282 for (
unsigned int face = range.first; face < range.second; ++face)
285 phi_m.gather_evaluate(src, face_evaluation_flags);
287 phi_p.gather_evaluate(src, face_evaluation_flags);
291 phi_m.integrate_scatter(face_evaluation_flags, dst);
292 phi_p.integrate_scatter(face_evaluation_flags, dst);
295 [&](
const auto &data,
auto &dst,
const auto &src,
const auto range) {
300 for (
unsigned int face = range.first; face < range.second; ++face)
303 phi_m.gather_evaluate(src, face_evaluation_flags);
307 phi_m.integrate_scatter(face_evaluation_flags, dst);
317matrix_free.template loop_cell_centric<VectorType, VectorType>(
318 [&](
const auto &data,
auto &dst,
const auto &src,
const auto range) {
323 for (
unsigned int cell = range.first; cell < range.second; ++cell)
326 phi.gather_evaluate(src, cell_evaluation_flags);
330 phi.integrate_scatter(cell_evaluation_flags, dst);
333 for (
unsigned int face = 0; face < GeometryInfo<dim>::faces_per_cell; ++face)
335 if (data.get_faces_by_cells_boundary_id(cell, face)[0] ==
339 phi_m.reinit(cell, face);
340 phi_m.gather_evaluate(src, face_evaluation_flags);
341 phi_p.reinit(cell, face);
342 phi_p.gather_evaluate(src, face_evaluation_flags);
346 phi_m.integrate_scatter(face_evaluation_flags, dst);
351 phi_m.reinit(cell, face);
352 phi_m.gather_evaluate(src, face_evaluation_flags);
356 phi_m.integrate_scatter(face_evaluation_flags, dst);
366the cell number and the local face number. The given example only
367highlights how to
transform face-centric loops into cell-centric loops and
368is by no means efficient, since data is read and written multiple times
369from and to the global vector as well as computations are performed
370redundantly. Below, we will discuss advanced techniques that target these issues.
381additional_data.mapping_update_flags_faces_by_cells =
382 additional_data.mapping_update_flags_inner_faces |
383 additional_data.mapping_update_flags_boundary_faces;
385data.reinit(mapping, dof_handler, constraint, quadrature, additional_data);
388In particular, these flags enable that the
internal data structures are set up
389for all faces of the cells.
391Currently, cell-centric loops in deal.II only work
for uniformly refined meshes
392and
if no constraints are applied (which is the standard
case DG is normally
396<a name=
"step_76-ProvidinglambdastoMatrixFreeloops"></a><h3>Providing lambdas to
MatrixFree loops</h3>
399The examples given above have already used lambdas, which have been provided to
401a version where a
class and a pointer to one of its methods are used and a
402variant where lambdas are utilized.
404In the following code, a class and a pointer to one of its methods, which should
411 const VectorType & src,
412 const std::pair<unsigned int, unsigned int> &range)
const
415 for (
unsigned int cell = range.first; cell < range.second; ++cell)
418 phi.gather_evaluate(src, cell_evaluation_flags);
422 phi.integrate_scatter(cell_evaluation_flags, dst);
428matrix_free.cell_loop(&Operator::local_apply_cell,
this, dst, src);
431However, it is also possible to pass an anonymous function via a
lambda function
435matrix_free.template cell_loop<VectorType, VectorType>(
436 [&](
const auto &data,
auto &dst,
const auto &src,
const auto range) {
438 for (
unsigned int cell = range.first; cell < range.second; ++cell)
441 phi.gather_evaluate(src, cell_evaluation_flags);
445 phi.integrate_scatter(cell_evaluation_flags, dst);
452<a name=
"step_76-VectorizedArrayType"></a><h3>VectorizedArrayType</h3>
455The
class VectorizedArray<Number> is a key component to achieve the high
456node-
level performance of the
matrix-free algorithms in deal.II.
457It is a wrapper class around a short vector of @f$n@f$ entries of type Number and
458maps arithmetic operations to appropriate single-instruction/multiple-data
463In the default case (<code>VectorizedArray<Number></code>), the vector length is
464set at compile time of the library to
465match the highest
value supported by the given processor architecture.
466However, also a
second optional template argument can be
467specified as <code>
VectorizedArray<Number, size></code>, where <code>size</code> explicitly
468controls the vector length within the capabilities of a particular instruction
469set. A full list of supported vector lengths is presented in the following table:
471<table align="center" class="doxtable">
480 <td>(auto-vectorization)</td>
485 <td>SSE2/AltiVec</td>
499This allows users to select the vector length/ISA and, as a consequence, the
500number of cells to be processed at once in
matrix-free operator evaluations,
501possibly reducing the pressure on the caches, an severe issue for very high
502degrees (and dimensions).
504A possible further reason to
reduce the number of filled lanes
505is to simplify debugging: instead of having to look at,
e.g., 8
506cells, one can concentrate on a single cell.
508The interface of
VectorizedArray also enables the replacement by any type with
509a
matching interface. Specifically, this prepares deal.II for the <code>std::simd</code>
510class that is planned to become part of the
C++23 standard. The following table
511compares the deal.II-specific SIMD classes and the equivalent
C++23 classes:
514<table align="center" class="doxtable">
517 <th>std::simd (C++23)</th>
521 <td><code>std::experimental::native_simd<Number></code></td>
525 <td><code>std::experimental::fixed_size_simd<Number, size></code></td>
530 * <a name="step_76-CommProg"></a>
531 * <h1> The commented program</h1>
534 * <a name="step_76-Parametersandutilityfunctions"></a>
535 * <h3>Parameters and utility
functions</h3>
539 * The same includes as in @ref step_67 "step-67":
542 * #include <deal.II/base/conditional_ostream.h>
543 * #include <deal.II/base/function.h>
544 * #include <deal.II/base/time_stepping.h>
545 * #include <deal.II/base/timer.h>
546 * #include <deal.II/base/utilities.h>
547 * #include <deal.II/base/vectorization.h>
549 * #include <deal.II/distributed/tria.h>
551 * #include <deal.II/dofs/dof_handler.h>
553 * #include <deal.II/fe/fe_dgq.h>
554 * #include <deal.II/fe/fe_system.h>
556 * #include <deal.II/grid/grid_generator.h>
557 * #include <deal.II/grid/tria.h>
558 * #include <deal.II/grid/tria_accessor.h>
559 * #include <deal.II/grid/tria_iterator.h>
561 * #include <deal.II/lac/affine_constraints.h>
562 * #include <deal.II/lac/la_parallel_vector.h>
564 * #include <deal.II/matrix_free/fe_evaluation.h>
565 * #include <deal.II/matrix_free/matrix_free.h>
566 * #include <deal.II/matrix_free/operators.h>
568 * #include <deal.II/numerics/data_out.h>
572 * #include <iostream>
576 * A new include for categorizing of cells according to their boundary IDs:
579 * #include <deal.II/matrix_free/tools.h>
589 * The same input parameters as in @ref step_67
"step-67":
592 *
constexpr unsigned int testcase = 1;
593 *
constexpr unsigned int dimension = 2;
594 *
constexpr unsigned int n_global_refinements = 2;
595 *
constexpr unsigned int fe_degree = 5;
596 *
constexpr unsigned int n_q_points_1d = fe_degree + 2;
600 * This parameter specifies the size of the shared-memory
group. Currently,
602 * to the options that the memory features can be turned off or all processes
603 * having access to the same shared-memory domain are grouped together.
608 *
using Number = double;
612 * Here, the type of the data structure is chosen
for vectorization. In the
614 * instruction-set-architecture extension available on the given hardware with
615 * the maximum number of vector lanes is used. However, one might
reduce
616 * the number of filled lanes,
e.g., by writing
625 * The following parameters have not changed:
628 *
constexpr double gamma = 1.4;
629 *
constexpr double final_time = testcase == 0 ? 10 : 2.0;
630 *
constexpr double output_tick = testcase == 0 ? 1 : 0.05;
632 *
const double courant_number = 0.15 /
std::pow(fe_degree, 1.5);
636 * Specify
max number of time steps useful
for performance studies.
643 * Runge-Kutta-related
functions copied from @ref step_67
"step-67" and slightly modified
644 * with the purpose to minimize global vector access:
647 *
enum LowStorageRungeKuttaScheme
654 *
constexpr LowStorageRungeKuttaScheme lsrk_scheme = stage_5_order_4;
658 *
class LowStorageRungeKuttaIntegrator
661 * LowStorageRungeKuttaIntegrator(
const LowStorageRungeKuttaScheme scheme)
666 *
case stage_3_order_3:
669 *
case stage_5_order_4:
672 *
case stage_7_order_4:
675 *
case stage_9_order_5:
684 * rk_integrator(lsrk);
685 * std::vector<double> ci;
686 * rk_integrator.get_coefficients(ai, bi, ci);
689 *
unsigned int n_stages() const
694 *
template <
typename VectorType,
typename Operator>
695 *
void perform_time_step(
const Operator &pde_operator,
696 *
const double current_time,
697 *
const double time_step,
698 * VectorType &solution,
699 * VectorType &vec_ri,
700 * VectorType &vec_ki)
const
704 * vec_ki.swap(solution);
706 *
double sum_previous_bi = 0;
707 *
for (
unsigned int stage = 0; stage < bi.size(); ++stage)
709 *
const double c_i = stage == 0 ? 0 : sum_previous_bi + ai[stage - 1];
711 * pde_operator.perform_stage(stage,
712 * current_time + c_i * time_step,
713 * bi[stage] * time_step,
714 * (stage == bi.size() - 1 ?
716 * ai[stage] * time_step),
717 * (stage % 2 == 0 ? vec_ki : vec_ri),
718 * (stage % 2 == 0 ? vec_ri : vec_ki),
722 * sum_previous_bi += bi[stage - 1];
727 * std::vector<double> bi;
728 * std::vector<double> ai;
734 * Euler-specific utility
functions from @ref step_67
"step-67":
737 *
enum EulerNumericalFlux
739 * lax_friedrichs_modified,
740 * harten_lax_vanleer,
742 *
constexpr EulerNumericalFlux numerical_flux_type = lax_friedrichs_modified;
747 *
class ExactSolution :
public Function<dim>
750 * ExactSolution(
const double time)
755 *
const unsigned int component = 0)
const override;
761 *
double ExactSolution<dim>::value(
const Point<dim> &x,
762 *
const unsigned int component)
const
764 *
const double t = this->
get_time();
770 *
Assert(dim == 2, ExcNotImplemented());
771 *
const double beta = 5;
775 *
const double radius_sqr =
776 * (x - x0).
norm_square() - 2. * (x[0] - x0[0]) * t + t * t;
777 *
const double factor =
779 *
const double density_log = std::log2(
780 *
std::abs(1. - (gamma - 1.) / gamma * 0.25 * factor * factor));
781 *
const double density = std::exp2(density_log * (1. / (gamma - 1.)));
782 *
const double u = 1. - factor * (x[1] - x0[1]);
783 *
const double v = factor * (x[0] - t - x0[0]);
785 *
if (component == 0)
787 *
else if (component == 1)
788 *
return density * u;
789 *
else if (component == 2)
790 *
return density * v;
793 *
const double pressure =
794 * std::exp2(density_log * (gamma / (gamma - 1.)));
795 *
return pressure / (
gamma - 1.) +
796 * 0.5 * (density * u * u + density * v * v);
802 *
if (component == 0)
804 *
else if (component == 1)
806 *
else if (component == dim + 1)
807 *
return 3.097857142857143;
820 *
template <
int dim,
typename Number>
825 *
const Number inverse_density = Number(1.) / conserved_variables[0];
828 *
for (
unsigned int d = 0;
d < dim; ++
d)
829 * velocity[d] = conserved_variables[1 + d] * inverse_density;
834 *
template <
int dim,
typename Number>
840 * euler_velocity<dim>(conserved_variables);
842 * Number rho_u_dot_u = conserved_variables[1] * velocity[0];
843 *
for (
unsigned int d = 1;
d < dim; ++
d)
844 * rho_u_dot_u += conserved_variables[1 + d] * velocity[d];
846 *
return (gamma - 1.) * (conserved_variables[dim + 1] - 0.5 * rho_u_dot_u);
849 *
template <
int dim,
typename Number>
855 * euler_velocity<dim>(conserved_variables);
856 *
const Number pressure = euler_pressure<dim>(conserved_variables);
859 *
for (
unsigned int d = 0;
d < dim; ++
d)
861 * flux[0][
d] = conserved_variables[1 +
d];
862 *
for (
unsigned int e = 0;
e < dim; ++
e)
863 * flux[e + 1][d] = conserved_variables[e + 1] * velocity[d];
864 * flux[
d + 1][
d] += pressure;
866 * velocity[
d] * (conserved_variables[dim + 1] + pressure);
872 *
template <
int n_components,
int dim,
typename Number>
879 *
for (
unsigned int d = 0;
d < n_components; ++
d)
880 * result[d] = matrix[d] * vector;
884 *
template <
int dim,
typename Number>
891 *
const auto velocity_m = euler_velocity<dim>(u_m);
892 *
const auto velocity_p = euler_velocity<dim>(u_p);
894 *
const auto pressure_m = euler_pressure<dim>(u_m);
895 *
const auto pressure_p = euler_pressure<dim>(u_p);
897 *
const auto flux_m = euler_flux<dim>(u_m);
898 *
const auto flux_p = euler_flux<dim>(u_p);
900 *
switch (numerical_flux_type)
902 *
case lax_friedrichs_modified:
906 * gamma * pressure_p * (1. / u_p[0]),
907 * velocity_m.norm_square() +
908 * gamma * pressure_m * (1. / u_m[0])));
910 *
return 0.5 * (flux_m * normal + flux_p * normal) +
911 * 0.5 * lambda * (u_m - u_p);
914 *
case harten_lax_vanleer:
916 *
const auto avg_velocity_normal =
917 * 0.5 * ((velocity_m + velocity_p) * normal);
920 * (pressure_p * (1. / u_p[0]) + pressure_m * (1. / u_m[0]))));
921 *
const Number s_pos =
922 *
std::max(Number(), avg_velocity_normal + avg_c);
923 *
const Number s_neg =
924 *
std::min(Number(), avg_velocity_normal - avg_c);
925 *
const Number inverse_s = Number(1.) / (s_pos - s_neg);
928 * ((s_pos * (flux_m * normal) - s_neg * (flux_p * normal)) -
929 * s_pos * s_neg * (u_m - u_p));
944 * General-purpose utility
functions from @ref step_67
"step-67":
947 *
template <
int dim,
typename VectorizedArrayType>
948 * VectorizedArrayType
951 *
const unsigned int component)
953 * VectorizedArrayType result;
954 *
for (
unsigned int v = 0; v < VectorizedArrayType::size(); ++v)
957 *
for (
unsigned int d = 0;
d < dim; ++
d)
958 * p[d] = p_vectorized[d][v];
959 * result[v] = function.value(p, component);
965 *
template <
int dim,
typename VectorizedArrayType,
int n_components = dim + 2>
972 *
for (
unsigned int v = 0; v < VectorizedArrayType::size(); ++v)
975 *
for (
unsigned int d = 0;
d < dim; ++
d)
976 * p[d] = p_vectorized[d][v];
977 *
for (
unsigned int d = 0;
d < n_components; ++
d)
978 * result[d][v] = function.value(p, d);
987 * <a name=
"step_76-EuleroperatorusingacellcentricloopandMPI30sharedmemory"></a>
988 * <h3>Euler
operator using a cell-centric
loop and
MPI-3.0 shared memory</h3>
992 * Euler
operator from @ref step_67
"step-67" with some changes as detailed below:
995 *
template <
int dim,
int degree,
int n_po
ints_1d>
996 *
class EulerOperator
999 *
static constexpr unsigned int n_quadrature_points_1d = n_points_1d;
1011 *
void set_subsonic_outflow_boundary(
1017 *
void set_body_force(std::unique_ptr<
Function<dim>> body_force);
1020 * perform_stage(
const unsigned int stage,
1021 *
const Number cur_time,
1031 * std::array<double, 3> compute_errors(
1035 *
double compute_cell_transport_speed(
1044 * Instance of SubCommunicatorWrapper containing the sub-communicator, which
1046 * shared-memory capabilities:
1051 *
MatrixFree<dim, Number, VectorizedArrayType> data;
1056 * inflow_boundaries;
1058 * subsonic_outflow_boundaries;
1059 *
std::set<
types::boundary_id> wall_boundaries;
1067 * New constructor, which creates a sub-communicator. The user can specify
1068 * the size of the sub-communicator via the global parameter group_size. If
1069 * the size is set to -1, all
MPI processes of a
1070 * shared-memory domain are combined to a group. The specified size is
1071 * decisive for the benefit of the shared-memory capabilities of
MatrixFree
1072 * and, therefore, setting the <code>size</code> to <code>-1</code> is a
1073 * reasonable choice. By setting, the size to <code>1</code> users explicitly
1074 * disable the
MPI-3.0 shared-memory features of
MatrixFree and rely
1075 * completely on
MPI-2.0 features, like <code>MPI_Isend</code> and
1076 * <code>MPI_Irecv</code>.
1079 * template <
int dim,
int degree,
int n_points_1d>
1080 * EulerOperator<dim, degree, n_points_1d>::EulerOperator(
TimerOutput &timer)
1083 * #ifdef DEAL_II_WITH_MPI
1084 *
if (group_size == 1)
1086 * this->subcommunicator = MPI_COMM_SELF;
1092 * MPI_Comm_split_type(MPI_COMM_WORLD,
1093 * MPI_COMM_TYPE_SHARED,
1096 * &subcommunicator);
1103 * (
void)subcommunicator;
1105 * this->subcommunicator = MPI_COMM_SELF;
1112 * New destructor responsible
for freeing of the sub-communicator.
1115 *
template <
int dim,
int degree,
int n_po
ints_1d>
1116 * EulerOperator<dim, degree, n_points_1d>::~EulerOperator()
1118 * #ifdef DEAL_II_WITH_MPI
1119 *
if (this->subcommunicator != MPI_COMM_SELF)
1120 * MPI_Comm_free(&subcommunicator);
1127 * Modified
reinit() function to set up the
internal data structures in
1128 *
MatrixFree in a way that it is usable by the cell-centric loops and
1129 * the
MPI-3.0 shared-memory capabilities are used:
1132 * template <
int dim,
int degree,
int n_points_1d>
1133 *
void EulerOperator<dim, degree, n_points_1d>::reinit(
1134 * const
Mapping<dim> &mapping,
1137 *
const std::vector<const DoFHandler<dim> *> dof_handlers = {&dof_handler};
1139 *
const std::vector<const AffineConstraints<double> *> constraints = {&dummy};
1140 *
const std::vector<Quadrature<1>> quadratures = {
QGauss<1>(n_q_points_1d),
1148 * additional_data.mapping_update_flags_inner_faces =
1151 * additional_data.mapping_update_flags_boundary_faces =
1154 * additional_data.tasks_parallel_scheme =
1159 * Categorize cells so that all lanes have the same boundary IDs
for each
1160 * face. This is strictly not necessary, however, allows to write simpler
1161 * code in EulerOperator::perform_stage() without masking, since it is
1163 * have to perform exactly the same operation also on the faces.
1166 *
MatrixFreeTools::categorize_by_boundary_ids(dof_handler.get_triangulation(),
1171 * Enable
MPI-3.0 shared-memory capabilities within
MatrixFree by providing
1172 * the sub-communicator:
1175 * additional_data.communicator_sm = subcommunicator;
1178 * mapping, dof_handlers, constraints, quadratures, additional_data);
1184 * The following function does an entire stage of a Runge--Kutta update
1185 * and is, alongside the slightly modified setup, the heart of this tutorial
1186 * compared to @ref step_67 "step-67".
1190 * In contrast to @ref step_67 "step-67", we are not executing the advection step
1191 * (using
MatrixFree::loop()) and the inverse mass-matrix step
1192 * (using
MatrixFree::cell_loop()) in sequence, but evaluate everything in
1193 * one go inside of
MatrixFree::loop_cell_centric(). This function expects
1194 * a single function that is executed on each locally-owned (macro) cell as
1195 * parameter so that we need to loop over all faces of that cell and perform
1196 * needed integration steps on our own.
1200 * The following function contains to a large extent copies of the following
1201 * functions from @ref step_67 "step-67" so that comments related the evaluation of the weak
1202 * form are skipped here:
1203 * - <code>EulerDG::EulerOperator::local_apply_cell</code>
1204 * - <code>EulerDG::EulerOperator::local_apply_face</code>
1205 * - <code>EulerDG::EulerOperator::local_apply_boundary_face</code>
1206 * - <code>EulerDG::EulerOperator::local_apply_inverse_mass_matrix</code>
1209 * template <
int dim,
int degree,
int n_points_1d>
1210 *
void EulerOperator<dim, degree, n_points_1d>::perform_stage(
1211 * const
unsigned int stage,
1212 * const Number current_time,
1219 *
for (
auto &i : inflow_boundaries)
1220 * i.
second->set_time(current_time);
1221 *
for (
auto &i : subsonic_outflow_boundaries)
1222 * i.
second->set_time(current_time);
1227 * providing a lambda containing the effects of the cell, face and
1228 * boundary-face integrals:
1233 * [&](const auto &data, auto &dst, const auto &src, const auto cell_range) {
1239 * VectorizedArrayType>;
1245 * VectorizedArrayType>;
1247 * FECellIntegral phi(data);
1248 * FECellIntegral phi_temp(data);
1249 * FEFaceIntegral phi_m(data,
true);
1250 * FEFaceIntegral phi_p(data,
false);
1256 *
if (constant_function)
1257 * constant_body_force =
1258 * evaluate_function<dim, VectorizedArrayType, dim>(
1266 * VectorizedArrayType,
1269 * data.get_shape_info().data[0].shape_gradients_collocation_eo,
1273 * phi.n_components);
1277 * Loop over all cell batches:
1280 *
for (
unsigned int cell = cell_range.first; cell < cell_range.second;
1285 *
if (ai != Number())
1286 * phi_temp.reinit(cell);
1290 * Read
values from global vector and compute the
values at the
1291 * quadrature points:
1294 *
if (ai != Number() && stage == 0)
1296 * phi.read_dof_values(src);
1298 *
for (
unsigned int i = 0;
1299 * i < phi.static_dofs_per_component * (dim + 2);
1301 * phi_temp.begin_dof_values()[i] = phi.begin_dof_values()[i];
1312 * Buffer the computed
values at the quadrature points, since
1314 * step, however, are needed later on for the face integrals:
1317 * for (
unsigned int i = 0; i < phi.static_n_q_points * (dim + 2); ++i)
1318 * buffer[i] = phi.begin_values()[i];
1322 * Apply the cell integral at the cell quadrature points. See also
1323 * the function <code>EulerOperator::local_apply_cell()</code> from
1324 * @ref step_67 "step-67":
1327 * for (const
unsigned int q : phi.quadrature_point_indices())
1329 *
const auto w_q = phi.get_value(q);
1330 * phi.submit_gradient(euler_flux<dim>(w_q), q);
1331 *
if (body_force.get() !=
nullptr)
1334 * constant_function ?
1335 * constant_body_force :
1336 * evaluate_function<dim, VectorizedArrayType, dim>(
1337 * *body_force, phi.quadrature_point(q));
1340 *
for (
unsigned int d = 0;
d < dim; ++
d)
1341 * forcing[d + 1] = w_q[0] * force[d];
1342 *
for (
unsigned int d = 0;
d < dim; ++
d)
1343 * forcing[dim + 1] += force[d] * w_q[d + 1];
1345 * phi.submit_value(forcing, q);
1352 * points. We skip the interpolation back to the support points
1353 * of the element, since we
first collect all contributions in the
1354 * cell quadrature points and only perform the interpolation back
1355 * as the
final step.
1359 *
auto *values_ptr = phi.begin_values();
1360 *
auto *gradient_ptr = phi.begin_gradients();
1362 *
for (
unsigned int c = 0; c < dim + 2; ++c)
1364 *
if (dim >= 1 && body_force.get() ==
nullptr)
1365 * eval.template gradients<0, false, false, dim>(gradient_ptr,
1367 *
else if (dim >= 1)
1368 * eval.template gradients<0, false, true, dim>(gradient_ptr,
1371 * eval.template gradients<1, false, true, dim>(gradient_ptr +
1375 * eval.template gradients<2, false, true, dim>(gradient_ptr +
1379 * values_ptr += phi.static_n_q_points;
1380 * gradient_ptr += phi.static_n_q_points * dim;
1386 * Loop over all faces of the current cell:
1389 *
for (
unsigned int face = 0;
1390 * face < GeometryInfo<dim>::faces_per_cell;
1395 * Determine the boundary ID of the current face. Since we have
1396 * set up
MatrixFree in a way that all filled lanes have
1397 * guaranteed the same boundary ID, we can select the
1398 * boundary ID of the
first lane.
1401 *
const auto boundary_ids =
1402 * data.get_faces_by_cells_boundary_id(cell, face);
1404 *
Assert(std::equal(boundary_ids.begin(),
1405 * boundary_ids.begin() +
1406 * data.n_active_entries_per_cell_batch(cell),
1407 * boundary_ids.begin()),
1408 * ExcMessage(
"Boundary IDs of lanes differ."));
1412 * phi_m.reinit(cell, face);
1416 * Interpolate the
values from the cell quadrature points to the
1417 * quadrature points of the current face via a simple 1
d
1423 * VectorizedArrayType>::
1424 *
template interpolate_quadrature<true, false>(
1427 * data.get_shape_info(),
1429 * phi_m.begin_values(),
1434 * Check
if the face is an
internal or a boundary face and
1435 * select a different code path based on
this information:
1442 * Process and
internal face. The following lines of code
1443 * are a
copy of the function
1444 * <code>EulerDG::EulerOperator::local_apply_face</code>
1445 * from @ref step_67
"step-67":
1448 * phi_p.reinit(cell, face);
1451 *
for (
const unsigned int q :
1452 * phi_m.quadrature_point_indices())
1454 * const auto numerical_flux =
1455 * euler_numerical_flux<dim>(phi_m.get_value(q),
1456 * phi_p.get_value(q),
1457 * phi_m.normal_vector(q));
1458 * phi_m.submit_value(-numerical_flux, q);
1465 * Process a boundary face. These following lines of code
1466 * are a
copy of the function
1467 * <code>EulerDG::EulerOperator::local_apply_boundary_face</code>
1468 * from @ref step_67
"step-67":
1471 *
for (
const unsigned int q :
1472 * phi_m.quadrature_point_indices())
1474 * const auto w_m = phi_m.get_value(q);
1475 *
const auto normal = phi_m.normal_vector(q);
1477 *
auto rho_u_dot_n = w_m[1] * normal[0];
1478 *
for (
unsigned int d = 1;
d < dim; ++
d)
1479 * rho_u_dot_n += w_m[1 + d] * normal[d];
1481 *
bool at_outflow =
false;
1485 *
if (wall_boundaries.find(boundary_id) !=
1486 * wall_boundaries.end())
1489 *
for (
unsigned int d = 0;
d < dim; ++
d)
1491 * w_m[d + 1] - 2. * rho_u_dot_n * normal[d];
1492 * w_p[dim + 1] = w_m[dim + 1];
1494 *
else if (inflow_boundaries.find(boundary_id) !=
1495 * inflow_boundaries.end())
1496 * w_p = evaluate_function(
1497 * *inflow_boundaries.find(boundary_id)->second,
1498 * phi_m.quadrature_point(q));
1499 *
else if (subsonic_outflow_boundaries.find(
1501 * subsonic_outflow_boundaries.end())
1505 * evaluate_function(*subsonic_outflow_boundaries
1506 * .find(boundary_id)
1508 * phi_m.quadrature_point(q),
1510 * at_outflow =
true;
1515 *
"Unknown boundary id, did "
1516 *
"you set a boundary condition for "
1517 *
"this part of the domain boundary?"));
1519 *
auto flux = euler_numerical_flux<dim>(w_m, w_p, normal);
1522 *
for (
unsigned int v = 0;
1523 * v < VectorizedArrayType::size();
1526 *
if (rho_u_dot_n[v] < -1e-12)
1527 *
for (
unsigned int d = 0;
d < dim; ++
d)
1528 * flux[d + 1][v] = 0.;
1531 * phi_m.submit_value(-flux, q);
1537 * Evaluate local integrals related to cell by quadrature and
1538 * add into cell contribution via a simple 1
d interpolation:
1543 * VectorizedArrayType>::
1544 *
template interpolate_quadrature<false, true>(
1547 * data.get_shape_info(),
1548 * phi_m.begin_values(),
1549 * phi.begin_values(),
1555 * Apply inverse mass
matrix in the cell quadrature points. See
1557 * <code>EulerDG::EulerOperator::local_apply_inverse_mass_matrix()</code>
1558 * from @ref step_67
"step-67":
1561 *
for (
unsigned int q = 0; q < phi.static_n_q_points; ++q)
1563 *
const auto factor = VectorizedArrayType(1.0) / phi.JxW(q);
1564 *
for (
unsigned int c = 0; c < dim + 2; ++c)
1565 * phi.begin_values()[c * phi.static_n_q_points + q] =
1566 * phi.begin_values()[c * phi.static_n_q_points + q] * factor;
1571 * Transform
values from collocation space to the original
1572 * Gauss-Lobatto space:
1580 * n_points_1d>::do_backward(dim + 2,
1581 * data.get_shape_info()
1583 * .inverse_shape_values_eo,
1585 * phi.begin_values(),
1586 * phi.begin_dof_values());
1590 * Perform Runge-Kutta update and write results back to global
1594 *
if (ai == Number())
1596 *
for (
unsigned int q = 0; q < phi.static_dofs_per_cell; ++q)
1597 * phi.begin_dof_values()[q] = bi * phi.begin_dof_values()[q];
1598 * phi.distribute_local_to_global(solution);
1603 * phi_temp.read_dof_values(solution);
1605 *
for (
unsigned int q = 0; q < phi.static_dofs_per_cell; ++q)
1607 *
const auto K_i = phi.begin_dof_values()[q];
1609 * phi.begin_dof_values()[q] =
1610 * phi_temp.begin_dof_values()[q] + (ai * K_i);
1612 * phi_temp.begin_dof_values()[q] += bi * K_i;
1614 * phi.set_dof_values(dst);
1615 * phi_temp.set_dof_values(solution);
1629 * From here, the code of @ref step_67
"step-67" has not changed.
1632 *
template <
int dim,
int degree,
int n_po
ints_1d>
1633 *
void EulerOperator<dim, degree, n_points_1d>::initialize_vector(
1636 * data.initialize_dof_vector(vector);
1641 *
template <
int dim,
int degree,
int n_po
ints_1d>
1642 *
void EulerOperator<dim, degree, n_points_1d>::set_inflow_boundary(
1646 *
AssertThrow(subsonic_outflow_boundaries.find(boundary_id) ==
1647 * subsonic_outflow_boundaries.end() &&
1648 * wall_boundaries.find(boundary_id) == wall_boundaries.end(),
1649 * ExcMessage(
"You already set the boundary with id " +
1650 * std::to_string(
static_cast<int>(boundary_id)) +
1651 *
" to another type of boundary before now setting " +
1653 *
AssertThrow(inflow_function->n_components == dim + 2,
1654 * ExcMessage(
"Expected function with dim+2 components"));
1656 * inflow_boundaries[
boundary_id] = std::move(inflow_function);
1661 *
template <
int dim,
int degree,
int n_po
ints_1d>
1662 *
void EulerOperator<dim, degree, n_points_1d>::set_subsonic_outflow_boundary(
1666 *
AssertThrow(inflow_boundaries.find(boundary_id) ==
1667 * inflow_boundaries.end() &&
1668 * wall_boundaries.find(boundary_id) == wall_boundaries.end(),
1669 * ExcMessage(
"You already set the boundary with id " +
1670 * std::to_string(
static_cast<int>(boundary_id)) +
1671 *
" to another type of boundary before now setting " +
1672 *
"it as subsonic outflow"));
1673 *
AssertThrow(outflow_function->n_components == dim + 2,
1674 * ExcMessage(
"Expected function with dim+2 components"));
1676 * subsonic_outflow_boundaries[
boundary_id] = std::move(outflow_function);
1681 *
template <
int dim,
int degree,
int n_po
ints_1d>
1682 *
void EulerOperator<dim, degree, n_points_1d>::set_wall_boundary(
1685 *
AssertThrow(inflow_boundaries.find(boundary_id) ==
1686 * inflow_boundaries.end() &&
1687 * subsonic_outflow_boundaries.find(boundary_id) ==
1688 * subsonic_outflow_boundaries.end(),
1689 * ExcMessage(
"You already set the boundary with id " +
1690 * std::to_string(
static_cast<int>(boundary_id)) +
1691 *
" to another type of boundary before now setting " +
1692 *
"it as wall boundary"));
1694 * wall_boundaries.insert(boundary_id);
1699 *
template <
int dim,
int degree,
int n_po
ints_1d>
1700 *
void EulerOperator<dim, degree, n_points_1d>::set_body_force(
1705 * this->body_force = std::move(body_force);
1710 *
template <
int dim,
int degree,
int n_po
ints_1d>
1711 *
void EulerOperator<dim, degree, n_points_1d>::project(
1721 * VectorizedArrayType>
1723 * solution.zero_out_ghost_values();
1724 *
for (
unsigned int cell = 0; cell < data.n_cell_batches(); ++cell)
1727 *
for (
const unsigned int q : phi.quadrature_point_indices())
1728 * phi.submit_dof_value(evaluate_function(function,
1729 * phi.quadrature_point(q)),
1731 * inverse.transform_from_q_points_to_basis(dim + 2,
1732 * phi.begin_dof_values(),
1733 * phi.begin_dof_values());
1734 * phi.set_dof_values(solution);
1740 *
template <
int dim,
int degree,
int n_po
ints_1d>
1741 * std::array<double, 3> EulerOperator<dim, degree, n_points_1d>::compute_errors(
1746 *
double errors_squared[3] = {};
1750 *
for (
unsigned int cell = 0; cell < data.n_cell_batches(); ++cell)
1754 * VectorizedArrayType local_errors_squared[3] = {};
1755 *
for (
const unsigned int q : phi.quadrature_point_indices())
1757 * const auto error =
1758 * evaluate_function(function, phi.quadrature_point(q)) -
1760 *
const auto JxW = phi.JxW(q);
1762 * local_errors_squared[0] += error[0] * error[0] * JxW;
1763 *
for (
unsigned int d = 0;
d < dim; ++
d)
1764 * local_errors_squared[1] += (error[d + 1] * error[d + 1]) * JxW;
1765 * local_errors_squared[2] += (error[dim + 1] * error[dim + 1]) * JxW;
1767 *
for (
unsigned int v = 0; v < data.n_active_entries_per_cell_batch(cell);
1769 *
for (
unsigned int d = 0;
d < 3; ++
d)
1770 * errors_squared[d] += local_errors_squared[d][v];
1775 * std::array<double, 3> errors;
1776 *
for (
unsigned int d = 0;
d < 3; ++
d)
1777 * errors[d] =
std::sqrt(errors_squared[d]);
1784 *
template <
int dim,
int degree,
int n_po
ints_1d>
1785 *
double EulerOperator<dim, degree, n_points_1d>::compute_cell_transport_speed(
1789 * Number max_transport = 0;
1793 *
for (
unsigned int cell = 0; cell < data.n_cell_batches(); ++cell)
1797 * VectorizedArrayType local_max = 0.;
1798 *
for (
const unsigned int q : phi.quadrature_point_indices())
1800 * const auto solution = phi.get_value(q);
1801 *
const auto velocity = euler_velocity<dim>(solution);
1802 *
const auto pressure = euler_pressure<dim>(solution);
1804 *
const auto inverse_jacobian = phi.inverse_jacobian(q);
1805 *
const auto convective_speed = inverse_jacobian * velocity;
1806 * VectorizedArrayType convective_limit = 0.;
1807 *
for (
unsigned int d = 0;
d < dim; ++
d)
1808 * convective_limit =
1811 *
const auto speed_of_sound =
1812 *
std::sqrt(gamma * pressure * (1. / solution[0]));
1815 *
for (
unsigned int d = 0;
d < dim; ++
d)
1816 * eigenvector[d] = 1.;
1817 *
for (
unsigned int i = 0; i < 5; ++i)
1819 * eigenvector =
transpose(inverse_jacobian) *
1820 * (inverse_jacobian * eigenvector);
1821 * VectorizedArrayType eigenvector_norm = 0.;
1822 *
for (
unsigned int d = 0;
d < dim; ++
d)
1823 * eigenvector_norm =
1825 * eigenvector /= eigenvector_norm;
1827 *
const auto jac_times_ev = inverse_jacobian * eigenvector;
1828 *
const auto max_eigenvalue =
std::sqrt(
1829 * (jac_times_ev * jac_times_ev) / (eigenvector * eigenvector));
1832 * max_eigenvalue * speed_of_sound + convective_limit);
1835 *
for (
unsigned int v = 0; v < data.n_active_entries_per_cell_batch(cell);
1837 * max_transport =
std::max(max_transport, local_max[v]);
1842 *
return max_transport;
1847 *
template <
int dim>
1848 *
class EulerProblem
1856 *
void make_grid_and_dofs();
1858 *
void output_results(
const unsigned int result_number);
1864 * #ifdef DEAL_II_WITH_P4EST
1876 * EulerOperator<dim, fe_degree, n_q_points_1d> euler_operator;
1878 *
double time, time_step;
1887 * std::vector<
Vector<double>> &computed_quantities)
const override;
1889 *
virtual std::vector<std::string>
get_names()
const override;
1891 *
virtual std::vector<
1898 *
const bool do_schlieren_plot;
1904 *
template <
int dim>
1905 * EulerProblem<dim>::Postprocessor::Postprocessor()
1906 * : do_schlieren_plot(dim == 2)
1911 *
template <
int dim>
1912 *
void EulerProblem<dim>::Postprocessor::evaluate_vector_field(
1916 *
const unsigned int n_evaluation_points = inputs.solution_values.size();
1918 *
if (do_schlieren_plot ==
true)
1919 *
Assert(inputs.solution_gradients.size() == n_evaluation_points,
1920 * ExcInternalError());
1922 *
Assert(computed_quantities.size() == n_evaluation_points,
1923 * ExcInternalError());
1924 *
Assert(inputs.solution_values[0].size() == dim + 2, ExcInternalError());
1925 *
Assert(computed_quantities[0].size() ==
1926 * dim + 2 + (do_schlieren_plot ==
true ? 1 : 0),
1927 * ExcInternalError());
1929 *
for (
unsigned int p = 0; p < n_evaluation_points; ++p)
1932 *
for (
unsigned int d = 0;
d < dim + 2; ++
d)
1933 * solution[d] = inputs.solution_values[p](d);
1935 *
const double density = solution[0];
1937 *
const double pressure = euler_pressure<dim>(solution);
1939 *
for (
unsigned int d = 0;
d < dim; ++
d)
1940 * computed_quantities[p](d) = velocity[
d];
1941 * computed_quantities[p](dim) = pressure;
1942 * computed_quantities[p](dim + 1) =
std::sqrt(gamma * pressure / density);
1944 *
if (do_schlieren_plot ==
true)
1945 * computed_quantities[p](dim + 2) =
1946 * inputs.solution_gradients[p][0] * inputs.solution_gradients[p][0];
1952 *
template <
int dim>
1953 * std::vector<std::string> EulerProblem<dim>::Postprocessor::get_names() const
1955 * std::vector<std::string> names;
1956 *
for (
unsigned int d = 0;
d < dim; ++
d)
1957 * names.emplace_back(
"velocity");
1958 * names.emplace_back(
"pressure");
1959 * names.emplace_back(
"speed_of_sound");
1961 *
if (do_schlieren_plot ==
true)
1962 * names.emplace_back(
"schlieren_plot");
1969 *
template <
int dim>
1970 * std::vector<DataComponentInterpretation::DataComponentInterpretation>
1971 * EulerProblem<dim>::Postprocessor::get_data_component_interpretation() const
1973 * std::vector<DataComponentInterpretation::DataComponentInterpretation>
1975 *
for (
unsigned int d = 0;
d < dim; ++
d)
1976 * interpretation.push_back(
1981 *
if (do_schlieren_plot ==
true)
1982 * interpretation.push_back(
1985 *
return interpretation;
1990 *
template <
int dim>
1991 *
UpdateFlags EulerProblem<dim>::Postprocessor::get_needed_update_flags() const
1993 *
if (do_schlieren_plot ==
true)
2001 *
template <
int dim>
2002 * EulerProblem<dim>::EulerProblem()
2004 * #ifdef DEAL_II_WITH_P4EST
2008 * , mapping(fe_degree)
2011 * , euler_operator(timer)
2018 *
template <
int dim>
2019 *
void EulerProblem<dim>::make_grid_and_dofs()
2026 *
for (
unsigned int d = 1;
d < dim; ++
d)
2027 * lower_left[d] = -5;
2030 * upper_right[0] = 10;
2031 *
for (
unsigned int d = 1;
d < dim; ++
d)
2032 * upper_right[d] = 5;
2039 * euler_operator.set_inflow_boundary(
2040 * 0, std::make_unique<ExactSolution<dim>>(0));
2050 * euler_operator.set_inflow_boundary(
2051 * 0, std::make_unique<ExactSolution<dim>>(0));
2052 * euler_operator.set_subsonic_outflow_boundary(
2053 * 1, std::make_unique<ExactSolution<dim>>(0));
2055 * euler_operator.set_wall_boundary(2);
2056 * euler_operator.set_wall_boundary(3);
2059 * euler_operator.set_body_force(
2061 * std::vector<double>({0., 0., -0.2})));
2072 * dof_handler.distribute_dofs(fe);
2074 * euler_operator.reinit(mapping, dof_handler);
2075 * euler_operator.initialize_vector(solution);
2077 * std::locale s = pcout.get_stream().getloc();
2078 * pcout.get_stream().imbue(std::locale(
""));
2079 * pcout <<
"Number of degrees of freedom: " << dof_handler.n_dofs()
2080 * <<
" ( = " << (dim + 2) <<
" [vars] x "
2084 * pcout.get_stream().imbue(s);
2089 *
template <
int dim>
2090 *
void EulerProblem<dim>::output_results(
const unsigned int result_number)
2092 *
const std::array<double, 3> errors =
2093 * euler_operator.compute_errors(ExactSolution<dim>(time), solution);
2094 *
const std::string quantity_name = testcase == 0 ?
"error" :
"norm";
2096 * pcout <<
"Time:" << std::setw(8) << std::setprecision(3) << time
2097 * <<
", dt: " << std::setw(8) << std::setprecision(2) << time_step
2098 * <<
", " << quantity_name <<
" rho: " << std::setprecision(4)
2099 * << std::setw(10) << errors[0] <<
", rho * u: " << std::setprecision(4)
2100 * << std::setw(10) << errors[1] <<
", energy:" << std::setprecision(4)
2101 * << std::setw(10) << errors[2] << std::endl;
2106 * Postprocessor postprocessor;
2111 * data_out.set_flags(flags);
2113 * data_out.attach_dof_handler(dof_handler);
2115 * std::vector<std::string> names;
2116 * names.emplace_back(
"density");
2117 *
for (
unsigned int d = 0;
d < dim; ++
d)
2118 * names.emplace_back(
"momentum");
2119 * names.emplace_back(
"energy");
2121 * std::vector<DataComponentInterpretation::DataComponentInterpretation>
2123 * interpretation.push_back(
2125 *
for (
unsigned int d = 0;
d < dim; ++
d)
2126 * interpretation.push_back(
2128 * interpretation.push_back(
2131 * data_out.add_data_vector(dof_handler, solution, names, interpretation);
2133 * data_out.add_data_vector(solution, postprocessor);
2136 *
if (testcase == 0 && dim == 2)
2138 * reference.
reinit(solution);
2139 * euler_operator.project(ExactSolution<dim>(time), reference);
2140 * reference.sadd(-1., 1, solution);
2141 * std::vector<std::string> names;
2142 * names.emplace_back(
"error_density");
2143 *
for (
unsigned int d = 0;
d < dim; ++
d)
2144 * names.emplace_back(
"error_momentum");
2145 * names.emplace_back(
"error_energy");
2147 * std::vector<DataComponentInterpretation::DataComponentInterpretation>
2149 * interpretation.push_back(
2151 *
for (
unsigned int d = 0;
d < dim; ++
d)
2152 * interpretation.push_back(
2154 * interpretation.push_back(
2157 * data_out.add_data_vector(dof_handler,
2165 * data_out.add_data_vector(mpi_owner,
"owner");
2167 * data_out.build_patches(mapping,
2171 *
const std::string filename =
2173 * data_out.write_vtu_in_parallel(filename, MPI_COMM_WORLD);
2179 *
template <
int dim>
2180 *
void EulerProblem<dim>::run()
2183 *
const unsigned int n_vect_number = VectorizedArrayType::size();
2184 *
const unsigned int n_vect_bits = 8 *
sizeof(Number) * n_vect_number;
2186 * pcout <<
"Running with "
2188 * <<
" MPI processes" << std::endl;
2189 * pcout <<
"Vectorization over " << n_vect_number <<
' '
2190 * << (std::is_same_v<Number, double> ?
"doubles" :
"floats") <<
" = "
2191 * << n_vect_bits <<
" bits ("
2196 * make_grid_and_dofs();
2198 *
const LowStorageRungeKuttaIntegrator integrator(lsrk_scheme);
2202 * rk_register_1.
reinit(solution);
2203 * rk_register_2.reinit(solution);
2205 * euler_operator.project(ExactSolution<dim>(time), solution);
2208 *
double min_vertex_distance = std::numeric_limits<double>::max();
2209 *
for (
const auto &cell :
triangulation.active_cell_iterators())
2210 * if (cell->is_locally_owned())
2211 * min_vertex_distance =
2212 *
std::
min(min_vertex_distance, cell->minimum_vertex_distance());
2213 * min_vertex_distance =
2216 * time_step = courant_number * integrator.n_stages() /
2217 * euler_operator.compute_cell_transport_speed(solution);
2218 * pcout <<
"Time step size: " << time_step
2219 * <<
", minimal h: " << min_vertex_distance
2220 * <<
", initial transport scaling: "
2221 * << 1. / euler_operator.compute_cell_transport_speed(solution)
2225 * output_results(0);
2227 *
unsigned int timestep_number = 0;
2229 *
while (time < final_time - 1e-12 && timestep_number < max_time_steps)
2231 * ++timestep_number;
2232 *
if (timestep_number % 5 == 0)
2234 * courant_number * integrator.n_stages() /
2236 * euler_operator.compute_cell_transport_speed(solution), 3);
2240 * integrator.perform_time_step(euler_operator,
2248 * time += time_step;
2250 *
if (
static_cast<int>(time / output_tick) !=
2251 *
static_cast<int>((time - time_step) / output_tick) ||
2252 * time >= final_time - 1e-12)
2254 *
static_cast<unsigned int>(std::round(time / output_tick)));
2257 * timer.print_wall_time_statistics(MPI_COMM_WORLD);
2258 * pcout << std::endl;
2264 *
int main(
int argc,
char **argv)
2266 *
using namespace Euler_DG;
2267 *
using namespace dealii;
2273 * EulerProblem<dimension> euler_problem;
2274 * euler_problem.run();
2276 *
catch (std::exception &exc)
2278 * std::cerr << std::endl
2280 * <<
"----------------------------------------------------"
2282 * std::cerr <<
"Exception on processing: " << std::endl
2283 * << exc.what() << std::endl
2284 * <<
"Aborting!" << std::endl
2285 * <<
"----------------------------------------------------"
2292 * std::cerr << std::endl
2294 * <<
"----------------------------------------------------"
2296 * std::cerr <<
"Unknown exception!" << std::endl
2297 * <<
"Aborting!" << std::endl
2298 * <<
"----------------------------------------------------"
2306<a name=
"step_76-Results"></a><h1>Results</h1>
2309Running the program with the
default settings on a machine with 40 processes
2310produces the following output:
2313Running with 40
MPI processes
2314Vectorization over 8 doubles = 512 bits (AVX512)
2315Number of degrees of freedom: 27.648.000 ( = 5 [vars] x 25.600 [cells] x 216 [dofs/cell/var] )
2316Time step size: 0.000295952, minimal h: 0.0075,
initial transport scaling: 0.00441179
2317Time: 0, dt: 0.0003,
norm rho: 5.385e-16, rho * u: 1.916e-16, energy: 1.547e-15
2318+--------------------------------------+------------------+------------+------------------+
2319| Total wallclock time elapsed | 17.52s 10 | 17.52s | 17.52s 11 |
2321| Section | no. calls |
min time rank |
avg time |
max time rank |
2322+--------------------------------------+------------------+------------+------------------+
2323| compute errors | 1 | 0.009594s 16 | 0.009705s | 0.009819s 8 |
2324| compute transport speed | 22 | 0.1366s 0 | 0.1367s | 0.1368s 18 |
2325| output | 1 | 1.233s 0 | 1.233s | 1.233s 32 |
2326| rk time stepping total | 100 | 8.746s 35 | 8.746s | 8.746s 0 |
2327| rk_stage - integrals L_h | 500 | 8.742s 36 | 8.742s | 8.743s 2 |
2328+--------------------------------------+------------------+------------+------------------+
2331and the following visual output:
2333<table align=
"center" class=
"doxtable" style=
"width:85%">
2336 <img src=
"https://www.dealii.org/images/steps/developer/step-67.pressure_010.png" alt=
"" width=
"100%">
2339 <img src=
"https://www.dealii.org/images/steps/developer/step-67.pressure_025.png" alt=
"" width=
"100%">
2344 <img src=
"https://www.dealii.org/images/steps/developer/step-67.pressure_050.png" alt=
"" width=
"100%">
2347 <img src=
"https://www.dealii.org/images/steps/developer/step-67.pressure_100.png" alt=
"" width=
"100%">
2352As a reference, the results of @ref step_67
"step-67" using FCL are:
2355Running with 40
MPI processes
2356Vectorization over 8 doubles = 512 bits (AVX512)
2357Number of degrees of freedom: 27.648.000 ( = 5 [vars] x 25.600 [cells] x 216 [dofs/cell/var] )
2358Time step size: 0.000295952, minimal h: 0.0075,
initial transport scaling: 0.00441179
2359Time: 0, dt: 0.0003,
norm rho: 5.385e-16, rho * u: 1.916e-16, energy: 1.547e-15
2360+-------------------------------------------+------------------+------------+------------------+
2361| Total wallclock time elapsed | 13.33s 0 | 13.34s | 13.35s 34 |
2363| Section | no. calls |
min time rank |
avg time |
max time rank |
2364+-------------------------------------------+------------------+------------+------------------+
2365| compute errors | 1 | 0.007977s 10 | 0.008053s | 0.008161s 30 |
2366| compute transport speed | 22 | 0.1228s 34 | 0.2227s | 0.3845s 0 |
2367| output | 1 | 1.255s 3 | 1.257s | 1.259s 27 |
2368| rk time stepping total | 100 | 11.15s 0 | 11.32s | 11.42s 34 |
2369| rk_stage - integrals L_h | 500 | 8.719s 10 | 8.932s | 9.196s 0 |
2370| rk_stage - inv mass + vec upd | 500 | 1.944s 0 | 2.377s | 2.55s 10 |
2371+-------------------------------------------+------------------+------------+------------------+
2374By the modifications shown in this tutorial, we were able to achieve a speedup of
237527% for the Runge-Kutta stages.
2377<a name=
"step_76-Possibilitiesforextensions"></a><h3>Possibilities for extensions</h3>
2380The algorithms are easily extendable to higher dimensions: a high-dimensional
2381<a href=
"https://github.com/hyperdeal/hyperdeal/blob/a9e67b4e625ff1dde2fed93ad91cdfacfaa3acdf/include/hyper.deal/operators/advection/advection_operation.h#L219-L569">advection operator based on cell-centric loops</a>
2382is part of the hyper.deal library. An extension of cell-centric loops
2383to locally-refined meshes is more involved.
2385<a name=
"step_76-ExtensiontothecompressibleNavierStokesequations"></a><h4>Extension to the compressible Navier-Stokes equations</h4>
2388The solver presented in this tutorial program can also be extended to the
2389compressible Navier–Stokes equations by adding viscous terms, as also
2390suggested in @ref step_67
"step-67". To keep as much of the performance obtained here despite
2391the additional cost of elliptic terms,
e.g. via an interior penalty method, that
2393the @ref step_59
"step-59" tutorial program. The reasoning behind this switch is that in the
2394case of
FE_DGQ all
values of neighboring cells (i.
e., @f$k+1@f$ layers) are needed,
2395whilst in the case of
FE_DGQHermite only 2 layers, making the latter
2396significantly more suitable for higher degrees. The additional layers have to be,
2397on the one hand, loaded from main memory during flux computation and, one the
2398other hand, have to be communicated. Using the shared-memory capabilities
2399introduced in this tutorial, the
second point can be eliminated on a single
2400compute node or its influence can be reduced in a
hybrid context.
2402<a name=
"step_76-BlockGaussSeidellikepreconditioners"></a><h4>Block Gauss-Seidel-like preconditioners</h4>
2405Cell-centric loops could be used to create block Gauss-Seidel preconditioners
2406that are multiplicative within one process and additive over processes. These
2407type of preconditioners use during flux computation, in contrast to Jacobi-type
2408preconditioners, already updated
values from neighboring cells. The following
2409pseudo-code visualizes how this could in principal be achieved:
2413Vector<Number> visit_flags(data.n_cell_batches () + data.n_ghost_cell_batches ());
2416data.template loop_cell_centric<VectorType, VectorType>(
2417 [&](
const auto &data,
auto &dst,
const auto &src,
const auto cell_range) {
2419 for (
unsigned int cell = cell_range.first; cell < cell_range.second; ++cell)
2423 for (
unsigned int face = 0; face < GeometryInfo<dim>::faces_per_cell; ++face)
2425 const auto boundary_id = data.get_faces_by_cells_boundary_id(cell, face)[0];
2429 phi_p.reinit(cell, face);
2431 const auto flags = phi_p.read_cell_data(visit_flags);
2432 const auto all_neighbors_have_been_updated =
2434 flags().
begin() + data.n_active_entries_per_cell_batch(cell) == 1;
2436 if(all_neighbors_have_been_updated)
2453 phi.set_cell_data(visit_flags, VectorizedArrayType(1.0));
2462For
this purpose, one can exploit the cell-data vector capabilities of
2465Please note that in the given example we process <code>VectorizedArrayType::size()</code>
2466number of blocks, since each lane corresponds to one block. We consider blocks
2467as updated
if all blocks processed by a vector
register have been updated. In
2468the
case of Cartesian meshes
this is a reasonable approach, however,
for
2469general unstructured meshes
this conservative approach might lead to a decrease in the
2470efficiency of the preconditioner. A reduction of cells processed in
parallel
2471by explicitly reducing the number of lanes used by <code>VectorizedArrayType</code>
2472might increase the quality of the preconditioner, but with the cost that each
2473iteration might be more expensive. This dilemma leads us to a further
2474"possibility for extension": vectorization within an element.
2477<a name=
"step_76-PlainProg"></a>
2478<h1> The plain program</h1>
2479@include
"step-76.cc"
virtual UpdateFlags get_needed_update_flags() const =0
virtual void evaluate_vector_field(const DataPostprocessorInputs::Vector< dim > &input_data, std::vector< Vector< double > > &computed_quantities) const
virtual std::vector< std::string > get_names() const =0
virtual std::vector< DataComponentInterpretation::DataComponentInterpretation > get_data_component_interpretation() const
void submit_value(const value_type val_in, const unsigned int q_point)
void reinit(const size_type size, const bool omit_zeroing_entries=false)
Abstract base class for mapping classes.
void loop_cell_centric(void(CLASS::*cell_operation)(const MatrixFree &, OutVector &, const InVector &, const std::pair< unsigned int, unsigned int > &) const, const CLASS *owning_class, OutVector &dst, const InVector &src, const bool zero_dst_vector=false, const DataAccessOnFaces src_vector_face_access=DataAccessOnFaces::unspecified) const
void reinit(const MappingType &mapping, const DoFHandler< dim > &dof_handler, const AffineConstraints< number2 > &constraint, const QuadratureType &quad, const AdditionalData &additional_data=AdditionalData())
constexpr numbers::NumberTraits< Number >::real_type norm_square() const
std::enable_if_t< std::is_floating_point_v< T > &&std::is_floating_point_v< U >, typename ProductType< std::complex< T >, std::complex< U > >::type > operator*(const std::complex< T > &left, const std::complex< U > &right)
#define DEAL_II_ALWAYS_INLINE
#define Assert(cond, exc)
#define AssertDimension(dim1, dim2)
#define AssertThrow(cond, exc)
void loop(IteratorType begin, std_cxx20::type_identity_t< IteratorType > end, DOFINFO &dinfo, INFOBOX &info, const std::function< void(std_cxx20::type_identity_t< DOFINFO > &, typename INFOBOX::CellInfo &)> &cell_worker, const std::function< void(std_cxx20::type_identity_t< DOFINFO > &, typename INFOBOX::CellInfo &)> &boundary_worker, const std::function< void(std_cxx20::type_identity_t< DOFINFO > &, std_cxx20::type_identity_t< DOFINFO > &, typename INFOBOX::CellInfo &, typename INFOBOX::CellInfo &)> &face_worker, AssemblerType &assembler, const LoopControl &lctrl=LoopControl())
@ update_values
Shape function values.
@ update_normal_vectors
Normal vectors.
@ update_JxW_values
Transformed quadrature weights.
@ update_gradients
Shape function gradients.
@ update_quadrature_points
Transformed quadrature points.
#define DEAL_II_NOT_IMPLEMENTED()
DataComponentInterpretation
@ component_is_part_of_vector
void hyper_rectangle(Triangulation< dim, spacedim > &tria, const Point< dim > &p1, const Point< dim > &p2, const bool colorize=false)
void channel_with_cylinder(Triangulation< dim > &tria, const double shell_region_width=0.03, const unsigned int n_shells=2, const double skewness=2.0, const bool colorize=false)
@ matrix
Contents is actually a matrix.
@ general
No special properties.
double norm(const FEValuesBase< dim > &fe, const ArrayView< const std::vector< Tensor< 1, dim > > > &Du)
Point< spacedim > point(const gp_Pnt &p, const double tolerance=1e-10)
SymmetricTensor< 2, dim, Number > C(const Tensor< 2, dim, Number > &F)
SymmetricTensor< 2, dim, Number > e(const Tensor< 2, dim, Number > &F)
SymmetricTensor< 2, dim, Number > d(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
@ LOW_STORAGE_RK_STAGE9_ORDER5
@ LOW_STORAGE_RK_STAGE3_ORDER3
@ LOW_STORAGE_RK_STAGE7_ORDER4
@ LOW_STORAGE_RK_STAGE5_ORDER4
VectorType::value_type * begin(VectorType &V)
T sum(const T &t, const MPI_Comm mpi_communicator)
unsigned int n_mpi_processes(const MPI_Comm mpi_communicator)
T max(const T &t, const MPI_Comm mpi_communicator)
T min(const T &t, const MPI_Comm mpi_communicator)
unsigned int this_mpi_process(const MPI_Comm mpi_communicator)
T reduce(const T &local_value, const MPI_Comm comm, const std::function< T(const T &, const T &)> &combiner, const unsigned int root_process=0)
std::string get_current_vectorization_level()
Number truncate_to_n_digits(const Number number, const unsigned int n_digits)
std::string int_to_string(const unsigned int value, const unsigned int digits=numbers::invalid_unsigned_int)
constexpr T pow(const T base, const int iexp)
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)
long double gamma(const unsigned int n)
void copy(const T *begin, const T *end, U *dest)
int(&) functions(const void *v1, const void *v2)
void reinit(MatrixBlock< MatrixType > &v, const BlockSparsityPattern &p)
static constexpr double PI
const types::boundary_id internal_face_boundary_id
static const unsigned int invalid_unsigned_int
void transform(const InputIterator &begin_in, const InputIterator &end_in, OutputIterator out, const Function &function, const unsigned int grainsize)
::VectorizedArray< Number, width > exp(const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > min(const ::VectorizedArray< Number, width > &, const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > max(const ::VectorizedArray< Number, width > &, const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > sqrt(const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > pow(const ::VectorizedArray< Number, width > &, const Number p)
::VectorizedArray< Number, width > abs(const ::VectorizedArray< Number, width > &)
const ::parallel::distributed::Triangulation< dim, spacedim > * triangulation
bool write_higher_order_cells
bool hold_all_faces_to_owned_cells
UpdateFlags mapping_update_flags