Reference documentation for deal.II version GIT relicensing-487-ge9eb5ab491 2024-04-25 07:20:02+00:00
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step-81.h
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741 *   types::material_id material)
742 *   {
743 *   return (material == 1 ? epsilon_1 : epsilon_2);
744 *   }
745 *  
746 *   template <int dim>
747 *   std::complex<double> Parameters<dim>::mu_inv(const Point<dim> & /*x*/,
748 *   types::material_id material)
749 *   {
750 *   return (material == 1 ? mu_inv_1 : mu_inv_2);
751 *   }
752 *  
753 *   template <int dim>
754 *   typename Parameters<dim>::rank2_type
755 *   Parameters<dim>::sigma(const Point<dim> & /*x*/,
756 *   types::material_id left,
757 *   types::material_id right)
758 *   {
759 *   return (left == right ? rank2_type() : sigma_tensor);
760 *   }
761 *  
762 *   template <int dim>
763 *   typename Parameters<dim>::rank1_type
764 *   Parameters<dim>::J_a(const Point<dim> &point, types::material_id /*id*/)
765 *   {
766 *   rank1_type J_a;
767 *   const auto distance = (dipole_position - point).norm() / dipole_radius;
768 *   if (distance > 1.)
769 *   return J_a;
770 *   double scale = std::cos(distance * M_PI / 2.) *
771 *   std::cos(distance * M_PI / 2.) / (M_PI / 2. - 2. / M_PI) /
772 *   dipole_radius / dipole_radius;
773 *   J_a = dipole_strength * dipole_orientation * scale;
774 *   return J_a;
775 *   }
776 *  
777 * @endcode
778 *
779 *
780 * <a name="step_81-PerfectlyMatchedLayerClass"></a>
781 * <h4>PerfectlyMatchedLayer Class</h4>
782 * The PerfectlyMatchedLayer class inherits ParameterAcceptor as well. It
783 * implements the transformation matrices used to modify the permittivity
784 * and permeability tensors supplied from the Parameters class. The
785 * actual transformation of the material tensors will be done in the
786 * assembly loop. The radii and the strength of the PML is specified, and
787 * the coefficients will be modified using transformation matrices within
788 * the PML region. The radii and strength of the PML are editable through
789 * a .prm file. The rotation function @f$T_{exer}@f$ is the same as
790 * introduced in the perfectly matched layer section of the introduction.
791 * Similarly, the matrices A, B and C are defined as follows
792 * @f[
793 * A = T_{e_xe_r}^{-1}
794 * \text{diag}\left(\frac{1}{\bar{d}^2},\frac{1}{d\bar{d}}\right)T_{e_xe_r},\qquad
795 * B = T_{e_xe_r}^{-1} \text{diag}\left(d,\bar{d}\right)T_{e_xe_r},\qquad
796 * C = T_{e_xe_r}^{-1} \text{diag}\left(\frac{1}{\bar{d}},\frac{1}{d}\right)
797 * T_{e_xe_r}.\qquad
798 * @f]
799 *
800
801 *
802 *
803 * @code
804 *   template <int dim>
805 *   class PerfectlyMatchedLayer : public ParameterAcceptor
806 *   {
807 *   public:
808 *   static_assert(dim == 2,
809 *   "The perfectly matched layer is only implemented in 2d.");
810 *  
811 *   Parameters<dim> parameters;
812 *  
813 *   using rank1_type = Tensor<1, dim, std::complex<double>>;
814 *  
815 *   using rank2_type = Tensor<2, dim, std::complex<double>>;
816 *  
817 *   PerfectlyMatchedLayer();
818 *  
819 *   std::complex<double> d(const Point<dim> point);
820 *  
821 *   std::complex<double> d_bar(const Point<dim> point);
822 *  
823 *  
824 *   rank2_type rotation(std::complex<double> d_1,
825 *   std::complex<double> d_2,
826 *   Point<dim> point);
827 *  
828 *   rank2_type a_matrix(const Point<dim> point);
829 *  
830 *   rank2_type b_matrix(const Point<dim> point);
831 *  
832 *   rank2_type c_matrix(const Point<dim> point);
833 *  
834 *   private:
835 *   double inner_radius;
836 *   double outer_radius;
837 *   double strength;
838 *   };
839 *  
840 *  
841 *   template <int dim>
842 *   PerfectlyMatchedLayer<dim>::PerfectlyMatchedLayer()
843 *   : ParameterAcceptor("PerfectlyMatchedLayer")
844 *   {
845 *   inner_radius = 12.;
846 *   add_parameter("inner radius",
847 *   inner_radius,
848 *   "inner radius of the PML shell");
849 *   outer_radius = 20.;
850 *   add_parameter("outer radius",
851 *   outer_radius,
852 *   "outer radius of the PML shell");
853 *   strength = 8.;
854 *   add_parameter("strength", strength, "strength of the PML");
855 *   }
856 *  
857 *  
858 *   template <int dim>
859 *   typename std::complex<double>
860 *   PerfectlyMatchedLayer<dim>::d(const Point<dim> point)
861 *   {
862 *   const auto radius = point.norm();
863 *   if (radius > inner_radius)
864 *   {
865 *   const double s =
866 *   strength * ((radius - inner_radius) * (radius - inner_radius)) /
867 *   ((outer_radius - inner_radius) * (outer_radius - inner_radius));
868 *   return {1.0, s};
869 *   }
870 *   else
871 *   {
872 *   return 1.0;
873 *   }
874 *   }
875 *  
876 *  
877 *   template <int dim>
878 *   typename std::complex<double>
879 *   PerfectlyMatchedLayer<dim>::d_bar(const Point<dim> point)
880 *   {
881 *   const auto radius = point.norm();
882 *   if (radius > inner_radius)
883 *   {
884 *   const double s_bar =
885 *   strength / 3. *
886 *   ((radius - inner_radius) * (radius - inner_radius) *
887 *   (radius - inner_radius)) /
888 *   (radius * (outer_radius - inner_radius) *
889 *   (outer_radius - inner_radius));
890 *   return {1.0, s_bar};
891 *   }
892 *   else
893 *   {
894 *   return 1.0;
895 *   }
896 *   }
897 *  
898 *  
899 *   template <int dim>
900 *   typename PerfectlyMatchedLayer<dim>::rank2_type
901 *   PerfectlyMatchedLayer<dim>::rotation(std::complex<double> d_1,
902 *   std::complex<double> d_2,
903 *   Point<dim> point)
904 *   {
905 *   rank2_type result;
906 *   result[0][0] = point[0] * point[0] * d_1 + point[1] * point[1] * d_2;
907 *   result[0][1] = point[0] * point[1] * (d_1 - d_2);
908 *   result[1][0] = point[0] * point[1] * (d_1 - d_2);
909 *   result[1][1] = point[1] * point[1] * d_1 + point[0] * point[0] * d_2;
910 *   return result;
911 *   }
912 *  
913 *  
914 *   template <int dim>
915 *   typename PerfectlyMatchedLayer<dim>::rank2_type
916 *   PerfectlyMatchedLayer<dim>::a_matrix(const Point<dim> point)
917 *   {
918 *   const auto d = this->d(point);
919 *   const auto d_bar = this->d_bar(point);
920 *   return invert(rotation(d * d, d * d_bar, point)) *
921 *   rotation(d * d, d * d_bar, point);
922 *   }
923 *  
924 *  
925 *   template <int dim>
926 *   typename PerfectlyMatchedLayer<dim>::rank2_type
927 *   PerfectlyMatchedLayer<dim>::b_matrix(const Point<dim> point)
928 *   {
929 *   const auto d = this->d(point);
930 *   const auto d_bar = this->d_bar(point);
931 *   return invert(rotation(d, d_bar, point)) * rotation(d, d_bar, point);
932 *   }
933 *  
934 *  
935 *   template <int dim>
936 *   typename PerfectlyMatchedLayer<dim>::rank2_type
937 *   PerfectlyMatchedLayer<dim>::c_matrix(const Point<dim> point)
938 *   {
939 *   const auto d = this->d(point);
940 *   const auto d_bar = this->d_bar(point);
941 *   return invert(rotation(1. / d_bar, 1. / d, point)) *
942 *   rotation(1. / d_bar, 1. / d, point);
943 *   }
944 *  
945 *  
946 * @endcode
947 *
948 *
949 * <a name="step_81-MaxwellClass"></a>
950 * <h4>Maxwell Class</h4>
951 * At this point we are ready to declare all the major building blocks of
952 * the finite element program which consists of the usual setup and
953 * assembly routines. Most of the structure has already been introduced
954 * in previous tutorial programs. The Maxwell class also holds private
955 * instances of the Parameters and PerfectlyMatchedLayers classes
956 * introduced above. The default values of these parameters are set to
957 * show us a standing wave with absorbing boundary conditions and a PML.
958 *
959
960 *
961 *
962 * @code
963 *   template <int dim>
964 *   class Maxwell : public ParameterAcceptor
965 *   {
966 *   public:
967 *   Maxwell();
968 *   void run();
969 *  
970 *   private:
971 *   /* run time parameters */
972 *   double scaling;
973 *   unsigned int refinements;
974 *   unsigned int fe_order;
975 *   unsigned int quadrature_order;
976 *   bool absorbing_boundary;
977 *  
978 *   void parse_parameters_callback();
979 *   void make_grid();
980 *   void setup_system();
981 *   void assemble_system();
982 *   void solve();
983 *   void output_results();
984 *  
985 *   Parameters<dim> parameters;
986 *   PerfectlyMatchedLayer<dim> perfectly_matched_layer;
987 *  
989 *   DoFHandler<dim> dof_handler;
990 *  
991 *   std::unique_ptr<FiniteElement<dim>> fe;
992 *  
993 *   AffineConstraints<double> constraints;
994 *   SparsityPattern sparsity_pattern;
995 *   SparseMatrix<double> system_matrix;
996 *   Vector<double> solution;
997 *   Vector<double> system_rhs;
998 *   };
999 *  
1000 * @endcode
1001 *
1002 *
1003 * <a name="step_81-ClassTemplateDefinitionsandImplementation"></a>
1004 * <h3>Class Template Definitions and Implementation</h3>
1005 *
1006
1007 *
1008 *
1009 * <a name="step_81-TheConstructor"></a>
1010 * <h4>The Constructor</h4>
1011 * The Constructor simply consists of default initialization a number of
1012 * discretization parameters (such as the domain size, mesh refinement,
1013 * and the order of finite elements and quadrature) and declaring a
1014 * corresponding entry via ParameterAcceptor::add_parameter(). All of
1015 * these can be modified by editing the .prm file. Absorbing boundary
1016 * conditions can be controlled with the absorbing_boundary boolean. If
1017 * absorbing boundary conditions are disabled we simply enforce
1018 * homogeneous Dirichlet conditions on the tangential component of the
1019 * electric field. In the context of time-harmonic Maxwell's equations
1020 * these are also known as perfectly conducting boundary conditions.
1021 *
1022
1023 *
1024 *
1025 * @code
1026 *   template <int dim>
1027 *   Maxwell<dim>::Maxwell()
1028 *   : ParameterAcceptor("Maxwell")
1029 *   , dof_handler(triangulation)
1030 *   {
1031 *   ParameterAcceptor::parse_parameters_call_back.connect(
1032 *   [&]() { parse_parameters_callback(); });
1033 *  
1034 *   scaling = 20;
1035 *   add_parameter("scaling", scaling, "scale of the hypercube geometry");
1036 *  
1037 *   refinements = 8;
1038 *   add_parameter("refinements",
1039 *   refinements,
1040 *   "number of refinements of the geometry");
1041 *  
1042 *   fe_order = 0;
1043 *   add_parameter("fe order", fe_order, "order of the finite element space");
1044 *  
1045 *   quadrature_order = 1;
1046 *   add_parameter("quadrature order",
1047 *   quadrature_order,
1048 *   "order of the quadrature");
1049 *  
1050 *   absorbing_boundary = true;
1051 *   add_parameter("absorbing boundary condition",
1052 *   absorbing_boundary,
1053 *   "use absorbing boundary conditions?");
1054 *   }
1055 *  
1056 *  
1057 *   template <int dim>
1058 *   void Maxwell<dim>::parse_parameters_callback()
1059 *   {
1060 *   fe = std::make_unique<FESystem<dim>>(FE_NedelecSZ<dim>(fe_order), 2);
1061 *   }
1062 *  
1063 * @endcode
1064 *
1065 * The Maxwell::make_grid() routine creates the mesh for the
1066 * computational domain which in our case is a scaled square domain.
1067 * Additionally, a material interface is introduced by setting the
1068 * material id of the upper half (@f$y>0@f$) to 1 and of the lower half
1069 * (@f$y<0@f$) of the computational domain to 2.
1070 * We are using a block decomposition into real and imaginary matrices
1071 * for the solution matrices. More details on this are available
1072 * under the Results section.
1073 *
1074
1075 *
1076 *
1077 * @code
1078 *   template <int dim>
1079 *   void Maxwell<dim>::make_grid()
1080 *   {
1081 *   GridGenerator::hyper_cube(triangulation, -scaling, scaling);
1082 *   triangulation.refine_global(refinements);
1083 *  
1084 *   if (!absorbing_boundary)
1085 *   {
1086 *   for (auto &face : triangulation.active_face_iterators())
1087 *   if (face->at_boundary())
1088 *   face->set_boundary_id(1);
1089 *   };
1090 *  
1091 *   for (auto &cell : triangulation.active_cell_iterators())
1092 *   if (cell->center()[1] > 0.)
1093 *   cell->set_material_id(1);
1094 *   else
1095 *   cell->set_material_id(2);
1096 *  
1097 *  
1098 *   std::cout << "Number of active cells: " << triangulation.n_active_cells()
1099 *   << std::endl;
1100 *   }
1101 *  
1102 * @endcode
1103 *
1104 * The Maxwell::setup_system() routine follows the usual routine of
1105 * enumerating all the degrees of freedom and setting up the matrix and
1106 * vector objects to hold the system data. Enumerating is done by using
1107 * DoFHandler::distribute_dofs().
1108 *
1109
1110 *
1111 *
1112 * @code
1113 *   template <int dim>
1114 *   void Maxwell<dim>::setup_system()
1115 *   {
1116 *   dof_handler.distribute_dofs(*fe);
1117 *   std::cout << "Number of degrees of freedom: " << dof_handler.n_dofs()
1118 *   << std::endl;
1119 *  
1120 *   solution.reinit(dof_handler.n_dofs());
1121 *   system_rhs.reinit(dof_handler.n_dofs());
1122 *  
1123 *   constraints.clear();
1124 *  
1125 *   DoFTools::make_hanging_node_constraints(dof_handler, constraints);
1126 *  
1127 *   VectorTools::project_boundary_values_curl_conforming_l2(
1128 *   dof_handler,
1129 *   0, /* real part */
1130 *   Functions::ZeroFunction<dim>(2 * dim),
1131 *   0, /* boundary id */
1132 *   constraints);
1133 *   VectorTools::project_boundary_values_curl_conforming_l2(
1134 *   dof_handler,
1135 *   dim, /* imaginary part */
1136 *   Functions::ZeroFunction<dim>(2 * dim),
1137 *   0, /* boundary id */
1138 *   constraints);
1139 *  
1140 *   constraints.close();
1141 *  
1142 *   DynamicSparsityPattern dsp(dof_handler.n_dofs(), dof_handler.n_dofs());
1143 *   DoFTools::make_sparsity_pattern(dof_handler,
1144 *   dsp,
1145 *   constraints,
1146 *   /* keep_constrained_dofs = */ true);
1147 *   sparsity_pattern.copy_from(dsp);
1148 *   system_matrix.reinit(sparsity_pattern);
1149 *   }
1150 *  
1151 * @endcode
1152 *
1153 * This is a helper function that takes the tangential component of a tensor.
1154 *
1155 * @code
1156 *   template <int dim>
1157 *   DEAL_II_ALWAYS_INLINE inline Tensor<1, dim, std::complex<double>>
1158 *   tangential_part(const Tensor<1, dim, std::complex<double>> &tensor,
1159 *   const Tensor<1, dim> &normal)
1160 *   {
1161 *   auto result = tensor;
1162 *   result[0] = normal[1] * (tensor[0] * normal[1] - tensor[1] * normal[0]);
1163 *   result[1] = -normal[0] * (tensor[0] * normal[1] - tensor[1] * normal[0]);
1164 *   return result;
1165 *   }
1166 *  
1167 *  
1168 * @endcode
1169 *
1170 * Assemble the stiffness matrix and the right-hand side:
1171 * \f{align*}{
1172 * A_{ij} = \int_\Omega (\mu_r^{-1}\nabla \times \varphi_j) \cdot
1173 * (\nabla\times\bar{\varphi}_i)\text{d}x
1174 * - \int_\Omega \varepsilon_r\varphi_j \cdot \bar{\varphi}_i\text{d}x
1175 * - i\int_\Sigma (\sigma_r^{\Sigma}(\varphi_j)_T) \cdot
1176 * (\bar{\varphi}_i)_T\text{do}x
1177 * - i\int_{\partial\Omega} (\sqrt{\mu_r^{-1}\varepsilon}(\varphi_j)_T) \cdot
1178 * (\nabla\times(\bar{\varphi}_i)_T)\text{d}x, \f} \f{align}{
1179 * F_i = i\int_\Omega J_a \cdot \bar{\varphi_i}\text{d}x - \int_\Omega
1180 * \mu_r^{-1} \cdot (\nabla \times \bar{\varphi_i}) \text{d}x.
1181 * \f}
1182 * In addition, we will be modifying the coefficients if the position of the
1183 * cell is within the PML region.
1184 *
1185
1186 *
1187 *
1188 * @code
1189 *   template <int dim>
1190 *   void Maxwell<dim>::assemble_system()
1191 *   {
1192 *   QGauss<dim> quadrature_formula(quadrature_order);
1193 *   QGauss<dim - 1> face_quadrature_formula(quadrature_order);
1194 *  
1195 *   FEValues<dim, dim> fe_values(*fe,
1196 *   quadrature_formula,
1197 *   update_values | update_gradients |
1198 *   update_quadrature_points |
1199 *   update_JxW_values);
1200 *   FEFaceValues<dim, dim> fe_face_values(*fe,
1201 *   face_quadrature_formula,
1202 *   update_values | update_gradients |
1203 *   update_quadrature_points |
1204 *   update_normal_vectors |
1205 *   update_JxW_values);
1206 *  
1207 *   const unsigned int dofs_per_cell = fe->dofs_per_cell;
1208 *  
1209 *   const unsigned int n_q_points = quadrature_formula.size();
1210 *   const unsigned int n_face_q_points = face_quadrature_formula.size();
1211 *  
1212 *   FullMatrix<double> cell_matrix(dofs_per_cell, dofs_per_cell);
1213 *   Vector<double> cell_rhs(dofs_per_cell);
1214 *   std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
1215 *  
1216 * @endcode
1217 *
1218 * Next, let us assemble on the interior of the domain on the left hand
1219 * side. So we are computing
1220 * \f{align*}{
1221 * \int_\Omega (\mu_r^{-1}\nabla \times \varphi_i) \cdot
1222 * (\nabla\times\bar{\varphi}_j)\text{d}x
1223 * -
1224 * \int_\Omega \varepsilon_r\varphi_i \cdot \bar{\varphi}_j\text{d}x
1225 * \f}
1226 * and
1227 * \f{align}{
1228 * i\int_\Omega J_a \cdot \bar{\varphi_i}\text{d}x
1229 * - \int_\Omega \mu_r^{-1} \cdot (\nabla \times \bar{\varphi_i})
1230 * \text{d}x.
1231 * \f}
1232 * In doing so, we need test functions @f$\varphi_i@f$ and @f$\varphi_j@f$, and the
1233 * curl of these test variables. We must be careful with the signs of the
1234 * imaginary parts of these complex test variables. Moreover, we have a
1235 * conditional that changes the parameters if the cell is in the PML region.
1236 *
1237 * @code
1238 *   const FEValuesExtractors::Vector real_part(0);
1239 *   const FEValuesExtractors::Vector imag_part(dim);
1240 *   for (const auto &cell : dof_handler.active_cell_iterators())
1241 *   {
1242 *   fe_values.reinit(cell);
1243 *  
1244 *   cell_matrix = 0.;
1245 *   cell_rhs = 0.;
1246 *  
1247 *   cell->get_dof_indices(local_dof_indices);
1248 *   const auto id = cell->material_id();
1249 *  
1250 *   const auto &quadrature_points = fe_values.get_quadrature_points();
1251 *  
1252 *   for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
1253 *   {
1254 *   const Point<dim> &position = quadrature_points[q_point];
1255 *  
1256 *   auto mu_inv = parameters.mu_inv(position, id);
1257 *   auto epsilon = parameters.epsilon(position, id);
1258 *   const auto J_a = parameters.J_a(position, id);
1259 *  
1260 *   const auto A = perfectly_matched_layer.a_matrix(position);
1261 *   const auto B = perfectly_matched_layer.b_matrix(position);
1262 *   const auto d = perfectly_matched_layer.d(position);
1263 *  
1264 *   mu_inv = mu_inv / d;
1265 *   epsilon = invert(A) * epsilon * invert(B);
1266 *  
1267 *   for (const auto i : fe_values.dof_indices())
1268 *   {
1269 *   constexpr std::complex<double> imag{0., 1.};
1270 *  
1271 *   const auto phi_i =
1272 *   fe_values[real_part].value(i, q_point) -
1273 *   imag * fe_values[imag_part].value(i, q_point);
1274 *   const auto curl_phi_i =
1275 *   fe_values[real_part].curl(i, q_point) -
1276 *   imag * fe_values[imag_part].curl(i, q_point);
1277 *  
1278 *   const auto rhs_value =
1279 *   (imag * scalar_product(J_a, phi_i)) * fe_values.JxW(q_point);
1280 *   cell_rhs(i) += rhs_value.real();
1281 *  
1282 *   for (const auto j : fe_values.dof_indices())
1283 *   {
1284 *   const auto phi_j =
1285 *   fe_values[real_part].value(j, q_point) +
1286 *   imag * fe_values[imag_part].value(j, q_point);
1287 *   const auto curl_phi_j =
1288 *   fe_values[real_part].curl(j, q_point) +
1289 *   imag * fe_values[imag_part].curl(j, q_point);
1290 *  
1291 *   const auto temp =
1292 *   (scalar_product(mu_inv * curl_phi_j, curl_phi_i) -
1293 *   scalar_product(epsilon * phi_j, phi_i)) *
1294 *   fe_values.JxW(q_point);
1295 *   cell_matrix(i, j) += temp.real();
1296 *   }
1297 *   }
1298 *   }
1299 *  
1300 * @endcode
1301 *
1302 * Now we assemble the face and the boundary. The following loops will
1303 * assemble
1304 * \f{align*}{
1305 * - i\int_\Sigma (\sigma_r^{\Sigma}(\varphi_i)_T) \cdot
1306 * (\bar{\varphi}_j)_T\text{do}x \f} and \f{align}{
1307 * - i\int_{\partial\Omega} (\sqrt{\mu_r^{-1}\varepsilon}(\varphi_i)_T)
1308 * \cdot (\nabla\times(\bar{\varphi}_j)_T)\text{d}x,
1309 * \f}
1310 * respectively. The test variables and the PML are implemented
1311 * similarly as the domain.
1312 *
1313 * @code
1314 *   const FEValuesExtractors::Vector real_part(0);
1315 *   const FEValuesExtractors::Vector imag_part(dim);
1316 *   for (const auto &face : cell->face_iterators())
1317 *   {
1318 *   if (face->at_boundary())
1319 *   {
1320 *   const auto id = face->boundary_id();
1321 *   if (id != 0)
1322 *   {
1323 *   fe_face_values.reinit(cell, face);
1324 *  
1325 *   for (unsigned int q_point = 0; q_point < n_face_q_points;
1326 *   ++q_point)
1327 *   {
1328 *   const auto &position = quadrature_points[q_point];
1329 *  
1330 *   auto mu_inv = parameters.mu_inv(position, id);
1331 *   auto epsilon = parameters.epsilon(position, id);
1332 *  
1333 *   const auto A =
1334 *   perfectly_matched_layer.a_matrix(position);
1335 *   const auto B =
1336 *   perfectly_matched_layer.b_matrix(position);
1337 *   const auto d = perfectly_matched_layer.d(position);
1338 *  
1339 *   mu_inv = mu_inv / d;
1340 *   epsilon = invert(A) * epsilon * invert(B);
1341 *  
1342 *   const auto normal =
1343 *   fe_face_values.normal_vector(q_point);
1344 *  
1345 *   for (const auto i : fe_face_values.dof_indices())
1346 *   {
1347 *   constexpr std::complex<double> imag{0., 1.};
1348 *  
1349 *   const auto phi_i =
1350 *   fe_face_values[real_part].value(i, q_point) -
1351 *   imag *
1352 *   fe_face_values[imag_part].value(i, q_point);
1353 *   const auto phi_i_T = tangential_part(phi_i, normal);
1354 *  
1355 *   for (const auto j : fe_face_values.dof_indices())
1356 *   {
1357 *   const auto phi_j =
1358 *   fe_face_values[real_part].value(j, q_point) +
1359 *   imag *
1360 *   fe_face_values[imag_part].value(j, q_point);
1361 *   const auto phi_j_T =
1362 *   tangential_part(phi_j, normal) *
1363 *   fe_face_values.JxW(q_point);
1364 *  
1365 *   const auto prod = mu_inv * epsilon;
1366 *   const auto sqrt_prod = prod;
1367 *  
1368 *   const auto temp =
1369 *   -imag * scalar_product((sqrt_prod * phi_j_T),
1370 *   phi_i_T);
1371 *   cell_matrix(i, j) += temp.real();
1372 *   } /* j */
1373 *   } /* i */
1374 *   } /* q_point */
1375 *   }
1376 *   }
1377 *   else
1378 *   {
1379 * @endcode
1380 *
1381 * We are on an interior face:
1382 *
1383 * @code
1384 *   const auto face_index = cell->face_iterator_to_index(face);
1385 *  
1386 *   const auto id1 = cell->material_id();
1387 *   const auto id2 = cell->neighbor(face_index)->material_id();
1388 *  
1389 *   if (id1 == id2)
1390 *   continue; /* skip this face */
1391 *  
1392 *   fe_face_values.reinit(cell, face);
1393 *  
1394 *   for (unsigned int q_point = 0; q_point < n_face_q_points;
1395 *   ++q_point)
1396 *   {
1397 *   const auto &position = quadrature_points[q_point];
1398 *  
1399 *   auto sigma = parameters.sigma(position, id1, id2);
1400 *  
1401 *   const auto B = perfectly_matched_layer.b_matrix(position);
1402 *   const auto C = perfectly_matched_layer.c_matrix(position);
1403 *   sigma = invert(C) * sigma * invert(B);
1404 *  
1405 *   const auto normal = fe_face_values.normal_vector(q_point);
1406 *  
1407 *   for (const auto i : fe_face_values.dof_indices())
1408 *   {
1409 *   constexpr std::complex<double> imag{0., 1.};
1410 *  
1411 *   const auto phi_i =
1412 *   fe_face_values[real_part].value(i, q_point) -
1413 *   imag * fe_face_values[imag_part].value(i, q_point);
1414 *   const auto phi_i_T = tangential_part(phi_i, normal);
1415 *  
1416 *   for (const auto j : fe_face_values.dof_indices())
1417 *   {
1418 *   const auto phi_j =
1419 *   fe_face_values[real_part].value(j, q_point) +
1420 *   imag *
1421 *   fe_face_values[imag_part].value(j, q_point);
1422 *   const auto phi_j_T = tangential_part(phi_j, normal);
1423 *  
1424 *   const auto temp =
1425 *   -imag *
1426 *   scalar_product((sigma * phi_j_T), phi_i_T) *
1427 *   fe_face_values.JxW(q_point);
1428 *   cell_matrix(i, j) += temp.real();
1429 *   } /* j */
1430 *   } /* i */
1431 *   } /* q_point */
1432 *   }
1433 *   }
1434 *  
1435 *   constraints.distribute_local_to_global(
1436 *   cell_matrix, cell_rhs, local_dof_indices, system_matrix, system_rhs);
1437 *   }
1438 *   }
1439 *  
1440 * @endcode
1441 *
1442 * We use a direct solver from the SparseDirectUMFPACK to solve the system
1443 *
1444 * @code
1445 *   template <int dim>
1446 *   void Maxwell<dim>::solve()
1447 *   {
1448 *   SparseDirectUMFPACK A_direct;
1449 *   A_direct.initialize(system_matrix);
1450 *   A_direct.vmult(solution, system_rhs);
1451 *   }
1452 *  
1453 * @endcode
1454 *
1455 * The output is written into a vtk file with 4 components
1456 *
1457 * @code
1458 *   template <int dim>
1459 *   void Maxwell<dim>::output_results()
1460 *   {
1461 *   DataOut<2> data_out;
1462 *   data_out.attach_dof_handler(dof_handler);
1463 *   data_out.add_data_vector(solution,
1464 *   {"real_Ex", "real_Ey", "imag_Ex", "imag_Ey"});
1465 *   data_out.build_patches();
1466 *   std::ofstream output("solution.vtk");
1467 *   data_out.write_vtk(output);
1468 *   }
1469 *  
1470 *  
1471 *   template <int dim>
1472 *   void Maxwell<dim>::run()
1473 *   {
1474 *   make_grid();
1475 *   setup_system();
1476 *   assemble_system();
1477 *   solve();
1478 *   output_results();
1479 *   }
1480 *  
1481 *   } // namespace Step81
1482 *  
1483 * @endcode
1484 *
1485 * The following main function calls the class @ref step_81 "step-81"(), initializes the
1486 * ParameterAcceptor, and calls the run() function.
1487 *
1488
1489 *
1490 *
1491 * @code
1492 *   int main()
1493 *   {
1494 *   try
1495 *   {
1496 *   using namespace dealii;
1497 *  
1498 *   Step81::Maxwell<2> maxwell_2d;
1499 *   ParameterAcceptor::initialize("parameters.prm");
1500 *   maxwell_2d.run();
1501 *   }
1502 *   catch (std::exception &exc)
1503 *   {
1504 *   std::cerr << std::endl
1505 *   << std::endl
1506 *   << "----------------------------------------------------"
1507 *   << std::endl;
1508 *   std::cerr << "Exception on processing: " << std::endl
1509 *   << exc.what() << std::endl
1510 *   << "Aborting!" << std::endl
1511 *   << "----------------------------------------------------"
1512 *   << std::endl;
1513 *   return 1;
1514 *   }
1515 *   catch (...)
1516 *   {
1517 *   std::cerr << std::endl
1518 *   << std::endl
1519 *   << "----------------------------------------------------"
1520 *   << std::endl;
1521 *   std::cerr << "Unknown exception!" << std::endl
1522 *   << "Aborting!" << std::endl
1523 *   << "----------------------------------------------------"
1524 *   << std::endl;
1525 *   return 1;
1526 *   }
1527 *   return 0;
1528 *   }
1529 * @endcode
1530<a name="step_81-Results"></a><h1>Results</h1>
1531
1532
1533The solution is written to a .vtk file with four components. These are the
1534real and imaginary parts of the @f$E_x@f$ and @f$E_y@f$ solution waves. With the
1535current setup, the output should read
1536
1537@code
1538Number of active cells: 4096
1539Number of degrees of freedom: 16640
1540Program ended with exit code: 0
1541@endcode
1542
1543<a name="step_81-AbsorbingboundaryconditionsandthePML"></a><h3> Absorbing boundary conditions and the PML </h3>
1544
1545
1546The following images are the outputs for the imaginary @f$E_x@f$ without the
1547interface and with the dipole centered at @f$(0,0)@f$. In order to remove the
1548interface, the surface conductivity is set to 0. First, we turn off the
1549absorbing boundary conditions and the PML. Second, we want to see the
1550effect of the PML when absorbing boundary conditions apply. So we set
1551absorbing boundary conditions to true and leave the PML strength to 0.
1552Lastly, we increase the strength of the PML to 4. Change the following in
1553the .prm file :
1554
1555@code
1556# use absorbing boundary conditions?
1557 set absorbing boundary condition = false
1558
1559# position of the dipole
1560 set dipole position = 0, 0
1561
1562# strength of the PML
1563 set strength = 0
1564
1565# surface conductivity between material 1 and material 2
1566 set sigma = 0, 0; 0, 0| 0, 0; 0, 0
1567@endcode
1568
1569Following are the output images:
1570
1571<table width="80%" align="center">
1572 <tr>
1573 <td align="center">
1574 <img src="https://www.dealii.org/images/steps/developer/step-81-nointerface_noabs_PML0.png" alt="Visualization of the solution of step-81 with no interface, Dirichlet boundary conditions and PML strength 0" height="210"/>
1575 <p> Solution with no interface, Dirichlet boundary conditions and PML strength 0.</p>
1576 </td>
1577 <td></td>
1578 <td align="center">
1579 <img src="https://www.dealii.org/images/steps/developer/step-81-nointerface_abs_PML0.png" alt="Visualization of the solution of step-81 with no interface, absorbing boundary conditions and PML strength 0" height="210">
1580 <p> Solution with no interface, absorbing boundary conditions and PML strength 0.</p>
1581 </td>
1582 <td></td>
1583 <td align="center">
1584 <img src="https://www.dealii.org/images/steps/developer/step-81-nointerface_abs_PML4.png" alt="Visualization of the solution of step-81 with no interface, absorbing boundary conditions and PML strength 4" height="210">
1585 <p> Solution with no interface, absorbing boundary conditions and PML strength 4.</p>
1586 </td>
1587 </tr>
1588</table>
1589
1590We observe that with absorbing boundary conditions and in absence of the
1591PML, there is a lot of distortion and resonance (the real parts will not be
1592generated without a PML). This is, as we stipulated, due to reflection from
1593infinity. As we see, a much more coherent image is generated with an
1594appropriate PML.
1595
1596<a name="step_81-SurfacePlasmonPolariton"></a><h3> Surface Plasmon Polariton </h3>
1597
1598Now, let's generate a standing wave by adding an interface at the center.
1599In order to observe this effect, we offset the center of the dipole to @f$(0,
16000.8)@f$ and set the surface conductivity back to @f$(0.001, 0.2)@f$:
1601
1602@code
1603# position of the dipole
1604 set dipole position = 0, 0.8
1605
1606# surface conductivity between material 1 and material 2
1607 set sigma = 0.001, 0.2; 0, 0| 0, 0; 0.001, 0.2
1608@endcode
1609
1610Once again, we will visualize the output with absorbing boundary conditions
1611and PML strength 0 and with absorbing boundary conditions and PML strength
16124. The following tables are the imaginary part of @f$E_x@f$ and the real part
1613of @f$E_x@f$.
1614
1615<table width="80%" align="center">
1616 <tr>
1617 <td align="center">
1618 <img src="https://www.dealii.org/images/steps/developer/step-81-imagEx_noabs_PML0.png" alt="Visualization of the solution of step-81 with an interface, absorbing boundary conditions and PML strength 0" height="210">
1619 <p> Solution with an interface, Dirichlet boundary conditions and PML strength 0.</p>
1620 </td>
1621 <td></td>
1622 <td align="center">
1623 <img src="https://www.dealii.org/images/steps/developer/step-81-imagEx_abs_PML0.png" alt="Visualization of the solution of step-81 with an interface, absorbing boundary conditions and PML strength 0" height="210">
1624 <p> Solution with an interface, absorbing boundary conditions and PML strength 0.</p>
1625 </td>
1626 <td></td>
1627 <td align="center">
1628 <img src="https://www.dealii.org/images/steps/developer/step-81-imagEx_abs_PML4.png" alt="Visualization of the solution of step-81 with an interface, absorbing boundary conditions and PML strength 4" height="210">
1629 <p> Solution with an interface, absorbing boundary conditions and PML strength 4.</p>
1630 </td>
1631 </tr>
1632</table>
1633
1634
1635<table width="80%" align="center">
1636 <tr>
1637 <td align="center">
1638 <img src="https://www.dealii.org/images/steps/developer/step-81-realEx_noabs_PML0.png" alt="Visualization of the solution of step-81 with an interface, absorbing boundary conditions and PML strength 0" height="210">
1639 <p> Solution with an interface, Dirichlet boundary conditions and PML strength 0.</p>
1640 </td>
1641 <td></td>
1642 <td align="center">
1643 <img src="https://www.dealii.org/images/steps/developer/step-81-realEx_abs_PML0.png" alt="Visualization of the solution of step-81 with an interface, absorbing boundary conditions and PML strength 0" height="210">
1644 <p> Solution with an interface, absorbing boundary conditions and PML strength 0.</p>
1645 </td>
1646 <td></td>
1647 <td align="center">
1648 <img src="https://www.dealii.org/images/steps/developer/step-81-realEx_abs_PML4.png" alt="Visualization of the solution of step-81 with an interface, absorbing boundary conditions and PML strength 4" height="210">
1649 <p> Solution with an interface, absorbing boundary conditions and PML strength 4.</p>
1650 </td>
1651 </tr>
1652</table>
1653
1654The SPP is confined near the interface that we created, however without
1655absorbing boundary conditions, we don't observe a dissipation effect. On
1656adding the absorbing boundary conditions, we observe distortion and
1657resonance and we still don't notice any dissipation. As expected, the PML
1658removes the distortion and resonance. The standing wave is also dissipating
1659and getting absorbed within the PML, and as we increase the PML strength,
1660the standing wave will dissipate more within the PML ring.
1661
1662Here are some animations to demonstrate the effect of the PML
1663<table width="80%" align="center">
1664 <tr>
1665 <td align="center">
1666 <img src="https://www.dealii.org/images/steps/developer/step-81-dirichlet_Ex.gif" alt="Visualization of the solution of step-81 with an interface, absorbing boundary conditions and PML strength 0" height="210">
1667 <p> Solution with an interface, Dirichlet boundary conditions and PML strength 0.</p>
1668 </td>
1669 <td></td>
1670 <td align="center">
1671 <img src="https://www.dealii.org/images/steps/developer/step-81-absorbing_Ex.gif" alt="Visualization of the solution of step-81 with an interface, absorbing boundary conditions and PML strength 0" height="210">
1672 <p> Solution with an interface, absorbing boundary conditions and PML strength 0.</p>
1673 </td>
1674 <td></td>
1675 <td align="center">
1676 <img src="https://www.dealii.org/images/steps/developer/step-81-perfectly_matched_layer_Ex.gif" alt="Visualization of the solution of step-81 with an interface, absorbing boundary conditions and PML strength 4" height="210">
1677 <p> Solution with an interface, absorbing boundary conditions and PML strength 4.</p>
1678 </td>
1679 </tr>
1680</table>
1681
1682
1683<table width="80%" align="center">
1684 <tr>
1685 <td align="center">
1686 <img src="https://www.dealii.org/images/steps/developer/step-81-dirichlet_Ey.gif" alt="Visualization of the solution of step-81 with an interface, absorbing boundary conditions and PML strength 0" height="210">
1687 <p> Solution with an interface, Dirichlet boundary conditions and PML strength 0.</p>
1688 </td>
1689 <td></td>
1690 <td align="center">
1691 <img src="https://www.dealii.org/images/steps/developer/step-81-absorbing_Ey.gif" alt="Visualization of the solution of step-81 with an interface, absorbing boundary conditions and PML strength 0" height="210">
1692 <p> Solution with an interface, absorbing boundary conditions and PML strength 0.</p>
1693 </td>
1694 <td></td>
1695 <td align="center">
1696 <img src="https://www.dealii.org/images/steps/developer/step-81-perfectly_matched_layer_Ey.gif" alt="Visualization of the solution of step-81 with an interface, absorbing boundary conditions and PML strength 4" height="210">
1697 <p> Solution with an interface, absorbing boundary conditions and PML strength 4.</p>
1698 </td>
1699 </tr>
1700</table>
1701
1702<a name="step_81-Notes"></a><h3> Notes </h3>
1703
1704
1705<a name="step_81-RealandComplexMatrices"></a><h4> Real and Complex Matrices </h4>
1706
1707As is evident from the results, we are splitting our solution matrices into
1708the real and the imaginary components. We started off using the @f$H^{curl}@f$
1709conforming Nédélec Elements, and we made two copies of the Finite Elements
1710in order to represent the real and the imaginary components of our input
1711(FE_NedelecSZ was used instead of FE_Nedelec to avoid the sign conflicts
1712issues present in traditional Nédélec elements). In the assembly, we create
1713two vectors of dimension @f$dim@f$ that assist us in extracting the real and
1714the imaginary components of our finite elements.
1715
1716
1717<a name="step_81-RotationsandScaling"></a><h4> Rotations and Scaling </h4>
1718
1719As we see in our assembly, our finite element is rotated and scaled as
1720follows:
1721
1722@code
1723const auto phi_i = real_part.value(i, q_point) - 1.0i * imag_part.value(i, q_point);
1724@endcode
1725
1726This @f$\phi_i@f$ variable doesn't need to be scaled in this way, we may choose
1727any arbitrary scaling constants @f$a@f$ and @f$b@f$. If we choose this scaling, the
1728@f$\phi_j@f$ must also be modified with the same scaling, as follows:
1729
1730@code
1731const auto phi_i = a*real_part.value(i, q_point) -
1732 bi * imag_part.value(i, q_point);
1733
1734const auto phi_j = a*real_part.value(i, q_point) +
1735 bi * imag_part.value(i, q_point);
1736@endcode
1737
1738Moreover, the cell_rhs need not be the real part of the rhs_value. Say if
1739we modify to take the imaginary part of the computed rhs_value, we must
1740also modify the cell_matrix accordingly to take the imaginary part of temp.
1741However, making these changes to both sides of the equation will not affect
1742our solution, and we will still be able to generate the surface plasmon
1743polariton.
1744
1745@code
1746cell_rhs(i) += rhs_value.imag();
1747
1748cell_matrix(i) += temp.imag();
1749@endcode
1750
1751<a name="step_81-Postprocessing"></a><h4> Postprocessing </h4>
1752
1753We will create a video demonstrating the wave in motion, which is
1754essentially an implementation of @f$e^{-i\omega t}(Re(E) + i*Im(E))@f$ as we
1755increment time. This is done by slightly changing the output function to
1756generate a series of .vtk files, which will represent out solution wave as
1757we increment time. Introduce an input variable @f$t@f$ in the output_results()
1758class as output_results(unsigned int t). Then change the class itself to
1759the following:
1760
1761@code
1762template <int dim>
1763void Maxwell<dim>::output_results(unsigned int t)
1764{
1765 std::cout << "Running step:" << t << std::endl;
1766 DataOut<2> data_out;
1767 data_out.attach_dof_handler(dof_handler);
1768 Vector<double> postprocessed;
1769 postprocessed.reinit(solution);
1770 for (unsigned int i = 0; i < dof_handler.n_dofs(); ++i)
1771 {
1772 if (i % 4 == 0)
1773 {
1774 postprocessed[i] = std::cos(2 * M_PI * 0.04 * t) * solution[i] -
1775 std::sin(2 * M_PI * 0.04 * t) * solution[i + 1];
1776 }
1777 else if (i % 4 == 2)
1778 {
1779 postprocessed[i] = std::cos(2 * M_PI * 0.04 * t) * solution[i] -
1780 std::sin(2 * M_PI * 0.04 * t) * solution[i + 1];
1781 }
1782 }
1783 data_out.add_data_vector(postprocessed, {"E_x", "E_y", "null0", "null1"});
1784 data_out.build_patches();
1785 const std::string filename =
1786 "solution-" + Utilities::int_to_string(t) + ".vtk";
1787 std::ofstream output(filename);
1788 data_out.write_vtk(output);
1789 std::cout << "Done running step:" << t << std::endl;
1790}
1791@endcode
1792
1793Finally, in the run() function, replace output_results() with
1794@code
1795for (int t = 0; t <= 100; t++)
1796 {
1797 output_results(t);
1798 }
1799@endcode
1800
1801This would generate 100 solution .vtk files, which can be opened in a group
1802on Paraview and then can be saved as an animation. We used FFMPEG to
1803generate gifs.
1804
1805<a name="step_81-PossibilitiesforExtension"></a><h3> Possibilities for Extension </h3>
1806
1807
1808The example step could be extended in a number of different directions.
1809<ul>
1810 <li>
1811 The current program uses a direct solver to solve the linear system.
1812 This is efficient for two spatial dimensions where scattering problems
1813 up to a few millions degrees of freedom can be solved. In 3D, however,
1814 the increased stencil size of the Nedelec element pose a severe
1815 limiting factor on the problem size that can be computed. As an
1816 alternative, the idea to use iterative solvers can be entertained.
1817 This, however requires specialized preconditioners. For example, just
1818 using an iterative Krylov space solver (such as SolverGMRES) on above
1819 problem will requires many thousands of iterations to converge.
1820 Unfortunately, time-harmonic Maxwell's equations lack the usual notion
1821 of local smoothing properties, which renders the usual suspects, such
1822 as a geometric multigrid (see the Multigrid class), largely useless. A
1823 possible extension would be to implement an additive Schwarz preconditioner
1824 (based on domain decomposition, see for example
1825 @cite Gopalakrishnan2003), or a sweeping preconditioner (see for
1826 example @cite Ying2012).
1827 </li>
1828 <li>
1829 Another possible extension of the current program is to introduce local
1830 mesh refinement (either based on a residual estimator, or based on the
1831 dual weighted residual method, see @ref step_14 "step-14"). This is in particular of
1832 interest to counter the increased computational cost caused by the
1833 scale separation between the SPP and the dipole.
1834 </li>
1835</ul>
1836 *
1837 *
1838<a name="step_81-PlainProg"></a>
1839<h1> The plain program</h1>
1840@include "step-81.cc"
1841*/
void add_parameter(const std::string &entry, ParameterType &parameter, const std::string &documentation="", ParameterHandler &prm_=prm, const Patterns::PatternBase &pattern= *Patterns::Tools::Convert< ParameterType >::to_pattern())
Definition point.h:111
numbers::NumberTraits< Number >::real_type norm() const
Point< 3 > center
__global__ void set(Number *val, const Number s, const size_type N)
void loop(IteratorType begin, std_cxx20::type_identity_t< IteratorType > 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, AssemblerType &assembler, const LoopControl &lctrl=LoopControl())
Definition loop.h:442
void scale(const double scaling_factor, Triangulation< dim, spacedim > &triangulation)
spacedim const Point< spacedim > & p
Definition grid_tools.h:990
const std::vector< bool > & used
spacedim & mesh
Definition grid_tools.h:989
double norm(const FEValuesBase< dim > &fe, const ArrayView< const std::vector< Tensor< 1, dim > > > &Du)
Definition divergence.h:471
Point< spacedim > point(const gp_Pnt &p, const double tolerance=1e-10)
Definition utilities.cc:191
SymmetricTensor< 2, dim, Number > C(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 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)
const InputIterator OutputIterator const Function & function
Definition parallel.h:168
::VectorizedArray< Number, width > cos(const ::VectorizedArray< Number, width > &)
const ::parallel::distributed::Triangulation< dim, spacedim > * triangulation
DEAL_II_HOST constexpr SymmetricTensor< 2, dim, Number > invert(const SymmetricTensor< 2, dim, Number > &)