]> https://gitweb.dealii.org/ - dealii.git/commitdiff
Step-68 with simplex 11575/head
authorBruno <blais.bruno@gmail.com>
Sun, 17 Jan 2021 17:40:21 +0000 (12:40 -0500)
committerBruno <blais.bruno@gmail.com>
Tue, 19 Jan 2021 17:29:23 +0000 (12:29 -0500)
Add a step-68 test using a simplex mesh
Add some small capacities to particle handler to support simplex meshes

source/particles/particle_handler.cc
tests/simplex/step-68.cc [new file with mode: 0644]
tests/simplex/step-68.mpirun=1.with_simplex_support=on.output [new file with mode: 0644]
tests/simplex/step-68.mpirun=4.with_simplex_support=on.output [new file with mode: 0644]

index 0f5c9b7ed4a24fdd86718cb5890d17c306f7e7db..7d014f07481ce6431c6fb7b17d1f4932afbd150d 100644 (file)
@@ -1208,7 +1208,7 @@ namespace Particles
         if (cell->is_locally_owned())
           {
             std::set<unsigned int> cell_to_neighbor_subdomain;
-            for (const unsigned int v : GeometryInfo<dim>::vertex_indices())
+            for (const unsigned int v : cell->vertex_indices())
               {
                 cell_to_neighbor_subdomain.insert(
                   vertex_to_neighbor_subdomain[cell->vertex_index(v)].begin(),
diff --git a/tests/simplex/step-68.cc b/tests/simplex/step-68.cc
new file mode 100644 (file)
index 0000000..8bfbc11
--- /dev/null
@@ -0,0 +1,579 @@
+/* ---------------------------------------------------------------------
+ *
+ * Copyright (C) 2020 by the deal.II authors
+ *
+ * This file is part of the deal.II library.
+ *
+ * The deal.II library is free software; you can use it, redistribute
+ * it, and/or modify it under the terms of the GNU Lesser General
+ * Public License as published by the Free Software Foundation; either
+ * version 2.1 of the License, or (at your option) any later version.
+ * The full text of the license can be found in the file LICENSE.md at
+ * the top level directory of deal.II.
+ *
+ * ---------------------------------------------------------------------
+ *
+ * This test is quasi identical to step-68, with the following exceptions:
+ * - There is no load balancing
+ * - A simplex mesh is used for the background mesh
+ * - The Euler Analytical integration using the analytically defined velocity
+ * field is not used because it would not test anything relevant
+ * - Step parameters are hardcoded instead of drawn from a parameter file
+ *
+ * Authors: Bruno Blais, Toni El Geitani Nehme, Rene Gassmoeller, Peter Munch
+ */
+
+// @sect3{Include files}
+
+#include <deal.II/base/bounding_box.h>
+#include <deal.II/base/conditional_ostream.h>
+#include <deal.II/base/discrete_time.h>
+#include <deal.II/base/mpi.h>
+#include <deal.II/base/parameter_acceptor.h>
+#include <deal.II/base/timer.h>
+
+#include <deal.II/distributed/cell_weights.h>
+#include <deal.II/distributed/fully_distributed_tria.h>
+#include <deal.II/distributed/solution_transfer.h>
+#include <deal.II/distributed/tria.h>
+
+#include <deal.II/dofs/dof_handler.h>
+#include <deal.II/dofs/dof_tools.h>
+
+#include <deal.II/fe/fe_q.h>
+#include <deal.II/fe/fe_system.h>
+#include <deal.II/fe/mapping_fe.h>
+
+#include <deal.II/grid/grid_generator.h>
+#include <deal.II/grid/grid_tools.h>
+#include <deal.II/grid/tria_accessor.h>
+#include <deal.II/grid/tria_iterator.h>
+
+#include <deal.II/lac/la_parallel_vector.h>
+#include <deal.II/lac/vector.h>
+
+#include <deal.II/numerics/data_out.h>
+#include <deal.II/numerics/vector_tools.h>
+
+#include <deal.II/particles/data_out.h>
+#include <deal.II/particles/generators.h>
+#include <deal.II/particles/particle_handler.h>
+
+#include <deal.II/simplex/fe_lib.h>
+#include <deal.II/simplex/grid_generator.h>
+#include <deal.II/simplex/quadrature_lib.h>
+
+#include <cmath>
+#include <iostream>
+
+#include "../tests.h"
+
+
+
+namespace Step68
+{
+  using namespace dealii;
+
+
+
+  template <int dim>
+  class Vortex : public Function<dim>
+  {
+  public:
+    Vortex()
+      : Function<dim>(dim)
+    {}
+
+
+    virtual void
+    vector_value(const Point<dim> &point,
+                 Vector<double> &  values) const override;
+  };
+
+
+  template <int dim>
+  void
+  Vortex<dim>::vector_value(const Point<dim> &point,
+                            Vector<double> &  values) const
+  {
+    const double T = 4;
+    const double t = this->get_time();
+
+    const double px = numbers::PI * point(0);
+    const double py = numbers::PI * point(1);
+    const double pt = numbers::PI / T * t;
+
+    values[0] = -2 * cos(pt) * pow(sin(px), 2) * sin(py) * cos(py);
+    values[1] = 2 * cos(pt) * pow(sin(py), 2) * sin(px) * cos(px);
+    if (dim == 3)
+      {
+        values[2] = 0;
+      }
+  }
+
+
+
+  template <int dim>
+  class ParticleTracking
+  {
+  public:
+    ParticleTracking();
+    void
+    run();
+
+  private:
+    void
+    generate_particles();
+
+    void
+    setup_background_dofs();
+
+    void
+    interpolate_function_to_field();
+
+    void
+    euler_step_interpolated(const double dt);
+    void
+    euler_step_analytical(const double dt);
+
+    void
+    output_particles(const unsigned int it);
+    void
+    output_background(const unsigned int it);
+
+    void
+    log_particles();
+
+    MPI_Comm                                       mpi_communicator;
+    parallel::fullydistributed::Triangulation<dim> background_triangulation;
+    Particles::ParticleHandler<dim>                particle_handler;
+
+    DoFHandler<dim>                            fluid_dh;
+    FESystem<dim>                              fluid_fe;
+    MappingFE<dim>                             mapping;
+    LinearAlgebra::distributed::Vector<double> velocity_field;
+
+    Vortex<dim> velocity;
+
+    // Simulation parameters. In step-68 drawn from prm file, here hardcoded.
+    std::string output_directory = "./";
+
+    static constexpr unsigned int velocity_degree  = 1;
+    static constexpr double       time_step        = 0.004;
+    static constexpr double       final_time       = 4.0;
+    static constexpr unsigned int output_frequency = 1000;
+    static constexpr unsigned int fluid_refinement = 8;
+  };
+
+  template <int dim>
+  ParticleTracking<dim>::ParticleTracking()
+    : mpi_communicator(MPI_COMM_WORLD)
+    , background_triangulation(mpi_communicator)
+    , fluid_dh(background_triangulation)
+    , fluid_fe(Simplex::FE_P<dim>(velocity_degree), dim)
+    , mapping(Simplex::FE_P<dim>(velocity_degree))
+  {}
+
+  // @sect4{Particles generation}
+
+  // This function generates the tracer particles and the background
+  // triangulation on which these particles evolve.
+  template <int dim>
+  void
+  ParticleTracking<dim>::generate_particles()
+  {
+    // We create a hyper cube triangulation which we globally refine. This
+    // triangulation covers the full trajectory of the particles.
+    // parallel::distributed::Triangulation<dim> tria_pdt(mpi_communicator);
+    //
+    // GridGenerator::hyper_cube(tria_pdt, 0, 1);
+    // tria_pdt.refine_global(fluid_refinement);
+
+    Triangulation<dim> temporary_triangulation;
+    GridGenerator::subdivided_hyper_cube_with_simplices(temporary_triangulation,
+                                                        fluid_refinement);
+
+
+    // extract relevant information from distributed triangulation
+    auto construction_data = TriangulationDescription::Utilities::
+      create_description_from_triangulation(temporary_triangulation,
+                                            mpi_communicator);
+    background_triangulation.create_triangulation(construction_data);
+
+
+    // This initializes the background triangulation where the particles are
+    // living and the number of properties of the particles.
+    particle_handler.initialize(background_triangulation, mapping, 1 + dim);
+
+    // We create a particle triangulation which is solely used to generate
+    // the points which will be used to insert the particles. This
+    // triangulation is a hyper shell which is offset from the
+    // center of the simulation domain. This will be used to generate a
+    // disk filled with particles which will allow an easy monitoring
+    // of the motion due to the vortex.
+    Point<dim> center;
+    center[0] = 0.5;
+    center[1] = 0.75;
+    if (dim == 3)
+      center[2] = 0.5;
+
+    const double outer_radius = 0.15;
+    const double inner_radius = 0.01;
+
+    Triangulation<dim> temporary_quad_particle_triangulation;
+
+    GridGenerator::hyper_shell(temporary_quad_particle_triangulation,
+                               center,
+                               inner_radius,
+                               outer_radius,
+                               6);
+
+
+    Triangulation<dim> temporary_tri_particle_triangulation;
+    GridGenerator::convert_hypercube_to_simplex_mesh(
+      temporary_quad_particle_triangulation,
+      temporary_tri_particle_triangulation);
+
+
+    // extract relevant information from distributed triangulation
+    auto particle_construction_data = TriangulationDescription::Utilities::
+      create_description_from_triangulation(
+        temporary_tri_particle_triangulation, mpi_communicator);
+
+
+
+    parallel::fullydistributed::Triangulation<dim> particle_triangulation(
+      mpi_communicator);
+    particle_triangulation.create_triangulation(particle_construction_data);
+
+    // We generate the necessary bounding boxes for the particles generator.
+    // These bounding boxes are required to quickly identify in which
+    // process's subdomain the inserted particle lies, and which cell owns it.
+
+    std::vector<BoundingBox<dim>> all_boxes;
+    all_boxes.reserve(background_triangulation.n_locally_owned_active_cells());
+    for (const auto cell : background_triangulation.active_cell_iterators())
+      if (cell->is_locally_owned())
+        all_boxes.emplace_back(cell->bounding_box());
+    const auto tree        = pack_rtree(all_boxes);
+    const auto local_boxes = extract_rtree_level(tree, 1);
+
+    std::vector<std::vector<BoundingBox<dim>>> global_bounding_boxes;
+    global_bounding_boxes =
+      Utilities::MPI::all_gather(mpi_communicator, local_boxes);
+
+    // We generate an empty vector of properties. We will attribute the
+    // properties to the particles once they are generated.
+    std::vector<std::vector<double>> properties(
+      particle_triangulation.n_locally_owned_active_cells(),
+      std::vector<double>(dim + 1, 0.));
+
+    // We generate the particles at the position of a single
+    // point quadrature. Consequently, one particle will be generated
+    // at the centroid of each cell.
+    Simplex::QGauss<dim> quadrature_formula(1);
+
+    Particles::Generators::quadrature_points(particle_triangulation,
+                                             quadrature_formula,
+                                             global_bounding_boxes,
+                                             particle_handler,
+                                             mapping,
+                                             properties);
+
+    if (Utilities::MPI::this_mpi_process(mpi_communicator) == 0)
+      deallog << "Number of particles inserted: "
+              << particle_handler.n_global_particles() << std::endl;
+  }
+
+
+
+  // @sect4{Background DOFs and interpolation}
+
+  // This function sets up the background degrees of freedom used for the
+  // velocity interpolation and allocates the field vector where the entire
+  // solution of the velocity field is stored.
+  template <int dim>
+  void
+  ParticleTracking<dim>::setup_background_dofs()
+  {
+    fluid_dh.distribute_dofs(fluid_fe);
+    const IndexSet locally_owned_dofs = fluid_dh.locally_owned_dofs();
+    IndexSet       locally_relevant_dofs;
+    DoFTools::extract_locally_relevant_dofs(fluid_dh, locally_relevant_dofs);
+
+    velocity_field.reinit(locally_owned_dofs,
+                          locally_relevant_dofs,
+                          mpi_communicator);
+  }
+
+
+
+  // This function takes care of interpolating the
+  // vortex velocity field to the field vector. This is achieved rather easily
+  // by using the VectorTools::interpolate() function.
+  template <int dim>
+  void
+  ParticleTracking<dim>::interpolate_function_to_field()
+  {
+    velocity_field.zero_out_ghosts();
+    VectorTools::interpolate(mapping, fluid_dh, velocity, velocity_field);
+    velocity_field.update_ghost_values();
+  }
+
+
+
+  // @sect4{Time integration of the trajectories}
+
+  // In contrast to the previous function in this function we
+  // integrate the particle trajectories by interpolating the value of
+  // the velocity field at the degrees of freedom to the position of
+  // the particles.
+  template <int dim>
+  void
+  ParticleTracking<dim>::euler_step_interpolated(const double dt)
+  {
+    Vector<double> local_dof_values(fluid_fe.dofs_per_cell);
+
+    // We loop over all the local particles. Although this could be achieved
+    // directly by looping over all the cells, this would force us
+    // to loop over numerous cells which do not contain particles.
+    // Rather, we loop over all the particles, but, we get the reference
+    // of the cell in which the particle lies and then loop over all particles
+    // within that cell. This enables us to gather the values of the velocity
+    // out of the `velocity_field` vector once and use them for all particles
+    // that lie within the cell.
+    auto particle = particle_handler.begin();
+    while (particle != particle_handler.end())
+      {
+        const auto cell =
+          particle->get_surrounding_cell(background_triangulation);
+        const auto dh_cell =
+          typename DoFHandler<dim>::cell_iterator(*cell, &fluid_dh);
+
+        dh_cell->get_dof_values(velocity_field, local_dof_values);
+
+        // Next, compute the velocity at the particle locations by evaluating
+        // the finite element solution at the position of the particles.
+        // This is essentially an optimized version of the particle advection
+        // functionality in step 19, but instead of creating quadrature
+        // objects and FEValues objects for each cell, we do the
+        // evaluation by hand, which is somewhat more efficient and only
+        // matters for this tutorial, because the particle work is the
+        // dominant cost of the whole program.
+        const auto pic = particle_handler.particles_in_cell(cell);
+        Assert(pic.begin() == particle, ExcInternalError());
+        for (auto &p : pic)
+          {
+            const Point<dim> reference_location = p.get_reference_location();
+            Tensor<1, dim>   particle_velocity;
+            for (unsigned int j = 0; j < fluid_fe.dofs_per_cell; ++j)
+              {
+                const auto comp_j = fluid_fe.system_to_component_index(j);
+
+                particle_velocity[comp_j.first] +=
+                  fluid_fe.shape_value(j, reference_location) *
+                  local_dof_values[j];
+              }
+
+            Point<dim> particle_location = particle->get_location();
+            for (int d = 0; d < dim; ++d)
+              particle_location[d] += particle_velocity[d] * dt;
+            p.set_location(particle_location);
+
+            // Again, we store the particle velocity and the processor id in the
+            // particle properties for visualization purposes.
+            ArrayView<double> properties = p.get_properties();
+            for (int d = 0; d < dim; ++d)
+              properties[d] = particle_velocity[d];
+
+            properties[dim] =
+              Utilities::MPI::this_mpi_process(mpi_communicator);
+
+            ++particle;
+          }
+      }
+  }
+
+
+
+  // @sect4{Data output}
+
+  // The next two functions take care of writing both the particles
+  // and the background mesh to vtu with a pvtu record. This ensures
+  // that the simulation results can be visualized when the simulation is
+  // launched in parallel.
+  template <int dim>
+  void
+  ParticleTracking<dim>::output_particles(const unsigned int it)
+  {
+    Particles::DataOut<dim, dim> particle_output;
+
+    std::vector<std::string> solution_names(dim, "velocity");
+    solution_names.push_back("process_id");
+
+    std::vector<DataComponentInterpretation::DataComponentInterpretation>
+      data_component_interpretation(
+        dim, DataComponentInterpretation::component_is_part_of_vector);
+    data_component_interpretation.push_back(
+      DataComponentInterpretation::component_is_scalar);
+
+    particle_output.build_patches(particle_handler,
+                                  solution_names,
+                                  data_component_interpretation);
+    const std::string output_folder(output_directory);
+    const std::string file_name("interpolated-particles");
+
+    particle_output.write_vtu_with_pvtu_record(
+      output_folder, file_name, it, mpi_communicator, 6);
+  }
+
+  template <int dim>
+  void
+  ParticleTracking<dim>::log_particles()
+  {
+    if (Utilities::MPI::this_mpi_process(mpi_communicator) == 0)
+      deallog << "Particles location" << std::endl;
+    for (unsigned int proc = 0;
+         proc < Utilities::MPI::n_mpi_processes(mpi_communicator);
+         ++proc)
+      {
+        if (Utilities::MPI::this_mpi_process(mpi_communicator) == proc)
+          {
+            for (auto part : particle_handler)
+              {
+                deallog << part.get_location() << std::endl;
+              }
+          }
+        MPI_Barrier(mpi_communicator);
+      }
+  }
+
+
+
+  template <int dim>
+  void
+  ParticleTracking<dim>::output_background(const unsigned int it)
+  {
+    std::vector<std::string> solution_names(dim, "velocity");
+    std::vector<DataComponentInterpretation::DataComponentInterpretation>
+      data_component_interpretation(
+        dim, DataComponentInterpretation::component_is_part_of_vector);
+
+    DataOut<dim> data_out;
+
+    // Attach the solution data to data_out object
+    data_out.attach_dof_handler(fluid_dh);
+    data_out.add_data_vector(velocity_field,
+                             solution_names,
+                             DataOut<dim>::type_dof_data,
+                             data_component_interpretation);
+    Vector<float> subdomain(background_triangulation.n_active_cells());
+    for (unsigned int i = 0; i < subdomain.size(); ++i)
+      subdomain(i) = background_triangulation.locally_owned_subdomain();
+    data_out.add_data_vector(subdomain, "subdomain");
+
+    data_out.build_patches(mapping);
+
+    const std::string output_folder(output_directory);
+    const std::string file_name("background");
+
+    data_out.write_vtu_with_pvtu_record(
+      output_folder, file_name, it, mpi_communicator, 6);
+  }
+
+
+  template <int dim>
+  void
+  ParticleTracking<dim>::run()
+  {
+    DiscreteTime discrete_time(0, final_time, time_step);
+
+    generate_particles();
+
+    // We set the initial property of the particles by doing an
+    // explicit Euler iteration with a time-step of 0 both in the case
+    // of the analytical and the interpolated approach.
+    setup_background_dofs();
+    interpolate_function_to_field();
+    euler_step_interpolated(0.);
+
+    output_particles(discrete_time.get_step_number());
+    output_background(discrete_time.get_step_number());
+
+    // The particles are advected by looping over time.
+    while (!discrete_time.is_at_end())
+      {
+        discrete_time.advance_time();
+        velocity.set_time(discrete_time.get_previous_time());
+
+        interpolate_function_to_field();
+        euler_step_interpolated(discrete_time.get_previous_step_size());
+
+        // After the particles have been moved, it is necessary to identify
+        // in which cell they now reside. This is achieved by calling
+        // <code>sort_particles_into_subdomains_and_cells</code>
+        particle_handler.sort_particles_into_subdomains_and_cells();
+
+        if ((discrete_time.get_step_number() % output_frequency) == 0)
+          {
+            output_particles(discrete_time.get_step_number());
+            output_background(discrete_time.get_step_number());
+          }
+      }
+    log_particles();
+  }
+
+} // namespace Step68
+
+
+
+// @sect3{The main() function}
+
+// The remainder of the code, the `main()` function, is standard.
+// We note that we run the particle tracking with the analytical velocity
+// and the interpolated velocity and produce both results
+int
+main(int argc, char *argv[])
+{
+  using namespace Step68;
+  using namespace dealii;
+  Utilities::MPI::MPI_InitFinalize mpi_initialization(argc, argv, 1);
+
+  initlog();
+  deallog.depth_console(1);
+
+
+  try
+    {
+      Step68::ParticleTracking<2> particle_tracking;
+      particle_tracking.run();
+    }
+  catch (std::exception &exc)
+    {
+      std::cerr << std::endl
+                << std::endl
+                << "----------------------------------------------------"
+                << std::endl;
+      std::cerr << "Exception on processing: " << std::endl
+                << exc.what() << std::endl
+                << "Aborting!" << std::endl
+                << "----------------------------------------------------"
+                << std::endl;
+
+      return 1;
+    }
+  catch (...)
+    {
+      std::cerr << std::endl
+                << std::endl
+                << "----------------------------------------------------"
+                << std::endl;
+      std::cerr << "Unknown exception!" << std::endl
+                << "Aborting!" << std::endl
+                << "----------------------------------------------------"
+                << std::endl;
+      return 1;
+    }
+
+  return 0;
+}
diff --git a/tests/simplex/step-68.mpirun=1.with_simplex_support=on.output b/tests/simplex/step-68.mpirun=1.with_simplex_support=on.output
new file mode 100644 (file)
index 0000000..427bb9e
--- /dev/null
@@ -0,0 +1,51 @@
+
+DEAL::Number of particles inserted: 48
+DEAL::Particles location
+DEAL::0.366577 0.681804
+DEAL::0.398361 0.691931
+DEAL::0.408783 0.708456
+DEAL::0.423868 0.706937
+DEAL::0.433550 0.698432
+DEAL::0.444141 0.713765
+DEAL::0.454362 0.697548
+DEAL::0.453680 0.706030
+DEAL::0.464383 0.709034
+DEAL::0.469644 0.709187
+DEAL::0.484716 0.715907
+DEAL::0.499111 0.711062
+DEAL::0.498868 0.710828
+DEAL::0.517719 0.673900
+DEAL::0.503282 0.705982
+DEAL::0.520884 0.707622
+DEAL::0.524944 0.701767
+DEAL::0.535494 0.683627
+DEAL::0.532509 0.716124
+DEAL::0.549286 0.705238
+DEAL::0.563363 0.673326
+DEAL::0.573466 0.689253
+DEAL::0.618983 0.657786
+DEAL::0.628548 0.625331
+DEAL::0.691290 0.744340
+DEAL::0.379293 0.831136
+DEAL::0.391377 0.827051
+DEAL::0.409176 0.839420
+DEAL::0.448930 0.832650
+DEAL::0.465573 0.816761
+DEAL::0.476410 0.833958
+DEAL::0.494476 0.826914
+DEAL::0.493642 0.819960
+DEAL::0.518139 0.836076
+DEAL::0.559598 0.806751
+DEAL::0.568815 0.811801
+DEAL::0.571248 0.776918
+DEAL::0.586805 0.806620
+DEAL::0.591790 0.794930
+DEAL::0.591050 0.794375
+DEAL::0.607814 0.783764
+DEAL::0.613584 0.807264
+DEAL::0.629515 0.787985
+DEAL::0.647557 0.779179
+DEAL::0.660445 0.781133
+DEAL::0.655199 0.775754
+DEAL::0.698558 0.758550
+DEAL::0.631046 0.755799
diff --git a/tests/simplex/step-68.mpirun=4.with_simplex_support=on.output b/tests/simplex/step-68.mpirun=4.with_simplex_support=on.output
new file mode 100644 (file)
index 0000000..427bb9e
--- /dev/null
@@ -0,0 +1,51 @@
+
+DEAL::Number of particles inserted: 48
+DEAL::Particles location
+DEAL::0.366577 0.681804
+DEAL::0.398361 0.691931
+DEAL::0.408783 0.708456
+DEAL::0.423868 0.706937
+DEAL::0.433550 0.698432
+DEAL::0.444141 0.713765
+DEAL::0.454362 0.697548
+DEAL::0.453680 0.706030
+DEAL::0.464383 0.709034
+DEAL::0.469644 0.709187
+DEAL::0.484716 0.715907
+DEAL::0.499111 0.711062
+DEAL::0.498868 0.710828
+DEAL::0.517719 0.673900
+DEAL::0.503282 0.705982
+DEAL::0.520884 0.707622
+DEAL::0.524944 0.701767
+DEAL::0.535494 0.683627
+DEAL::0.532509 0.716124
+DEAL::0.549286 0.705238
+DEAL::0.563363 0.673326
+DEAL::0.573466 0.689253
+DEAL::0.618983 0.657786
+DEAL::0.628548 0.625331
+DEAL::0.691290 0.744340
+DEAL::0.379293 0.831136
+DEAL::0.391377 0.827051
+DEAL::0.409176 0.839420
+DEAL::0.448930 0.832650
+DEAL::0.465573 0.816761
+DEAL::0.476410 0.833958
+DEAL::0.494476 0.826914
+DEAL::0.493642 0.819960
+DEAL::0.518139 0.836076
+DEAL::0.559598 0.806751
+DEAL::0.568815 0.811801
+DEAL::0.571248 0.776918
+DEAL::0.586805 0.806620
+DEAL::0.591790 0.794930
+DEAL::0.591050 0.794375
+DEAL::0.607814 0.783764
+DEAL::0.613584 0.807264
+DEAL::0.629515 0.787985
+DEAL::0.647557 0.779179
+DEAL::0.660445 0.781133
+DEAL::0.655199 0.775754
+DEAL::0.698558 0.758550
+DEAL::0.631046 0.755799

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