_mask(velocities));
341 * After we provided the mesh with the necessary information
for the
342 * periodicity constraints, we are now able to actual create them. For
343 * describing the
matching we are
using the same approach as before, i.e.,
344 * the @f$\text{
vertices}_2@f$ of a face on the left boundary should be
346 * @f$\text{
vertices}_1@f$ of a face on the lower boundary given by
348 *
matrix @f$R@f$ and the offset @f$b@f$ are given by
356 * These two objects not only describe how faces should be matched but
357 * also in which sense the solution should be transformed from
358 * @f$\text{face}_2@f$ to
359 * @f$\text{face}_1@f$.
363 * rotation_matrix[0][1] = 1.;
364 * rotation_matrix[1][0] = -1.;
370 * For setting up the constraints, we
first store the periodicity
371 * information in an auxiliary
object of type
373 *
DoFHandler@<dim@>::%cell_iterator@> </code>. The periodic boundaries
374 * have the boundary indicators 2 (x=0) and 3 (y=0). All the other
375 * parameters we have set up before. In
this case the direction does not
377 * exactly what we want.
382 * periodicity_vector;
384 *
const unsigned int direction = 1;
390 * periodicity_vector,
396 * Next, we need to provide information on which vector valued components
397 * of the solution should be rotated. Since we choose here to just
398 * constraint the velocity and
this starts at the
first component of the
399 * solution vector, we simply
insert a 0:
402 * std::vector<unsigned int> first_vector_components;
403 * first_vector_components.push_back(0);
407 * After setting up all the information in periodicity_vector all we have
412 * DoFTools::make_periodicity_constraints<dim, dim>(periodicity_vector,
416 * first_vector_components);
419 * constraints.close();
423 * owned_partitioning,
424 * relevant_partitioning,
428 *
for (
unsigned int c = 0; c < dim + 1; ++c)
429 *
for (
unsigned int d = 0;
d < dim + 1; ++
d)
430 *
if (!((c == dim) && (d == dim)))
441 * mpi_communicator));
445 * system_matrix.reinit(bsp);
450 * owned_partitioning,
451 * owned_partitioning,
452 * relevant_partitioning,
456 *
for (
unsigned int c = 0; c < dim + 1; ++c)
457 *
for (
unsigned int d = 0;
d < dim + 1; ++
d)
458 *
if ((c == dim) && (d == dim))
464 * preconditioner_coupling,
465 * preconditioner_bsp,
469 * mpi_communicator));
471 * preconditioner_bsp.compress();
473 * preconditioner_matrix.reinit(preconditioner_bsp);
476 * system_rhs.reinit(owned_partitioning, mpi_communicator);
477 * solution.reinit(owned_partitioning,
478 * relevant_partitioning,
484 * The rest of the program is then again identical to @ref step_22
"step-22". We will omit
485 * it here now, but as before, you can find these parts in the
"Plain program"
490<a name=
"step_45-Results"></a><h1>Results</h1>
493The created output is not very surprising. We simply see that the solution is
494periodic with respect to the left and lower boundary:
496<img src=
"https://www.dealii.org/images/steps/developer/step-45.periodic.png" alt=
"">
498Without the periodicity constraints we would have ended up with the following solution:
500<img src=
"https://www.dealii.org/images/steps/developer/step-45.non_periodic.png" alt=
"">
503<a name=
"step_45-PlainProg"></a>
504<h1> The plain program</h1>
virtual void add_periodicity(const std::vector<::GridTools::PeriodicFacePair< cell_iterator > > &) override
typename ::Triangulation< dim, spacedim >::cell_iterator cell_iterator
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={}, 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 >())
void parallelogram(Triangulation< dim > &tria, const Point< dim >(&corners)[dim], const bool colorize=false)
void quarter_hyper_shell(Triangulation< dim > &tria, const Point< dim > ¢er, const double inner_radius, const double outer_radius, const unsigned int n_cells=0, const bool colorize=false)
@ matrix
Contents is actually a matrix.
Point< spacedim > point(const gp_Pnt &p, const double tolerance=1e-10)
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)
int(&) functions(const void *v1, const void *v2)
const ::parallel::distributed::Triangulation< dim, spacedim > * triangulation