553 *
const double m = 0.5;
554 *
const double c1 = 0.;
555 *
const double c2 = 0.;
556 *
const double factor =
558 *
double result = 1.;
559 *
for (
unsigned int d = 0;
d < dim; ++
d)
570 * <a name=
"SineGordonProblemclass"></a>
571 * <h3>SineGordonProblem
class</h3>
575 * This is the main
class that builds on the class in @ref step_25 "step-25". However, we
577 * the geometry data. Also, we use a distributed
triangulation in
this
582 *
class SineGordonProblem
585 * SineGordonProblem();
591 *
void make_grid_and_dofs();
592 *
void output_results(
const unsigned int timestep_number);
594 * #ifdef DEAL_II_WITH_P4EST
612 *
const unsigned int n_global_refinements;
613 *
double time, time_step;
614 *
const double final_time;
615 *
const double cfl_number;
616 *
const unsigned int output_timestep_skip;
623 * <a name=
"SineGordonProblemSineGordonProblem"></a>
624 * <h4>SineGordonProblem::SineGordonProblem</h4>
628 * This is the constructor of the SineGordonProblem
class. The time interval
629 * and time step size are defined here. Moreover, we use the degree of the
630 * finite element that we defined at the top of the program to initialize a
631 *
FE_Q finite element based on Gauss-Lobatto support points. These points
632 * are convenient because in conjunction with a
QGaussLobatto quadrature
634 * compromising accuracy too much (note that the integration is inexact,
635 * though), see also the discussion in the introduction. Note that
FE_Q
636 * selects the Gauss-Lobatto nodal points by
default due to their improved
637 * conditioning versus equidistant points. To make things more
explicit, we
638 * state the selection of the nodal points nonetheless.
642 * SineGordonProblem<dim>::SineGordonProblem()
645 * #ifdef DEAL_II_WITH_P4EST
651 * , n_global_refinements(10 - 2 * dim)
655 * , cfl_number(.1 / fe_degree)
656 * , output_timestep_skip(200)
662 * <a name=
"SineGordonProblemmake_grid_and_dofs"></a>
663 * <h4>SineGordonProblem::make_grid_and_dofs</h4>
667 * As in @ref step_25
"step-25" this functions sets up a cube grid in <code>dim</code>
668 * dimensions of extent @f$[-15,15]@f$. We
refine the mesh more in the
center of
669 * the domain since the solution is concentrated there. We
first refine all
670 * cells whose
center is within a radius of 11, and then
refine once more
671 *
for a radius 6. This simple ad hoc refinement could be done better by
672 * adapting the mesh to the solution
using error estimators during the time
673 * stepping as done in other example programs, and
using
679 *
void SineGordonProblem<dim>::make_grid_and_dofs()
687 *
for (; cell != end_cell; ++cell)
688 *
if (cell->is_locally_owned())
689 *
if (cell->center().norm() < 11)
690 * cell->set_refine_flag();
695 *
for (; cell != end_cell; ++cell)
696 *
if (cell->is_locally_owned())
697 *
if (cell->center().norm() < 6)
698 * cell->set_refine_flag();
702 * pcout <<
" Number of global active cells: "
703 * #ifdef DEAL_II_WITH_P4EST
710 * dof_handler.distribute_dofs(fe);
712 * pcout <<
" Number of degrees of freedom: " << dof_handler.n_dofs()
718 * We generate hanging node constraints
for ensuring continuity of the
719 * solution. As in @ref step_40
"step-40", we need to equip the constraint
matrix with
720 * the
IndexSet of locally relevant degrees of freedom to avoid it to
721 * consume too much memory
for big problems. Next, the <code>
MatrixFree
722 * </code>
object for the problem is
set up. Note that we specify a
723 * particular scheme
for shared-memory parallelization (hence
one would
724 * use multithreading
for intra-node parallelism and not MPI; we here
725 * choose the standard option —
if we wanted to disable shared
726 * memory parallelization even in
case where there is more than
one TBB
727 * thread available in the program, we would choose
729 * instead of
using the
default QGauss quadrature argument, we supply a
731 * behavior. Finally, three solution vectors are initialized.
MatrixFree
732 * expects a particular layout of ghost indices (as it handles index
733 * access in MPI-local
numbers that need to match between the vector and
734 *
MatrixFree), so we just ask it to initialize the vectors to be sure the
735 * ghost exchange is properly handled.
739 * constraints.clear();
740 * constraints.reinit(locally_relevant_dofs);
742 * constraints.close();
748 * matrix_free_data.
reinit(mapping,
754 * matrix_free_data.initialize_dof_vector(solution);
755 * old_solution.reinit(solution);
756 * old_old_solution.reinit(solution);
764 * <a name=
"SineGordonProblemoutput_results"></a>
765 * <h4>SineGordonProblem::output_results</h4>
769 * This function prints the
norm of the solution and writes the solution
770 * vector to a file. The
norm is standard (except
for the fact that we need
771 * to accumulate the norms over all processors
for the
parallel grid which
773 *
second is similar to what we did in @ref step_40
"step-40" or @ref step_37
"step-37". Note that we can
774 * use the same vector
for output as the
one used during computations: The
776 * all locally owned cells (
this is what is needed in the local evaluations,
777 * too), including ghost vector entries on these cells. This is the only
779 * as well as in
DataOut. The only action to take at this
point is to make
780 * sure that the vector updates its ghost
values before we read from
781 * them, and to reset ghost
values once done. This is a feature present only
783 * PETSc and Trilinos, on the other hand, need to be copied to special
784 * vectors including ghost
values (see the relevant section in @ref step_40 "step-40"). If
785 * we also wanted to access all degrees of freedom on ghost cells (
e.g. when
786 * computing error estimators that use the jump of solution over cell
787 * boundaries), we would need more information and create a vector
788 * initialized with locally relevant dofs just as in @ref step_40 "step-40". Observe also
789 * that we need to distribute constraints for output - they are not filled
790 * during computations (rather, they are interpolated on the fly in the
796 * SineGordonProblem<dim>::output_results(const
unsigned int timestep_number)
798 * constraints.distribute(solution);
801 * solution.update_ghost_values();
809 *
const double solution_norm =
814 * pcout <<
" Time:" << std::setw(8) << std::setprecision(3) << time
815 * <<
", solution norm: " << std::setprecision(5) << std::setw(7)
816 * << solution_norm << std::endl;
825 *
"./",
"solution", timestep_number, MPI_COMM_WORLD, 3);
827 * solution.zero_out_ghost_values();
834 * <a name=
"SineGordonProblemrun"></a>
839 * This function is called by the main function and steps into the
840 * subroutines of the
class.
844 * After printing some information about the
parallel setup, the
first
845 * action is to
set up the grid and the cell
operator. Then, the time step
846 * is computed from the CFL number given in the constructor and the finest
847 * mesh size. The finest mesh size is computed as the
diameter of the last
849 * the mesh. This is only possible
for meshes where all elements on a
level
850 * have the same size, otherwise,
one needs to
loop over all cells. Note
851 * that we need to query all the processors
for their finest cell since
852 * not all processors might hold a region where the mesh is at the finest
853 *
level. Then, we readjust the time step a little to hit the
final time
861 * pcout <<
"Number of MPI ranks: "
863 * pcout <<
"Number of threads on each rank: "
866 *
const unsigned int n_vect_bits = 8 *
sizeof(double) * n_vect_doubles;
867 * pcout <<
"Vectorization over " << n_vect_doubles
868 * <<
" doubles = " << n_vect_bits <<
" bits ("
873 * make_grid_and_dofs();
875 *
const double local_min_cell_diameter =
877 *
const double global_min_cell_diameter =
879 * time_step = cfl_number * global_min_cell_diameter;
880 * time_step = (final_time - time) / (
int((final_time - time) / time_step));
881 * pcout <<
" Time step size: " << time_step
882 * <<
", finest cell: " << global_min_cell_diameter << std::endl
887 * Next the
initial value is
set. Since we have a two-step time stepping
888 * method, we also need a
value of the solution at time-time_step. For
889 * accurate results,
one would need to compute
this from the time
890 * derivative of the solution at
initial time, but here we ignore
this
896 * We then go on by writing the
initial state to file and collecting
897 * the two starting solutions in a <tt>std::vector</tt> of pointers that
898 * get later consumed by the SineGordonOperation::apply() function. Next,
899 * an instance of the <code> SineGordonOperation class </code> based on
900 * the finite element degree specified at the top of this file is
set up.
905 * InitialCondition<dim>(1, time),
909 * InitialCondition<dim>(1, time - time_step),
914 * previous_solutions({&old_solution, &old_old_solution});
916 * SineGordonOperation<dim, fe_degree> sine_gordon_op(matrix_free_data,
921 * Now
loop over the time steps. In each iteration, we
shift the solution
922 * vectors by
one and
call the `apply` function of the
923 * `SineGordonOperator`
class. Then, we write the solution to a file. We
924 * clock the wall times
for the computational time needed as wall as the
925 * time needed to create the output and report the
numbers when the time
926 * stepping is finished.
930 * Note how
this shift is implemented: We simply
call the
swap method on
931 * the two vectors which swaps only some pointers without the need to
copy
932 * data around, a relatively expensive operation within an
explicit time
933 * stepping method. Let us see what happens in more detail: First, we
934 * exchange <code>old_solution</code> with <code>old_old_solution</code>,
935 * which means that <code>old_old_solution</code> gets
936 * <code>old_solution</code>, which is what we expect. Similarly,
937 * <code>old_solution</code> gets the content from <code>solution</code>
938 * in the next step. After
this, <code>solution</code> holds
939 * <code>old_old_solution</code>, but that will be overwritten during
this
943 *
unsigned int timestep_number = 1;
947 *
double output_time = 0;
948 *
for (time += time_step; time <= final_time;
949 * time += time_step, ++timestep_number)
952 * old_old_solution.swap(old_solution);
953 * old_solution.swap(solution);
954 * sine_gordon_op.apply(solution, previous_solutions);
958 *
if (timestep_number % output_timestep_skip == 0)
959 * output_results(timestep_number / output_timestep_skip);
964 * output_results(timestep_number / output_timestep_skip + 1);
968 * <<
" Performed " << timestep_number <<
" time steps." << std::endl;
970 * pcout <<
" Average wallclock time per time step: "
971 * << wtime / timestep_number <<
"s" << std::endl;
973 * pcout <<
" Spent " << output_time <<
"s on output and " << wtime
974 * <<
"s on computations." << std::endl;
983 * <a name=
"Thecodemaincodefunction"></a>
984 * <h3>The <code>main</code> function</h3>
988 * As in @ref step_40
"step-40", we initialize MPI at the start of the program. Since we will
989 * in
general mix MPI parallelization with threads, we also
set the third
990 * argument in MPI_InitFinalize that controls the number of threads to an
991 *
invalid number, which means that the TBB library chooses the number of
992 * threads automatically, typically to the number of available cores in the
993 * system. As an alternative, you can also
set this number manually
if you
994 * want to
set a specific number of threads (
e.g. when MPI-only is required).
997 *
int main(
int argc,
char **argv)
999 *
using namespace Step48;
1000 *
using namespace dealii;
1007 * SineGordonProblem<dimension> sg_problem;
1010 *
catch (std::exception &exc)
1012 * std::cerr << std::endl
1014 * <<
"----------------------------------------------------"
1016 * std::cerr <<
"Exception on processing: " << std::endl
1017 * << exc.what() << std::endl
1018 * <<
"Aborting!" << std::endl
1019 * <<
"----------------------------------------------------"
1026 * std::cerr << std::endl
1028 * <<
"----------------------------------------------------"
1030 * std::cerr <<
"Unknown exception!" << std::endl
1031 * <<
"Aborting!" << std::endl
1032 * <<
"----------------------------------------------------"
1040<a name=
"Results"></a><h1>Results</h1>
1043<a name=
"Comparisonwithasparsematrix"></a><h3>Comparison with a sparse
matrix</h3>
1046In order to demonstrate the gain in
using the
MatrixFree class instead of
1047the standard <code>deal.II</code> assembly routines for evaluating the
1048information from old time steps, we study a simple serial
run of the code on a
1049nonadaptive mesh. Since much time is spent on evaluating the sine function, we
1050do not only show the
numbers of the full sine-Gordon equation but also for the
1051wave equation (the sine-term skipped from the sine-Gordon equation). We use
1052both
second and fourth order elements. The results are summarized in the
1055<table align=
"center" class=
"doxtable">
1058 <th colspan=
"3">wave equation</th>
1059 <th colspan=
"2">sine-Gordon</th>
1070 <td>2D, @f$\mathcal{Q}_2@f$</td>
1071 <td align=
"right"> 0.0106</td>
1072 <td align=
"right"> 0.00971</td>
1073 <td align=
"right"> 0.109</td>
1074 <td align=
"right"> 0.0243</td>
1075 <td align=
"right"> 0.124</td>
1078 <td>2D, @f$\mathcal{Q}_4@f$</td>
1079 <td align=
"right"> 0.0328</td>
1080 <td align=
"right"> 0.0706</td>
1081 <td align=
"right"> 0.528</td>
1082 <td align=
"right"> 0.0714</td>
1083 <td align=
"right"> 0.502</td>
1086 <td>3D, @f$\mathcal{Q}_2@f$</td>
1087 <td align=
"right"> 0.0151</td>
1088 <td align=
"right"> 0.0320</td>
1089 <td align=
"right"> 0.331</td>
1090 <td align=
"right"> 0.0376</td>
1091 <td align=
"right"> 0.364</td>
1094 <td>3D, @f$\mathcal{Q}_4@f$</td>
1095 <td align=
"right"> 0.0918</td>
1096 <td align=
"right"> 0.844</td>
1097 <td align=
"right"> 6.83</td>
1098 <td align=
"right"> 0.194</td>
1099 <td align=
"right"> 6.95</td>
1103It is apparent that the
matrix-
free code outperforms the standard assembly
1104routines in deal.II by far. In 3D and
for fourth order elements,
one operator
1105evaluation is also almost ten times as fast as a sparse
matrix-vector
1108<a name=
"Parallelrunin2Dand3D"></a><h3>Parallel
run in 2D and 3D</h3>
1111We start with the program output obtained on a workstation with 12 cores / 24
1112threads (
one Intel Xeon E5-2687W v4 CPU running at 3.2 GHz, hyperthreading
1113enabled), running the program in release mode:
1116Number of MPI ranks: 1
1117Number of threads on each rank: 24
1118Vectorization over 4 doubles = 256 bits (AVX)
1120 Number of global active cells: 15412
1121 Number of degrees of freedom: 249065
1122 Time step size: 0.00292997, finest cell: 0.117188
1124 Time: -10, solution
norm: 9.5599
1125 Time: -9.41, solution
norm: 17.678
1126 Time: -8.83, solution
norm: 23.504
1127 Time: -8.24, solution
norm: 27.5
1128 Time: -7.66, solution
norm: 29.513
1129 Time: -7.07, solution
norm: 29.364
1130 Time: -6.48, solution
norm: 27.23
1131 Time: -5.9, solution
norm: 23.527
1132 Time: -5.31, solution
norm: 18.439
1133 Time: -4.73, solution
norm: 11.935
1134 Time: -4.14, solution
norm: 5.5284
1135 Time: -3.55, solution
norm: 8.0354
1136 Time: -2.97, solution
norm: 14.707
1137 Time: -2.38, solution
norm: 20
1138 Time: -1.8, solution
norm: 22.834
1139 Time: -1.21, solution
norm: 22.771
1140 Time: -0.624, solution
norm: 20.488
1141 Time: -0.0381, solution
norm: 16.697
1142 Time: 0.548, solution
norm: 11.221
1143 Time: 1.13, solution
norm: 5.3912
1144 Time: 1.72, solution
norm: 8.4528
1145 Time: 2.31, solution
norm: 14.335
1146 Time: 2.89, solution
norm: 18.555
1147 Time: 3.48, solution
norm: 20.894
1148 Time: 4.06, solution
norm: 21.305
1149 Time: 4.65, solution
norm: 19.903
1150 Time: 5.24, solution
norm: 16.864
1151 Time: 5.82, solution
norm: 12.223
1152 Time: 6.41, solution
norm: 6.758
1153 Time: 6.99, solution
norm: 7.2423
1154 Time: 7.58, solution
norm: 12.888
1155 Time: 8.17, solution
norm: 17.273
1156 Time: 8.75, solution
norm: 19.654
1157 Time: 9.34, solution
norm: 19.838
1158 Time: 9.92, solution
norm: 17.964
1159 Time: 10, solution
norm: 17.595
1161 Performed 6826 time steps.
1162 Average wallclock time per time step: 0.0013453s
1163 Spent 14.976s on output and 9.1831s on computations.
1166In 3D, the respective output looks like
1169Number of MPI ranks: 1
1170Number of threads on each rank: 24
1171Vectorization over 4 doubles = 256 bits (AVX)
1173 Number of global active cells: 17592
1174 Number of degrees of freedom: 1193881
1175 Time step size: 0.0117233, finest cell: 0.46875
1177 Time: -10, solution
norm: 29.558
1178 Time: -7.66, solution
norm: 129.13
1179 Time: -5.31, solution
norm: 67.753
1180 Time: -2.97, solution
norm: 79.245
1181 Time: -0.621, solution
norm: 123.52
1182 Time: 1.72, solution
norm: 43.525
1183 Time: 4.07, solution
norm: 93.285
1184 Time: 6.41, solution
norm: 97.722
1185 Time: 8.76, solution
norm: 36.734
1186 Time: 10, solution
norm: 94.115
1188 Performed 1706 time steps.
1189 Average wallclock time per time step: 0.0084542s
1190 Spent 16.766s on output and 14.423s on computations.
1193It takes 0.008 seconds
for one time step with more than a million
1194degrees of freedom (note that we would need many processors to reach such
1195numbers when solving linear systems).
1197If we replace the thread-parallelization by a pure MPI parallelization, the
1200$ mpirun -n 24 ./step-48
1201Number of MPI ranks: 24
1202Number of threads on each rank: 1
1203Vectorization over 4 doubles = 256 bits (AVX)
1205 Performed 1706 time steps.
1206 Average wallclock time per time step: 0.0051747s
1207 Spent 2.0535s on output and 8.828s on computations.
1210We observe a dramatic speedup
for the output (which makes sense, given that
1211most code of the output is not parallelized via threads, whereas it is
for
1212MPI), but less than the theoretical factor of 12 we would expect from the
1213parallelism. More interestingly, the computations also get faster when
1214switching from the threads-only variant to the MPI-only variant. This is a
12162019). The main reason is that the decisions regarding work on conflicting
1217cell batches made to enable execution in
parallel are overly pessimistic:
1218While they ensure that no work on neighboring cells is done on different
1219threads at the same time,
this conservative setting implies that data from
1220neighboring cells is also evicted from caches by the time neighbors get
1221touched. Furthermore, the current scheme is not able to provide a constant
1222load
for all 24 threads
for the given mesh with 17,592 cells.
1224The current program allows to also mix MPI parallelization with thread
1225parallelization. This is most beneficial when running programs on clusters
1226with multiple nodes,
using MPI
for the inter-node parallelization and threads
1227for the intra-node parallelization. On the workstation used above, we can
run
1228threads in the hyperthreading region (i.e.,
using 2 threads
for each of the 12
1229MPI ranks). An important setting
for mixing MPI with threads is to ensure
1230proper binning of tasks to CPUs. On many clusters the placing is either
1231automatically via the `mpirun/mpiexec` environment, or there can be manual
1232settings. Here, we simply report the
run times the plain version of the
1233program (noting that things could be improved towards the timings of the
1234MPI-only program when proper pinning is done):
1236$ mpirun -n 12 ./step-48
1237Number of MPI ranks: 12
1238Number of threads on each rank: 2
1239Vectorization over 4 doubles = 256 bits (AVX)
1241 Performed 1706 time steps.
1242 Average wallclock time per time step: 0.0056651s
1243 Spent 2.5175s on output and 9.6646s on computations.
1248<a name=
"Possibilitiesforextensions"></a><h3>Possibilities for extensions</h3>
1251There are several things in this program that could be improved to make it
1252even more efficient (besides improved boundary conditions and physical
1253stuff as discussed in @ref step_25
"step-25"):
1255<ul> <li> <
b>Faster evaluation of sine terms:</
b> As becomes obvious
1256 from the comparison of the plain wave equation and the sine-Gordon
1257 equation above, the evaluation of the sine terms dominates the total
1258 time for the finite element operator application. There are a few
1259 reasons for this: Firstly, the deal.II sine computation of a
1261 the operator application). This could be cured by handing the sine
1262 computation to a library with vectorized sine computations like
1263 Intel
's math kernel library (MKL). By using the function
1264 <code>vdSin</code> in MKL, the program uses half the computing time
1265 in 2D and 40 percent less time in 3D. On the other hand, the sine
1266 computation is structurally much more complicated than the simple
1267 arithmetic operations like additions and multiplications in the rest
1268 of the local operation.
1270 <li> <b>Higher order time stepping:</b> While the implementation allows for
1271 arbitrary order in the spatial part (by adjusting the degree of the finite
1272 element), the time stepping scheme is a standard second-order leap-frog
1273 scheme. Since solutions in wave propagation problems are usually very
1274 smooth, the error is likely dominated by the time stepping part. Of course,
1275 this could be cured by using smaller time steps (at a fixed spatial
1276 resolution), but it would be more efficient to use higher order time
1277 stepping as well. While it would be straight-forward to do so for a
1278 first-order system (use some Runge–Kutta scheme of higher order,
1279 probably combined with adaptive time step selection like the <a
1280 href="http://en.wikipedia.org/wiki/Dormand%E2%80%93Prince_method">Dormand–Prince
1281 method</a>), it is more challenging for the second-order formulation. At
1282 least in the finite difference community, people usually use the PDE to find
1283 spatial correction terms that improve the temporal error.
1288<a name="PlainProg"></a>
1289<h1> The plain program</h1>
1290@include "step-48.cc"
void swap(BlockIndices &u, BlockIndices &v)
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 MappingType &mapping, const DoFHandler< dim > &dof_handler, const AffineConstraints< number2 > &constraint, const QuadratureType &quad, const AdditionalData &additional_data=AdditionalData())
static unsigned int n_threads()
static constexpr std::size_t size()
__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
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())
void make_hanging_node_constraints(const DoFHandler< dim, spacedim > &dof_handler, AffineConstraints< number > &constraints)
Expression cosh(const Expression &x)
Expression atan(const Expression &x)
void hyper_cube(Triangulation< dim, spacedim > &tria, const double left=0., const double right=1., 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)
@ matrix
Contents is actually a matrix.
@ diagonal
Matrix is diagonal.
@ general
No special properties.
static const types::blas_int one
double norm(const FEValuesBase< dim > &fe, const ArrayView< const std::vector< Tensor< 1, dim > > > &Du)
Point< spacedim > point(const gp_Pnt &p, const double tolerance=1e-10)
SymmetricTensor< 2, dim, Number > e(const Tensor< 2, dim, Number > &F)
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)
void call(const std::function< RT()> &function, internal::return_value< RT > &ret_val)
unsigned int this_mpi_process(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)
const std::string get_current_vectorization_level()
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)
void copy(const T *begin, const T *end, U *dest)
int(&) functions(const void *v1, const void *v2)
static const unsigned int invalid_unsigned_int
::VectorizedArray< Number, width > sin(const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > sqrt(const ::VectorizedArray< Number, width > &)
const ::parallel::distributed::Triangulation< dim, spacedim > * triangulation
TasksParallelScheme tasks_parallel_scheme