deal.II version GIT relicensing-1721-g8100761196 2024-08-31 12:30:00+00:00
\(\newcommand{\dealvcentcolon}{\mathrel{\mathop{:}}}\) \(\newcommand{\dealcoloneq}{\dealvcentcolon\mathrel{\mkern-1.2mu}=}\) \(\newcommand{\jump}[1]{\left[\!\left[ #1 \right]\!\right]}\) \(\newcommand{\average}[1]{\left\{\!\left\{ #1 \right\}\!\right\}}\)
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step-48.h
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1) const override
550 *   {
551 *   double t = this->get_time();
552 *  
553 *   const double m = 0.5;
554 *   const double c1 = 0.;
555 *   const double c2 = 0.;
556 *   const double factor =
557 *   (m / std::sqrt(1. - m * m) * std::sin(std::sqrt(1. - m * m) * t + c2));
558 *   double result = 1.;
559 *   for (unsigned int d = 0; d < dim; ++d)
560 *   result *= -4. * std::atan(factor / std::cosh(m * p[d] + c1));
561 *   return result;
562 *   }
563 *   };
564 *  
565 *  
566 *  
567 * @endcode
568 *
569 *
570 * <a name="step_48-SineGordonProblemclass"></a>
571 * <h3>SineGordonProblem class</h3>
572 *
573
574 *
575 * This is the main class that builds on the class in @ref step_25 "step-25". However, we
576 * replaced the SparseMatrix<double> class by the MatrixFree class to store
577 * the geometry data. Also, we use a distributed triangulation in this
578 * example.
579 *
580 * @code
581 *   template <int dim>
582 *   class SineGordonProblem
583 *   {
584 *   public:
585 *   SineGordonProblem();
586 *   void run();
587 *  
588 *   private:
589 *   ConditionalOStream pcout;
590 *  
591 *   void make_grid_and_dofs();
592 *   void output_results(const unsigned int timestep_number);
593 *  
594 *   #ifdef DEAL_II_WITH_P4EST
596 *   #else
598 *   #endif
599 *   const FE_Q<dim> fe;
600 *   DoFHandler<dim> dof_handler;
601 *  
602 *   const MappingQ1<dim> mapping;
603 *  
604 *   AffineConstraints<double> constraints;
605 *   IndexSet locally_relevant_dofs;
606 *  
607 *   MatrixFree<dim, double> matrix_free_data;
608 *  
609 *   LinearAlgebra::distributed::Vector<double> solution, old_solution,
610 *   old_old_solution;
611 *  
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;
617 *   };
618 *  
619 *  
620 * @endcode
621 *
622 *
623 * <a name="step_48-SineGordonProblemSineGordonProblem"></a>
624 * <h4>SineGordonProblem::SineGordonProblem</h4>
625 *
626
627 *
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
633 * rule of the same order they give a diagonal mass matrix without
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.
639 *
640 * @code
641 *   template <int dim>
642 *   SineGordonProblem<dim>::SineGordonProblem()
643 *   : pcout(std::cout, Utilities::MPI::this_mpi_process(MPI_COMM_WORLD) == 0)
644 *   #ifdef DEAL_II_WITH_P4EST
645 *   , triangulation(MPI_COMM_WORLD)
646 *   #endif
647 *   , fe(QGaussLobatto<1>(fe_degree + 1))
648 *   , dof_handler(triangulation)
649 *   , n_global_refinements(10 - 2 * dim)
650 *   , time(-10)
651 *   , time_step(10.)
652 *   , final_time(10.)
653 *   , cfl_number(.1 / fe_degree)
654 *   , output_timestep_skip(200)
655 *   {}
656 *  
657 * @endcode
658 *
659 *
660 * <a name="step_48-SineGordonProblemmake_grid_and_dofs"></a>
661 * <h4>SineGordonProblem::make_grid_and_dofs</h4>
662 *
663
664 *
665 * As in @ref step_25 "step-25" this functions sets up a cube grid in <code>dim</code>
666 * dimensions of extent @f$[-15,15]@f$. We refine the mesh more in the center of
667 * the domain since the solution is concentrated there. We first refine all
668 * cells whose center is within a radius of 11, and then refine once more
669 * for a radius 6. This simple ad hoc refinement could be done better by
670 * adapting the mesh to the solution using error estimators during the time
671 * stepping as done in other example programs, and using
672 * parallel::distributed::SolutionTransfer to transfer the solution to the
673 * new mesh.
674 *
675 * @code
676 *   template <int dim>
677 *   void SineGordonProblem<dim>::make_grid_and_dofs()
678 *   {
680 *   triangulation.refine_global(n_global_refinements);
681 *   {
682 *   for (const auto &cell : triangulation.active_cell_iterators())
683 *   if (cell->is_locally_owned())
684 *   if (cell->center().norm() < 11)
685 *   cell->set_refine_flag();
686 *   triangulation.execute_coarsening_and_refinement();
687 *  
688 *   for (const auto &cell : triangulation.active_cell_iterators())
689 *   if (cell->is_locally_owned())
690 *   if (cell->center().norm() < 6)
691 *   cell->set_refine_flag();
692 *   triangulation.execute_coarsening_and_refinement();
693 *   }
694 *  
695 *   pcout << " Number of global active cells: "
696 *   << triangulation.n_global_active_cells() << std::endl;
697 *  
698 *   dof_handler.distribute_dofs(fe);
699 *  
700 *   pcout << " Number of degrees of freedom: " << dof_handler.n_dofs()
701 *   << std::endl;
702 *  
703 *  
704 * @endcode
705 *
706 * We generate hanging node constraints for ensuring continuity of the
707 * solution. As in @ref step_40 "step-40", we need to equip the constraint matrix with
708 * the IndexSet of locally active and locally relevant degrees of freedom
709 * to avoid it consuming too much memory for big problems. Next, the
710 * MatrixFree object for the problem is set up. Note that we specify a
711 * particular scheme for shared-memory parallelization (hence one would
712 * use multithreading for intra-node parallelism and not MPI; we here
713 * choose the standard option &mdash; if we wanted to disable shared
714 * memory parallelization even in case where there is more than one TBB
715 * thread available in the program, we would choose
717 * instead of using the default QGauss quadrature argument, we supply a
718 * QGaussLobatto quadrature formula to enable the desired
719 * behavior. Finally, three solution vectors are initialized. MatrixFree
720 * expects a particular layout of ghost indices (as it handles index
721 * access in MPI-local numbers that need to match between the vector and
722 * MatrixFree), so we just ask it to initialize the vectors to be sure the
723 * ghost exchange is properly handled.
724 *
725 * @code
726 *   locally_relevant_dofs =
728 *   constraints.clear();
729 *   constraints.reinit(dof_handler.locally_owned_dofs(), locally_relevant_dofs);
730 *   DoFTools::make_hanging_node_constraints(dof_handler, constraints);
731 *   constraints.close();
732 *  
733 *   typename MatrixFree<dim>::AdditionalData additional_data;
734 *   additional_data.tasks_parallel_scheme =
736 *  
737 *   matrix_free_data.reinit(mapping,
738 *   dof_handler,
739 *   constraints,
740 *   QGaussLobatto<1>(fe_degree + 1),
741 *   additional_data);
742 *  
743 *   matrix_free_data.initialize_dof_vector(solution);
744 *   old_solution.reinit(solution);
745 *   old_old_solution.reinit(solution);
746 *   }
747 *  
748 *  
749 *  
750 * @endcode
751 *
752 *
753 * <a name="step_48-SineGordonProblemoutput_results"></a>
754 * <h4>SineGordonProblem::output_results</h4>
755 *
756
757 *
758 * This function prints the norm of the solution and writes the solution
759 * vector to a file. The norm is standard (except for the fact that we need
760 * to accumulate the norms over all processors for the parallel grid which
761 * we do via the VectorTools::compute_global_error() function), and the
762 * second is similar to what we did in @ref step_40 "step-40" or @ref step_37 "step-37". Note that we can
763 * use the same vector for output as the one used during computations: The
764 * vectors in the matrix-free framework always provide full information on
765 * all locally owned cells (this is what is needed in the local evaluations,
766 * too), including ghost vector entries on these cells. This is the only
767 * data that is needed in the VectorTools::integrate_difference() function
768 * as well as in DataOut. The only action to take at this point is to make
769 * sure that the vector updates its ghost values before we read from
770 * them, and to reset ghost values once done. This is a feature present only
771 * in the LinearAlgebra::distributed::Vector class. Distributed vectors with
772 * PETSc and Trilinos, on the other hand, need to be copied to special
773 * vectors including ghost values (see the relevant section in @ref step_40 "step-40"). If
774 * we also wanted to access all degrees of freedom on ghost cells (e.g. when
775 * computing error estimators that use the jump of solution over cell
776 * boundaries), we would need more information and create a vector
777 * initialized with locally relevant dofs just as in @ref step_40 "step-40". Observe also
778 * that we need to distribute constraints for output - they are not filled
779 * during computations (rather, they are interpolated on the fly in the
780 * matrix-free method FEEvaluation::read_dof_values()).
781 *
782 * @code
783 *   template <int dim>
784 *   void
785 *   SineGordonProblem<dim>::output_results(const unsigned int timestep_number)
786 *   {
787 *   constraints.distribute(solution);
788 *  
789 *   Vector<float> norm_per_cell(triangulation.n_active_cells());
790 *   solution.update_ghost_values();
792 *   dof_handler,
793 *   solution,
795 *   norm_per_cell,
796 *   QGauss<dim>(fe_degree + 1),
798 *   const double solution_norm =
800 *   norm_per_cell,
802 *  
803 *   pcout << " Time:" << std::setw(8) << std::setprecision(3) << time
804 *   << ", solution norm: " << std::setprecision(5) << std::setw(7)
805 *   << solution_norm << std::endl;
806 *  
807 *   DataOut<dim> data_out;
808 *  
809 *   data_out.attach_dof_handler(dof_handler);
810 *   data_out.add_data_vector(solution, "solution");
811 *   data_out.build_patches(mapping);
812 *  
813 *   data_out.write_vtu_with_pvtu_record(
814 *   "./", "solution", timestep_number, MPI_COMM_WORLD, 3);
815 *  
816 *   solution.zero_out_ghost_values();
817 *   }
818 *  
819 *  
820 * @endcode
821 *
822 *
823 * <a name="step_48-SineGordonProblemrun"></a>
824 * <h4>SineGordonProblem::run</h4>
825 *
826
827 *
828 * This function is called by the main function and steps into the
829 * subroutines of the class.
830 *
831
832 *
833 * After printing some information about the parallel setup, the first
834 * action is to set up the grid and the cell operator. Then, the time step
835 * is computed from the CFL number given in the constructor and the finest
836 * mesh size. The finest mesh size is computed as the diameter of the last
837 * cell in the triangulation, which is the last cell on the finest level of
838 * the mesh. This is only possible for meshes where all elements on a level
839 * have the same size, otherwise, one needs to loop over all cells. Note
840 * that we need to query all the processors for their finest cell since
841 * not all processors might hold a region where the mesh is at the finest
842 * level. Then, we readjust the time step a little to hit the final time
843 * exactly.
844 *
845 * @code
846 *   template <int dim>
847 *   void SineGordonProblem<dim>::run()
848 *   {
849 *   {
850 *   pcout << "Number of MPI ranks: "
851 *   << Utilities::MPI::n_mpi_processes(MPI_COMM_WORLD) << std::endl;
852 *   pcout << "Number of threads on each rank: "
853 *   << MultithreadInfo::n_threads() << std::endl;
854 *   const unsigned int n_vect_doubles = VectorizedArray<double>::size();
855 *   const unsigned int n_vect_bits = 8 * sizeof(double) * n_vect_doubles;
856 *   pcout << "Vectorization over " << n_vect_doubles
857 *   << " doubles = " << n_vect_bits << " bits ("
859 *   << std::endl
860 *   << std::endl;
861 *   }
862 *   make_grid_and_dofs();
863 *  
864 *   const double local_min_cell_diameter =
865 *   triangulation.last()->diameter() / std::sqrt(dim);
866 *   const double global_min_cell_diameter =
867 *   -Utilities::MPI::max(-local_min_cell_diameter, MPI_COMM_WORLD);
868 *   time_step = cfl_number * global_min_cell_diameter;
869 *   time_step = (final_time - time) / (int((final_time - time) / time_step));
870 *   pcout << " Time step size: " << time_step
871 *   << ", finest cell: " << global_min_cell_diameter << std::endl
872 *   << std::endl;
873 *  
874 * @endcode
875 *
876 * Next the initial value is set. Since we have a two-step time stepping
877 * method, we also need a value of the solution at time-time_step. For
878 * accurate results, one would need to compute this from the time
879 * derivative of the solution at initial time, but here we ignore this
880 * difficulty and just set it to the initial value function at that
881 * artificial time.
882 *
883
884 *
885 * We then go on by writing the initial state to file and collecting
886 * the two starting solutions in a <tt>std::vector</tt> of pointers that
887 * get later consumed by the SineGordonOperation::apply() function. Next,
888 * an instance of the <code> SineGordonOperation class </code> based on
889 * the finite element degree specified at the top of this file is set up.
890 *
891 * @code
892 *   VectorTools::interpolate(mapping,
893 *   dof_handler,
894 *   InitialCondition<dim>(1, time),
895 *   solution);
896 *   VectorTools::interpolate(mapping,
897 *   dof_handler,
898 *   InitialCondition<dim>(1, time - time_step),
899 *   old_solution);
900 *   output_results(0);
901 *  
902 *   std::vector<LinearAlgebra::distributed::Vector<double> *>
903 *   previous_solutions({&old_solution, &old_old_solution});
904 *  
905 *   SineGordonOperation<dim, fe_degree> sine_gordon_op(matrix_free_data,
906 *   time_step);
907 *  
908 * @endcode
909 *
910 * Now loop over the time steps. In each iteration, we shift the solution
911 * vectors by one and call the `apply` function of the
912 * `SineGordonOperator` class. Then, we write the solution to a file. We
913 * clock the wall times for the computational time needed as wall as the
914 * time needed to create the output and report the numbers when the time
915 * stepping is finished.
916 *
917
918 *
919 * Note how this shift is implemented: We simply call the swap method on
920 * the two vectors which swaps only some pointers without the need to copy
921 * data around, a relatively expensive operation within an explicit time
922 * stepping method. Let us see what happens in more detail: First, we
923 * exchange <code>old_solution</code> with <code>old_old_solution</code>,
924 * which means that <code>old_old_solution</code> gets
925 * <code>old_solution</code>, which is what we expect. Similarly,
926 * <code>old_solution</code> gets the content from <code>solution</code>
927 * in the next step. After this, <code>solution</code> holds
928 * <code>old_old_solution</code>, but that will be overwritten during this
929 * step.
930 *
931 * @code
932 *   unsigned int timestep_number = 1;
933 *  
934 *   Timer timer;
935 *   double wtime = 0;
936 *   double output_time = 0;
937 *   for (time += time_step; time <= final_time;
938 *   time += time_step, ++timestep_number)
939 *   {
940 *   timer.restart();
941 *   old_old_solution.swap(old_solution);
942 *   old_solution.swap(solution);
943 *   sine_gordon_op.apply(solution, previous_solutions);
944 *   wtime += timer.wall_time();
945 *  
946 *   timer.restart();
947 *   if (timestep_number % output_timestep_skip == 0)
948 *   output_results(timestep_number / output_timestep_skip);
949 *  
950 *   output_time += timer.wall_time();
951 *   }
952 *   timer.restart();
953 *   output_results(timestep_number / output_timestep_skip + 1);
954 *   output_time += timer.wall_time();
955 *  
956 *   pcout << std::endl
957 *   << " Performed " << timestep_number << " time steps." << std::endl;
958 *  
959 *   pcout << " Average wallclock time per time step: "
960 *   << wtime / timestep_number << 's' << std::endl;
961 *  
962 *   pcout << " Spent " << output_time << "s on output and " << wtime
963 *   << "s on computations." << std::endl;
964 *   }
965 *   } // namespace Step48
966 *  
967 *  
968 *  
969 * @endcode
970 *
971 *
972 * <a name="step_48-Thecodemaincodefunction"></a>
973 * <h3>The <code>main</code> function</h3>
974 *
975
976 *
977 * As in @ref step_40 "step-40", we initialize MPI at the start of the program. Since we will
978 * in general mix MPI parallelization with threads, we also set the third
979 * argument in MPI_InitFinalize that controls the number of threads to an
980 * invalid number, which means that the TBB library chooses the number of
981 * threads automatically, typically to the number of available cores in the
982 * system. As an alternative, you can also set this number manually if you
983 * want to set a specific number of threads (e.g. when MPI-only is required).
984 *
985 * @code
986 *   int main(int argc, char **argv)
987 *   {
988 *   using namespace Step48;
989 *   using namespace dealii;
990 *  
991 *   Utilities::MPI::MPI_InitFinalize mpi_initialization(
992 *   argc, argv, numbers::invalid_unsigned_int);
993 *  
994 *   try
995 *   {
996 *   SineGordonProblem<dimension> sg_problem;
997 *   sg_problem.run();
998 *   }
999 *   catch (std::exception &exc)
1000 *   {
1001 *   std::cerr << std::endl
1002 *   << std::endl
1003 *   << "----------------------------------------------------"
1004 *   << std::endl;
1005 *   std::cerr << "Exception on processing: " << std::endl
1006 *   << exc.what() << std::endl
1007 *   << "Aborting!" << std::endl
1008 *   << "----------------------------------------------------"
1009 *   << std::endl;
1010 *  
1011 *   return 1;
1012 *   }
1013 *   catch (...)
1014 *   {
1015 *   std::cerr << std::endl
1016 *   << std::endl
1017 *   << "----------------------------------------------------"
1018 *   << std::endl;
1019 *   std::cerr << "Unknown exception!" << std::endl
1020 *   << "Aborting!" << std::endl
1021 *   << "----------------------------------------------------"
1022 *   << std::endl;
1023 *   return 1;
1024 *   }
1025 *  
1026 *   return 0;
1027 *   }
1028 * @endcode
1029<a name="step_48-Results"></a><h1>Results</h1>
1030
1031
1032<a name="step_48-Comparisonwithasparsematrix"></a><h3>Comparison with a sparse matrix</h3>
1033
1034
1035In order to demonstrate the gain in using the MatrixFree class instead of
1036the standard <code>deal.II</code> assembly routines for evaluating the
1037information from old time steps, we study a simple serial run of the code on a
1038nonadaptive mesh. Since much time is spent on evaluating the sine function, we
1039do not only show the numbers of the full sine-Gordon equation but also for the
1040wave equation (the sine-term skipped from the sine-Gordon equation). We use
1041both second and fourth order elements. The results are summarized in the
1042following table.
1043
1044<table align="center" class="doxtable">
1045 <tr>
1046 <th>&nbsp;</th>
1047 <th colspan="3">wave equation</th>
1048 <th colspan="2">sine-Gordon</th>
1049 </tr>
1050 <tr>
1051 <th>&nbsp;</th>
1052 <th>MF</th>
1053 <th>SpMV</th>
1054 <th>dealii</th>
1055 <th>MF</th>
1056 <th>dealii</th>
1057 </tr>
1058 <tr>
1059 <td>2D, @f$\mathcal{Q}_2@f$</td>
1060 <td align="right"> 0.0106</td>
1061 <td align="right"> 0.00971</td>
1062 <td align="right"> 0.109</td>
1063 <td align="right"> 0.0243</td>
1064 <td align="right"> 0.124</td>
1065 </tr>
1066 <tr>
1067 <td>2D, @f$\mathcal{Q}_4@f$</td>
1068 <td align="right"> 0.0328</td>
1069 <td align="right"> 0.0706</td>
1070 <td align="right"> 0.528</td>
1071 <td align="right"> 0.0714</td>
1072 <td align="right"> 0.502</td>
1073 </tr>
1074 <tr>
1075 <td>3D, @f$\mathcal{Q}_2@f$</td>
1076 <td align="right"> 0.0151</td>
1077 <td align="right"> 0.0320</td>
1078 <td align="right"> 0.331</td>
1079 <td align="right"> 0.0376</td>
1080 <td align="right"> 0.364</td>
1081 </tr>
1082 <tr>
1083 <td>3D, @f$\mathcal{Q}_4@f$</td>
1084 <td align="right"> 0.0918</td>
1085 <td align="right"> 0.844</td>
1086 <td align="right"> 6.83</td>
1087 <td align="right"> 0.194</td>
1088 <td align="right"> 6.95</td>
1089 </tr>
1090</table>
1091
1092It is apparent that the matrix-free code outperforms the standard assembly
1093routines in deal.II by far. In 3D and for fourth order elements, one operator
1094evaluation is also almost ten times as fast as a sparse matrix-vector
1095product.
1096
1097<a name="step_48-Parallelrunin2Dand3D"></a><h3>Parallel run in 2D and 3D</h3>
1098
1099
1100We start with the program output obtained on a workstation with 12 cores / 24
1101threads (one Intel Xeon E5-2687W v4 CPU running at 3.2 GHz, hyperthreading
1102enabled), running the program in release mode:
1103@code
1104\$ make run
1105Number of MPI ranks: 1
1106Number of threads on each rank: 24
1107Vectorization over 4 doubles = 256 bits (AVX)
1108
1109 Number of global active cells: 15412
1110 Number of degrees of freedom: 249065
1111 Time step size: 0.00292997, finest cell: 0.117188
1112
1113 Time: -10, solution norm: 9.5599
1114 Time: -9.41, solution norm: 17.678
1115 Time: -8.83, solution norm: 23.504
1116 Time: -8.24, solution norm: 27.5
1117 Time: -7.66, solution norm: 29.513
1118 Time: -7.07, solution norm: 29.364
1119 Time: -6.48, solution norm: 27.23
1120 Time: -5.9, solution norm: 23.527
1121 Time: -5.31, solution norm: 18.439
1122 Time: -4.73, solution norm: 11.935
1123 Time: -4.14, solution norm: 5.5284
1124 Time: -3.55, solution norm: 8.0354
1125 Time: -2.97, solution norm: 14.707
1126 Time: -2.38, solution norm: 20
1127 Time: -1.8, solution norm: 22.834
1128 Time: -1.21, solution norm: 22.771
1129 Time: -0.624, solution norm: 20.488
1130 Time: -0.0381, solution norm: 16.697
1131 Time: 0.548, solution norm: 11.221
1132 Time: 1.13, solution norm: 5.3912
1133 Time: 1.72, solution norm: 8.4528
1134 Time: 2.31, solution norm: 14.335
1135 Time: 2.89, solution norm: 18.555
1136 Time: 3.48, solution norm: 20.894
1137 Time: 4.06, solution norm: 21.305
1138 Time: 4.65, solution norm: 19.903
1139 Time: 5.24, solution norm: 16.864
1140 Time: 5.82, solution norm: 12.223
1141 Time: 6.41, solution norm: 6.758
1142 Time: 6.99, solution norm: 7.2423
1143 Time: 7.58, solution norm: 12.888
1144 Time: 8.17, solution norm: 17.273
1145 Time: 8.75, solution norm: 19.654
1146 Time: 9.34, solution norm: 19.838
1147 Time: 9.92, solution norm: 17.964
1148 Time: 10, solution norm: 17.595
1149
1150 Performed 6826 time steps.
1151 Average wallclock time per time step: 0.0013453s
1152 Spent 14.976s on output and 9.1831s on computations.
1153@endcode
1154
1155In 3D, the respective output looks like
1156@code
1157\$ make run
1158Number of MPI ranks: 1
1159Number of threads on each rank: 24
1160Vectorization over 4 doubles = 256 bits (AVX)
1161
1162 Number of global active cells: 17592
1163 Number of degrees of freedom: 1193881
1164 Time step size: 0.0117233, finest cell: 0.46875
1165
1166 Time: -10, solution norm: 29.558
1167 Time: -7.66, solution norm: 129.13
1168 Time: -5.31, solution norm: 67.753
1169 Time: -2.97, solution norm: 79.245
1170 Time: -0.621, solution norm: 123.52
1171 Time: 1.72, solution norm: 43.525
1172 Time: 4.07, solution norm: 93.285
1173 Time: 6.41, solution norm: 97.722
1174 Time: 8.76, solution norm: 36.734
1175 Time: 10, solution norm: 94.115
1176
1177 Performed 1706 time steps.
1178 Average wallclock time per time step: 0.0084542s
1179 Spent 16.766s on output and 14.423s on computations.
1180@endcode
1181
1182It takes 0.008 seconds for one time step with more than a million
1183degrees of freedom (note that we would need many processors to reach such
1184numbers when solving linear systems).
1185
1186If we replace the thread-parallelization by a pure MPI parallelization, the
1187timings change into:
1188@code
1189\$ mpirun -n 24 ./step-48
1190Number of MPI ranks: 24
1191Number of threads on each rank: 1
1192Vectorization over 4 doubles = 256 bits (AVX)
1193...
1194 Performed 1706 time steps.
1195 Average wallclock time per time step: 0.0051747s
1196 Spent 2.0535s on output and 8.828s on computations.
1197@endcode
1198
1199We observe a dramatic speedup for the output (which makes sense, given that
1200most code of the output is not parallelized via threads, whereas it is for
1201MPI), but less than the theoretical factor of 12 we would expect from the
1202parallelism. More interestingly, the computations also get faster when
1203switching from the threads-only variant to the MPI-only variant. This is a
1204general observation for the MatrixFree framework (as of updating this data in
12052019). The main reason is that the decisions regarding work on conflicting
1206cell batches made to enable execution in parallel are overly pessimistic:
1207While they ensure that no work on neighboring cells is done on different
1208threads at the same time, this conservative setting implies that data from
1209neighboring cells is also evicted from caches by the time neighbors get
1210touched. Furthermore, the current scheme is not able to provide a constant
1211load for all 24 threads for the given mesh with 17,592 cells.
1212
1213The current program allows to also mix MPI parallelization with thread
1214parallelization. This is most beneficial when running programs on clusters
1215with multiple nodes, using MPI for the inter-node parallelization and threads
1216for the intra-node parallelization. On the workstation used above, we can run
1217threads in the hyperthreading region (i.e., using 2 threads for each of the 12
1218MPI ranks). An important setting for mixing MPI with threads is to ensure
1219proper binning of tasks to CPUs. On many clusters the placing is either
1220automatically via the `mpirun/mpiexec` environment, or there can be manual
1221settings. Here, we simply report the run times the plain version of the
1222program (noting that things could be improved towards the timings of the
1223MPI-only program when proper pinning is done):
1224@code
1225\$ mpirun -n 12 ./step-48
1226Number of MPI ranks: 12
1227Number of threads on each rank: 2
1228Vectorization over 4 doubles = 256 bits (AVX)
1229...
1230 Performed 1706 time steps.
1231 Average wallclock time per time step: 0.0056651s
1232 Spent 2.5175s on output and 9.6646s on computations.
1233@endcode
1234
1235
1236
1237<a name="step_48-Possibilitiesforextensions"></a><h3>Possibilities for extensions</h3>
1238
1239
1240There are several things in this program that could be improved to make it
1241even more efficient (besides improved boundary conditions and physical
1242stuff as discussed in @ref step_25 "step-25"):
1243
1244<ul> <li> <b>Faster evaluation of sine terms:</b> As becomes obvious
1245 from the comparison of the plain wave equation and the sine-Gordon
1246 equation above, the evaluation of the sine terms dominates the total
1247 time for the finite element operator application. There are a few
1248 reasons for this: Firstly, the deal.II sine computation of a
1249 VectorizedArray field is not vectorized (as opposed to the rest of
1250 the operator application). This could be cured by handing the sine
1251 computation to a library with vectorized sine computations like
1252 Intel's math kernel library (MKL). By using the function
1253 <code>vdSin</code> in MKL, the program uses half the computing time
1254 in 2D and 40 percent less time in 3D. On the other hand, the sine
1255 computation is structurally much more complicated than the simple
1256 arithmetic operations like additions and multiplications in the rest
1257 of the local operation.
1258
1259 <li> <b>Higher order time stepping:</b> While the implementation allows for
1260 arbitrary order in the spatial part (by adjusting the degree of the finite
1261 element), the time stepping scheme is a standard second-order leap-frog
1262 scheme. Since solutions in wave propagation problems are usually very
1263 smooth, the error is likely dominated by the time stepping part. Of course,
1264 this could be cured by using smaller time steps (at a fixed spatial
1265 resolution), but it would be more efficient to use higher order time
1266 stepping as well. While it would be straight-forward to do so for a
1267 first-order system (use some Runge&ndash;Kutta scheme of higher order,
1268 probably combined with adaptive time step selection like the <a
1269 href="http://en.wikipedia.org/wiki/Dormand%E2%80%93Prince_method">Dormand&ndash;Prince
1270 method</a>), it is more challenging for the second-order formulation. At
1271 least in the finite difference community, people usually use the PDE to find
1272 spatial correction terms that improve the temporal error.
1273
1274</ul>
1275 *
1276 *
1277<a name="step_48-PlainProg"></a>
1278<h1> The plain program</h1>
1279@include "step-48.cc"
1280*/
void attach_dof_handler(const DoFHandler< dim, spacedim > &)
Definition fe_q.h:554
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()
Definition timer.h:117
void restart()
Definition timer.h:896
Point< 3 > center
Point< 2 > second
Definition grid_out.cc:4624
Point< 2 > first
Definition grid_out.cc:4623
unsigned int level
Definition grid_out.cc:4626
void loop(IteratorType begin, std_cxx20::type_identity_t< IteratorType > end, DOFINFO &dinfo, INFOBOX &info, const std::function< void(std_cxx20::type_identity_t< DOFINFO > &, typename INFOBOX::CellInfo &)> &cell_worker, const std::function< void(std_cxx20::type_identity_t< DOFINFO > &, typename INFOBOX::CellInfo &)> &boundary_worker, const std::function< void(std_cxx20::type_identity_t< DOFINFO > &, std_cxx20::type_identity_t< DOFINFO > &, typename INFOBOX::CellInfo &, typename INFOBOX::CellInfo &)> &face_worker, AssemblerType &assembler, const LoopControl &lctrl=LoopControl())
Definition loop.h:564
void make_hanging_node_constraints(const DoFHandler< dim, spacedim > &dof_handler, AffineConstraints< number > &constraints)
const Event initial
Definition event.cc:64
IndexSet extract_locally_relevant_dofs(const DoFHandler< dim, spacedim > &dof_handler)
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)
void shift(const Tensor< 1, spacedim > &shift_vector, Triangulation< dim, spacedim > &triangulation)
double diameter(const Triangulation< dim, spacedim > &tria)
@ matrix
Contents is actually a matrix.
@ diagonal
Matrix is diagonal.
@ general
No special properties.
double norm(const FEValuesBase< dim > &fe, const ArrayView< const std::vector< Tensor< 1, dim > > > &Du)
Definition divergence.h:471
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 apply(const Kokkos::TeamPolicy< MemorySpace::Default::kokkos_space::execution_space >::member_type &team_member, const Kokkos::View< Number *, MemorySpace::Default::kokkos_space > shape_data, const ViewTypeIn in, ViewTypeOut out)
std::vector< unsigned int > serial(const std::vector< unsigned int > &targets, const std::function< RequestType(const unsigned int)> &create_request, const std::function< AnswerType(const unsigned int, const RequestType &)> &answer_request, const std::function< void(const unsigned int, const AnswerType &)> &process_answer, const MPI_Comm comm)
unsigned int n_mpi_processes(const MPI_Comm mpi_communicator)
Definition mpi.cc:92
T max(const T &t, const MPI_Comm mpi_communicator)
unsigned int this_mpi_process(const MPI_Comm mpi_communicator)
Definition mpi.cc:107
std::string get_time()
std::string get_current_vectorization_level()
Definition utilities.cc:938
double compute_global_error(const Triangulation< dim, spacedim > &tria, const InVector &cellwise_error, const NormType &norm, const double exponent=2.)
void integrate_difference(const Mapping< dim, spacedim > &mapping, const DoFHandler< dim, spacedim > &dof, const ReadVector< Number > &fe_function, const Function< spacedim, Number > &exact_solution, OutVector &difference, const Quadrature< dim > &q, const NormType &norm, const Function< spacedim, double > *weight=nullptr, const double exponent=2.)
void run(const Iterator &begin, const std_cxx20::type_identity_t< Iterator > &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
Definition types.h:220
::SolutionTransfer< dim, VectorType, spacedim > SolutionTransfer
STL namespace.
::VectorizedArray< Number, width > sin(const ::VectorizedArray< Number, width > &)
inline ::VectorizedArray< Number, width > cosh(const ::VectorizedArray< Number, width > &x)
::VectorizedArray< Number, width > sqrt(const ::VectorizedArray< Number, width > &)
inline ::VectorizedArray< Number, width > atan(const ::VectorizedArray< Number, width > &x)
const ::parallel::distributed::Triangulation< dim, spacedim > * triangulation
void swap(SmartPointer< T, P > &t1, SmartPointer< T, Q > &t2)
TasksParallelScheme tasks_parallel_scheme