Reference documentation for deal.II version 9.5.0
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step-23.h
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1,
659 *   const unsigned int component = 0) const override
660 *   {
661 *   (void)component;
662 *   Assert(component == 0, ExcIndexRange(component, 0, 1));
663 *   return 0;
664 *   }
665 *   };
666 *  
667 *  
668 *  
669 *   template <int dim>
670 *   class InitialValuesV : public Function<dim>
671 *   {
672 *   public:
673 *   virtual double value(const Point<dim> & /*p*/,
674 *   const unsigned int component = 0) const override
675 *   {
676 *   (void)component;
677 *   Assert(component == 0, ExcIndexRange(component, 0, 1));
678 *   return 0;
679 *   }
680 *   };
681 *  
682 *  
683 *  
684 * @endcode
685 *
686 * Secondly, we have the right hand side forcing term. Boring as we are, we
687 * choose zero here as well:
688 *
689 * @code
690 *   template <int dim>
691 *   class RightHandSide : public Function<dim>
692 *   {
693 *   public:
694 *   virtual double value(const Point<dim> & /*p*/,
695 *   const unsigned int component = 0) const override
696 *   {
697 *   (void)component;
698 *   Assert(component == 0, ExcIndexRange(component, 0, 1));
699 *   return 0;
700 *   }
701 *   };
702 *  
703 *  
704 *  
705 * @endcode
706 *
707 * Finally, we have boundary values for @f$u@f$ and @f$v@f$. They are as described
708 * in the introduction, one being the time derivative of the other:
709 *
710 * @code
711 *   template <int dim>
712 *   class BoundaryValuesU : public Function<dim>
713 *   {
714 *   public:
715 *   virtual double value(const Point<dim> & p,
716 *   const unsigned int component = 0) const override
717 *   {
718 *   (void)component;
719 *   Assert(component == 0, ExcIndexRange(component, 0, 1));
720 *  
721 *   if ((this->get_time() <= 0.5) && (p[0] < 0) && (p[1] < 1. / 3) &&
722 *   (p[1] > -1. / 3))
723 *   return std::sin(this->get_time() * 4 * numbers::PI);
724 *   else
725 *   return 0;
726 *   }
727 *   };
728 *  
729 *  
730 *  
731 *   template <int dim>
732 *   class BoundaryValuesV : public Function<dim>
733 *   {
734 *   public:
735 *   virtual double value(const Point<dim> & p,
736 *   const unsigned int component = 0) const override
737 *   {
738 *   (void)component;
739 *   Assert(component == 0, ExcIndexRange(component, 0, 1));
740 *  
741 *   if ((this->get_time() <= 0.5) && (p[0] < 0) && (p[1] < 1. / 3) &&
742 *   (p[1] > -1. / 3))
743 *   return (std::cos(this->get_time() * 4 * numbers::PI) * 4 * numbers::PI);
744 *   else
745 *   return 0;
746 *   }
747 *   };
748 *  
749 *  
750 *  
751 * @endcode
752 *
753 *
754 * <a name="ImplementationofthecodeWaveEquationcodeclass"></a>
755 * <h3>Implementation of the <code>WaveEquation</code> class</h3>
756 *
757
758 *
759 * The implementation of the actual logic is actually fairly short, since we
760 * relegate things like assembling the matrices and right hand side vectors
761 * to the library. The rest boils down to not much more than 130 lines of
762 * actual code, a significant fraction of which is boilerplate code that can
763 * be taken from previous example programs (e.g. the functions that solve
764 * linear systems, or that generate output).
765 *
766
767 *
768 * Let's start with the constructor (for an explanation of the choice of
769 * time step, see the section on Courant, Friedrichs, and Lewy in the
770 * introduction):
771 *
772 * @code
773 *   template <int dim>
774 *   WaveEquation<dim>::WaveEquation()
775 *   : fe(1)
776 *   , dof_handler(triangulation)
777 *   , time_step(1. / 64)
778 *   , time(time_step)
779 *   , timestep_number(1)
780 *   , theta(0.5)
781 *   {}
782 *  
783 *  
784 * @endcode
785 *
786 *
787 * <a name="WaveEquationsetup_system"></a>
788 * <h4>WaveEquation::setup_system</h4>
789 *
790
791 *
792 * The next function is the one that sets up the mesh, DoFHandler, and
793 * matrices and vectors at the beginning of the program, i.e. before the
794 * first time step. The first few lines are pretty much standard if you've
795 * read through the tutorial programs at least up to @ref step_6 "step-6":
796 *
797 * @code
798 *   template <int dim>
799 *   void WaveEquation<dim>::setup_system()
800 *   {
802 *   triangulation.refine_global(7);
803 *  
804 *   std::cout << "Number of active cells: " << triangulation.n_active_cells()
805 *   << std::endl;
806 *  
807 *   dof_handler.distribute_dofs(fe);
808 *  
809 *   std::cout << "Number of degrees of freedom: " << dof_handler.n_dofs()
810 *   << std::endl
811 *   << std::endl;
812 *  
813 *   DynamicSparsityPattern dsp(dof_handler.n_dofs(), dof_handler.n_dofs());
814 *   DoFTools::make_sparsity_pattern(dof_handler, dsp);
815 *   sparsity_pattern.copy_from(dsp);
816 *  
817 * @endcode
818 *
819 * Then comes a block where we have to initialize the 3 matrices we need
820 * in the course of the program: the mass matrix, the Laplace matrix, and
821 * the matrix @f$M+k^2\theta^2A@f$ used when solving for @f$U^n@f$ in each time
822 * step.
823 *
824
825 *
826 * When setting up these matrices, note that they all make use of the same
827 * sparsity pattern object. Finally, the reason why matrices and sparsity
828 * patterns are separate objects in deal.II (unlike in many other finite
829 * element or linear algebra classes) becomes clear: in a significant
830 * fraction of applications, one has to hold several matrices that happen
831 * to have the same sparsity pattern, and there is no reason for them not
832 * to share this information, rather than re-building and wasting memory
833 * on it several times.
834 *
835
836 *
837 * After initializing all of these matrices, we call library functions
838 * that build the Laplace and mass matrices. All they need is a DoFHandler
839 * object and a quadrature formula object that is to be used for numerical
840 * integration. Note that in many respects these functions are better than
841 * what we would usually do in application programs, for example because
842 * they automatically parallelize building the matrices if multiple
843 * processors are available in a machine: for more information see the
844 * documentation of WorkStream or the
845 * @ref threads "Parallel computing with multiple processors"
846 * module. The matrices for solving linear systems will be filled in the
847 * run() method because we need to re-apply boundary conditions every time
848 * step.
849 *
850 * @code
851 *   mass_matrix.reinit(sparsity_pattern);
852 *   laplace_matrix.reinit(sparsity_pattern);
853 *   matrix_u.reinit(sparsity_pattern);
854 *   matrix_v.reinit(sparsity_pattern);
855 *  
856 *   MatrixCreator::create_mass_matrix(dof_handler,
857 *   QGauss<dim>(fe.degree + 1),
858 *   mass_matrix);
859 *   MatrixCreator::create_laplace_matrix(dof_handler,
860 *   QGauss<dim>(fe.degree + 1),
861 *   laplace_matrix);
862 *  
863 * @endcode
864 *
865 * The rest of the function is spent on setting vector sizes to the
866 * correct value. The final line closes the hanging node constraints
867 * object. Since we work on a uniformly refined mesh, no constraints exist
868 * or have been computed (i.e. there was no need to call
869 * DoFTools::make_hanging_node_constraints as in other programs), but we
870 * need a constraints object in one place further down below anyway.
871 *
872 * @code
873 *   solution_u.reinit(dof_handler.n_dofs());
874 *   solution_v.reinit(dof_handler.n_dofs());
875 *   old_solution_u.reinit(dof_handler.n_dofs());
876 *   old_solution_v.reinit(dof_handler.n_dofs());
877 *   system_rhs.reinit(dof_handler.n_dofs());
878 *  
879 *   constraints.close();
880 *   }
881 *  
882 *  
883 *  
884 * @endcode
885 *
886 *
887 * <a name="WaveEquationsolve_uandWaveEquationsolve_v"></a>
888 * <h4>WaveEquation::solve_u and WaveEquation::solve_v</h4>
889 *
890
891 *
892 * The next two functions deal with solving the linear systems associated
893 * with the equations for @f$U^n@f$ and @f$V^n@f$. Both are not particularly
894 * interesting as they pretty much follow the scheme used in all the
895 * previous tutorial programs.
896 *
897
898 *
899 * One can make little experiments with preconditioners for the two matrices
900 * we have to invert. As it turns out, however, for the matrices at hand
901 * here, using Jacobi or SSOR preconditioners reduces the number of
902 * iterations necessary to solve the linear system slightly, but due to the
903 * cost of applying the preconditioner it is no win in terms of run-time. It
904 * is not much of a loss either, but let's keep it simple and just do
905 * without:
906 *
907 * @code
908 *   template <int dim>
909 *   void WaveEquation<dim>::solve_u()
910 *   {
911 *   SolverControl solver_control(1000, 1e-8 * system_rhs.l2_norm());
912 *   SolverCG<Vector<double>> cg(solver_control);
913 *  
914 *   cg.solve(matrix_u, solution_u, system_rhs, PreconditionIdentity());
915 *  
916 *   std::cout << " u-equation: " << solver_control.last_step()
917 *   << " CG iterations." << std::endl;
918 *   }
919 *  
920 *  
921 *  
922 *   template <int dim>
923 *   void WaveEquation<dim>::solve_v()
924 *   {
925 *   SolverControl solver_control(1000, 1e-8 * system_rhs.l2_norm());
926 *   SolverCG<Vector<double>> cg(solver_control);
927 *  
928 *   cg.solve(matrix_v, solution_v, system_rhs, PreconditionIdentity());
929 *  
930 *   std::cout << " v-equation: " << solver_control.last_step()
931 *   << " CG iterations." << std::endl;
932 *   }
933 *  
934 *  
935 *  
936 * @endcode
937 *
938 *
939 * <a name="WaveEquationoutput_results"></a>
940 * <h4>WaveEquation::output_results</h4>
941 *
942
943 *
944 * Likewise, the following function is pretty much what we've done
945 * before. The only thing worth mentioning is how here we generate a string
946 * representation of the time step number padded with leading zeros to 3
947 * character length using the Utilities::int_to_string function's second
948 * argument.
949 *
950 * @code
951 *   template <int dim>
952 *   void WaveEquation<dim>::output_results() const
953 *   {
954 *   DataOut<dim> data_out;
955 *  
956 *   data_out.attach_dof_handler(dof_handler);
957 *   data_out.add_data_vector(solution_u, "U");
958 *   data_out.add_data_vector(solution_v, "V");
959 *  
960 *   data_out.build_patches();
961 *  
962 *   const std::string filename =
963 *   "solution-" + Utilities::int_to_string(timestep_number, 3) + ".vtu";
964 * @endcode
965 *
966 * Like @ref step_15 "step-15", since we write output at every time step (and the system
967 * we have to solve is relatively easy), we instruct DataOut to use the
968 * zlib compression algorithm that is optimized for speed instead of disk
969 * usage since otherwise plotting the output becomes a bottleneck:
970 *
971 * @code
972 *   DataOutBase::VtkFlags vtk_flags;
974 *   data_out.set_flags(vtk_flags);
975 *   std::ofstream output(filename);
976 *   data_out.write_vtu(output);
977 *   }
978 *  
979 *  
980 *  
981 * @endcode
982 *
983 *
984 * <a name="WaveEquationrun"></a>
985 * <h4>WaveEquation::run</h4>
986 *
987
988 *
989 * The following is really the only interesting function of the program. It
990 * contains the loop over all time steps, but before we get to that we have
991 * to set up the grid, DoFHandler, and matrices. In addition, we have to
992 * somehow get started with initial values. To this end, we use the
993 * VectorTools::project function that takes an object that describes a
994 * continuous function and computes the @f$L^2@f$ projection of this function
995 * onto the finite element space described by the DoFHandler object. Can't
996 * be any simpler than that:
997 *
998 * @code
999 *   template <int dim>
1000 *   void WaveEquation<dim>::run()
1001 *   {
1002 *   setup_system();
1003 *  
1004 *   VectorTools::project(dof_handler,
1005 *   constraints,
1006 *   QGauss<dim>(fe.degree + 1),
1007 *   InitialValuesU<dim>(),
1008 *   old_solution_u);
1009 *   VectorTools::project(dof_handler,
1010 *   constraints,
1011 *   QGauss<dim>(fe.degree + 1),
1012 *   InitialValuesV<dim>(),
1013 *   old_solution_v);
1014 *  
1015 * @endcode
1016 *
1017 * The next thing is to loop over all the time steps until we reach the
1018 * end time (@f$T=5@f$ in this case). In each time step, we first have to
1019 * solve for @f$U^n@f$, using the equation @f$(M^n + k^2\theta^2 A^n)U^n =@f$
1020 * @f$(M^{n,n-1} - k^2\theta(1-\theta) A^{n,n-1})U^{n-1} + kM^{n,n-1}V^{n-1}
1021 * +@f$ @f$k\theta \left[k \theta F^n + k(1-\theta) F^{n-1} \right]@f$. Note
1022 * that we use the same mesh for all time steps, so that @f$M^n=M^{n,n-1}=M@f$
1023 * and @f$A^n=A^{n,n-1}=A@f$. What we therefore have to do first is to add up
1024 * @f$MU^{n-1} - k^2\theta(1-\theta) AU^{n-1} + kMV^{n-1}@f$ and the forcing
1025 * terms, and put the result into the <code>system_rhs</code> vector. (For
1026 * these additions, we need a temporary vector that we declare before the
1027 * loop to avoid repeated memory allocations in each time step.)
1028 *
1029
1030 *
1031 * The one thing to realize here is how we communicate the time variable
1032 * to the object describing the right hand side: each object derived from
1033 * the Function class has a time field that can be set using the
1034 * Function::set_time and read by Function::get_time. In essence, using
1035 * this mechanism, all functions of space and time are therefore
1036 * considered functions of space evaluated at a particular time. This
1037 * matches well what we typically need in finite element programs, where
1038 * we almost always work on a single time step at a time, and where it
1039 * never happens that, for example, one would like to evaluate a
1040 * space-time function for all times at any given spatial location.
1041 *
1042 * @code
1043 *   Vector<double> tmp(solution_u.size());
1044 *   Vector<double> forcing_terms(solution_u.size());
1045 *  
1046 *   for (; time <= 5; time += time_step, ++timestep_number)
1047 *   {
1048 *   std::cout << "Time step " << timestep_number << " at t=" << time
1049 *   << std::endl;
1050 *  
1051 *   mass_matrix.vmult(system_rhs, old_solution_u);
1052 *  
1053 *   mass_matrix.vmult(tmp, old_solution_v);
1054 *   system_rhs.add(time_step, tmp);
1055 *  
1056 *   laplace_matrix.vmult(tmp, old_solution_u);
1057 *   system_rhs.add(-theta * (1 - theta) * time_step * time_step, tmp);
1058 *  
1059 *   RightHandSide<dim> rhs_function;
1060 *   rhs_function.set_time(time);
1061 *   VectorTools::create_right_hand_side(dof_handler,
1062 *   QGauss<dim>(fe.degree + 1),
1063 *   rhs_function,
1064 *   tmp);
1065 *   forcing_terms = tmp;
1066 *   forcing_terms *= theta * time_step;
1067 *  
1068 *   rhs_function.set_time(time - time_step);
1069 *   VectorTools::create_right_hand_side(dof_handler,
1070 *   QGauss<dim>(fe.degree + 1),
1071 *   rhs_function,
1072 *   tmp);
1073 *  
1074 *   forcing_terms.add((1 - theta) * time_step, tmp);
1075 *  
1076 *   system_rhs.add(theta * time_step, forcing_terms);
1077 *  
1078 * @endcode
1079 *
1080 * After so constructing the right hand side vector of the first
1081 * equation, all we have to do is apply the correct boundary
1082 * values. As for the right hand side, this is a space-time function
1083 * evaluated at a particular time, which we interpolate at boundary
1084 * nodes and then use the result to apply boundary values as we
1085 * usually do. The result is then handed off to the solve_u()
1086 * function:
1087 *
1088 * @code
1089 *   {
1090 *   BoundaryValuesU<dim> boundary_values_u_function;
1091 *   boundary_values_u_function.set_time(time);
1092 *  
1093 *   std::map<types::global_dof_index, double> boundary_values;
1094 *   VectorTools::interpolate_boundary_values(dof_handler,
1095 *   0,
1096 *   boundary_values_u_function,
1097 *   boundary_values);
1098 *  
1099 * @endcode
1100 *
1101 * The matrix for solve_u() is the same in every time steps, so one
1102 * could think that it is enough to do this only once at the
1103 * beginning of the simulation. However, since we need to apply
1104 * boundary values to the linear system (which eliminate some matrix
1105 * rows and columns and give contributions to the right hand side),
1106 * we have to refill the matrix in every time steps before we
1107 * actually apply boundary data. The actual content is very simple:
1108 * it is the sum of the mass matrix and a weighted Laplace matrix:
1109 *
1110 * @code
1111 *   matrix_u.copy_from(mass_matrix);
1112 *   matrix_u.add(theta * theta * time_step * time_step, laplace_matrix);
1113 *   MatrixTools::apply_boundary_values(boundary_values,
1114 *   matrix_u,
1115 *   solution_u,
1116 *   system_rhs);
1117 *   }
1118 *   solve_u();
1119 *  
1120 *  
1121 * @endcode
1122 *
1123 * The second step, i.e. solving for @f$V^n@f$, works similarly, except
1124 * that this time the matrix on the left is the mass matrix (which we
1125 * copy again in order to be able to apply boundary conditions, and
1126 * the right hand side is @f$MV^{n-1} - k\left[ \theta A U^n +
1127 * (1-\theta) AU^{n-1}\right]@f$ plus forcing terms. Boundary values
1128 * are applied in the same way as before, except that now we have to
1129 * use the BoundaryValuesV class:
1130 *
1131 * @code
1132 *   laplace_matrix.vmult(system_rhs, solution_u);
1133 *   system_rhs *= -theta * time_step;
1134 *  
1135 *   mass_matrix.vmult(tmp, old_solution_v);
1136 *   system_rhs += tmp;
1137 *  
1138 *   laplace_matrix.vmult(tmp, old_solution_u);
1139 *   system_rhs.add(-time_step * (1 - theta), tmp);
1140 *  
1141 *   system_rhs += forcing_terms;
1142 *  
1143 *   {
1144 *   BoundaryValuesV<dim> boundary_values_v_function;
1145 *   boundary_values_v_function.set_time(time);
1146 *  
1147 *   std::map<types::global_dof_index, double> boundary_values;
1148 *   VectorTools::interpolate_boundary_values(dof_handler,
1149 *   0,
1150 *   boundary_values_v_function,
1151 *   boundary_values);
1152 *   matrix_v.copy_from(mass_matrix);
1153 *   MatrixTools::apply_boundary_values(boundary_values,
1154 *   matrix_v,
1155 *   solution_v,
1156 *   system_rhs);
1157 *   }
1158 *   solve_v();
1159 *  
1160 * @endcode
1161 *
1162 * Finally, after both solution components have been computed, we
1163 * output the result, compute the energy in the solution, and go on to
1164 * the next time step after shifting the present solution into the
1165 * vectors that hold the solution at the previous time step. Note the
1166 * function SparseMatrix::matrix_norm_square that can compute
1167 * @f$\left<V^n,MV^n\right>@f$ and @f$\left<U^n,AU^n\right>@f$ in one step,
1168 * saving us the expense of a temporary vector and several lines of
1169 * code:
1170 *
1171 * @code
1172 *   output_results();
1173 *  
1174 *   std::cout << " Total energy: "
1175 *   << (mass_matrix.matrix_norm_square(solution_v) +
1176 *   laplace_matrix.matrix_norm_square(solution_u)) /
1177 *   2
1178 *   << std::endl;
1179 *  
1180 *   old_solution_u = solution_u;
1181 *   old_solution_v = solution_v;
1182 *   }
1183 *   }
1184 *   } // namespace Step23
1185 *  
1186 *  
1187 * @endcode
1188 *
1189 *
1190 * <a name="Thecodemaincodefunction"></a>
1191 * <h3>The <code>main</code> function</h3>
1192 *
1193
1194 *
1195 * What remains is the main function of the program. There is nothing here
1196 * that hasn't been shown in several of the previous programs:
1197 *
1198 * @code
1199 *   int main()
1200 *   {
1201 *   try
1202 *   {
1203 *   using namespace Step23;
1204 *  
1205 *   WaveEquation<2> wave_equation_solver;
1206 *   wave_equation_solver.run();
1207 *   }
1208 *   catch (std::exception &exc)
1209 *   {
1210 *   std::cerr << std::endl
1211 *   << std::endl
1212 *   << "----------------------------------------------------"
1213 *   << std::endl;
1214 *   std::cerr << "Exception on processing: " << std::endl
1215 *   << exc.what() << std::endl
1216 *   << "Aborting!" << std::endl
1217 *   << "----------------------------------------------------"
1218 *   << std::endl;
1219 *  
1220 *   return 1;
1221 *   }
1222 *   catch (...)
1223 *   {
1224 *   std::cerr << std::endl
1225 *   << std::endl
1226 *   << "----------------------------------------------------"
1227 *   << std::endl;
1228 *   std::cerr << "Unknown exception!" << std::endl
1229 *   << "Aborting!" << std::endl
1230 *   << "----------------------------------------------------"
1231 *   << std::endl;
1232 *   return 1;
1233 *   }
1234 *  
1235 *   return 0;
1236 *   }
1237 * @endcode
1238<a name="Results"></a><h1>Results</h1>
1239
1240
1241When the program is run, it produces the following output:
1242@code
1243Number of active cells: 16384
1244Number of degrees of freedom: 16641
1245
1246Time step 1 at t=0.015625
1247 u-equation: 8 CG iterations.
1248 v-equation: 22 CG iterations.
1249 Total energy: 1.17887
1250Time step 2 at t=0.03125
1251 u-equation: 8 CG iterations.
1252 v-equation: 20 CG iterations.
1253 Total energy: 2.9655
1254Time step 3 at t=0.046875
1255 u-equation: 8 CG iterations.
1256 v-equation: 21 CG iterations.
1257 Total energy: 4.33761
1258Time step 4 at t=0.0625
1259 u-equation: 7 CG iterations.
1260 v-equation: 21 CG iterations.
1261 Total energy: 5.35499
1262Time step 5 at t=0.078125
1263 u-equation: 7 CG iterations.
1264 v-equation: 21 CG iterations.
1265 Total energy: 6.18652
1266Time step 6 at t=0.09375
1267 u-equation: 7 CG iterations.
1268 v-equation: 20 CG iterations.
1269 Total energy: 6.6799
1270
1271...
1272
1273Time step 31 at t=0.484375
1274 u-equation: 7 CG iterations.
1275 v-equation: 20 CG iterations.
1276 Total energy: 21.9068
1277Time step 32 at t=0.5
1278 u-equation: 7 CG iterations.
1279 v-equation: 20 CG iterations.
1280 Total energy: 23.3394
1281Time step 33 at t=0.515625
1282 u-equation: 7 CG iterations.
1283 v-equation: 20 CG iterations.
1284 Total energy: 23.1019
1285
1286...
1287
1288Time step 319 at t=4.98438
1289 u-equation: 7 CG iterations.
1290 v-equation: 20 CG iterations.
1291 Total energy: 23.1019
1292Time step 320 at t=5
1293 u-equation: 7 CG iterations.
1294 v-equation: 20 CG iterations.
1295 Total energy: 23.1019
1296@endcode
1297
1298What we see immediately is that the energy is a constant at least after
1299@f$t=\frac 12@f$ (until which the boundary source term @f$g@f$ is nonzero, injecting
1300energy into the system).
1301
1302In addition to the screen output, the program writes the solution of each time
1303step to an output file. If we process them adequately and paste them into a
1304movie, we get the following:
1305
1306<img src="https://www.dealii.org/images/steps/developer/step-23.movie.gif" alt="Animation of the solution of step 23.">
1307
1308The movie shows the generated wave nice traveling through the domain and back,
1309being reflected at the clamped boundary. Some numerical noise is trailing the
1310wave, an artifact of a too-large mesh size that can be reduced by reducing the
1311mesh width and the time step.
1312
1313
1314<a name="extensions"></a>
1315<a name="Possibilitiesforextensions"></a><h3>Possibilities for extensions</h3>
1316
1317
1318If you want to explore a bit, try out some of the following things:
1319<ul>
1320 <li>Varying @f$\theta@f$. This gives different time stepping schemes, some of
1321 which are stable while others are not. Take a look at how the energy
1322 evolves.
1323
1324 <li>Different initial and boundary conditions, right hand sides.
1325
1326 <li>More complicated domains or more refined meshes. Remember that the time
1327 step needs to be bounded by the mesh width, so changing the mesh should
1328 always involve also changing the time step. We will come back to this issue
1329 in @ref step_24 "step-24".
1330
1331 <li>Variable coefficients: In real media, the wave speed is often
1332 variable. In particular, the "real" wave equation in realistic media would
1333 read
1334 @f[
1335 \rho(x) \frac{\partial^2 u}{\partial t^2}
1336 -
1337 \nabla \cdot
1338 a(x) \nabla u = f,
1339 @f]
1340 where @f$\rho(x)@f$ is the density of the material, and @f$a(x)@f$ is related to the
1341 stiffness coefficient. The wave speed is then @f$c=\sqrt{a/\rho}@f$.
1342
1343 To make such a change, we would have to compute the mass and Laplace
1344 matrices with a variable coefficient. Fortunately, this isn't too hard: the
1345 functions MatrixCreator::create_laplace_matrix and
1346 MatrixCreator::create_mass_matrix have additional default parameters that can
1347 be used to pass non-constant coefficient functions to them. The required
1348 changes are therefore relatively small. On the other hand, care must be
1349 taken again to make sure the time step is within the allowed range.
1350
1351 <li>In the in-code comments, we discussed the fact that the matrices for
1352 solving for @f$U^n@f$ and @f$V^n@f$ need to be reset in every time because of
1353 boundary conditions, even though the actual content does not change. It is
1354 possible to avoid copying by not eliminating columns in the linear systems,
1355 which is implemented by appending a @p false argument to the call:
1356 @code
1357 MatrixTools::apply_boundary_values(boundary_values,
1358 matrix_u,
1359 solution_u,
1360 system_rhs,
1361 false);
1362 @endcode
1363
1364 <li>deal.II being a library that supports adaptive meshes it would of course be
1365 nice if this program supported change the mesh every few time steps. Given the
1366 structure of the solution &mdash; a wave that travels through the domain &mdash;
1367 it would seem appropriate if we only refined the mesh where the wave currently is,
1368 and not simply everywhere. It is intuitively clear that we should be able to
1369 save a significant amount of cells this way. (Though upon further thought one
1370 realizes that this is really only the case in the initial stages of the simulation.
1371 After some time, for wave phenomena, the domain is filled with reflections of
1372 the initial wave going in every direction and filling every corner of the domain.
1373 At this point, there is in general little one can gain using local mesh
1374 refinement.)
1375
1376 To make adaptively changing meshes possible, there are basically two routes.
1377 The "correct" way would be to go back to the weak form we get using Rothe's
1378 method. For example, the first of the two equations to be solved in each time
1379 step looked like this:
1380 \f{eqnarray*}
1381 (u^n,\varphi) + k^2\theta^2(\nabla u^n,\nabla \varphi) &=&
1382 (u^{n-1},\varphi) - k^2\theta(1-\theta)(\nabla u^{n-1},\nabla \varphi)
1383 +
1384 k(v^{n-1},\varphi)
1385 + k^2\theta
1386 \left[
1387 \theta (f^n,\varphi) + (1-\theta) (f^{n-1},\varphi)
1388 \right].
1389 \f}
1390 Now, note that we solve for @f$u^n@f$ on mesh @f${\mathbb T}^n@f$, and
1391 consequently the test functions @f$\varphi@f$ have to be from the space
1392 @f$V_h^n@f$ as well. As discussed in the introduction, terms like
1393 @f$(u^{n-1},\varphi)@f$ then require us to integrate the solution of the
1394 previous step (which may have been computed on a different mesh
1395 @f${\mathbb T}^{n-1}@f$) against the test functions of the current mesh,
1396 leading to a matrix @f$M^{n,n-1}@f$. This process of integrating shape
1397 functions from different meshes is, at best, awkward. It can be done
1398 but because it is difficult to ensure that @f${\mathbb T}^{n-1}@f$ and
1399 @f${\mathbb T}^{n}@f$ differ by at most one level of refinement, one
1400 has to recursively match cells from both meshes. It is feasible to
1401 do this, but it leads to lengthy and not entirely obvious code.
1402
1403 The second approach is the following: whenever we change the mesh,
1404 we simply interpolate the solution from the last time step on the old
1405 mesh to the new mesh, using the SolutionTransfer class. In other words,
1406 instead of the equation above, we would solve
1407 \f{eqnarray*}
1408 (u^n,\varphi) + k^2\theta^2(\nabla u^n,\nabla \varphi) &=&
1409 (I^n u^{n-1},\varphi) - k^2\theta(1-\theta)(\nabla I^n u^{n-1},\nabla \varphi)
1410 +
1411 k(I^n v^{n-1},\varphi)
1412 + k^2\theta
1413 \left[
1414 \theta (f^n,\varphi) + (1-\theta) (f^{n-1},\varphi)
1415 \right],
1416 \f}
1417 where @f$I^n@f$ interpolates a given function onto mesh @f${\mathbb T}^n@f$.
1418 This is a much simpler approach because, in each time step, we no
1419 longer have to worry whether @f$u^{n-1},v^{n-1}@f$ were computed on the
1420 same mesh as we are using now or on a different mesh. Consequently,
1421 the only changes to the code necessary are the addition of a function
1422 that computes the error, marks cells for refinement, sets up a
1423 SolutionTransfer object, transfers the solution to the new mesh, and
1424 rebuilds matrices and right hand side vectors on the new mesh. Neither
1425 the functions building the matrices and right hand sides, nor the
1426 solvers need to be changed.
1427
1428 While this second approach is, strictly speaking,
1429 not quite correct in the Rothe framework (it introduces an addition source
1430 of error, namely the interpolation), it is nevertheless what
1431 almost everyone solving time dependent equations does. We will use this
1432 method in @ref step_31 "step-31", for example.
1433</ul>
1434 *
1435 *
1436<a name="PlainProg"></a>
1437<h1> The plain program</h1>
1438@include "step-23.cc"
1439*/
void attach_dof_handler(const DoFHandler< dim, spacedim > &)
Number get_time() const
virtual RangeNumberType value(const Point< dim > &p, const unsigned int component=0) const
Definition point.h:112
Point< 2 > second
Definition grid_out.cc:4616
Point< 2 > first
Definition grid_out.cc:4615
unsigned int level
Definition grid_out.cc:4618
__global__ void set(Number *val, const Number s, const size_type N)
#define Assert(cond, exc)
void loop(ITERATOR begin, std_cxx20::type_identity_t< ITERATOR > 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())
Definition loop.h:439
void make_sparsity_pattern(const DoFHandler< dim, spacedim > &dof_handler, SparsityPatternBase &sparsity_pattern, const AffineConstraints< number > &constraints=AffineConstraints< number >(), const bool keep_constrained_dofs=true, const types::subdomain_id subdomain_id=numbers::invalid_subdomain_id)
const Event initial
Definition event.cc:65
void interpolate(const DoFHandler< dim, spacedim > &dof1, const InVector &u1, const DoFHandler< dim, spacedim > &dof2, OutVector &u2)
void hyper_cube(Triangulation< dim, spacedim > &tria, const double left=0., const double right=1., const bool colorize=false)
@ matrix
Contents is actually a matrix.
SymmetricTensor< 2, dim, Number > e(const Tensor< 2, dim, Number > &F)
void call(const std::function< RT()> &function, internal::return_value< RT > &ret_val)
VectorType::value_type * end(VectorType &V)
std::string int_to_string(const unsigned int value, const unsigned int digits=numbers::invalid_unsigned_int)
Definition utilities.cc:471
void project(const Mapping< dim, spacedim > &mapping, const DoFHandler< dim, spacedim > &dof, const AffineConstraints< typename VectorType::value_type > &constraints, const Quadrature< dim > &quadrature, const Function< spacedim, typename VectorType::value_type > &function, VectorType &vec, const bool enforce_zero_boundary=false, const Quadrature< dim - 1 > &q_boundary=(dim > 1 ? QGauss< dim - 1 >(2) :Quadrature< dim - 1 >(0)), const bool project_to_boundary_first=false)
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)
int(&) functions(const void *v1, const void *v2)
static constexpr double PI
Definition numbers.h:259
::VectorizedArray< Number, width > cos(const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > sin(const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > sqrt(const ::VectorizedArray< Number, width > &)
const ::parallel::distributed::Triangulation< dim, spacedim > * triangulation
DataOutBase::CompressionLevel compression_level
DEAL_II_HOST constexpr SymmetricTensor< 2, dim, Number > invert(const SymmetricTensor< 2, dim, Number > &)