464 *
const std::array<double, dim> p_sphere =
467 *
constexpr const double alpha = 2. / 3.;
477 * <a name=
"Parameters"></a>
478 * <h3>Parameters</h3>
482 * For
this tutorial, we will use a simplified
set of parameters. It is also
484 *
short we decided on
using simple structs. The actual intention of all these
485 * parameters will be described in the upcoming classes at their respective
486 * location where they are used.
490 * The following parameter
set controls the coarse-grid solver, the smoothers,
491 * and the inter-grid transfer scheme of the multigrid mechanism.
492 * We populate it with
default parameters.
495 *
struct MultigridParameters
499 * std::string type =
"cg_with_amg";
500 *
unsigned int maxiter = 10000;
501 *
double abstol = 1
e-20;
502 *
double reltol = 1
e-4;
503 *
unsigned int smoother_sweeps = 1;
504 *
unsigned int n_cycles = 1;
505 * std::string smoother_type =
"ILU";
510 * std::string type =
"chebyshev";
511 *
double smoothing_range = 20;
512 *
unsigned int degree = 5;
513 *
unsigned int eig_cg_n_iterations = 20;
520 * PolynomialCoarseningSequenceType::decrease_by_one;
521 *
bool perform_h_transfer =
true;
529 * This is the
general parameter
struct for the problem class. You will find
530 *
this struct divided into several categories, including
general runtime
532 * parameters
for cell weighting. It also contains an instance of the above
533 *
struct for multigrid parameters which will be passed to the multigrid
539 *
unsigned int n_cycles = 8;
540 *
double tolerance_factor = 1
e-12;
542 * MultigridParameters mg_data;
544 *
unsigned int min_h_level = 5;
545 *
unsigned int max_h_level = 12;
546 *
unsigned int min_p_degree = 2;
547 *
unsigned int max_p_degree = 6;
548 *
unsigned int max_p_level_difference = 1;
550 *
double refine_fraction = 0.3;
551 *
double coarsen_fraction = 0.03;
552 *
double p_refine_fraction = 0.9;
553 *
double p_coarsen_fraction = 0.9;
555 *
double weighting_factor = 1e6;
556 *
double weighting_exponent = 1.;
564 * <a name=
"MatrixfreeLaplaceoperator"></a>
565 * <h3>Matrix-
free Laplace
operator</h3>
569 * This is a
matrix-
free implementation of the Laplace
operator that will
570 * basically take over the part of the `assemble_system()` function from other
571 * tutorials. The meaning of all member
functions will be explained at their
576 * We will use the
FEEvaluation class to evaluate the solution vector
577 * at the quadrature points and to perform the integration. In contrast to
578 * other tutorials, the
template arguments `degree` is
set to @f$-1@f$ and
579 * `number of quadrature in 1D` to @f$0@f$. In
this case,
FEEvaluation selects
580 * dynamically the correct polynomial degree and number of quadrature
581 * points. Here, we introduce an alias to
FEEvaluation with the correct
582 *
template parameters so that we
do not have to worry about them later on.
585 *
template <
int dim,
typename number>
591 *
using FECellIntegrator =
FEEvaluation<dim, -1, 0, 1, number>;
593 * LaplaceOperator() =
default;
599 * VectorType & system_rhs);
605 * VectorType & system_rhs);
609 * number el(
unsigned int,
unsigned int)
const;
611 *
void initialize_dof_vector(VectorType &vec)
const;
613 *
void vmult(VectorType &dst,
const VectorType &src)
const;
615 *
void Tvmult(VectorType &dst,
const VectorType &src)
const;
619 *
void compute_inverse_diagonal(VectorType &
diagonal)
const;
622 *
void do_cell_integral_local(FECellIntegrator &integrator)
const;
624 *
void do_cell_integral_global(FECellIntegrator &integrator,
626 *
const VectorType &src)
const;
629 *
void do_cell_integral_range(
632 *
const VectorType & src,
633 *
const std::pair<unsigned int, unsigned int> &range)
const;
639 * To solve the equation system on the coarsest
level with an AMG
640 * preconditioner, we need an actual system
matrix on the coarsest
level.
641 * For
this purpose, we provide a mechanism that optionally computes a
644 * empty. Once `get_system_matrix()` is called,
this matrix is filled (lazy
645 * allocation). Since
this is a `
const` function, we need the
"mutable"
646 * keyword here. We also need a the constraints
object to build the
matrix.
657 * The following section contains
functions to initialize and reinitialize
659 *
MatrixFree instance. For sake of simplicity, we also compute the system
660 * right-hand-side vector.
663 *
template <
int dim,
typename number>
664 * LaplaceOperator<dim, number>::LaplaceOperator(
669 * VectorType & system_rhs)
671 * this->
reinit(mapping, dof_handler, quad, constraints, system_rhs);
676 *
template <
int dim,
typename number>
682 * VectorType & system_rhs)
686 * Clear
internal data structures (in the
case that the
operator is reused).
689 * this->system_matrix.clear();
693 * Copy the constraints, since they might be needed
for computation of the
697 * this->constraints.copy_from(constraints);
701 * Set up
MatrixFree. At the quadrature points, we only need to evaluate
709 * matrix_free.reinit(mapping, dof_handler, constraints, quad, data);
713 * Compute the right-hand side vector. For
this purpose, we
set up a
second
715 * the constraints due to Dirichlet-boundary conditions. This modified
716 *
operator is applied to a vector with only the Dirichlet
values set. The
717 * result is the negative right-hand-side vector.
725 * locally_relevant_dofs);
726 * constraints_without_dbc.
reinit(locally_relevant_dofs);
729 * constraints_without_dbc);
730 * constraints_without_dbc.
close();
734 * this->initialize_dof_vector(system_rhs);
738 * mapping, dof_handler, constraints_without_dbc, quad, data);
743 * constraints.distribute(x);
745 * matrix_free.
cell_loop(&LaplaceOperator::do_cell_integral_range,
750 * constraints.set_zero(
b);
760 * The following
functions are implicitly needed by the multigrid algorithm,
761 * including the smoothers.
766 * degrees of freedom.
769 *
template <
int dim,
typename number>
779 * Access a particular element in the
matrix. This function is neither
780 * needed nor implemented, however, is required to compile the program.
783 *
template <
int dim,
typename number>
784 * number LaplaceOperator<dim, number>::el(
unsigned int,
unsigned int)
const
794 * Initialize the given vector. We simply delegate the task to the
798 *
template <
int dim,
typename number>
800 * LaplaceOperator<dim, number>::initialize_dof_vector(VectorType &vec)
const
809 * Perform an
operator evaluation by looping with the help of
MatrixFree
810 * over all cells and evaluating the effect of the cell integrals (see also:
811 * `do_cell_integral_local()` and `do_cell_integral_global()`).
814 *
template <
int dim,
typename number>
815 *
void LaplaceOperator<dim, number>::vmult(VectorType & dst,
816 *
const VectorType &src)
const
819 * &LaplaceOperator::do_cell_integral_range,
this, dst, src,
true);
826 * Perform the transposed
operator evaluation. Since we are considering
827 *
symmetric "matrices",
this function can simply delegate it task to vmult().
830 *
template <
int dim,
typename number>
831 *
void LaplaceOperator<dim, number>::Tvmult(VectorType & dst,
832 *
const VectorType &src)
const
834 * this->vmult(dst, src);
841 * Since we
do not have a system
matrix, we cannot
loop over the the
843 * performing a sequence of
operator evaluations to unit basis vectors.
845 *
namespace is used. The inversion is performed manually afterwards.
848 *
template <
int dim,
typename number>
849 *
void LaplaceOperator<dim, number>::compute_inverse_diagonal(
854 * &LaplaceOperator::do_cell_integral_local,
858 * i = (
std::abs(i) > 1.0e-10) ? (1.0 / i) : 1.0;
866 * initialization of
this class. As a consequence, it has to be computed
867 * here
if it should be requested. Since the
matrix is only computed in
868 *
this tutorial
for linear elements (on the coarse grid),
this is
870 * The
matrix entries are obtained via sequence of
operator evaluations.
872 * is used. The
matrix will only be computed
if it has not been
set up yet
876 * template <int dim, typename number>
878 * LaplaceOperator<dim, number>::get_system_matrix() const
880 *
if (system_matrix.m() == 0 && system_matrix.n() == 0)
885 * dof_handler.locally_owned_dofs(),
886 * dof_handler.get_triangulation().get_communicator());
891 * system_matrix.reinit(dsp);
897 * &LaplaceOperator::do_cell_integral_local,
901 *
return this->system_matrix;
908 * Perform cell integral on a cell batch without gathering and scattering
913 *
template <
int dim,
typename number>
914 *
void LaplaceOperator<dim, number>::do_cell_integral_local(
915 * FECellIntegrator &integrator)
const
919 *
for (
unsigned int q = 0; q < integrator.n_q_points; ++q)
920 * integrator.submit_gradient(integrator.get_gradient(q), q);
929 * Same as above but with access to the global vectors.
932 *
template <
int dim,
typename number>
933 *
void LaplaceOperator<dim, number>::do_cell_integral_global(
934 * FECellIntegrator &integrator,
936 *
const VectorType &src)
const
940 *
for (
unsigned int q = 0; q < integrator.n_q_points; ++q)
941 * integrator.submit_gradient(integrator.get_gradient(q), q);
950 * This function loops over all cell batches within a cell-batch range and
951 * calls the above function.
954 *
template <
int dim,
typename number>
955 *
void LaplaceOperator<dim, number>::do_cell_integral_range(
958 *
const VectorType & src,
959 *
const std::pair<unsigned int, unsigned int> &range)
const
961 * FECellIntegrator integrator(matrix_free, range);
963 *
for (
unsigned cell = range.first; cell < range.second; ++cell)
965 * integrator.reinit(cell);
967 * do_cell_integral_global(integrator, dst, src);
976 * <a name=
"Solverandpreconditioner"></a>
977 * <h3>Solver and preconditioner</h3>
982 * <a name=
"Conjugategradientsolverwithmultigridpreconditioner"></a>
983 * <h4>Conjugate-
gradient solver with multigrid preconditioner</h4>
987 * This function solves the equation system with a sequence of provided
988 * multigrid objects. It is meant to be treated as
general as possible, hence
989 * the multitude of
template parameters.
992 *
template <
typename VectorType,
994 *
typename SystemMatrixType,
995 *
typename LevelMatrixType,
996 *
typename MGTransferType>
1000 *
const VectorType & src,
1001 *
const MultigridParameters &mg_data,
1003 *
const SystemMatrixType & fine_matrix,
1004 *
const MGLevelObject<std::unique_ptr<LevelMatrixType>> &mg_matrices,
1005 *
const MGTransferType & mg_transfer)
1007 *
AssertThrow(mg_data.coarse_solver.type ==
"cg_with_amg",
1011 *
const unsigned int min_level = mg_matrices.min_level();
1012 *
const unsigned int max_level = mg_matrices.max_level();
1017 * SmootherPreconditionerType>;
1022 * We initialize
level operators and Chebyshev smoothers here.
1028 * min_level, max_level);
1032 * smoother_data[
level].preconditioner =
1033 * std::make_shared<SmootherPreconditionerType>();
1034 * mg_matrices[
level]->compute_inverse_diagonal(
1035 * smoother_data[
level].preconditioner->get_vector());
1036 * smoother_data[
level].smoothing_range = mg_data.smoother.smoothing_range;
1037 * smoother_data[
level].degree = mg_data.smoother.degree;
1038 * smoother_data[
level].eig_cg_n_iterations =
1039 * mg_data.smoother.eig_cg_n_iterations;
1044 * mg_smoother.
initialize(mg_matrices, smoother_data);
1048 * Next, we initialize the coarse-grid solver. We use conjugate-
gradient
1049 * method with AMG as preconditioner.
1052 *
ReductionControl coarse_grid_solver_control(mg_data.coarse_solver.maxiter,
1053 * mg_data.coarse_solver.abstol,
1054 * mg_data.coarse_solver.reltol,
1059 * std::unique_ptr<MGCoarseGridBase<VectorType>> mg_coarse;
1064 * amg_data.
n_cycles = mg_data.coarse_solver.n_cycles;
1065 * amg_data.
smoother_type = mg_data.coarse_solver.smoother_type.c_str();
1067 * precondition_amg.
initialize(mg_matrices[min_level]->get_system_matrix(),
1074 *
decltype(precondition_amg)>>(
1075 * coarse_grid_solver, *mg_matrices[min_level], precondition_amg);
1079 * Finally, we create the
Multigrid object, convert it to a preconditioner,
1080 * and use it inside of a conjugate-
gradient solver to solve the linear
1081 * system of equations.
1085 * mg_matrix, *mg_coarse, mg_transfer, mg_smoother, mg_smoother);
1087 * PreconditionerType preconditioner(dof,
mg, mg_transfer);
1090 * .
solve(fine_matrix, dst, src, preconditioner);
1098 * <a name=
"Hybridpolynomialgeometricglobalcoarseningmultigridpreconditioner"></a>
1099 * <h4>Hybrid polynomial/geometric-global-coarsening multigrid preconditioner</h4>
1103 * The above function deals with the actual solution
for a given sequence of
1104 * multigrid objects. This
functions creates the actual multigrid levels, in
1105 * particular the operators, and the transfer
operator as a
1109 *
template <
typename VectorType,
typename OperatorType,
int dim>
1111 *
const OperatorType & system_matrix,
1113 *
const VectorType & src,
1114 *
const MultigridParameters & mg_data,
1122 * as well as, create transfer operators. To be able to
1124 * via global coarsening of p or h. For latter, we need also a sequence
1130 * In
case no h-transfer is requested, we provide an empty deleter
for the
1132 * an external field and its destructor is called somewhere
else.
1139 * std::vector<std::shared_ptr<const Triangulation<dim>>>
1140 * coarse_grid_triangulations;
1141 *
if (mg_data.transfer.perform_h_transfer)
1142 * coarse_grid_triangulations =
1144 * dof_handler.get_triangulation());
1146 * coarse_grid_triangulations.emplace_back(
1152 * Determine the total number of levels
for the multigrid operation and
1153 * allocate sufficient memory
for all levels.
1156 *
const unsigned int n_h_levels = coarse_grid_triangulations.size() - 1;
1158 *
const auto get_max_active_fe_degree = [&](
const auto &dof_handler) {
1159 *
unsigned int max = 0;
1161 *
for (
auto &cell : dof_handler.active_cell_iterators())
1162 *
if (cell->is_locally_owned())
1164 *
std::max(
max, dof_handler.get_fe(cell->active_fe_index()).degree);
1169 *
const unsigned int n_p_levels =
1171 * get_max_active_fe_degree(dof_handler), mg_data.transfer.p_sequence)
1174 * std::map<unsigned int, unsigned int> fe_index_for_degree;
1175 *
for (
unsigned int i = 0; i < dof_handler.get_fe_collection().size(); ++i)
1177 *
const unsigned int degree = dof_handler.get_fe(i).degree;
1178 *
Assert(fe_index_for_degree.find(degree) == fe_index_for_degree.end(),
1179 *
ExcMessage(
"FECollection does not contain unique degrees."));
1180 * fe_index_for_degree[degree] = i;
1183 *
unsigned int minlevel = 0;
1184 *
unsigned int minlevel_p = n_h_levels;
1185 *
unsigned int maxlevel = n_h_levels + n_p_levels - 1;
1187 * dof_handlers.resize(minlevel, maxlevel);
1188 * operators.resize(minlevel, maxlevel);
1189 * transfers.
resize(minlevel, maxlevel);
1193 * Loop from the minimum (coarsest) to the maximum (finest)
level and
set up
1194 *
DoFHandler accordingly. We start with the h-levels, where we distribute
1195 * on increasingly finer meshes linear elements.
1198 *
for (
unsigned int l = 0;
l < n_h_levels; ++
l)
1200 * dof_handlers[
l].
reinit(*coarse_grid_triangulations[
l]);
1201 * dof_handlers[
l].distribute_dofs(dof_handler.get_fe_collection());
1206 * After we reached the finest mesh, we will adjust the polynomial degrees
1207 * on each
level. We reverse iterate over our data structure and start at
1208 * the finest mesh that contains all information about the active FE
1209 * indices. We then lower the polynomial degree of each cell
level by
level.
1212 *
for (
unsigned int i = 0,
l = maxlevel; i < n_p_levels; ++i, --
l)
1214 * dof_handlers[
l].reinit(dof_handler.get_triangulation());
1216 *
if (
l == maxlevel)
1218 *
auto &dof_handler_mg = dof_handlers[
l];
1220 *
auto cell_other = dof_handler.begin_active();
1221 *
for (
auto &cell : dof_handler_mg.active_cell_iterators())
1223 *
if (cell->is_locally_owned())
1224 * cell->set_active_fe_index(cell_other->active_fe_index());
1230 *
auto &dof_handler_fine = dof_handlers[
l + 1];
1231 *
auto &dof_handler_coarse = dof_handlers[
l + 0];
1233 *
auto cell_other = dof_handler_fine.begin_active();
1234 *
for (
auto &cell : dof_handler_coarse.active_cell_iterators())
1236 *
if (cell->is_locally_owned())
1238 *
const unsigned int next_degree =
1241 * cell_other->get_fe().degree,
1242 * mg_data.transfer.p_sequence);
1243 *
Assert(fe_index_for_degree.find(next_degree) !=
1244 * fe_index_for_degree.end(),
1245 *
ExcMessage(
"Next polynomial degree in sequence "
1246 *
"does not exist in FECollection."));
1248 * cell->set_active_fe_index(fe_index_for_degree[next_degree]);
1254 * dof_handlers[
l].distribute_dofs(dof_handler.get_fe_collection());
1259 * Next, we will create all data structures additionally needed on each
1260 * multigrid
level. This involves determining constraints with homogeneous
1261 * Dirichlet boundary conditions, and building the
operator just like on the
1266 * constraints(minlevel, maxlevel);
1270 *
const auto &dof_handler = dof_handlers[
level];
1271 *
auto & constraint = constraints[
level];
1275 * locally_relevant_dofs);
1276 * constraint.reinit(locally_relevant_dofs);
1284 * constraint.close();
1288 * operators[
level] = std::make_unique<OperatorType>(mapping_collection,
1290 * quadrature_collection,
1297 * Set up intergrid operators and collect transfer operators within a single
1298 *
operator as needed by the
Multigrid solver
class.
1302 * transfers[
level + 1].reinit_geometric_transfer(dof_handlers[
level + 1],
1303 * dof_handlers[
level],
1304 * constraints[
level + 1],
1305 * constraints[
level]);
1308 * transfers[
level + 1].reinit_polynomial_transfer(dof_handlers[
level + 1],
1309 * dof_handlers[
level],
1310 * constraints[
level + 1],
1311 * constraints[
level]);
1314 * transfers, [&](
const auto l,
auto &vec) {
1315 * operators[
l]->initialize_dof_vector(vec);
1320 * Finally, proceed to solve the problem with multigrid.
1323 * mg_solve(solver_control,
1338 * <a name=
"ThecodeLaplaceProblemcodeclasstemplate"></a>
1339 * <h3>The <code>LaplaceProblem</code>
class template</h3>
1343 * Now we will
finally declare the main
class of this program, which solves
1344 * the Laplace equation on subsequently refined function spaces. Its structure
1345 * will look familiar as it is similar to the main classes of @ref step_27
"step-27" and
1346 * @ref step_40
"step-40". There are basically just two additions:
1348 * replaced by an
object of the LaplaceOperator
class for the
MatrixFree
1351 * balancing, has been added.
1354 *
template <
int dim>
1355 *
class LaplaceProblem
1358 * LaplaceProblem(
const Parameters ¶meters);
1363 *
void initialize_grid();
1364 *
void setup_system();
1365 *
void print_diagnostics();
1366 *
void solve_system();
1367 *
void compute_indicators();
1368 *
void adapt_resolution();
1369 *
void output_results(
const unsigned int cycle);
1373 *
const Parameters prm;
1388 * LaplaceOperator<dim, double> laplace_operator;
1392 * std::unique_ptr<FESeries::Legendre<dim>>
legendre;
1393 * std::unique_ptr<parallel::CellWeights<dim>> cell_weights;
1407 * <a name=
"ThecodeLaplaceProblemcodeclassimplementation"></a>
1408 * <h3>The <code>LaplaceProblem</code>
class implementation</h3>
1413 * <a name=
"Constructor"></a>
1414 * <h4>Constructor</h4>
1418 * The constructor starts with an initializer list that looks similar to the
1420 * only the
first process to output anything over the console, and initialize
1421 * the computing timer properly.
1424 *
template <
int dim>
1425 * LaplaceProblem<dim>::LaplaceProblem(
const Parameters ¶meters)
1426 * : mpi_communicator(MPI_COMM_WORLD)
1430 * , pcout(std::cout,
1432 * , computing_timer(mpi_communicator,
1437 *
Assert(prm.min_h_level <= prm.max_h_level,
1439 *
"Triangulation level limits have been incorrectly set up."));
1440 *
Assert(prm.min_p_degree <= prm.max_p_degree,
1441 *
ExcMessage(
"FECollection degrees have been incorrectly set up."));
1445 * We need to prepare the data structures
for the
hp-functionality in the
1446 * actual body of the constructor, and create corresponding objects
for
1447 * every degree in the specified range from the parameter
struct. As we are
1448 * only dealing with non-distorted rectangular cells, a linear mapping
1449 *
object is sufficient in
this context.
1453 * In the Parameters
struct, we provide ranges
for levels on which the
1454 * function space is operating with a reasonable resolution. The multigrid
1455 * algorithm
requires linear elements on the coarsest possible
level. So we
1456 * start with the lowest polynomial degree and fill the collection with
1457 * consecutively higher degrees until the user-specified maximum is
1463 *
for (
unsigned int degree = 1; degree <= prm.max_p_degree; ++degree)
1465 * fe_collection.push_back(
FE_Q<dim>(degree));
1466 * quadrature_collection.push_back(
QGauss<dim>(degree + 1));
1472 * As our FECollection contains more finite elements than we want to use
for
1473 * the finite element approximation of our solution, we would like to limit
1474 * the range on which active FE indices can operate on. For
this, the
1475 * FECollection
class allows to register a hierarchy that determines the
1476 * succeeding and preceding finite element in
case of of p-refinement and
1478 * consult
this hierarchy to determine future FE indices. We will
register
1479 * such a hierarchy that only works on finite elements with polynomial
1480 * degrees in the proposed range <code>[min_p_degree, max_p_degree]</code>.
1483 *
const unsigned int min_fe_index = prm.min_p_degree - 1;
1484 * fe_collection.set_hierarchy(
1487 *
const unsigned int fe_index) ->
unsigned int {
1488 *
return ((fe_index + 1) < fe_collection.
size()) ? fe_index + 1 :
1493 *
const unsigned int fe_index) ->
unsigned int {
1494 *
Assert(fe_index >= min_fe_index,
1495 *
ExcMessage(
"Finite element is not part of hierarchy!"));
1496 *
return (fe_index > min_fe_index) ? fe_index - 1 : fe_index;
1502 *
for smoothness estimation.
1505 *
legendre = std::make_unique<FESeries::Legendre<dim>>(
1510 * The next part is going to be tricky. During execution of refinement, a
1511 * few
hp-algorithms need to interfere with the actual refinement process on
1514 * the actual refinement process and trigger all connected
functions. We
1515 * require
this functionality
for load balancing and to limit the polynomial
1516 * degrees of neighboring cells.
1520 * For the former, we would like to assign a weight to every cell that is
1521 * proportional to the number of degrees of freedom of its future finite
1523 * easily attach individual weights at the right place during the refinement
1524 * process, i.e., after all
refine and
coarsen flags have been
set correctly
1525 *
for hp-adaptation and right before repartitioning
for load balancing is
1526 * about to happen.
Functions can be registered that will attach weights in
1527 * the form that @f$a (n_\text{dofs})^
b@f$ with a provided pair of parameters
1528 * @f$(a,
b)@f$. We
register such a function in the following. Every cell will be
1529 * charged with a constant weight at creation, which is a
value of 1000 (see
1534 * For load balancing, efficient solvers like the
one we use should
scale
1535 * linearly with the number of degrees of freedom owned. Further, to
1536 * increase the impact of the weights we would like to attach, make sure
1537 * that the individual weight will exceed
this base weight by orders of
1538 * magnitude. We
set the parameters
for cell weighting correspondingly:
A
1539 * large weighting factor of @f$10^6@f$ and an exponent of @f$1@f$.
1545 * {prm.weighting_factor, prm.weighting_exponent}));
1549 * In h-adaptive applications, we ensure a 2:1 mesh balance by limiting the
1550 * difference of refinement levels of neighboring cells to
one. With the
1551 *
second call in the following code snippet, we will ensure the same
for
1552 * p-levels on neighboring cells: levels of future finite elements are not
1553 * allowed to differ by more than a specified difference. The function
1555 * be connected to a very specific signal in the
parallel context. The issue
1556 * is that we need to know how the mesh will be actually refined to
set
1557 * future FE indices accordingly. As we ask the p4est oracle to perform
1558 * refinement, we need to ensure that the
Triangulation has been updated
1559 * with the adaptation flags of the oracle
first. An instantiation of
1561 * that
for the duration of its life. Thus, we will create an
object of
this
1562 *
class right before limiting the p-
level difference, and connect the
1563 * corresponding
lambda function to the signal
1565 * after the oracle got refined, but before the
Triangulation is refined.
1566 * Furthermore, we specify that
this function will be connected to the front
1567 * of the signal, to ensure that the modification is performed before any
1568 * other function connected to the same signal.
1572 * [&, min_fe_index]() {
1576 * prm.max_p_level_difference,
1579 * boost::signals2::at_front);
1587 * <a name=
"LaplaceProbleminitialize_grid"></a>
1588 * <h4>LaplaceProblem::initialize_grid</h4>
1593 * as demonstrated in @ref step_50
"step-50". However in the 2D
case, that particular
1594 * function removes the
first quadrant,
while we need the fourth quadrant
1595 * removed in our scenario. Thus, we will use a different function
1597 * the mesh. Furthermore, we formulate that function in a way that it also
1598 * generates a 3D mesh: the 2D
L-shaped domain will basically elongated by 1
1599 * in the positive z-direction.
1604 * The parameters that we need to provide are
Point objects for the lower left
1605 * and top right corners, as well as the number of repetitions that the base
1606 * mesh will have in each direction. We provide them for the
first two
1607 * dimensions and treat the higher third dimension separately.
1611 * To create a
L-shaped domain, we need to remove the excess cells. For this,
1612 * we specify the <code>cells_to_remove</code> accordingly. We would like to
1613 * remove
one cell in every cell from the negative direction, but remove
one
1614 * from the positive x-direction.
1618 * In the
end, we supply the number of
initial refinements that corresponds to
1619 * the supplied minimal grid refinement
level. Further, we
set the
initial
1620 * active FE indices accordingly.
1623 * template <
int dim>
1624 *
void LaplaceProblem<dim>::initialize_grid()
1628 * std::vector<unsigned int> repetitions(dim);
1630 *
for (
unsigned int d = 0;
d < dim; ++
d)
1633 * repetitions[
d] = 2;
1634 * bottom_left[
d] = -1.;
1635 * top_right[
d] = 1.;
1639 * repetitions[
d] = 1;
1640 * bottom_left[
d] = 0.;
1641 * top_right[
d] = 1.;
1644 * std::vector<int> cells_to_remove(dim, 1);
1645 * cells_to_remove[0] = -1;
1648 *
triangulation, repetitions, bottom_left, top_right, cells_to_remove);
1652 *
const unsigned int min_fe_index = prm.min_p_degree - 1;
1653 *
for (
const auto &cell : dof_handler.active_cell_iterators())
1654 *
if (cell->is_locally_owned())
1655 * cell->set_active_fe_index(min_fe_index);
1663 * <a name=
"LaplaceProblemsetup_system"></a>
1664 * <h4>LaplaceProblem::setup_system</h4>
1668 * This function looks exactly the same to the
one of @ref step_40
"step-40", but you will
1669 * notice the absence of the system
matrix as well as the scaffold that
1670 * surrounds it. Instead, we will initialize the
MatrixFree formulation of the
1671 * <code>laplace_operator</code> here. For boundary conditions, we will use
1672 * the Solution
class introduced earlier in this tutorial.
1675 *
template <
int dim>
1676 *
void LaplaceProblem<dim>::setup_system()
1680 * dof_handler.distribute_dofs(fe_collection);
1682 * locally_owned_dofs = dof_handler.locally_owned_dofs();
1685 * locally_relevant_solution.reinit(locally_owned_dofs,
1686 * locally_relevant_dofs,
1687 * mpi_communicator);
1688 * system_rhs.reinit(locally_owned_dofs, mpi_communicator);
1690 * constraints.clear();
1691 * constraints.reinit(locally_relevant_dofs);
1694 * mapping_collection, dof_handler, 0, Solution<dim>(), constraints);
1695 * constraints.close();
1697 * laplace_operator.reinit(mapping_collection,
1699 * quadrature_collection,
1709 * <a name=
"LaplaceProblemprint_diagnostics"></a>
1710 * <h4>LaplaceProblem::print_diagnostics</h4>
1714 * This is a function that prints additional diagnostics about the equation
1715 * system and its partitioning. In addition to the usual global number of
1716 * active cells and degrees of freedom, we also output their local
1717 * equivalents. For a regulated output, we will communicate the local
1719 * which will then output all information. Output of local quantities is
1720 * limited to the
first 8 processes to avoid cluttering the terminal.
1724 * Furthermore, we would like to print the frequencies of the polynomial
1725 * degrees in the numerical discretization. Since
this information is only
1726 * stored locally, we will count the finite elements on locally owned cells
1730 *
template <
int dim>
1731 *
void LaplaceProblem<dim>::print_diagnostics()
1733 *
const unsigned int first_n_processes =
1734 * std::min<unsigned int>(8,
1736 *
const bool output_cropped =
1740 * pcout <<
" Number of active cells: "
1742 * <<
" by partition: ";
1744 * std::vector<unsigned int> n_active_cells_per_subdomain =
1747 *
for (
unsigned int i = 0; i < first_n_processes; ++i)
1748 * pcout <<
' ' << n_active_cells_per_subdomain[i];
1749 *
if (output_cropped)
1751 * pcout << std::endl;
1755 * pcout <<
" Number of degrees of freedom: " << dof_handler.n_dofs()
1757 * <<
" by partition: ";
1759 * std::vector<types::global_dof_index> n_dofs_per_subdomain =
1761 * dof_handler.n_locally_owned_dofs());
1762 *
for (
unsigned int i = 0; i < first_n_processes; ++i)
1763 * pcout <<
' ' << n_dofs_per_subdomain[i];
1764 *
if (output_cropped)
1766 * pcout << std::endl;
1770 * std::vector<types::global_dof_index> n_constraints_per_subdomain =
1773 * pcout <<
" Number of constraints: "
1774 * << std::accumulate(n_constraints_per_subdomain.begin(),
1775 * n_constraints_per_subdomain.end(),
1778 * <<
" by partition: ";
1779 *
for (
unsigned int i = 0; i < first_n_processes; ++i)
1780 * pcout <<
' ' << n_constraints_per_subdomain[i];
1781 *
if (output_cropped)
1783 * pcout << std::endl;
1787 * std::vector<unsigned int> n_fe_indices(fe_collection.
size(), 0);
1788 *
for (
const auto &cell : dof_handler.active_cell_iterators())
1789 *
if (cell->is_locally_owned())
1790 * n_fe_indices[cell->active_fe_index()]++;
1794 * pcout <<
" Frequencies of poly. degrees:";
1795 *
for (
unsigned int i = 0; i < fe_collection.
size(); ++i)
1796 *
if (n_fe_indices[i] > 0)
1797 * pcout <<
' ' << fe_collection[i].degree <<
":" << n_fe_indices[i];
1798 * pcout << std::endl;
1807 * <a name=
"LaplaceProblemsolve_system"></a>
1808 * <h4>LaplaceProblem::solve_system</h4>
1812 * The scaffold around the solution is similar to the
one of @ref step_40
"step-40". We
1813 * prepare a vector that matches the requirements of
MatrixFree and collect
1814 * the locally-relevant degrees of freedoms we solved the equation system. The
1815 * solution happens with the function introduced earlier.
1818 *
template <
int dim>
1819 *
void LaplaceProblem<dim>::solve_system()
1824 * laplace_operator.initialize_dof_vector(completely_distributed_solution);
1827 * prm.tolerance_factor * system_rhs.l2_norm());
1829 * solve_with_gmg(solver_control,
1831 * completely_distributed_solution,
1834 * mapping_collection,
1836 * quadrature_collection);
1838 * pcout <<
" Solved in " << solver_control.last_step() <<
" iterations."
1841 * constraints.distribute(completely_distributed_solution);
1843 * locally_relevant_solution.copy_locally_owned_data_from(
1844 * completely_distributed_solution);
1845 * locally_relevant_solution.update_ghost_values();
1853 * <a name=
"LaplaceProblemcompute_indicators"></a>
1854 * <h4>LaplaceProblem::compute_indicators</h4>
1858 * This function contains only a part of the typical <code>refine_grid</code>
1859 * function from other tutorials and is
new in that sense. Here, we will only
1860 * calculate all indicators
for adaptation with actually refining the grid. We
1861 *
do this for the purpose of writing all indicators to the file system, so we
1862 * store them
for later.
1866 * Since we are dealing the an elliptic problem, we will make use of the
1868 * scaling factor of the underlying face integrals to be dependent on the
1869 * actual polynomial degree of the neighboring elements is favorable in
1870 *
hp-adaptive applications @cite davydov2017hp. We can
do this by specifying
1871 * the very last parameter from the additional ones you notices. The others
1872 * are actually just the defaults.
1876 * For the purpose of
hp-adaptation, we will calculate smoothness estimates
1877 * with the strategy presented in the tutorial introduction and use the
1879 * we
set the minimal polynomial degree to 2 as it seems that the smoothness
1880 * estimation algorithms have trouble with linear elements.
1883 *
template <
int dim>
1884 *
void LaplaceProblem<dim>::compute_indicators()
1888 * estimated_error_per_cell.grow_or_shrink(
triangulation.n_active_cells());
1891 * face_quadrature_collection,
1893 * locally_relevant_solution,
1894 * estimated_error_per_cell,
1903 * hp_decision_indicators.grow_or_shrink(
triangulation.n_active_cells());
1906 * locally_relevant_solution,
1907 * hp_decision_indicators);
1915 * <a name=
"LaplaceProblemadapt_resolution"></a>
1916 * <h4>LaplaceProblem::adapt_resolution</h4>
1920 * With the previously calculated indicators, we will
finally flag all cells
1921 *
for adaptation and also execute refinement in
this function. As in previous
1922 * tutorials, we will use the
"fixed number" strategy, but now
for
1926 *
template <
int dim>
1927 *
void LaplaceProblem<dim>::adapt_resolution()
1934 * on each cell. There is
nothing new here.
1939 * elaborated in the other deal.II tutorials:
using the fixed number
1940 * strategy, we will flag 30% of all cells
for refinement and 3%
for
1941 * coarsening, as provided in the Parameters
struct.
1946 * estimated_error_per_cell,
1947 * prm.refine_fraction,
1948 * prm.coarsen_fraction);
1952 * Next, we will make all adjustments
for hp-adaptation. We want to
refine
1953 * and
coarsen those cells flagged in the previous step, but need to decide
1954 *
if we would like to
do it by adjusting the grid resolution or the
1955 * polynomial degree.
1959 * The next function
call sets future FE indices according to the previously
1960 * calculated smoothness indicators as p-adaptation indicators. These
1961 * indices will only be
set on those cells that have
refine or
coarsen flags
1966 * For the p-adaptation fractions, we will take an educated guess. Since we
1967 * only expect a single singularity in our scenario, i.e., in the origin of
1968 * the domain, and a smooth solution anywhere
else, we would like to
1969 * strongly prefer to use p-adaptation over h-adaptation. This reflects in
1970 * our choice of a fraction of 90%
for both p-refinement and p-coarsening.
1974 * hp_decision_indicators,
1975 * prm.p_refine_fraction,
1976 * prm.p_coarsen_fraction);
1980 * At
this stage, we have both the future FE indices and the classic
refine
1981 * and
coarsen flags
set, from which the latter will be interpreted by
1983 * We would like to only impose
one type of adaptation on cells, which is
1984 * what the next function will sort out for us. In
short, on cells which
1985 * have both
types of indicators assigned, we will favor the p-adaptation
1986 *
one and remove the h-adaptation
one.
1993 * After setting all indicators, we will remove those that exceed the
1994 * specified limits of the provided
level ranges in the Parameters struct.
1995 * This limitation naturally arises for p-adaptation as the number of
1996 * supplied finite elements is limited. In addition, we registered a custom
1997 * hierarchy for p-adaptation in the constructor. Now, we need to do this
1998 * manually in the h-adaptive context like in @ref step_31 "step-31".
2002 * We will iterate over all cells on the designated
min and
max levels and
2003 * remove the corresponding flags. As an alternative, we could also flag
2004 * these cells for p-adaptation by setting future FE indices accordingly
2013 * for (const auto &cell :
2014 *
triangulation.active_cell_iterators_on_level(prm.max_h_level))
2015 * cell->clear_refine_flag();
2017 * for (const auto &cell :
2018 *
triangulation.active_cell_iterators_on_level(prm.min_h_level))
2019 * cell->clear_coarsen_flag();
2023 * In the
end, we are left to execute coarsening and refinement. Here, not
2024 * only the grid will be updated, but also all previous future FE indices
2025 * will become active.
2030 * constructor, will be triggered in this function
call. So there is even
2031 * more happening: weighted repartitioning will be performed to ensure load
2032 * balancing, as well as we will limit the difference of p-levels between
2033 * neighboring cells.
2044 * <a name="LaplaceProblemoutput_results"></a>
2045 * <h4>LaplaceProblem::output_results</h4>
2049 * Writing results to the file system in
parallel applications works exactly
2050 * like in @ref step_40 "step-40". In addition to the data containers that we prepared
2051 * throughout the tutorial, we would also like to write out the polynomial
2052 * degree of each finite element on the grid as well as the subdomain each
2053 * cell belongs to. We prepare necessary containers for this in the scope of
2057 * template <
int dim>
2058 *
void LaplaceProblem<dim>::output_results(const
unsigned int cycle)
2063 *
for (
const auto &cell : dof_handler.active_cell_iterators())
2064 *
if (cell->is_locally_owned())
2065 * fe_degrees(cell->active_cell_index()) = cell->get_fe().degree;
2068 *
for (
auto &subd : subdomain)
2081 *
"./",
"solution", cycle, mpi_communicator, 2, 1);
2089 * <a name=
"LaplaceProblemrun"></a>
2094 * The actual
run function again looks very familiar to @ref step_40
"step-40". The only
2095 * addition is the bracketed section that precedes the actual cycle
loop.
2096 * Here, we will pre-calculate the Legendre transformation matrices. In
2097 *
general, these will be calculated on the fly via lazy allocation whenever a
2098 * certain
matrix is needed. For timing purposes however, we would like to
2099 * calculate them all at once before the actual time measurement begins. We
2100 * will thus designate their calculation to their own scope.
2103 *
template <
int dim>
2106 * pcout <<
"Running with Trilinos on "
2108 * <<
" MPI rank(s)..." << std::endl;
2111 * pcout <<
"Calculating transformation matrices..." << std::endl;
2113 *
legendre->precalculate_all_transformation_matrices();
2116 *
for (
unsigned int cycle = 0; cycle < prm.n_cycles; ++cycle)
2118 * pcout <<
"Cycle " << cycle <<
':' << std::endl;
2121 * initialize_grid();
2123 * adapt_resolution();
2127 * print_diagnostics();
2131 * compute_indicators();
2134 * output_results(cycle);
2136 * computing_timer.print_summary();
2137 * computing_timer.reset();
2139 * pcout << std::endl;
2149 * <a name=
"main"></a>
2154 * The
final function is the <code>main</code> function that will ultimately
2155 * create and
run a LaplaceOperator instantiation. Its structure is similar to
2156 * most other tutorial programs.
2159 *
int main(
int argc,
char *argv[])
2163 *
using namespace dealii;
2164 *
using namespace Step75;
2169 * LaplaceProblem<2> laplace_problem(prm);
2170 * laplace_problem.run();
2172 *
catch (std::exception &exc)
2174 * std::cerr << std::endl
2176 * <<
"----------------------------------------------------"
2178 * std::cerr <<
"Exception on processing: " << std::endl
2179 * << exc.what() << std::endl
2180 * <<
"Aborting!" << std::endl
2181 * <<
"----------------------------------------------------"
2188 * std::cerr << std::endl
2190 * <<
"----------------------------------------------------"
2192 * std::cerr <<
"Unknown exception!" << std::endl
2193 * <<
"Aborting!" << std::endl
2194 * <<
"----------------------------------------------------"
2202<a name=
"Results"></a><h1>Results</h1>
2205When you
run the program with the given parameters on four processes in
2206release mode, your terminal output should look like
this:
2208Running with Trilinos on 4 MPI rank(s)...
2209Calculating transformation matrices...
2211 Number of active cells: 3072
2213 Number of degrees of freedom: 12545
2215 Number of constraints: 542
2217 Frequencies of poly. degrees: 2:3072
2218 Solved in 7 iterations.
2221+---------------------------------------------+------------+------------+
2222| Total wallclock time elapsed since start | 0.598s | |
2224| Section | no. calls | wall time | % of total |
2225+---------------------------------+-----------+------------+------------+
2226| calculate transformation | 1 | 0.0533s | 8.9% |
2227| compute indicators | 1 | 0.0177s | 3% |
2228| initialize grid | 1 | 0.0397s | 6.6% |
2229| output results | 1 | 0.0844s | 14% |
2230| setup system | 1 | 0.0351s | 5.9% |
2231| solve system | 1 | 0.362s | 61% |
2232+---------------------------------+-----------+------------+------------+
2236 Number of active cells: 3351
2238 Number of degrees of freedom: 18223
2240 Number of constraints: 1202
2242 Frequencies of poly. degrees: 2:2523 3:828
2243 Solved in 7 iterations.
2246+---------------------------------------------+------------+------------+
2247| Total wallclock time elapsed since start | 0.442s | |
2249| Section | no. calls | wall time | % of total |
2250+---------------------------------+-----------+------------+------------+
2251| adapt resolution | 1 | 0.0189s | 4.3% |
2252| compute indicators | 1 | 0.0135s | 3% |
2253| output results | 1 | 0.064s | 14% |
2254| setup system | 1 | 0.0232s | 5.2% |
2255| solve system | 1 | 0.322s | 73% |
2256+---------------------------------+-----------+------------+------------+
2263 Number of active cells: 5610
2265 Number of degrees of freedom: 82062
2267 Number of constraints: 14383
2269 Frequencies of poly. degrees: 2:1130 3:1283 4:2727 5:465 6:5
2270 Solved in 7 iterations.
2273+---------------------------------------------+------------+------------+
2274| Total wallclock time elapsed since start | 0.932s | |
2276| Section | no. calls | wall time | % of total |
2277+---------------------------------+-----------+------------+------------+
2278| adapt resolution | 1 | 0.0182s | 1.9% |
2279| compute indicators | 1 | 0.0173s | 1.9% |
2280| output results | 1 | 0.0572s | 6.1% |
2281| setup system | 1 | 0.0252s | 2.7% |
2282| solve system | 1 | 0.813s | 87% |
2283+---------------------------------+-----------+------------+------------+
2286When running the code with more processes, you will notice slight
2287differences in the number of active cells and degrees of freedom. This
2288is due to the fact that solver and preconditioner depend on the
2289partitioning of the problem, which might yield to slight differences of
2290the solution in the last digits and ultimately yields to different
2293Furthermore, the number of iterations
for the solver stays about the
2294same in all cycles despite
hp-adaptation, indicating the robustness of
2295the proposed algorithms and promising good scalability
for even larger
2296problem sizes and on more processes.
2298Let us have a look at the graphical output of the program. After all
2299refinement cycles in the given parameter configuration, the actual
2300discretized function space looks like the following with its
2301partitioning on twelve processes on the left and the polynomial degrees
2302of finite elements on the right. In the left picture, each color
2303represents a unique subdomain. In the right picture, the lightest color
2304corresponds to the polynomial degree two and the darkest
one corresponds
2307<div
class=
"twocolumn" style=
"width: 80%; text-align: center;">
2309 <img src=
"https://www.dealii.org/images/steps/developer/step-75.subdomains-07.svg"
2310 alt=
"Partitioning after seven refinements.">
2313 <img src=
"https://www.dealii.org/images/steps/developer/step-75.fedegrees-07.svg"
2314 alt=
"Local approximation degrees after seven refinements.">
2320<a name=
"extensions"></a>
2321<a name=
"Possibilitiesforextensions"></a><h3>Possibilities
for extensions</h3>
2324<a name=
"Differenthpdecisionstrategies"></a><h4>Different
hp-decision strategies</h4>
2327The deal.II library offers multiple strategies to decide which type of
2328adaptation to impose on cells: either adjust the grid resolution or
2329change the polynomial degree. We only presented the <i>Legendre
2330coefficient decay</i> strategy in
this tutorial,
while @ref step_27
"step-27"
2331demonstrated the <i>Fourier</i> equivalent of the same idea.
2333See the
"possibilities for extensions" section of @ref step_27
"step-27" for an
2334overview over these strategies, or the corresponding documentation
2335for a detailed description.
2337There, another strategy is mentioned that has not been shown in any
2338tutorial so far: the strategy based on <i>refinement history</i>. The
2339usage of
this method
for parallel distributed applications is more
2340tricky than the others, so we will highlight the challenges that come
2341along with it. We need information about the
final state of refinement
2342flags, and we need to transfer the solution across refined meshes. For
2344function to the
Triangulation::Signals::post_p4est_refinement signal in
2345a way that it will be called <i>after</i> the
2347refinement flags and future FE indices are terminally
set and a reliable
2348prediction of the error is possible. The predicted error then needs to
2349be transferred across refined meshes with the aid of
2352Try implementing
one of these strategies into this tutorial and observe
2353the subtle changes to the results. You will notice that all strategies
2354are capable of identifying the singularities near the reentrant corners
2355and will perform @f$h@f$-refinement in these regions, while preferring
2356@f$p@f$-refinement in the bulk domain.
A detailed comparison of these
2357strategies is presented in @cite fehling2020 .
2360<a name="Solvewithmatrixbasedmethods"></a><h4>Solve with
matrix-based methods</h4>
2363This tutorial focuses solely on
matrix-
free strategies. All
hp-adaptive
2364algorithms however also work with
matrix-based approaches in the
2367To create a system
matrix, you can either use the
2368LaplaceOperator::get_system_matrix() function, or use an
2369<code>assemble_system()</code> function similar to the
one of @ref step_27 "step-27".
2370You can then pass the system
matrix to the solver as usual.
2373implementations, quantify the speed-up, and convince yourself which
2377<a name="Multigridvariants"></a><h4>
Multigrid variants</h4>
2380For sake of simplicity, we have restricted ourselves to a single type of
2381coarse-grid solver (CG with AMG), smoother (Chebyshev smoother with
2382point Jacobi preconditioner), and geometric-coarsening scheme (global
2383coarsening) within the multigrid algorithm. Feel
free to try out
2384alternatives and investigate their performance and robustness.
2387<a name="PlainProg"></a>
2388<h1> The plain program</h1>
2389@include "step-75.cc"
void reinit(const IndexSet &local_constraints=IndexSet())
void attach_dof_handler(const DoFHandlerType &)
void add_data_vector(const VectorType &data, const std::vector< std::string > &names, const DataVectorType type=type_automatic, const std::vector< DataComponentInterpretation::DataComponentInterpretation > &data_component_interpretation=std::vector< DataComponentInterpretation::DataComponentInterpretation >())
virtual void build_patches(const unsigned int n_subdivisions=0)
void reinit(const Triangulation< dim, spacedim > &tria)
void evaluate(const EvaluationFlags::EvaluationFlags evaluation_flag)
static void estimate(const Mapping< dim, spacedim > &mapping, const DoFHandler< dim, spacedim > &dof, const Quadrature< dim - 1 > &quadrature, const std::map< types::boundary_id, const Function< spacedim, typename InputVector::value_type > * > &neumann_bc, const InputVector &solution, Vector< float > &error, const ComponentMask &component_mask=ComponentMask(), const Function< spacedim > *coefficients=nullptr, const unsigned int n_threads=numbers::invalid_unsigned_int, const types::subdomain_id subdomain_id=numbers::invalid_subdomain_id, const types::material_id material_id=numbers::invalid_material_id, const Strategy strategy=cell_diameter_over_24)
void resize(const unsigned int new_minlevel, const unsigned int new_maxlevel, Args &&... args)
void initialize(const MGLevelObject< MatrixType2 > &matrices, const typename PreconditionerType::AdditionalData &additional_data=typename PreconditionerType::AdditionalData())
const DoFHandler< dim > & get_dof_handler(const unsigned int dof_handler_index=0) const
void initialize_dof_vector(VectorType &vec, const unsigned int dof_handler_index=0) const
void cell_loop(const std::function< void(const MatrixFree< dim, Number, VectorizedArrayType > &, OutVector &, const InVector &, const std::pair< unsigned int, unsigned int > &)> &cell_operation, OutVector &dst, const InVector &src, const bool zero_dst_vector=false) const
void reinit(const MappingType &mapping, const DoFHandler< dim > &dof_handler, const AffineConstraints< number2 > &constraint, const QuadratureType &quad, const AdditionalData &additional_data=AdditionalData())
void solve(const MatrixType &A, VectorType &x, const VectorType &b, const PreconditionerType &preconditioner)
void coarsen_global(const unsigned int times=1)
virtual void execute_coarsening_and_refinement()
VectorizedArray< Number, width > max(const ::VectorizedArray< Number, width > &x, const ::VectorizedArray< Number, width > &y)
VectorizedArray< Number, width > min(const ::VectorizedArray< Number, width > &x, const ::VectorizedArray< Number, width > &y)
unsigned int size() const
SmartPointer< const ::DoFHandler< dim, spacedim >, CellWeights > dof_handler
@ update_gradients
Shape function gradients.
__global__ void set(Number *val, const Number s, const size_type N)
std::string write_vtu_with_pvtu_record(const std::string &directory, const std::string &filename_without_extension, const unsigned int counter, const MPI_Comm &mpi_communicator, const unsigned int n_digits_for_counter=numbers::invalid_unsigned_int, const unsigned int n_groups=0) const
static ::ExceptionBase & ExcNotImplemented()
#define Assert(cond, exc)
static ::ExceptionBase & ExcInternalError()
static ::ExceptionBase & ExcMessage(std::string arg1)
#define AssertThrow(cond, exc)
void loop(ITERATOR begin, typename identity< ITERATOR >::type end, DOFINFO &dinfo, INFOBOX &info, const std::function< void(DOFINFO &, typename INFOBOX::CellInfo &)> &cell_worker, const std::function< void(DOFINFO &, typename INFOBOX::CellInfo &)> &boundary_worker, const std::function< void(DOFINFO &, DOFINFO &, typename INFOBOX::CellInfo &, typename INFOBOX::CellInfo &)> &face_worker, ASSEMBLER &assembler, const LoopControl &lctrl=LoopControl())
unsigned int smoother_sweeps
void initialize(const SparseMatrix &matrix, const AdditionalData &additional_data=AdditionalData())
const char * smoother_type
void make_hanging_node_constraints(const DoFHandler< dim, spacedim > &dof_handler, AffineConstraints< number > &constraints)
void make_sparsity_pattern(const DoFHandler< dim, spacedim > &dof_handler, SparsityPatternType &sparsity_pattern, const AffineConstraints< number > &constraints=AffineConstraints< number >(), const bool keep_constrained_dofs=true, const types::subdomain_id subdomain_id=numbers::invalid_subdomain_id)
std::array< double, dim > to_spherical(const Point< dim > &point)
void hyper_L(Triangulation< dim > &tria, const double left=-1., const double right=1., const bool colorize=false)
void subdivided_hyper_L(Triangulation< dim, spacedim > &tria, const std::vector< unsigned int > &repetitions, const Point< dim > &bottom_left, const Point< dim > &top_right, const std::vector< int > &n_cells_to_remove)
void subdivided_hyper_rectangle(Triangulation< dim, spacedim > &tria, const std::vector< unsigned int > &repetitions, const Point< dim > &p1, const Point< dim > &p2, const bool colorize=false)
void refine(Triangulation< dim, spacedim > &tria, const Vector< Number > &criteria, const double threshold, const unsigned int max_to_mark=numbers::invalid_unsigned_int)
void coarsen(Triangulation< dim, spacedim > &tria, const Vector< Number > &criteria, const double threshold)
@ matrix
Contents is actually a matrix.
@ symmetric
Matrix is symmetric.
@ diagonal
Matrix is diagonal.
@ general
No special properties.
static const types::blas_int one
Point< spacedim > point(const gp_Pnt &p, const double tolerance=1e-10)
SymmetricTensor< 2, dim, Number > e(const Tensor< 2, dim, Number > &F)
Tensor< 2, dim, Number > l(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
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)
FESeries::Legendre< dim, spacedim > default_fe_series(const hp::FECollection< dim, spacedim > &fe_collection, const unsigned int component=numbers::invalid_unsigned_int)
void coefficient_decay(FESeries::Legendre< dim, spacedim > &fe_legendre, const DoFHandler< dim, spacedim > &dof_handler, const VectorType &solution, Vector< float > &smoothness_indicators, const VectorTools::NormType regression_strategy=VectorTools::Linfty_norm, const double smallest_abs_coefficient=1e-10, const bool only_flagged_cells=false)
void call(const std::function< RT()> &function, internal::return_value< RT > &ret_val)
VectorType::value_type * end(VectorType &V)
unsigned int this_mpi_process(const MPI_Comm &mpi_communicator)
std::vector< T > gather(const MPI_Comm &comm, const T &object_to_send, const unsigned int root_process=0)
T sum(const T &t, const MPI_Comm &mpi_communicator)
unsigned int n_mpi_processes(const MPI_Comm &mpi_communicator)
T max(const T &t, const MPI_Comm &mpi_communicator)
void run(const Iterator &begin, const typename identity< Iterator >::type &end, Worker worker, Copier copier, const ScratchData &sample_scratch_data, const CopyData &sample_copy_data, const unsigned int queue_length, const unsigned int chunk_size)
bool limit_p_level_difference(const ::DoFHandler< dim, spacedim > &dof_handler, const unsigned int max_difference=1, const unsigned int contains_fe_index=0)
void choose_p_over_h(const ::DoFHandler< dim, spacedim > &dof_handler)
void p_adaptivity_fixed_number(const ::DoFHandler< dim, spacedim > &dof_handler, const Vector< Number > &criteria, const double p_refine_fraction=0.5, const double p_coarsen_fraction=0.5, const ComparisonFunction< typename identity< Number >::type > &compare_refine=std::greater_equal< Number >(), const ComparisonFunction< typename identity< Number >::type > &compare_coarsen=std::less_equal< Number >())
void predict_error(const ::DoFHandler< dim, spacedim > &dof_handler, const Vector< Number > &error_indicators, Vector< Number > &predicted_errors, const double gamma_p=std::sqrt(0.4), const double gamma_h=2., const double gamma_n=1.)
int(&) functions(const void *v1, const void *v2)
void reinit(MatrixBlock< MatrixType > &v, const BlockSparsityPattern &p)
const types::material_id invalid_material_id
const types::subdomain_id invalid_subdomain_id
static const unsigned int invalid_unsigned_int
void refine_and_coarsen_fixed_number(parallel::distributed::Triangulation< dim, spacedim > &tria, const ::Vector< Number > &criteria, const double top_fraction_of_cells, const double bottom_fraction_of_cells, const types::global_cell_index max_n_cells=std::numeric_limits< types::global_cell_index >::max())
double legendre(unsigned int l, double x)
::VectorizedArray< Number, width > max(const ::VectorizedArray< Number, width > &, const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > sin(const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > pow(const ::VectorizedArray< Number, width > &, const Number p)
::VectorizedArray< Number, width > abs(const ::VectorizedArray< Number, width > &)
const ::parallel::distributed::Triangulation< dim, spacedim > * triangulation
UpdateFlags mapping_update_flags
boost::signals2::signal< void()> post_p4est_refinement
boost::signals2::signal< unsigned int(const cell_iterator &, const CellStatus), CellWeightSum< unsigned int > > cell_weight