825 * Since the Laplacian is
symmetric, the `Tvmult()` (needed by the multigrid
826 * smoother interfaces) operation is simply forwarded to the `vmult()`
case.
832 *
template <
int dim,
int fe_degree,
typename number>
833 *
void LaplaceOperator<dim, fe_degree, number>::Tvmult(
844 * The cell operation is very similar to @ref step_37
"step-37". We
do not use a
845 * coefficient here, though. The
second difference is that we replaced the
848 *
FEEvaluation::gather_evaluate() which internally calls the sequence of
849 * the two individual methods. Likewise,
FEEvaluation::integrate_scatter()
850 * implements the sequence of
FEEvaluation::integrate() followed by
851 *
FEEvaluation::distribute_local_to_global(). In this case, these new
852 *
functions merely save two lines of code. However, we use them for the
860 * template <
int dim,
int fe_degree, typename number>
861 *
void LaplaceOperator<dim, fe_degree, number>::apply_cell(
865 * const
std::pair<
unsigned int,
unsigned int> & cell_range) const
868 *
for (
unsigned int cell = cell_range.first;
cell < cell_range.second; ++
cell)
872 *
for (
unsigned int q = 0; q < phi.n_q_points; ++q)
873 * phi.submit_gradient(phi.get_gradient(q), q);
882 * The face operation implements the terms of the interior penalty method in
883 * analogy to @ref step_39
"step-39", as explained in the introduction. We need two
884 * evaluator objects
for this task,
one for handling the solution that comes
885 * from the cell on
one of the two sides of an interior face, and
one for
886 * handling the solution from the other side. The evaluators
for face
888 *
second slot of the constructor to indicate which of the two sides the
890 *
one of the two sides the `interior`
one and the other the `exterior`
891 *
one. The name `exterior` refers to the fact that the evaluator from both
892 * sides will
return the same normal vector. For the `interior` side, the
893 * normal vector points outwards, whereas it points inwards on the other
894 * side, and is opposed to the outer normal vector of that cell. Apart from
895 * the
new class name, we again get a range of items to work with in
896 * analogy to what was discussed in @ref step_37
"step-37", but
for the interior faces in
897 *
this case. Note that the data structure of
MatrixFree forms batches of
898 * faces that are analogous to the batches of cells
for the cell
899 * integrals. All faces within a batch involve different cell
numbers but
900 * have the face number within the reference cell, have the same refinement
901 * configuration (no refinement or the same subface), and the same
902 * orientation, to keep SIMD operations simple and efficient.
906 * Note that there is no implied meaning in interior versus exterior except
907 * the logic decision of the orientation of the normal, which is pretty
908 *
random internally. One can in no way rely on a certain pattern of
909 * assigning interior versus exterior flags, as the decision is made
for the
910 * sake of access regularity and uniformity in the
MatrixFree setup
911 * routines. Since most sane DG methods are conservative, i.e., fluxes look
912 * the same from both sides of an interface, the mathematics are unaltered
913 *
if the interior/exterior flags are switched and normal vectors get the
920 *
template <
int dim,
int fe_degree,
typename number>
921 *
void LaplaceOperator<dim, fe_degree, number>::apply_face(
925 *
const std::pair<unsigned int, unsigned int> & face_range)
const
931 *
for (
unsigned int face = face_range.first; face < face_range.second; ++face)
935 * On a given batch of faces, we
first update the pointers to the
936 * current face and then access the vector. As mentioned above, we
937 * combine the vector access with the evaluation. In the
case of face
938 * integrals, the data access into the vector can be reduced
for the
939 * special
case of an
FE_DGQHermite basis as explained
for the data
940 * exchange above: Since only @f$2(k+1)^{
d-1}@f$ out of the @f$(k+1)^
d@f$ cell
941 * degrees of freedom get multiplied by a non-
zero value or derivative
942 * of a shape function,
this structure can be utilized
for the
943 * evaluation, significantly reducing the data access. The
reduction
944 * of the data access is not only beneficial because it reduces the
945 * data in flight and thus helps caching, but also because the data
946 * access to faces is often more irregular than
for cell integrals when
947 * gathering
values from cells that are farther apart in the index
951 * phi_inner.reinit(face);
952 * phi_inner.gather_evaluate(src,
955 * phi_outer.reinit(face);
956 * phi_outer.gather_evaluate(src,
962 * The next two statements compute the penalty parameter
for the
963 * interior penalty method. As explained in the introduction, we would
964 * like to have a scaling like @f$\frac{1}{h_\text{i}}@f$ of the length
965 * @f$h_\text{i}@f$ normal to the face. For a
general non-Cartesian mesh,
966 *
this length must be computed by the product of the inverse Jacobian
967 * times the normal vector in real coordinates. From
this vector of
968 * `dim` components, we must
finally pick the component that is
969 * oriented normal to the reference cell. In the geometry data stored
970 * in
MatrixFree, a permutation of the components in the Jacobian is
971 * applied such that
this latter direction is
always the last
972 * component `dim-1` (
this is beneficial because reference-cell
973 * derivative sorting can be made agnostic of the direction of the
974 * face). This means that we can simply access the last entry `dim-1`
975 * and must not look up the local face number in
976 * `data.get_face_info(face).interior_face_no` and
977 * `data.get_face_info(face).exterior_face_no`. Finally, we must also
978 * take the absolute
value of these factors as the normal could
point
979 * into either positive or negative direction.
983 * 0.5 * (
std::abs((phi_inner.get_normal_vector(0) *
984 * phi_inner.inverse_jacobian(0))[dim - 1]) +
985 *
std::abs((phi_outer.get_normal_vector(0) *
986 * phi_outer.inverse_jacobian(0))[dim - 1]));
988 * inverse_length_normal_to_face * get_penalty_factor();
992 * In the
loop over the quadrature points, we eventually compute all
993 * contributions to the interior penalty scheme. According to the
994 * formulas in the introduction, the
value of the test function gets
995 * multiplied by the difference of the jump in the solution times the
996 * penalty parameter and the average of the normal derivative in real
997 * space. Since the two evaluators
for interior and exterior sides get
998 * different signs due to the jump, we pass the result with a
999 * different
sign here. The normal derivative of the test function
1000 * gets multiplied by the negative jump in the solution between the
1001 * interior and exterior side. This term, coined adjoint consistency
1002 * term, must also include the factor of @f$\frac{1}{2}@f$ in the code in
1003 * accordance with its relation to the primal consistency term that
1004 * gets the factor of
one half due to the average in the test function
1008 *
for (
unsigned int q = 0; q < phi_inner.n_q_points; ++q)
1011 * (phi_inner.get_value(q) - phi_outer.get_value(q));
1013 * (phi_inner.get_normal_derivative(q) +
1014 * phi_outer.get_normal_derivative(q)) *
1017 * solution_jump * sigma - average_normal_derivative;
1019 * phi_inner.submit_value(test_by_value, q);
1020 * phi_outer.submit_value(-test_by_value, q);
1022 * phi_inner.submit_normal_derivative(-solution_jump * number(0.5), q);
1023 * phi_outer.submit_normal_derivative(-solution_jump * number(0.5), q);
1028 * Once we are done with the
loop over quadrature points, we can
do
1029 * the
sum factorization operations
for the integration loops on faces
1030 * and
sum the results into the result vector,
using the
1031 * `integrate_scatter` function. The name `scatter` reflects the
1032 * distribution of the vector data into scattered positions in the
1033 * vector
using the same pattern as in `gather_evaluate`. Like before,
1034 * the combined integrate + write operation allows us to reduce the
1051 * The boundary face function follows by and large the interior face
1052 * function. The only difference is the fact that we
do not have a separate
1054 * we must define them from the boundary conditions and interior
values
1055 * @f$u^-@f$. As explained in the introduction, we use @f$u^+ = -u^- + 2
1056 * g_\text{D}@f$ and @f$\mathbf{n}^-\cdot \nabla u^+ = \mathbf{n}^-\cdot \nabla
1057 * u^-@f$ on Dirichlet boundaries and @f$u^+=u^-@f$ and @f$\mathbf{n}^-\cdot \nabla
1058 * u^+ = -\mathbf{n}^-\cdot \nabla u^- + 2 g_\text{
N}@f$ on Neumann
1059 * boundaries. Since
this operation implements the homogeneous part, i.e.,
1061 * @f$g_\text{D}@f$ and @f$g_\text{
N}@f$ here, and added them to the right hand side
1062 * in `LaplaceProblem::compute_rhs()`. Note that due to extension of the
1063 * solution @f$u^-@f$ to the exterior via @f$u^+@f$, we can keep all factors @f$0.5@f$
1064 * the same as in the inner face function, see also the discussion in
1065 * @ref step_39
"step-39".
1069 * There is
one catch at
this point: The implementation below uses a
boolean
1070 * variable `is_dirichlet` to
switch between the Dirichlet and the Neumann
1071 * cases. However, we solve a problem where we also want to impose periodic
1072 * boundary conditions on some boundaries, namely along those in the @f$x@f$
1073 * direction. One might wonder how those conditions should be handled
1074 * here. The answer is that
MatrixFree automatically treats periodic
1075 * boundaries as what they are technically, namely an inner face where the
1076 * solution
values of two adjacent cells meet and must be treated by proper
1077 * numerical fluxes. Thus, all the faces on the periodic boundaries will
1078 * appear in the `apply_face()` function and not in
this one.
1084 *
template <
int dim,
int fe_degree,
typename number>
1085 *
void LaplaceOperator<dim, fe_degree, number>::apply_boundary(
1089 *
const std::pair<unsigned int, unsigned int> & face_range)
const
1093 *
for (
unsigned int face = face_range.first; face < face_range.second; ++face)
1095 * phi_inner.reinit(face);
1096 * phi_inner.gather_evaluate(src,
1101 *
std::abs((phi_inner.get_normal_vector(0) *
1102 * phi_inner.inverse_jacobian(0))[dim - 1]);
1104 * inverse_length_normal_to_face * get_penalty_factor();
1106 *
const bool is_dirichlet = (data.get_boundary_id(face) == 0);
1108 *
for (
unsigned int q = 0; q < phi_inner.n_q_points; ++q)
1112 * is_dirichlet ? -u_inner : u_inner;
1114 * phi_inner.get_normal_derivative(q);
1116 * is_dirichlet ? normal_derivative_inner : -normal_derivative_inner;
1119 * (normal_derivative_inner + normal_derivative_outer) * number(0.5);
1121 * solution_jump * sigma - average_normal_derivative;
1122 * phi_inner.submit_normal_derivative(-solution_jump * number(0.5), q);
1123 * phi_inner.submit_value(test_by_value, q);
1135 * Next we turn to the preconditioner initialization. As explained in the
1136 * introduction, we want to construct an (
approximate) inverse of the cell
1137 * matrices from a product of 1D mass and Laplace matrices. Our
first task
1138 * is to compute the 1D matrices, which we
do by
first creating a 1D finite
1139 * element. Instead of anticipating
FE_DGQHermite<1> here, we get the finite
1140 * element
's name from DoFHandler, replace the @p dim argument (2 or 3) by 1
1141 * to create a 1D name, and construct the 1D element by using FETools.
1147 * template <int dim, int fe_degree, typename number>
1148 * void PreconditionBlockJacobi<dim, fe_degree, number>::initialize(
1149 * const LaplaceOperator<dim, fe_degree, number> &op)
1151 * data = op.get_matrix_free();
1153 * std::string name = data->get_dof_handler().get_fe().get_name();
1154 * name.replace(name.find('<
') + 1, 1, "1");
1155 * std::unique_ptr<FiniteElement<1>> fe_1d = FETools::get_fe_by_name<1>(name);
1159 * As for computing the 1D matrices on the unit element, we simply write
1160 * down what a typical assembly procedure over rows and columns of the
1161 * matrix as well as the quadrature points would do. We select the same
1162 * Laplace matrices once and for all using the coefficients 0.5 for
1163 * interior faces (but possibly scaled differently in different directions
1164 * as a result of the mesh). Thus, we make a slight mistake at the
1165 * Dirichlet boundary (where the correct factor would be 1 for the
1166 * derivative terms and 2 for the penalty term, see @ref step_39 "step-39") or at the
1167 * Neumann boundary where the factor should be zero. Since we only use
1168 * this class as a smoother inside a multigrid scheme, this error is not
1169 * going to have any significant effect and merely affects smoothing
1173 * const unsigned int N = fe_degree + 1;
1174 * FullMatrix<double> laplace_unscaled(N, N);
1175 * std::array<Table<2, VectorizedArray<number>>, dim> mass_matrices;
1176 * std::array<Table<2, VectorizedArray<number>>, dim> laplace_matrices;
1177 * for (unsigned int d = 0; d < dim; ++d)
1179 * mass_matrices[d].reinit(N, N);
1180 * laplace_matrices[d].reinit(N, N);
1183 * QGauss<1> quadrature(N);
1184 * for (unsigned int i = 0; i < N; ++i)
1185 * for (unsigned int j = 0; j < N; ++j)
1187 * double sum_mass = 0, sum_laplace = 0;
1188 * for (unsigned int q = 0; q < quadrature.size(); ++q)
1190 * sum_mass += (fe_1d->shape_value(i, quadrature.point(q)) *
1191 * fe_1d->shape_value(j, quadrature.point(q))) *
1192 * quadrature.weight(q);
1193 * sum_laplace += (fe_1d->shape_grad(i, quadrature.point(q))[0] *
1194 * fe_1d->shape_grad(j, quadrature.point(q))[0]) *
1195 * quadrature.weight(q);
1197 * for (unsigned int d = 0; d < dim; ++d)
1198 * mass_matrices[d](i, j) = sum_mass;
1202 * The left and right boundary terms assembled by the next two
1203 * statements appear to have somewhat arbitrary signs, but those are
1204 * correct as can be verified by looking at @ref step_39 "step-39" and inserting
1205 * the value -1 and 1 for the normal vector in the 1D case.
1209 * (1. * fe_1d->shape_value(i, Point<1>()) *
1210 * fe_1d->shape_value(j, Point<1>()) * op.get_penalty_factor() +
1211 * 0.5 * fe_1d->shape_grad(i, Point<1>())[0] *
1212 * fe_1d->shape_value(j, Point<1>()) +
1213 * 0.5 * fe_1d->shape_grad(j, Point<1>())[0] *
1214 * fe_1d->shape_value(i, Point<1>()));
1217 * (1. * fe_1d->shape_value(i, Point<1>(1.0)) *
1218 * fe_1d->shape_value(j, Point<1>(1.0)) * op.get_penalty_factor() -
1219 * 0.5 * fe_1d->shape_grad(i, Point<1>(1.0))[0] *
1220 * fe_1d->shape_value(j, Point<1>(1.0)) -
1221 * 0.5 * fe_1d->shape_grad(j, Point<1>(1.0))[0] *
1222 * fe_1d->shape_value(i, Point<1>(1.0)));
1224 * laplace_unscaled(i, j) = sum_laplace;
1229 * Next, we go through the cells and pass the scaled matrices to
1230 * TensorProductMatrixSymmetricSum to actually compute the generalized
1231 * eigenvalue problem for representing the inverse: Since the matrix
1232 * approximation is constructed as @f$A\otimes M + M\otimes A@f$ and the
1233 * weights are constant for each element, we can apply all weights on the
1234 * Laplace matrix and simply keep the mass matrices unscaled. In the loop
1235 * over cells, we want to make use of the geometry compression provided by
1236 * the MatrixFree class and check if the current geometry is the same as
1237 * on the last cell batch, in which case there is nothing to do. This
1238 * compression can be accessed by
1239 * FEEvaluation::get_mapping_data_index_offset() once `reinit()` has been
1244 * Once we have accessed the inverse Jacobian through the FEEvaluation
1245 * access function (we take the one for the zeroth quadrature point as
1246 * they should be the same on all quadrature points for a Cartesian cell),
1247 * we check that it is diagonal and then extract the determinant of the
1248 * original Jacobian, i.e., the inverse of the determinant of the inverse
1249 * Jacobian, and set the weight as @f$\text{det}(J) / h_d^2@f$ according to
1250 * the 1D Laplacian times @f$d-1@f$ copies of the mass matrix.
1253 * cell_matrices.clear();
1254 * FEEvaluation<dim, fe_degree, fe_degree + 1, 1, number> phi(*data);
1255 * unsigned int old_mapping_data_index = numbers::invalid_unsigned_int;
1256 * for (unsigned int cell = 0; cell < data->n_cell_batches(); ++cell)
1260 * if (phi.get_mapping_data_index_offset() == old_mapping_data_index)
1263 * Tensor<2, dim, VectorizedArray<number>> inverse_jacobian =
1264 * phi.inverse_jacobian(0);
1266 * for (unsigned int d = 0; d < dim; ++d)
1267 * for (unsigned int e = 0; e < dim; ++e)
1269 * for (unsigned int v = 0; v < VectorizedArray<number>::size(); ++v)
1270 * AssertThrow(inverse_jacobian[d][e][v] == 0.,
1271 * ExcNotImplemented());
1273 * VectorizedArray<number> jacobian_determinant = inverse_jacobian[0][0];
1274 * for (unsigned int e = 1; e < dim; ++e)
1275 * jacobian_determinant *= inverse_jacobian[e][e];
1276 * jacobian_determinant = 1. / jacobian_determinant;
1278 * for (unsigned int d = 0; d < dim; ++d)
1280 * const VectorizedArray<number> scaling_factor =
1281 * inverse_jacobian[d][d] * inverse_jacobian[d][d] *
1282 * jacobian_determinant;
1286 * Once we know the factor by which we should scale the Laplace
1287 * matrix, we apply this weight to the unscaled DG Laplace matrix
1288 * and send the array to the class TensorProductMatrixSymmetricSum
1289 * for computing the generalized eigenvalue problem mentioned in
1296 * for (unsigned int i = 0; i < N; ++i)
1297 * for (unsigned int j = 0; j < N; ++j)
1298 * laplace_matrices[d](i, j) =
1299 * scaling_factor * laplace_unscaled(i, j);
1301 * if (cell_matrices.size() <= phi.get_mapping_data_index_offset())
1302 * cell_matrices.resize(phi.get_mapping_data_index_offset() + 1);
1303 * cell_matrices[phi.get_mapping_data_index_offset()].reinit(
1304 * mass_matrices, laplace_matrices);
1312 * The vmult function for the approximate block-Jacobi preconditioner is
1313 * very simple in the DG context: We simply need to read the values of the
1314 * current cell batch, apply the inverse for the given entry in the array of
1315 * tensor product matrix, and write the result back. In this loop, we
1316 * overwrite the content in `dst` rather than first setting the entries to
1317 * zero. This is legitimate for a DG method because every cell has
1318 * independent degrees of freedom. Furthermore, we manually write out the
1319 * loop over all cell batches, rather than going through
1320 * MatrixFree::cell_loop(). We do this because we know that we are not going
1321 * to need data exchange over the MPI network here as all computations are
1322 * done on the cells held locally on each processor.
1328 * template <int dim, int fe_degree, typename number>
1329 * void PreconditionBlockJacobi<dim, fe_degree, number>::vmult(
1330 * LinearAlgebra::distributed::Vector<number> & dst,
1331 * const LinearAlgebra::distributed::Vector<number> &src) const
1333 * adjust_ghost_range_if_necessary(*data, dst);
1334 * adjust_ghost_range_if_necessary(*data, src);
1336 * FEEvaluation<dim, fe_degree, fe_degree + 1, 1, number> phi(*data);
1337 * for (unsigned int cell = 0; cell < data->n_cell_batches(); ++cell)
1340 * phi.read_dof_values(src);
1341 * cell_matrices[phi.get_mapping_data_index_offset()].apply_inverse(
1342 * ArrayView<VectorizedArray<number>>(phi.begin_dof_values(),
1343 * phi.dofs_per_cell),
1344 * ArrayView<const VectorizedArray<number>>(phi.begin_dof_values(),
1345 * phi.dofs_per_cell));
1346 * phi.set_dof_values(dst);
1354 * The definition of the LaplaceProblem class is very similar to
1355 * @ref step_37 "step-37". One difference is the fact that we add the element degree as a
1356 * template argument to the class, which would allow us to more easily
1357 * include more than one degree in the same program by creating different
1358 * instances in the `main()` function. The second difference is the
1359 * selection of the element, FE_DGQHermite, which is specialized for this
1360 * kind of equations.
1366 * template <int dim, int fe_degree>
1367 * class LaplaceProblem
1374 * void setup_system();
1375 * void compute_rhs();
1377 * void analyze_results() const;
1379 * #ifdef DEAL_II_WITH_P4EST
1380 * parallel::distributed::Triangulation<dim> triangulation;
1382 * Triangulation<dim> triangulation;
1385 * FE_DGQHermite<dim> fe;
1386 * DoFHandler<dim> dof_handler;
1388 * MappingQ1<dim> mapping;
1390 * using SystemMatrixType = LaplaceOperator<dim, fe_degree, double>;
1391 * SystemMatrixType system_matrix;
1393 * using LevelMatrixType = LaplaceOperator<dim, fe_degree, float>;
1394 * MGLevelObject<LevelMatrixType> mg_matrices;
1396 * LinearAlgebra::distributed::Vector<double> solution;
1397 * LinearAlgebra::distributed::Vector<double> system_rhs;
1399 * double setup_time;
1400 * ConditionalOStream pcout;
1401 * ConditionalOStream time_details;
1406 * template <int dim, int fe_degree>
1407 * LaplaceProblem<dim, fe_degree>::LaplaceProblem()
1409 * #ifdef DEAL_II_WITH_P4EST
1412 * Triangulation<dim>::limit_level_difference_at_vertices,
1413 * parallel::distributed::Triangulation<dim>::construct_multigrid_hierarchy)
1416 * triangulation(Triangulation<dim>::limit_level_difference_at_vertices)
1420 * , dof_handler(triangulation)
1422 * , pcout(std::cout, Utilities::MPI::this_mpi_process(MPI_COMM_WORLD) == 0)
1423 * , time_details(std::cout,
1425 * Utilities::MPI::this_mpi_process(MPI_COMM_WORLD) == 0)
1432 * The setup function differs in two aspects from @ref step_37 "step-37". The first is that
1433 * we do not need to interpolate any constraints for the discontinuous
1434 * ansatz space, and simply pass a dummy AffineConstraints object into
1435 * Matrixfree::reinit(). The second change arises because we need to tell
1436 * MatrixFree to also initialize the data structures for faces. We do this
1437 * by setting update flags for the inner and boundary faces,
1438 * respectively. On the boundary faces, we need both the function values,
1439 * their gradients, JxW values (for integration), the normal vectors, and
1440 * quadrature points (for the evaluation of the boundary conditions),
1441 * whereas we only need shape function values, gradients, JxW values, and
1442 * normal vectors for interior faces. The face data structures in MatrixFree
1443 * are always built as soon as one of `mapping_update_flags_inner_faces` or
1444 * `mapping_update_flags_boundary_faces` are different from the default
1445 * value `update_default` of UpdateFlags.
1451 * template <int dim, int fe_degree>
1452 * void LaplaceProblem<dim, fe_degree>::setup_system()
1457 * system_matrix.clear();
1458 * mg_matrices.clear_elements();
1460 * dof_handler.distribute_dofs(fe);
1461 * dof_handler.distribute_mg_dofs();
1463 * pcout << "Number of degrees of freedom: " << dof_handler.n_dofs()
1466 * setup_time += time.wall_time();
1467 * time_details << "Distribute DoFs " << time.wall_time() << " s"
1471 * AffineConstraints<double> dummy;
1475 * typename MatrixFree<dim, double>::AdditionalData additional_data;
1476 * additional_data.tasks_parallel_scheme =
1477 * MatrixFree<dim, double>::AdditionalData::none;
1478 * additional_data.mapping_update_flags =
1479 * (update_gradients | update_JxW_values | update_quadrature_points);
1480 * additional_data.mapping_update_flags_inner_faces =
1481 * (update_gradients | update_JxW_values | update_normal_vectors);
1482 * additional_data.mapping_update_flags_boundary_faces =
1483 * (update_gradients | update_JxW_values | update_normal_vectors |
1484 * update_quadrature_points);
1485 * const auto system_mf_storage =
1486 * std::make_shared<MatrixFree<dim, double>>();
1487 * system_mf_storage->reinit(
1488 * mapping, dof_handler, dummy, QGauss<1>(fe.degree + 1), additional_data);
1489 * system_matrix.initialize(system_mf_storage);
1492 * system_matrix.initialize_dof_vector(solution);
1493 * system_matrix.initialize_dof_vector(system_rhs);
1495 * setup_time += time.wall_time();
1496 * time_details << "Setup matrix-free system " << time.wall_time() << " s"
1500 * const unsigned int nlevels = triangulation.n_global_levels();
1501 * mg_matrices.resize(0, nlevels - 1);
1503 * for (unsigned int level = 0; level < nlevels; ++level)
1505 * typename MatrixFree<dim, float>::AdditionalData additional_data;
1506 * additional_data.tasks_parallel_scheme =
1507 * MatrixFree<dim, float>::AdditionalData::none;
1508 * additional_data.mapping_update_flags =
1509 * (update_gradients | update_JxW_values);
1510 * additional_data.mapping_update_flags_inner_faces =
1511 * (update_gradients | update_JxW_values);
1512 * additional_data.mapping_update_flags_boundary_faces =
1513 * (update_gradients | update_JxW_values);
1514 * additional_data.mg_level = level;
1515 * const auto mg_mf_storage_level =
1516 * std::make_shared<MatrixFree<dim, float>>();
1517 * mg_mf_storage_level->reinit(mapping,
1520 * QGauss<1>(fe.degree + 1),
1523 * mg_matrices[level].initialize(mg_mf_storage_level);
1525 * setup_time += time.wall_time();
1526 * time_details << "Setup matrix-free levels " << time.wall_time() << " s"
1534 * The computation of the right hand side is a bit more complicated than in
1535 * @ref step_37 "step-37". The cell term now consists of the negative Laplacian of the
1536 * analytical solution, `RightHandSide`, for which we need to first split up
1537 * the Point of VectorizedArray fields, i.e., a batch of points, into a
1538 * single point by evaluating all lanes in the VectorizedArray
1539 * separately. Remember that the number of lanes depends on the hardware; it
1540 * could be 1 for systems that do not offer vectorization (or where deal.II
1541 * does not have intrinsics), but it could also be 8 or 16 on AVX-512 of
1542 * recent Intel architectures.
1545 * template <int dim, int fe_degree>
1546 * void LaplaceProblem<dim, fe_degree>::compute_rhs()
1550 * const MatrixFree<dim, double> &data = *system_matrix.get_matrix_free();
1551 * FEEvaluation<dim, fe_degree> phi(data);
1552 * RightHandSide<dim> rhs_func;
1553 * Solution<dim> exact_solution;
1554 * for (unsigned int cell = 0; cell < data.n_cell_batches(); ++cell)
1557 * for (unsigned int q = 0; q < phi.n_q_points; ++q)
1559 * VectorizedArray<double> rhs_val = VectorizedArray<double>();
1560 * Point<dim, VectorizedArray<double>> point_batch =
1561 * phi.quadrature_point(q);
1562 * for (unsigned int v = 0; v < VectorizedArray<double>::size(); ++v)
1564 * Point<dim> single_point;
1565 * for (unsigned int d = 0; d < dim; ++d)
1566 * single_point[d] = point_batch[d][v];
1567 * rhs_val[v] = rhs_func.value(single_point);
1569 * phi.submit_value(rhs_val, q);
1571 * phi.integrate_scatter(EvaluationFlags::values, system_rhs);
1576 * Secondly, we also need to apply the Dirichlet and Neumann boundary
1577 * conditions. This function is the missing part of to the function
1578 * `LaplaceOperator::apply_boundary()` function once the exterior solution
1579 * values @f$u^+ = -u^- + 2 g_\text{D}@f$ and @f$\mathbf{n}^-\cdot \nabla u^+ =
1580 * \mathbf{n}^-\cdot \nabla u^-@f$ on Dirichlet boundaries and @f$u^+=u^-@f$ and
1581 * @f$\mathbf{n}^-\cdot \nabla u^+ = -\mathbf{n}^-\cdot \nabla u^- + 2
1582 * g_\text{N}@f$ on Neumann boundaries are inserted and expanded in terms of
1583 * the boundary functions @f$g_\text{D}@f$ and @f$g_\text{N}@f$. One thing to
1584 * remember is that we move the boundary conditions to the right hand
1585 * side, so the sign is the opposite from what we imposed on the solution
1590 * We could have issued both the cell and the boundary part through a
1591 * MatrixFree::loop part, but we choose to manually write the full loop
1592 * over all faces to learn how the index layout of face indices is set up
1593 * in MatrixFree: Both the inner faces and the boundary faces share the
1594 * index range, and all batches of inner faces have lower numbers than the
1595 * batches of boundary cells. A single index for both variants allows us
1596 * to easily use the same data structure FEFaceEvaluation for both cases
1597 * that attaches to the same data field, just at different positions. The
1598 * number of inner face batches (where a batch is due to the combination
1599 * of several faces into one for vectorization) is given by
1600 * MatrixFree::n_inner_face_batches(), whereas the number of boundary face
1601 * batches is given by MatrixFree::n_boundary_face_batches().
1604 * FEFaceEvaluation<dim, fe_degree> phi_face(data, true);
1605 * for (unsigned int face = data.n_inner_face_batches();
1606 * face < data.n_inner_face_batches() + data.n_boundary_face_batches();
1609 * phi_face.reinit(face);
1611 * const VectorizedArray<double> inverse_length_normal_to_face =
1612 * std::abs((phi_face.get_normal_vector(0) *
1613 * phi_face.inverse_jacobian(0))[dim - 1]);
1614 * const VectorizedArray<double> sigma =
1615 * inverse_length_normal_to_face * system_matrix.get_penalty_factor();
1617 * for (unsigned int q = 0; q < phi_face.n_q_points; ++q)
1619 * VectorizedArray<double> test_value = VectorizedArray<double>(),
1620 * test_normal_derivative =
1621 * VectorizedArray<double>();
1622 * Point<dim, VectorizedArray<double>> point_batch =
1623 * phi_face.quadrature_point(q);
1625 * for (unsigned int v = 0; v < VectorizedArray<double>::size(); ++v)
1627 * Point<dim> single_point;
1628 * for (unsigned int d = 0; d < dim; ++d)
1629 * single_point[d] = point_batch[d][v];
1633 * The MatrixFree class lets us query the boundary_id of the
1634 * current face batch. Remember that MatrixFree sets up the
1635 * batches for vectorization such that all faces within a
1636 * batch have the same properties, which includes their
1637 * `boundary_id`. Thus, we can query that id here for the
1638 * current face index `face` and either impose the Dirichlet
1639 * case (where we add something to the function value) or the
1640 * Neumann case (where we add something to the normal
1644 * if (data.get_boundary_id(face) == 0)
1645 * test_value[v] = 2.0 * exact_solution.value(single_point);
1648 * Tensor<1, dim> normal;
1649 * for (unsigned int d = 0; d < dim; ++d)
1650 * normal[d] = phi_face.get_normal_vector(q)[d][v];
1651 * test_normal_derivative[v] =
1652 * -normal * exact_solution.gradient(single_point);
1655 * phi_face.submit_value(test_value * sigma - test_normal_derivative,
1657 * phi_face.submit_normal_derivative(-0.5 * test_value, q);
1659 * phi_face.integrate_scatter(EvaluationFlags::values |
1660 * EvaluationFlags::gradients,
1666 * Since we have manually run the loop over cells rather than using
1667 * MatrixFree::loop(), we must not forget to perform the data exchange
1668 * with MPI - or actually, we would not need that for DG elements here
1669 * because each cell carries its own degrees of freedom and cell and
1670 * boundary integrals only evaluate quantities on the locally owned
1671 * cells. The coupling to neighboring subdomain only comes in by the inner
1672 * face integrals, which we have not done here. That said, it does not
1673 * hurt to call this function here, so we do it as a reminder of what
1674 * happens inside MatrixFree::loop().
1677 * system_rhs.compress(VectorOperation::add);
1678 * setup_time += time.wall_time();
1679 * time_details << "Compute right hand side " << time.wall_time()
1687 * The `solve()` function is copied almost verbatim from @ref step_37 "step-37". We set up
1688 * the same multigrid ingredients, namely the level transfer, a smoother,
1689 * and a coarse grid solver. The only difference is the fact that we do not
1690 * use the diagonal of the Laplacian for the preconditioner of the Chebyshev
1691 * iteration used for smoothing, but instead our newly resolved class
1692 * `%PreconditionBlockJacobi`. The mechanisms are the same, though.
1695 * template <int dim, int fe_degree>
1696 * void LaplaceProblem<dim, fe_degree>::solve()
1699 * MGTransferMatrixFree<dim, float> mg_transfer;
1700 * mg_transfer.build(dof_handler);
1701 * setup_time += time.wall_time();
1702 * time_details << "MG build transfer time " << time.wall_time()
1706 * using SmootherType =
1707 * PreconditionChebyshev<LevelMatrixType,
1708 * LinearAlgebra::distributed::Vector<float>,
1709 * PreconditionBlockJacobi<dim, fe_degree, float>>;
1710 * mg::SmootherRelaxation<SmootherType,
1711 * LinearAlgebra::distributed::Vector<float>>
1713 * MGLevelObject<typename SmootherType::AdditionalData> smoother_data;
1714 * smoother_data.resize(0, triangulation.n_global_levels() - 1);
1715 * for (unsigned int level = 0; level < triangulation.n_global_levels();
1720 * smoother_data[level].smoothing_range = 15.;
1721 * smoother_data[level].degree = 3;
1722 * smoother_data[level].eig_cg_n_iterations = 10;
1726 * smoother_data[0].smoothing_range = 2e-2;
1727 * smoother_data[0].degree = numbers::invalid_unsigned_int;
1728 * smoother_data[0].eig_cg_n_iterations = mg_matrices[0].m();
1730 * smoother_data[level].preconditioner =
1731 * std::make_shared<PreconditionBlockJacobi<dim, fe_degree, float>>();
1732 * smoother_data[level].preconditioner->initialize(mg_matrices[level]);
1734 * mg_smoother.initialize(mg_matrices, smoother_data);
1736 * MGCoarseGridApplySmoother<LinearAlgebra::distributed::Vector<float>>
1738 * mg_coarse.initialize(mg_smoother);
1740 * mg::Matrix<LinearAlgebra::distributed::Vector<float>> mg_matrix(
1743 * Multigrid<LinearAlgebra::distributed::Vector<float>> mg(
1744 * mg_matrix, mg_coarse, mg_transfer, mg_smoother, mg_smoother);
1746 * PreconditionMG<dim,
1747 * LinearAlgebra::distributed::Vector<float>,
1748 * MGTransferMatrixFree<dim, float>>
1749 * preconditioner(dof_handler, mg, mg_transfer);
1751 * SolverControl solver_control(10000, 1e-12 * system_rhs.l2_norm());
1752 * SolverCG<LinearAlgebra::distributed::Vector<double>> cg(solver_control);
1753 * setup_time += time.wall_time();
1754 * time_details << "MG build smoother time " << time.wall_time()
1756 * pcout << "Total setup time " << setup_time << " s\n";
1760 * cg.solve(system_matrix, solution, system_rhs, preconditioner);
1762 * pcout << "Time solve (" << solver_control.last_step() << " iterations) "
1763 * << time.wall_time() << " s" << std::endl;
1770 * Since we have solved a problem with analytic solution, we want to verify
1771 * the correctness of our implementation by computing the L2 error of the
1772 * numerical result against the analytic solution.
1778 * template <int dim, int fe_degree>
1779 * void LaplaceProblem<dim, fe_degree>::analyze_results() const
1781 * Vector<float> error_per_cell(triangulation.n_active_cells());
1782 * VectorTools::integrate_difference(mapping,
1787 * QGauss<dim>(fe.degree + 2),
1788 * VectorTools::L2_norm);
1789 * pcout << "Verification via L2 error: "
1791 * Utilities::MPI::sum(error_per_cell.norm_sqr(), MPI_COMM_WORLD))
1799 * The `run()` function sets up the initial grid and then runs the multigrid
1800 * program in the usual way. As a domain, we choose a rectangle with
1801 * periodic boundary conditions in the @f$x@f$-direction, a Dirichlet condition
1802 * on the front face in @f$y@f$ direction (i.e., the face with index number 2,
1803 * with boundary id equal to 0), and Neumann conditions on the back face as
1804 * well as the two faces in @f$z@f$ direction for the 3D case (with boundary id
1805 * equal to 1). The extent of the domain is a bit different in the @f$x@f$
1806 * direction (where we want to achieve a periodic solution given the
1807 * definition of `Solution`) as compared to the @f$y@f$ and @f$z@f$ directions.
1813 * template <int dim, int fe_degree>
1814 * void LaplaceProblem<dim, fe_degree>::run()
1816 * const unsigned int n_ranks =
1817 * Utilities::MPI::n_mpi_processes(MPI_COMM_WORLD);
1818 * pcout << "Running with " << n_ranks << " MPI process"
1819 * << (n_ranks > 1 ? "es" : "") << ", element " << fe.get_name()
1822 * for (unsigned int cycle = 0; cycle < 9 - dim; ++cycle)
1824 * pcout << "Cycle " << cycle << std::endl;
1828 * Point<dim> upper_right;
1829 * upper_right[0] = 2.5;
1830 * for (unsigned int d = 1; d < dim; ++d)
1831 * upper_right[d] = 2.8;
1832 * GridGenerator::hyper_rectangle(triangulation,
1835 * triangulation.begin_active()->face(0)->set_boundary_id(10);
1836 * triangulation.begin_active()->face(1)->set_boundary_id(11);
1837 * triangulation.begin_active()->face(2)->set_boundary_id(0);
1838 * for (unsigned int f = 3;
1839 * f < triangulation.begin_active()->n_faces();
1841 * triangulation.begin_active()->face(f)->set_boundary_id(1);
1843 * std::vector<GridTools::PeriodicFacePair<
1844 * typename Triangulation<dim>::cell_iterator>>
1846 * GridTools::collect_periodic_faces(
1847 * triangulation, 10, 11, 0, periodic_faces);
1848 * triangulation.add_periodicity(periodic_faces);
1850 * triangulation.refine_global(6 - 2 * dim);
1852 * triangulation.refine_global(1);
1856 * analyze_results();
1857 * pcout << std::endl;
1860 * } // namespace Step59
1866 * There is nothing unexpected in the `main()` function. We call `MPI_Init()`
1867 * through the `MPI_InitFinalize` class, pass on the two parameters on the
1868 * dimension and the degree set at the top of the file, and run the Laplace
1875 * int main(int argc, char *argv[])
1879 * using namespace Step59;
1881 * Utilities::MPI::MPI_InitFinalize mpi_init(argc, argv, 1);
1883 * LaplaceProblem<dimension, degree_finite_element> laplace_problem;
1884 * laplace_problem.run();
1886 * catch (std::exception &exc)
1888 * std::cerr << std::endl
1890 * << "----------------------------------------------------"
1892 * std::cerr << "Exception on processing: " << std::endl
1893 * << exc.what() << std::endl
1894 * << "Aborting!" << std::endl
1895 * << "----------------------------------------------------"
1901 * std::cerr << std::endl
1903 * << "----------------------------------------------------"
1905 * std::cerr << "Unknown exception!" << std::endl
1906 * << "Aborting!" << std::endl
1907 * << "----------------------------------------------------"
1915<a name="Results"></a><h1>Results</h1>
1918<a name="Programoutput"></a><h3>Program output</h3>
1921Like in @ref step_37 "step-37", we evaluate the multigrid solver in terms of run time. In
1922two space dimensions with elements of degree 8, a possible output could look
1925Running with 12 MPI processes, element FE_DGQHermite<2>(8)
1928Number of degrees of freedom: 5184
1929Total setup time 0.0282445 s
1930Time solve (14 iterations) 0.0110712 s
1931Verification via L2 error: 1.66232e-07
1934Number of degrees of freedom: 20736
1935Total setup time 0.0126282 s
1936Time solve (14 iterations) 0.0157021 s
1937Verification via L2 error: 2.91505e-10
1940Number of degrees of freedom: 82944
1941Total setup time 0.0227573 s
1942Time solve (14 iterations) 0.026568 s
1943Verification via L2 error: 6.64514e-13
1946Number of degrees of freedom: 331776
1947Total setup time 0.0604685 s
1948Time solve (14 iterations) 0.0628356 s
1949Verification via L2 error: 5.57513e-13
1952Number of degrees of freedom: 1327104
1953Total setup time 0.154359 s
1954Time solve (13 iterations) 0.219555 s
1955Verification via L2 error: 3.08139e-12
1958Number of degrees of freedom: 5308416
1959Total setup time 0.467764 s
1960Time solve (13 iterations) 1.1821 s
1961Verification via L2 error: 3.90334e-12
1964Number of degrees of freedom: 21233664
1965Total setup time 1.73263 s
1966Time solve (13 iterations) 5.21054 s
1967Verification via L2 error: 4.94543e-12
1970Like in @ref step_37 "step-37", the number of CG iterations remains constant with increasing
1971problem size. The iteration counts are a bit higher, which is because we use a
1972lower degree of the Chebyshev polynomial (2 vs 5 in @ref step_37 "step-37") and because the
1973interior penalty discretization has a somewhat larger spread in
1974eigenvalues. Nonetheless, 13 iterations to reduce the residual by 12 orders of
1975magnitude, or almost a factor of 9 per iteration, indicates an overall very
1976efficient method. In particular, we can solve a system with 21 million degrees
1977of freedom in 5 seconds when using 12 cores, which is a very good
1978efficiency. Of course, in 2D we are well inside the regime of roundoff for a
1979polynomial degree of 8 – as a matter of fact, around 83k DoFs or 0.025s
1980would have been enough to fully converge this (simple) analytic solution
1983Not much changes if we run the program in three spatial dimensions, except for
1984the fact that we now use do something more useful with the higher polynomial
1985degree and increasing mesh sizes, as the roundoff errors are only obtained at
1986the finest mesh. Still, it is remarkable that we can solve a 3D Laplace
1987problem with a wave of three periods to roundoff accuracy on a twelve-core
1988machine pretty easily - using about 3.5 GB of memory in total for the second
1989to largest case with 24m DoFs, taking not more than eight seconds. The largest
1990case uses 30GB of memory with 191m DoFs.
1993Running with 12 MPI processes, element FE_DGQHermite<3>(8)
1996Number of degrees of freedom: 5832
1997Total setup time 0.0210681 s
1998Time solve (15 iterations) 0.0956945 s
1999Verification via L2 error: 0.0297194
2002Number of degrees of freedom: 46656
2003Total setup time 0.0452428 s
2004Time solve (15 iterations) 0.113827 s
2005Verification via L2 error: 9.55733e-05
2008Number of degrees of freedom: 373248
2009Total setup time 0.190423 s
2010Time solve (15 iterations) 0.218309 s
2011Verification via L2 error: 2.6868e-07
2014Number of degrees of freedom: 2985984
2015Total setup time 0.627914 s
2016Time solve (15 iterations) 1.0595 s
2017Verification via L2 error: 4.6918e-10
2020Number of degrees of freedom: 23887872
2021Total setup time 2.85215 s
2022Time solve (15 iterations) 8.30576 s
2023Verification via L2 error: 9.38583e-13
2026Number of degrees of freedom: 191102976
2027Total setup time 16.1324 s
2028Time solve (15 iterations) 65.57 s
2029Verification via L2 error: 3.17875e-13
2032<a name="Comparisonofefficiencyatdifferentpolynomialdegrees"></a><h3>Comparison of efficiency at different polynomial degrees</h3>
2035In the introduction and in-code comments, it was mentioned several times that
2036high orders are treated very efficiently with the FEEvaluation and
2037FEFaceEvaluation evaluators. Now, we want to substantiate these claims by
2038looking at the throughput of the 3D multigrid solver for various polynomial
2039degrees. We collect the times as follows: We first run a solver at problem
2040size close to ten million, indicated in the first four table rows, and record
2041the timings. Then, we normalize the throughput by recording the number of
2042million degrees of freedom solved per second (MDoFs/s) to be able to compare
2043the efficiency of the different degrees, which is computed by dividing the
2044number of degrees of freedom by the solver time.
2046<table align="center" class="doxtable">
2063 <th>Number of DoFs</th>
2078 <th>Number of iterations</th>
2093 <th>Solver time [s]</th>
2124We clearly see how the efficiency per DoF initially improves until it reaches
2125a maximum for the polynomial degree @f$k=4@f$. This effect is surprising, not only
2126because higher polynomial degrees often yield a vastly better solution, but
2127especially also when having matrix-based schemes in mind where the denser
2128coupling at higher degree leads to a monotonously decreasing throughput (and a
2129drastic one in 3D, with @f$k=4@f$ being more than ten times slower than
2130@f$k=1@f$!). For higher degrees, the throughput decreases a bit, which is both due
2131to an increase in the number of iterations (going from 12 at @f$k=2,3,4@f$ to 19
2132at @f$k=10@f$) and due to the @f$\mathcal O(k)@f$ complexity of operator
2133evaluation. Nonetheless, efficiency as the time to solution would be still
2134better for higher polynomial degrees because they have better convergence rates (at least
2135for problems as simple as this one): For @f$k=12@f$, we reach roundoff accuracy
2136already with 1 million DoFs (solver time less than a second), whereas for @f$k=8@f$
2137we need 24 million DoFs and 8 seconds. For @f$k=5@f$, the error is around
2138@f$10^{-9}@f$ with 57m DoFs and thus still far away from roundoff, despite taking 16
2141Note that the above numbers are a bit pessimistic because they include the
2142time it takes the Chebyshev smoother to compute an eigenvalue estimate, which
2143is around 10 percent of the solver time. If the system is solved several times
2144(as e.g. common in fluid dynamics), this eigenvalue cost is only paid once and
2145faster times become available.
2147<a name="Evaluationofefficiencyofingredients"></a><h3>Evaluation of efficiency of ingredients</h3>
2150Finally, we take a look at some of the special ingredients presented in this
2151tutorial program, namely the FE_DGQHermite basis in particular and the
2152specification of MatrixFree::DataAccessOnFaces. In the following table, the
2153third row shows the optimized solver above, the fourth row shows the timings
2154with only the MatrixFree::DataAccessOnFaces set to `unspecified` rather than
2155the optimal `gradients`, and the last one with replacing FE_DGQHermite by the
2156basic FE_DGQ elements where both the MPI exchange are more expensive and the
2157operations done by FEFaceEvaluation::gather_evaluate() and
2158FEFaceEvaluation::integrate_scatter().
2160<table align="center" class="doxtable">
2177 <th>Number of DoFs</th>
2192 <th>Solver time optimized as in tutorial [s]</th>
2207 <th>Solver time MatrixFree::DataAccessOnFaces::unspecified [s]</th>
2222 <th>Solver time FE_DGQ [s]</th>
2238The data in the table shows that not using MatrixFree::DataAccessOnFaces
2239increases costs by around 10% for higher polynomial degrees. For lower
2240degrees, the difference is obviously less pronounced because the
2241volume-to-surface ratio is more beneficial and less data needs to be
2242exchanged. The difference is larger when looking at the matrix-vector product
2243only, rather than the full multigrid solver shown here, with around 20% worse
2244timings just because of the MPI communication.
2246For @f$k=1@f$ and @f$k=2@f$, the Hermite-like basis functions do obviously not really
2247pay off (indeed, for @f$k=1@f$ the polynomials are exactly the same as for FE_DGQ)
2248and the results are similar as with the FE_DGQ basis. However, for degrees
2249starting at three, we see an increasing advantage for FE_DGQHermite, showing
2250the effectiveness of these basis functions.
2252<a name="Possibilitiesforextension"></a><h3>Possibilities for extension</h3>
2255As mentioned in the introduction, the fast diagonalization method is tied to a
2256Cartesian mesh with constant coefficients. If we wanted to solve
2257variable-coefficient problems, we would need to invest a bit more time in the
2258design of the smoother parameters by selecting proper generalizations (e.g.,
2259approximating the inverse on the nearest box-shaped element).
2261Another way of extending the program would be to include support for adaptive
2262meshes, for which interface operations at edges of different refinement
2263level become necessary, as discussed in @ref step_39 "step-39".
2266<a name="PlainProg"></a>
2267<h1> The plain program</h1>
2268@include "step-59.cc"
const internal::MatrixFreeFunctions::ShapeInfo< VectorizedArrayType > * data
void read_dof_values(const VectorType &src, const unsigned int first_index=0)
SymmetricTensor< rank, dim, Number > sum(const SymmetricTensor< rank, dim, Number > &local, const MPI_Comm &mpi_communicator)
__global__ void reduction(Number *result, const Number *v, const size_type N)
void loop(ITERATOR begin, typename identity< ITERATOR >::type end, DOFINFO &dinfo, INFOBOX &info, const std::function< void(DOFINFO &, typename INFOBOX::CellInfo &)> &cell_worker, const std::function< void(DOFINFO &, typename INFOBOX::CellInfo &)> &boundary_worker, const std::function< void(DOFINFO &, DOFINFO &, typename INFOBOX::CellInfo &, typename INFOBOX::CellInfo &)> &face_worker, ASSEMBLER &assembler, const LoopControl &lctrl=LoopControl())
void approximate(SynchronousIterators< std::tuple< typename DoFHandler< dim, spacedim >::active_cell_iterator, Vector< float >::iterator > > const &cell, const Mapping< dim, spacedim > &mapping, const DoFHandler< dim, spacedim > &dof_handler, const InputVector &solution, const unsigned int component)
Expression sign(const Expression &x)
void random(DoFHandler< dim, spacedim > &dof_handler)
static const types::blas_int zero
@ matrix
Contents is actually a matrix.
@ symmetric
Matrix is symmetric.
@ general
No special properties.
static const types::blas_int one
Point< spacedim > point(const gp_Pnt &p, const double tolerance=1e-10)
SymmetricTensor< 2, dim, Number > d(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
void call(const std::function< RT()> &function, internal::return_value< RT > &ret_val)
int(&) functions(const void *v1, const void *v2)
::VectorizedArray< Number, width > abs(const ::VectorizedArray< Number, width > &)