deal.II version GIT relicensing-1721-g8100761196 2024-08-31 12:30:00+00:00
\(\newcommand{\dealvcentcolon}{\mathrel{\mathop{:}}}\) \(\newcommand{\dealcoloneq}{\dealvcentcolon\mathrel{\mkern-1.2mu}=}\) \(\newcommand{\jump}[1]{\left[\!\left[ #1 \right]\!\right]}\) \(\newcommand{\average}[1]{\left\{\!\left\{ #1 \right\}\!\right\}}\)
Loading...
Searching...
No Matches
step-65.h
Go to the documentation of this file.
1 = 0) const override
588 *   {
589 *   if (p.norm_square() < 0.25)
590 *   return p.norm_square();
591 *   else
592 *   return 0.1 * p.norm_square() + (0.25 - 0.025);
593 *   }
594 *  
595 *   virtual Tensor<1, dim>
596 *   gradient(const Point<dim> &p,
597 *   const unsigned int /*component*/ = 0) const override
598 *   {
599 *   if (p.norm_square() < 0.25)
600 *   return 2. * p;
601 *   else
602 *   return 0.2 * p;
603 *   }
604 *   };
605 *  
606 *  
607 *   template <int dim>
608 *   double coefficient(const Point<dim> &p)
609 *   {
610 *   if (p.norm_square() < 0.25)
611 *   return 0.5;
612 *   else
613 *   return 5.0;
614 *   }
615 *  
616 *  
617 *  
618 * @endcode
619 *
620 *
621 * <a name="step_65-ThePoissonProblemclass"></a>
622 * <h3>The PoissonProblem class</h3>
623 *
624
625 *
626 * The implementation of the Poisson problem is very similar to what
627 * we used in the @ref step_5 "step-5" tutorial program. The two main differences
628 * are that we pass a mapping object to the various steps in the
629 * program in order to switch between two mapping representations as
630 * explained in the introduction, and the `timer` object (of type
631 * TimerOutput) that will be used for measuring the run times in the
632 * various cases. (The concept of mapping objects was first
633 * introduced in @ref step_10 "step-10" and @ref step_11 "step-11", in case you want to look up
634 * the use of these classes.)
635 *
636 * @code
637 *   template <int dim>
638 *   class PoissonProblem
639 *   {
640 *   public:
641 *   PoissonProblem();
642 *   void run();
643 *  
644 *   private:
645 *   void create_grid();
646 *   void setup_system(const Mapping<dim> &mapping);
647 *   void assemble_system(const Mapping<dim> &mapping);
648 *   void solve();
649 *   void postprocess(const Mapping<dim> &mapping);
650 *  
652 *   const FE_Q<dim> fe;
653 *   DoFHandler<dim> dof_handler;
654 *  
655 *   AffineConstraints<double> constraints;
656 *   SparsityPattern sparsity_pattern;
657 *   SparseMatrix<double> system_matrix;
658 *   Vector<double> solution;
659 *   Vector<double> system_rhs;
660 *  
661 *   TimerOutput timer;
662 *   };
663 *  
664 *  
665 *  
666 * @endcode
667 *
668 * In the constructor, we set up the timer object to record wall times but
669 * be quiet during the normal execution. We will query it for timing details
670 * in the `PoissonProblem::run()` function. Furthermore, we select a
671 * relatively high polynomial degree of three for the finite element in use.
672 *
673 * @code
674 *   template <int dim>
675 *   PoissonProblem<dim>::PoissonProblem()
676 *   : fe(3)
677 *   , dof_handler(triangulation)
678 *   , timer(std::cout, TimerOutput::never, TimerOutput::wall_times)
679 *   {}
680 *  
681 *  
682 *  
683 * @endcode
684 *
685 *
686 * <a name="step_65-Gridcreationandinitializationofthemanifolds"></a>
687 * <h3>Grid creation and initialization of the manifolds</h3>
688 *
689
690 *
691 * The next function presents the typical usage of
692 * TransfiniteInterpolationManifold. The first step is to create the desired
693 * grid, which can be done by composition of two grids from
694 * GridGenerator. The inner ball mesh is simple enough: We run
695 * GridGenerator::hyper_cube() centered at the origin with radius 0.5 (third
696 * function argument). The second mesh is more interesting and constructed
697 * as follows: We want to have a mesh that is spherical in the interior but
698 * flat on the outer surface. Furthermore, the mesh topology of the inner
699 * ball should be compatible with the outer grid in the sense that their
700 * vertices coincide so as to allow the two grid to be merged. The grid coming
701 * out of GridGenerator::hyper_shell fulfills the requirements on the inner
702 * side in case it is created with @f$2d@f$ coarse cells (6 coarse cells in 3d
703 * which we are going to use) &ndash; this is the same number of cells as
704 * there are boundary faces for the ball. For the outer surface, we use the
705 * fact that the 6 faces on the surface of the shell without a manifold
706 * attached would degenerate to the surface of a cube. What we are still
707 * missing is the radius of the outer shell boundary. Since we desire a cube
708 * of extent
709 * @f$[-1, 1]@f$ and the 6-cell shell puts its 8 outer vertices at the 8
710 * opposing diagonals, we must translate the points @f$(\pm 1, \pm 1, \pm 1)@f$
711 * into a radius: Clearly, the radius must be @f$\sqrt{d}@f$ in @f$d@f$ dimensions,
712 * i.e., @f$\sqrt{3}@f$ for the three-dimensional case we want to consider.
713 *
714
715 *
716 * Thus, we have a plan: After creating the inner triangulation for
717 * the ball and the one for the outer shell, we merge those two
718 * grids but remove all manifolds that the functions in
719 * GridGenerator may have set from the resulting triangulation, to
720 * ensure that we have full control over manifolds. In particular,
721 * we want additional points added on the boundary during refinement
722 * to follow a flat manifold description. To start the process of
723 * adding more appropriate manifold ids, we assign the manifold id 0
724 * to all mesh entities (cells, faces, lines), which will later be
725 * associated with the TransfiniteInterpolationManifold. Then, we
726 * must identify the faces and lines that are along the sphere of
727 * radius 0.5 and mark them with a different manifold id, so as to then
728 * assign a SphericalManifold to those. We will choose the manifold
729 * id of 1. Since we have thrown away all manifolds that pre-existed
730 * after calling GridGenerator::hyper_ball(), we manually go through
731 * the cells of the mesh and all their faces. We have found a face
732 * on the sphere if all four vertices have a radius of 0.5, or, as
733 * we write in the program, have @f$r^2-0.25 \approx 0@f$. Note that we call
734 * `cell->face(f)->set_all_manifold_ids(1)` to set the manifold id
735 * both on the faces and the surrounding lines. Furthermore, we want
736 * to distinguish the cells inside the ball and outside the ball by
737 * a material id for visualization, corresponding to the picture in the
738 * introduction.
739 *
740 * @code
741 *   template <int dim>
742 *   void PoissonProblem<dim>::create_grid()
743 *   {
744 *   Triangulation<dim> tria_inner;
745 *   GridGenerator::hyper_ball(tria_inner, Point<dim>(), 0.5);
746 *  
747 *   Triangulation<dim> tria_outer;
749 *   tria_outer, Point<dim>(), 0.5, std::sqrt(dim), 2 * dim);
750 *  
751 *   GridGenerator::merge_triangulations(tria_inner, tria_outer, triangulation);
752 *  
753 *   triangulation.reset_all_manifolds();
754 *   triangulation.set_all_manifold_ids(0);
755 *  
756 *   for (const auto &cell : triangulation.cell_iterators())
757 *   {
758 *   for (const auto &face : cell->face_iterators())
759 *   {
760 *   bool face_at_sphere_boundary = true;
761 *   for (const auto v : face->vertex_indices())
762 *   {
763 *   if (std::abs(face->vertex(v).norm_square() - 0.25) > 1e-12)
764 *   face_at_sphere_boundary = false;
765 *   }
766 *   if (face_at_sphere_boundary)
767 *   face->set_all_manifold_ids(1);
768 *   }
769 *   if (cell->center().norm_square() < 0.25)
770 *   cell->set_material_id(1);
771 *   else
772 *   cell->set_material_id(0);
773 *   }
774 *  
775 * @endcode
776 *
777 * With all cells, faces and lines marked appropriately, we can
778 * attach the Manifold objects to those numbers. The entities with
779 * manifold id 1 will get a spherical manifold, whereas the other
780 * entities, which have the manifold id 0, will be assigned the
781 * TransfiniteInterpolationManifold. As mentioned in the
782 * introduction, we must explicitly initialize the manifold with
783 * the current mesh using a call to
785 * up the coarse mesh cells and the manifolds attached to the
786 * boundaries of those cells. We also note that the manifold
787 * objects we create locally in this function are allowed to go
788 * out of scope (as they do at the end of the function scope),
789 * because the Triangulation object internally copies them.
790 *
791
792 *
793 * With all manifolds attached, we will finally go about and refine the
794 * mesh a few times to create a sufficiently large test case.
795 *
796 * @code
797 *   triangulation.set_manifold(1, SphericalManifold<dim>());
798 *  
799 *   TransfiniteInterpolationManifold<dim> transfinite_manifold;
800 *   transfinite_manifold.initialize(triangulation);
801 *   triangulation.set_manifold(0, transfinite_manifold);
802 *  
803 *   triangulation.refine_global(9 - 2 * dim);
804 *   }
805 *  
806 *  
807 *  
808 * @endcode
809 *
810 *
811 * <a name="step_65-Setupofdatastructures"></a>
812 * <h3>Setup of data structures</h3>
813 *
814
815 *
816 * The following function is well-known from other tutorials in that
817 * it enumerates the degrees of freedom, creates a constraint object
818 * and sets up a sparse matrix for the linear system. The only thing
819 * worth mentioning is the fact that the function receives a
820 * reference to a mapping object that we then pass to the
821 * VectorTools::interpolate_boundary_values() function to ensure
822 * that our boundary values are evaluated on the high-order mesh
823 * used for assembly. In the present example, it does not really
824 * matter because the outer surfaces are flat, but for curved outer
825 * cells this leads to more accurate approximation of the boundary
826 * values.
827 *
828 * @code
829 *   template <int dim>
830 *   void PoissonProblem<dim>::setup_system(const Mapping<dim> &mapping)
831 *   {
832 *   dof_handler.distribute_dofs(fe);
833 *   std::cout << " Number of active cells: "
834 *   << triangulation.n_global_active_cells() << std::endl;
835 *   std::cout << " Number of degrees of freedom: " << dof_handler.n_dofs()
836 *   << std::endl;
837 *  
838 *   {
839 *   TimerOutput::Scope scope(timer, "Compute constraints");
840 *  
841 *   constraints.clear();
842 *  
843 *   DoFTools::make_hanging_node_constraints(dof_handler, constraints);
845 *   mapping, dof_handler, 0, ExactSolution<dim>(), constraints);
846 *  
847 *   constraints.close();
848 *   }
849 *  
850 *   DynamicSparsityPattern dsp(dof_handler.n_dofs());
851 *   DoFTools::make_sparsity_pattern(dof_handler, dsp, constraints, false);
852 *  
853 *   sparsity_pattern.copy_from(dsp);
854 *   system_matrix.reinit(sparsity_pattern);
855 *  
856 *   solution.reinit(dof_handler.n_dofs());
857 *   system_rhs.reinit(dof_handler.n_dofs());
858 *   }
859 *  
860 *  
861 * @endcode
862 *
863 *
864 * <a name="step_65-Assemblyofthesystemmatrixandrighthandside"></a>
865 * <h3>Assembly of the system matrix and right hand side</h3>
866 *
867
868 *
869 * The function that assembles the linear system is also well known
870 * from the previous tutorial programs. One thing to note is that we
871 * set the number of quadrature points to the polynomial degree plus
872 * two, not the degree plus one as in most other tutorials. This is
873 * because we expect some extra accuracy as the mapping also
874 * involves a degree one more than the polynomials for the solution.
875 *
876
877 *
878 * The only somewhat unusual code in the assembly is the way we compute the
879 * cell matrix. Rather than using three nested loop over the quadrature
880 * point index, the row, and the column of the matrix, we first collect the
881 * derivatives of the shape function, multiplied by the square root of the
882 * product of the coefficient and the integration factor `JxW` in a separate
883 * matrix `partial_matrix`. To compute the cell matrix, we then execute
884 * `cell_matrix = partial_matrix * transpose(partial_matrix)` in the line
885 * `partial_matrix.mTmult(cell_matrix, partial_matrix);`. To understand why
886 * this works, we realize that the matrix-matrix multiplication performs a
887 * summation over the columns of `partial_matrix`. If we denote the
888 * coefficient by @f$a(\mathbf{x}_q)@f$, the entries in the temporary matrix are
889 * @f$\sqrt{\text{det}(J) w_q a(x)} \frac{\partial \varphi_i(\boldsymbol
890 * \xi_q)}{\partial x_k}@f$. If we take the product of the <i>i</i>th row with
891 * the <i>j</i>th column of that matrix, we compute a nested sum involving
892 * @f$\sum_q \sum_{k=1}^d \sqrt{\text{det}(J) w_q a(x)} \frac{\partial
893 * \varphi_i(\boldsymbol \xi_q)}{\partial x_k} \sqrt{\text{det}(J) w_q a(x)}
894 * \frac{\partial \varphi_j(\boldsymbol \xi_q)}{\partial x_k} = \sum_q
895 * \sum_{k=1}^d\text{det}(J) w_q a(x)\frac{\partial \varphi_i(\boldsymbol
896 * \xi_q)}{\partial x_k} \frac{\partial \varphi_j(\boldsymbol
897 * \xi_q)}{\partial x_k}@f$, which is exactly the terms needed for the
898 * bilinear form of the Laplace equation.
899 *
900
901 *
902 * The reason for choosing this somewhat unusual scheme is due to the heavy
903 * work involved in computing the cell matrix for a relatively high
904 * polynomial degree in 3d. As we want to highlight the cost of the mapping
905 * in this tutorial program, we better do the assembly in an optimized way
906 * in order to not chase bottlenecks that have been solved by the community
907 * already. Matrix-matrix multiplication is one of the best optimized
908 * kernels in the HPC context, and the FullMatrix::mTmult() function will
909 * call into those optimized BLAS functions. If the user has provided a good
910 * BLAS library when configuring deal.II (like OpenBLAS or Intel's MKL), the
911 * computation of the cell matrix will execute close to the processor's peak
912 * arithmetic performance. As a side note, we mention that despite an
913 * optimized matrix-matrix multiplication, the current strategy is
914 * sub-optimal in terms of complexity as the work to be done is proportional
915 * to @f$(p+1)^9@f$ operations for degree @f$p@f$ (this also applies to the usual
916 * evaluation with FEValues). One could compute the cell matrix with
917 * @f$\mathcal O((p+1)^7)@f$ operations by utilizing the tensor product
918 * structure of the shape functions, as is done by the matrix-free framework
919 * in deal.II. We refer to @ref step_37 "step-37" and the documentation of the
920 * tensor-product-aware evaluators FEEvaluation for details on how an even
921 * more efficient cell matrix computation could be realized.
922 *
923 * @code
924 *   template <int dim>
925 *   void PoissonProblem<dim>::assemble_system(const Mapping<dim> &mapping)
926 *   {
927 *   TimerOutput::Scope scope(timer, "Assemble linear system");
928 *  
929 *   const QGauss<dim> quadrature_formula(fe.degree + 2);
930 *   FEValues<dim> fe_values(mapping,
931 *   fe,
932 *   quadrature_formula,
935 *  
936 *   const unsigned int dofs_per_cell = fe.n_dofs_per_cell();
937 *   const unsigned int n_q_points = quadrature_formula.size();
938 *  
939 *   FullMatrix<double> cell_matrix(dofs_per_cell, dofs_per_cell);
940 *   Vector<double> cell_rhs(dofs_per_cell);
941 *   FullMatrix<double> partial_matrix(dofs_per_cell, dim * n_q_points);
942 *   std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
943 *  
944 *   for (const auto &cell : dof_handler.active_cell_iterators())
945 *   {
946 *   cell_rhs = 0.;
947 *   fe_values.reinit(cell);
948 *  
949 *   for (unsigned int q_index = 0; q_index < n_q_points; ++q_index)
950 *   {
951 *   const double current_coefficient =
952 *   coefficient(fe_values.quadrature_point(q_index));
953 *   for (unsigned int i = 0; i < dofs_per_cell; ++i)
954 *   {
955 *   for (unsigned int d = 0; d < dim; ++d)
956 *   partial_matrix(i, q_index * dim + d) =
957 *   std::sqrt(fe_values.JxW(q_index) * current_coefficient) *
958 *   fe_values.shape_grad(i, q_index)[d];
959 *   cell_rhs(i) +=
960 *   (fe_values.shape_value(i, q_index) * // phi_i(x_q)
961 *   (-dim) * // f(x_q)
962 *   fe_values.JxW(q_index)); // dx
963 *   }
964 *   }
965 *  
966 *   partial_matrix.mTmult(cell_matrix, partial_matrix);
967 *  
968 *   cell->get_dof_indices(local_dof_indices);
969 *   constraints.distribute_local_to_global(
970 *   cell_matrix, cell_rhs, local_dof_indices, system_matrix, system_rhs);
971 *   }
972 *   }
973 *  
974 *  
975 *  
976 * @endcode
977 *
978 *
979 * <a name="step_65-Solutionofthelinearsystem"></a>
980 * <h3>Solution of the linear system</h3>
981 *
982
983 *
984 * For solving the linear system, we pick a simple Jacobi-preconditioned
985 * conjugate gradient solver, similar to the settings in the early tutorials.
986 *
987 * @code
988 *   template <int dim>
989 *   void PoissonProblem<dim>::solve()
990 *   {
991 *   TimerOutput::Scope scope(timer, "Solve linear system");
992 *  
993 *   SolverControl solver_control(1000, 1e-12);
994 *   SolverCG<Vector<double>> solver(solver_control);
995 *  
996 *   PreconditionJacobi<SparseMatrix<double>> preconditioner;
997 *   preconditioner.initialize(system_matrix);
998 *  
999 *   solver.solve(system_matrix, solution, system_rhs, preconditioner);
1000 *   constraints.distribute(solution);
1001 *  
1002 *   std::cout << " Number of solver iterations: "
1003 *   << solver_control.last_step() << std::endl;
1004 *   }
1005 *  
1006 *  
1007 *  
1008 * @endcode
1009 *
1010 *
1011 * <a name="step_65-Outputofthesolutionandcomputationoferrors"></a>
1012 * <h3>Output of the solution and computation of errors</h3>
1013 *
1014
1015 *
1016 * In the next function we do various post-processing steps with the
1017 * solution, all of which involve the mapping in one way or the other.
1018 *
1019
1020 *
1021 * The first operation we do is to write the solution as well as the
1022 * material ids to a VTU file. This is similar to what was done in many
1023 * other tutorial programs. The new ingredient presented in this tutorial
1024 * program is that we want to ensure that the data written to the file
1025 * used for visualization is actually a faithful representation of what
1026 * is used internally by deal.II. That is because most of the visualization
1027 * data formats only represent cells by their vertex coordinates, but
1028 * have no way of representing the curved boundaries that are used
1029 * in deal.II when using higher order mappings -- in other words, what
1030 * you see in the visualization tool is not actually what you are computing
1031 * on. (The same, incidentally, is true when using higher order shape
1032 * functions: Most visualization tools only render bilinear/trilinear
1033 * representations. This is discussed in detail in DataOut::build_patches().)
1034 *
1035
1036 *
1037 * So we need to ensure that a high-order representation is written
1038 * to the file. We need to consider two particular topics. Firstly, we tell
1039 * the DataOut object via the DataOutBase::VtkFlags that we intend to
1040 * interpret the subdivisions of the elements as a high-order Lagrange
1041 * polynomial rather than a collection of bilinear patches.
1042 * Recent visualization programs, like ParaView version 5.5
1043 * or newer, can then render a high-order solution (see a <a
1044 * href="https://github.com/dealii/dealii/wiki/Notes-on-visualizing-high-order-output">wiki
1045 * page</a> for more details).
1046 * Secondly, we need to make sure that the mapping is passed to the
1047 * DataOut::build_patches() method. Finally, the DataOut class only prints
1048 * curved faces for <i>boundary</i> cells by default, so we need to ensure
1049 * that also inner cells are printed in a curved representation via the
1050 * mapping.
1051 *
1052
1053 *
1054 * @note As of 2023, Visit 3.3.3 can still not deal with higher-order cells.
1055 * Rather, it simply reports that there is no data to show. To view the
1056 * results of this program with Visit, you will want to comment out the
1057 * line that sets `flags.write_higher_order_cells = true;`. On the other
1058 * hand, Paraview is able to understand VTU files with higher order cells
1059 * just fine.
1060 *
1061 * @code
1062 *   template <int dim>
1063 *   void PoissonProblem<dim>::postprocess(const Mapping<dim> &mapping)
1064 *   {
1065 *   {
1066 *   TimerOutput::Scope scope(timer, "Write output");
1067 *  
1068 *   DataOut<dim> data_out;
1069 *  
1070 *   DataOutBase::VtkFlags flags;
1071 *   flags.write_higher_order_cells = true;
1072 *   data_out.set_flags(flags);
1073 *  
1074 *   data_out.attach_dof_handler(dof_handler);
1075 *   data_out.add_data_vector(solution, "solution");
1076 *  
1077 *   Vector<double> material_ids(triangulation.n_active_cells());
1078 *   for (const auto &cell : triangulation.active_cell_iterators())
1079 *   material_ids[cell->active_cell_index()] = cell->material_id();
1080 *   data_out.add_data_vector(material_ids, "material_ids");
1081 *  
1082 *   data_out.build_patches(mapping,
1083 *   fe.degree,
1085 *  
1086 *   std::ofstream file(
1087 *   ("solution-" +
1088 *   std::to_string(triangulation.n_global_levels() - 10 + 2 * dim) +
1089 *   ".vtu")
1090 *   .c_str());
1091 *  
1092 *   data_out.write_vtu(file);
1093 *   }
1094 *  
1095 * @endcode
1096 *
1097 * The next operation in the postprocessing function is to compute the @f$L_2@f$
1098 * and @f$H^1@f$ errors against the analytical solution. As the analytical
1099 * solution is a quadratic polynomial, we expect a very accurate result at
1100 * this point. If we were solving on a simple mesh with planar faces and a
1101 * coefficient whose jumps are aligned with the faces between cells, then
1102 * we would expect the numerical result to coincide with the
1103 * analytical solution up to roundoff accuracy. However, since we are using
1104 * deformed cells following a sphere, which are only tracked by
1105 * polynomials of degree 4 (one more than the degree for the finite
1106 * elements), we will see that there is an error around @f$10^{-7}@f$. We could
1107 * get more accuracy by increasing the polynomial degree or refining the
1108 * mesh.
1109 *
1110 * @code
1111 *   {
1112 *   TimerOutput::Scope scope(timer, "Compute error norms");
1113 *  
1114 *   Vector<double> norm_per_cell_p(triangulation.n_active_cells());
1115 *  
1117 *   dof_handler,
1118 *   solution,
1119 *   ExactSolution<dim>(),
1120 *   norm_per_cell_p,
1121 *   QGauss<dim>(fe.degree + 2),
1123 *   std::cout << " L2 error vs exact solution: "
1124 *   << norm_per_cell_p.l2_norm() << std::endl;
1125 *  
1127 *   dof_handler,
1128 *   solution,
1129 *   ExactSolution<dim>(),
1130 *   norm_per_cell_p,
1131 *   QGauss<dim>(fe.degree + 2),
1133 *   std::cout << " H1 error vs exact solution: "
1134 *   << norm_per_cell_p.l2_norm() << std::endl;
1135 *   }
1136 *  
1137 * @endcode
1138 *
1139 * The final post-processing operation we do here is to compute an error
1140 * estimate with the KellyErrorEstimator. We use the exact same settings
1141 * as in the @ref step_6 "step-6" tutorial program, except for the fact that we also
1142 * hand in the mapping to ensure that errors are evaluated along the
1143 * curved element, consistent with the remainder of the program. However,
1144 * we do not really use the result here to drive a mesh adaptation step
1145 * (that would refine the mesh around the material interface along the
1146 * sphere), as the focus here is on the cost of this operation.
1147 *
1148 * @code
1149 *   {
1150 *   TimerOutput::Scope scope(timer, "Compute error estimator");
1151 *  
1152 *   Vector<float> estimated_error_per_cell(triangulation.n_active_cells());
1154 *   mapping,
1155 *   dof_handler,
1156 *   QGauss<dim - 1>(fe.degree + 1),
1157 *   std::map<types::boundary_id, const Function<dim> *>(),
1158 *   solution,
1159 *   estimated_error_per_cell);
1160 *   std::cout << " Max cell-wise error estimate: "
1161 *   << estimated_error_per_cell.linfty_norm() << std::endl;
1162 *   }
1163 *   }
1164 *  
1165 *  
1166 *  
1167 * @endcode
1168 *
1169 *
1170 * <a name="step_65-ThePoissonProblemrunfunction"></a>
1171 * <h3>The PoissonProblem::run() function</h3>
1172 *
1173
1174 *
1175 * Finally, we define the `run()` function that controls how we want to
1176 * execute this program (which is called by the main() function in the usual
1177 * way). We start by calling the `create_grid()` function that sets up our
1178 * geometry with the appropriate manifolds. We then run two instances of a
1179 * solver chain, starting from the setup of the equations, the assembly of
1180 * the linear system, its solution with a simple iterative solver, and the
1181 * postprocessing discussed above. The two instances differ in the way they
1182 * use the mapping. The first uses a conventional MappingQ mapping
1183 * object which we initialize to a degree one more than we use for the
1184 * finite element &ndash; after all, we expect the geometry representation
1185 * to be the bottleneck as the analytic solution is only a quadratic
1186 * polynomial. (In reality, things are interlinked to quite some extent
1187 * because the evaluation of the polynomials in real coordinates involves
1188 * the mapping of a higher-degree polynomials, which represent some smooth
1189 * rational functions. As a consequence, higher-degree polynomials still pay
1190 * off, so it does not make sense to increase the degree of the mapping
1191 * further.) Once the first pass is completed, we let the timer print a
1192 * summary of the compute times of the individual stages.
1193 *
1194 * @code
1195 *   template <int dim>
1196 *   void PoissonProblem<dim>::run()
1197 *   {
1198 *   create_grid();
1199 *  
1200 *   {
1201 *   std::cout << std::endl
1202 *   << "====== Running with the basic MappingQ class ====== "
1203 *   << std::endl
1204 *   << std::endl;
1205 *  
1206 *   const MappingQ<dim> mapping(fe.degree + 1);
1207 *   setup_system(mapping);
1208 *   assemble_system(mapping);
1209 *   solve();
1210 *   postprocess(mapping);
1211 *  
1212 *   timer.print_summary();
1213 *   timer.reset();
1214 *   }
1215 *  
1216 * @endcode
1217 *
1218 * For the second instance, we instead set up the MappingQCache class. Its
1219 * use is very simple: After constructing it (with the degree, given that
1220 * we want it to show the correct degree functionality in other contexts),
1221 * we fill the cache via the MappingQCache::initialize() function. At this
1222 * stage, we specify which mapping we want to use (obviously, the same
1223 * MappingQ as previously in order to repeat the same computations)
1224 * for the cache, and then run through the same functions again, now
1225 * handing in the modified mapping. In the end, we again print the
1226 * accumulated wall times since the reset to see how the times compare to
1227 * the original setting.
1228 *
1229 * @code
1230 *   {
1231 *   std::cout
1232 *   << "====== Running with the optimized MappingQCache class ====== "
1233 *   << std::endl
1234 *   << std::endl;
1235 *  
1236 *   MappingQCache<dim> mapping(fe.degree + 1);
1237 *   {
1238 *   TimerOutput::Scope scope(timer, "Initialize mapping cache");
1239 *   mapping.initialize(MappingQ<dim>(fe.degree + 1), triangulation);
1240 *   }
1241 *   std::cout << " Memory consumption cache: "
1242 *   << 1e-6 * mapping.memory_consumption() << " MB" << std::endl;
1243 *  
1244 *   setup_system(mapping);
1245 *   assemble_system(mapping);
1246 *   solve();
1247 *   postprocess(mapping);
1248 *  
1249 *   timer.print_summary();
1250 *   }
1251 *   }
1252 *   } // namespace Step65
1253 *  
1254 *  
1255 *  
1256 *   int main()
1257 *   {
1258 *   Step65::PoissonProblem<3> test_program;
1259 *   test_program.run();
1260 *   return 0;
1261 *   }
1262 * @endcode
1263
1264<a name="step_65-Results"></a><h1>Results</h1>
1265
1266
1267<a name="step_65-Programoutput"></a><h3>Program output</h3>
1268
1269
1270If we run the three-dimensional version of this program with polynomials of
1271degree three, we get the following program output:
1272
1273@code
1274> make run
1275Scanning dependencies of target step-65
1276[ 33%] Building CXX object CMakeFiles/step-65.dir/step-65.cc.o
1277[ 66%] Linking CXX executable step-65
1278[ 66%] Built target step-65
1279[100%] Run step-65 with Release configuration
1280
1281====== Running with the basic MappingQ class ======
1282
1283 Number of active cells: 6656
1284 Number of degrees of freedom: 181609
1285 Number of solver iterations: 285
1286 L2 error vs exact solution: 8.99339e-08
1287 H1 error vs exact solution: 6.45341e-06
1288 Max cell-wise error estimate: 0.00743406
1289
1290
1291+---------------------------------------------+------------+------------+
1292| Total wallclock time elapsed since start | 49.4s | |
1293| | | |
1294| Section | no. calls | wall time | % of total |
1295+---------------------------------+-----------+------------+------------+
1296| Assemble linear system | 1 | 5.8s | 12% |
1297| Compute constraints | 1 | 0.109s | 0.22% |
1298| Compute error estimator | 1 | 16.5s | 33% |
1299| Compute error norms | 1 | 9.11s | 18% |
1300| Solve linear system | 1 | 9.92s | 20% |
1301| Write output | 1 | 4.85s | 9.8% |
1302+---------------------------------+-----------+------------+------------+
1303
1304====== Running with the optimized MappingQCache class ======
1305
1306 Memory consumption cache: 22.9981 MB
1307 Number of active cells: 6656
1308 Number of degrees of freedom: 181609
1309 Number of solver iterations: 285
1310 L2 error vs exact solution: 8.99339e-08
1311 H1 error vs exact solution: 6.45341e-06
1312 Max cell-wise error estimate: 0.00743406
1313
1314
1315+---------------------------------------------+------------+------------+
1316| Total wallclock time elapsed since start | 18.4s | |
1317| | | |
1318| Section | no. calls | wall time | % of total |
1319+---------------------------------+-----------+------------+------------+
1320| Assemble linear system | 1 | 1.44s | 7.8% |
1321| Compute constraints | 1 | 0.00336s | 0% |
1322| Compute error estimator | 1 | 0.476s | 2.6% |
1323| Compute error norms | 1 | 0.505s | 2.7% |
1324| Initialize mapping cache | 1 | 4.96s | 27% |
1325| Solve linear system | 1 | 9.95s | 54% |
1326| Write output | 1 | 0.875s | 4.8% |
1327+---------------------------------+-----------+------------+------------+
1328
1329[100%] Built target run
1330@endcode
1331
1332Before discussing the timings, we look at the memory consumption for the
1333MappingQCache object: Our program prints that it utilizes 23 MB of
1334memory. If we relate this number to the memory consumption of a single
1335(solution or right hand side) vector,
1336which is 1.5 MB (namely, 181,609 elements times 8 bytes per entry in
1337double precision), or to the memory consumed by the
1338system matrix and the sparsity pattern (which is 274 MB), we realize that it is
1339not an overly heavy data structure, given its benefits.
1340
1341With respect to the timers, we see a clear improvement in the overall run time
1342of the program by a factor of 2.7. If we disregard the iterative solver, which
1343is the same in both cases (and not optimal, given the simple preconditioner we
1344use, and the fact that sparse matrix-vector products waste operations for
1345cubic polynomials), the advantage is a factor of almost 5. This is pretty
1346impressive for a linear stationary problem, and cost savings would indeed be
1347much more prominent for time-dependent and nonlinear problems where assembly
1348is called several times. If we look into the individual components, we get a
1349clearer picture of what is going on and why the cache is so efficient: In the
1350MappingQ case, essentially every operation that involves a mapping take
1351at least 5 seconds to run. The norm computation runs two
1352VectorTools::integrate_difference() functions, which each take almost 5
1353seconds. (The computation of constraints is cheaper because it only evaluates
1354the mapping in cells at the boundary for the interpolation of boundary
1355conditions.) If we compare these 5 seconds to the time it takes to fill the
1356MappingQCache, which is 5.2 seconds (for all cells, not just the active ones),
1357it becomes obvious that the computation of the mapping support points
1358dominates over everything else in the MappingQ case. Perhaps the most
1359striking result is the time for the error estimator, labeled "Compute error
1360estimator", where the MappingQ implementation takes 17.3 seconds and
1361the MappingQCache variant less than 0.5 seconds. The reason why the former is
1362so expensive (three times more expensive than the assembly, for instance) is
1363that the error estimation involves evaluation of quantities over faces, where
1364each face in the mesh requests additional points of the mapping that in turn
1365go through the very expensive TransfiniteInterpolationManifold class. As there
1366are six faces per cell, this happens much more often than in assembly. Again,
1367MappingQCache nicely eliminates the repeated evaluation, aggregating all the
1368expensive steps involving the manifold in a single initialization call that
1369gets repeatedly used.
1370 *
1371 *
1372<a name="step_65-PlainProg"></a>
1373<h1> The plain program</h1>
1374@include "step-65.cc"
1375*/
SmartPointer< const Triangulation< dim, spacedim > > triangulation
virtual void build_patches(const unsigned int n_subdivisions=0)
Definition data_out.cc:1062
Definition fe_q.h:554
void mTmult(FullMatrix< number2 > &C, const FullMatrix< number2 > &B, const bool adding=false) const
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, Number > * > &neumann_bc, const ReadVector< Number > &solution, Vector< float > &error, const ComponentMask &component_mask={}, 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 initialize(const Mapping< dim, spacedim > &mapping, const Triangulation< dim, spacedim > &triangulation)
Abstract base class for mapping classes.
Definition mapping.h:318
Definition point.h:111
void initialize(const MatrixType &A, const AdditionalData &parameters=AdditionalData())
constexpr numbers::NumberTraits< Number >::real_type norm_square() const
@ wall_times
Definition timer.h:651
const Triangulation< dim, spacedim > * triangulation
void initialize(const Triangulation< dim, spacedim > &triangulation)
virtual types::global_cell_index n_global_active_cells() const
Point< 3 > vertices[4]
DerivativeForm< 1, spacedim, dim, Number > transpose(const DerivativeForm< 1, dim, spacedim, Number > &DF)
Point< 2 > second
Definition grid_out.cc:4624
Point< 2 > first
Definition grid_out.cc:4623
unsigned int vertex_indices[2]
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)
void make_sparsity_pattern(const DoFHandler< dim, spacedim > &dof_handler, SparsityPatternBase &sparsity_pattern, const AffineConstraints< number > &constraints={}, const bool keep_constrained_dofs=true, const types::subdomain_id subdomain_id=numbers::invalid_subdomain_id)
@ update_values
Shape function values.
@ update_JxW_values
Transformed quadrature weights.
@ update_gradients
Shape function gradients.
@ update_quadrature_points
Transformed quadrature points.
void hyper_shell(Triangulation< dim, spacedim > &tria, const Point< spacedim > &center, const double inner_radius, const double outer_radius, const unsigned int n_cells=0, bool colorize=false)
void hyper_ball(Triangulation< dim > &tria, const Point< dim > &center=Point< dim >(), const double radius=1., const bool attach_spherical_manifold_on_boundary_cells=false)
void hyper_cube(Triangulation< dim, spacedim > &tria, const double left=0., const double right=1., const bool colorize=false)
void merge_triangulations(const Triangulation< dim, spacedim > &triangulation_1, const Triangulation< dim, spacedim > &triangulation_2, Triangulation< dim, spacedim > &result, const double duplicated_vertex_tolerance=1.0e-12, const bool copy_manifold_ids=false, const bool copy_boundary_ids=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.
void cell_matrix(FullMatrix< double > &M, const FEValuesBase< dim > &fe, const FEValuesBase< dim > &fetest, const ArrayView< const std::vector< double > > &velocity, const double factor=1.)
Definition advection.h:74
double norm(const FEValuesBase< dim > &fe, const ArrayView< const std::vector< Tensor< 1, dim > > > &Du)
Definition divergence.h:471
void L2(Vector< number > &result, const FEValuesBase< dim > &fe, const std::vector< double > &input, const double factor=1.)
Definition l2.h:159
Point< spacedim > point(const gp_Pnt &p, const double tolerance=1e-10)
Definition utilities.cc:191
SymmetricTensor< 2, dim, Number > e(const Tensor< 2, dim, Number > &F)
SymmetricTensor< 2, dim, Number > d(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
T sum(const T &t, const MPI_Comm mpi_communicator)
void interpolate_boundary_values(const Mapping< dim, spacedim > &mapping, const DoFHandler< dim, spacedim > &dof, const std::map< types::boundary_id, const Function< spacedim, number > * > &function_map, std::map< types::global_dof_index, number > &boundary_values, const ComponentMask &component_mask={})
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)
DEAL_II_HOST constexpr TableIndices< 2 > merge(const TableIndices< 2 > &previous_indices, const unsigned int new_index, const unsigned int position)
int(&) functions(const void *v1, const void *v2)
STL namespace.
::VectorizedArray< Number, width > sqrt(const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > abs(const ::VectorizedArray< Number, width > &)
unsigned int material_id
Definition types.h:167
const ::parallel::distributed::Triangulation< dim, spacedim > * triangulation