From: Peter Munch <peterrmuench@gmail.com>
Date: Thu, 2 Jul 2020 06:24:13 +0000 (+0200)
Subject: Add Simplex::FE_P and Simplex::FE_DGP
X-Git-Tag: v9.3.0-rc1~1248^2
X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=1b0ef492789c212d5e6d21f5c364da423469f025;p=dealii.git

Add Simplex::FE_P and Simplex::FE_DGP
---

diff --git a/include/deal.II/simplex/fe_lib.h b/include/deal.II/simplex/fe_lib.h
new file mode 100644
index 0000000000..2573906922
--- /dev/null
+++ b/include/deal.II/simplex/fe_lib.h
@@ -0,0 +1,126 @@
+// ---------------------------------------------------------------------
+//
+// 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.
+//
+// ---------------------------------------------------------------------
+
+#ifndef dealii_simplex_fe_lib_h
+#define dealii_simplex_fe_lib_h
+
+#include <deal.II/base/config.h>
+
+#include <deal.II/fe/fe_poly.h>
+
+#include <deal.II/simplex/polynomials.h>
+
+DEAL_II_NAMESPACE_OPEN
+
+namespace Simplex
+{
+  /**
+   * Base class of FE_P and FE_DGP.
+   *
+   * @note Only implemented for 2D and 3D.
+   *
+   * @ingroup simplex
+   */
+  template <int dim, int spacedim = dim>
+  class FE_Poly : public dealii::FE_Poly<dim, spacedim>
+  {
+  public:
+    /**
+     * Constructor.
+     */
+    FE_Poly(const unsigned int               degree,
+            const std::vector<unsigned int> &dpo_vector);
+
+  private:
+    /**
+     * @copydoc dealii::FiniteElement::convert_generalized_support_point_values_to_dof_values()
+     */
+    void
+    convert_generalized_support_point_values_to_dof_values(
+      const std::vector<Vector<double>> &support_point_values,
+      std::vector<double> &              nodal_values) const override;
+  };
+
+
+
+  /**
+   * Implementation of a scalar Lagrange finite element Pp that yields
+   * the finite element space of continuous, piecewise polynomials of
+   * degree p.
+   *
+   * @ingroup simplex
+   */
+  template <int dim, int spacedim = dim>
+  class FE_P : public FE_Poly<dim, spacedim>
+  {
+  public:
+    /**
+     * Constructor.
+     */
+    FE_P(const unsigned int degree);
+
+    /**
+     * @copydoc dealii::FiniteElement::clone()
+     */
+    std::unique_ptr<FiniteElement<dim, spacedim>>
+    clone() const override;
+
+    /**
+     * Return a string that uniquely identifies a finite element. This class
+     * returns <tt>Simplex::FE_P<dim>(degree)</tt>, with @p dim and @p degree
+     * replaced by appropriate values.
+     */
+    std::string
+    get_name() const override;
+  };
+
+
+
+  /**
+   * Implementation of a scalar Lagrange finite element Pp that yields
+   * the finite element space of discontinuous, piecewise polynomials of
+   * degree p.
+   *
+   * @ingroup simplex
+   */
+  template <int dim, int spacedim = dim>
+  class FE_DGP : public FE_Poly<dim, spacedim>
+  {
+  public:
+    /**
+     * Constructor.
+     */
+    FE_DGP(const unsigned int degree);
+
+    /**
+     * @copydoc dealii::FiniteElement::clone()
+     */
+    std::unique_ptr<FiniteElement<dim, spacedim>>
+    clone() const override;
+
+    /**
+     * Return a string that uniquely identifies a finite element. This class
+     * returns <tt>Simplex::FE_DGP<dim>(degree)</tt>, with @p dim and @p degree
+     * replaced by appropriate values.
+     */
+    std::string
+    get_name() const override;
+  };
+
+} // namespace Simplex
+
+DEAL_II_NAMESPACE_CLOSE
+
+#endif
diff --git a/source/simplex/CMakeLists.txt b/source/simplex/CMakeLists.txt
index f0c06e1d0d..604ce1d1f2 100644
--- a/source/simplex/CMakeLists.txt
+++ b/source/simplex/CMakeLists.txt
@@ -16,6 +16,7 @@
 INCLUDE_DIRECTORIES(BEFORE ${CMAKE_CURRENT_BINARY_DIR})
 
 SET(_unity_include_src
+  fe_lib.cc
   polynomials.cc
   quadrature_lib.cc
   )
@@ -33,6 +34,7 @@ SETUP_SOURCE_LIST("${_unity_include_src}"
   )
 
 SET(_inst
+  fe_lib.inst.in
 )
 
 FILE(GLOB _header
diff --git a/source/simplex/fe_lib.cc b/source/simplex/fe_lib.cc
new file mode 100644
index 0000000000..75af427581
--- /dev/null
+++ b/source/simplex/fe_lib.cc
@@ -0,0 +1,275 @@
+// ---------------------------------------------------------------------
+//
+// 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.
+//
+// ---------------------------------------------------------------------
+
+#include <deal.II/base/config.h>
+
+#include <deal.II/simplex/fe_lib.h>
+
+DEAL_II_NAMESPACE_OPEN
+
+namespace Simplex
+{
+  namespace
+  {
+    /**
+     * Helper function to set up the dpo vector of FE_P for a given @p dim and
+     * @p degree.
+     */
+    std::vector<unsigned int>
+    get_dpo_vector_fe_p(const unsigned int dim, const unsigned int degree)
+    {
+      std::vector<unsigned int> dpo(dim + 1, 0U);
+
+      if (degree == 1)
+        {
+          // one dof at each vertex
+          dpo[0] = 1;
+        }
+      else if (degree == 2)
+        {
+          // one dof at each vertex and in the middle of each line
+          dpo[0] = 1;
+          dpo[1] = 1;
+          dpo[2] = 0;
+        }
+      else
+        {
+          Assert(false, ExcNotImplemented());
+        }
+
+      return dpo;
+    }
+
+    /**
+     * Helper function to set up the dpo vector of FE_DGP for a given @p dim and
+     * @p degree.
+     */
+    std::vector<unsigned int>
+    get_dpo_vector_fe_dgp(const unsigned int dim, const unsigned int degree)
+    {
+      std::vector<unsigned int> dpo(dim + 1, 0U);
+
+      // all dofs are internal
+      if (dim == 2 && degree == 1)
+        dpo[dim] = 3;
+      else if (dim == 2 && degree == 2)
+        dpo[dim] = 6;
+      else if (dim == 3 && degree == 1)
+        dpo[dim] = 4;
+      else if (dim == 3 && degree == 2)
+        dpo[dim] = 10;
+      else
+        {
+          Assert(false, ExcNotImplemented());
+        }
+
+      return dpo;
+    }
+  } // namespace
+
+
+
+  template <int dim, int spacedim>
+  FE_Poly<dim, spacedim>::FE_Poly(const unsigned int               degree,
+                                  const std::vector<unsigned int> &dpo_vector)
+    : dealii::FE_Poly<dim, spacedim>(
+        Simplex::ScalarPolynomial<dim>(degree),
+        FiniteElementData<dim>(dpo_vector,
+                               dim == 2 ? ReferenceCell::Type::Tri :
+                                          ReferenceCell::Type::Tet,
+                               1,
+                               degree,
+                               FiniteElementData<dim>::L2),
+        std::vector<bool>(FiniteElementData<dim>(dpo_vector,
+                                                 dim == 2 ?
+                                                   ReferenceCell::Type::Tri :
+                                                   ReferenceCell::Type::Tet,
+                                                 1,
+                                                 degree)
+                            .dofs_per_cell,
+                          true),
+        std::vector<ComponentMask>(
+          FiniteElementData<dim>(dpo_vector,
+                                 dim == 2 ? ReferenceCell::Type::Tri :
+                                            ReferenceCell::Type::Tet,
+                                 1,
+                                 degree)
+            .dofs_per_cell,
+          std::vector<bool>(1, true)))
+  {
+    this->unit_support_points.clear();
+
+    if (dim == 2)
+      {
+        if (degree == 1)
+          {
+            this->unit_support_points.emplace_back(1.0, 0.0);
+            this->unit_support_points.emplace_back(0.0, 1.0);
+            this->unit_support_points.emplace_back(0.0, 0.0);
+
+            // TODO
+            this->unit_face_support_points.emplace_back(0.0);
+            this->unit_face_support_points.emplace_back(1.0);
+          }
+        else if (degree == 2)
+          {
+            this->unit_support_points.emplace_back(1.0, 0.0);
+            this->unit_support_points.emplace_back(0.0, 1.0);
+            this->unit_support_points.emplace_back(0.0, 0.0);
+            this->unit_support_points.emplace_back(0.5, 0.5);
+            this->unit_support_points.emplace_back(0.0, 0.5);
+            this->unit_support_points.emplace_back(0.5, 0.0);
+
+            // TODO
+            this->unit_face_support_points.emplace_back(0.0);
+            this->unit_face_support_points.emplace_back(1.0);
+            this->unit_face_support_points.emplace_back(0.5);
+          }
+        else
+          {
+            Assert(false, ExcNotImplemented());
+          }
+      }
+    else if (dim == 3)
+      {
+        if (degree == 1)
+          {
+            this->unit_support_points.emplace_back(0.0, 0.0, 0.0);
+            this->unit_support_points.emplace_back(1.0, 0.0, 0.0);
+            this->unit_support_points.emplace_back(0.0, 1.0, 0.0);
+            this->unit_support_points.emplace_back(0.0, 0.0, 1.0);
+
+            // TODO
+            this->unit_face_support_points.emplace_back(1.0, 0.0);
+            this->unit_face_support_points.emplace_back(0.0, 1.0);
+            this->unit_face_support_points.emplace_back(0.0, 0.0);
+          }
+        else if (degree == 2)
+          {
+            this->unit_support_points.emplace_back(0.0, 0.0, 0.0);
+            this->unit_support_points.emplace_back(1.0, 0.0, 0.0);
+            this->unit_support_points.emplace_back(0.0, 1.0, 0.0);
+            this->unit_support_points.emplace_back(0.0, 0.0, 1.0);
+            this->unit_support_points.emplace_back(0.5, 0.0, 0.0);
+            this->unit_support_points.emplace_back(0.5, 0.5, 0.0);
+            this->unit_support_points.emplace_back(0.0, 0.5, 0.0);
+            this->unit_support_points.emplace_back(0.0, 0.0, 0.5);
+            this->unit_support_points.emplace_back(0.5, 0.0, 0.5);
+            this->unit_support_points.emplace_back(0.0, 0.5, 0.5);
+
+            // TODO
+            this->unit_face_support_points.emplace_back(1.0, 0.0);
+            this->unit_face_support_points.emplace_back(0.0, 1.0);
+            this->unit_face_support_points.emplace_back(0.0, 0.0);
+            this->unit_face_support_points.emplace_back(0.5, 0.5);
+            this->unit_face_support_points.emplace_back(0.0, 0.5);
+            this->unit_face_support_points.emplace_back(0.5, 0.0);
+          }
+        else
+          {
+            Assert(false, ExcNotImplemented());
+          }
+      }
+    else
+      {
+        Assert(false, ExcNotImplemented());
+      }
+  }
+
+
+
+  template <int dim, int spacedim>
+  void
+  FE_Poly<dim, spacedim>::
+    convert_generalized_support_point_values_to_dof_values(
+      const std::vector<Vector<double>> &support_point_values,
+      std::vector<double> &              nodal_values) const
+  {
+    AssertDimension(support_point_values.size(),
+                    this->get_unit_support_points().size());
+    AssertDimension(support_point_values.size(), nodal_values.size());
+    AssertDimension(this->dofs_per_cell, nodal_values.size());
+
+    for (unsigned int i = 0; i < this->dofs_per_cell; ++i)
+      {
+        AssertDimension(support_point_values[i].size(), 1);
+
+        nodal_values[i] = support_point_values[i](0);
+      }
+  }
+
+
+
+  template <int dim, int spacedim>
+  FE_P<dim, spacedim>::FE_P(const unsigned int degree)
+    : FE_Poly<dim, spacedim>(degree, get_dpo_vector_fe_p(dim, degree))
+  {}
+
+
+
+  template <int dim, int spacedim>
+  std::unique_ptr<FiniteElement<dim, spacedim>>
+  FE_P<dim, spacedim>::clone() const
+  {
+    return std::make_unique<FE_P<dim, spacedim>>(*this);
+  }
+
+
+
+  template <int dim, int spacedim>
+  std::string
+  FE_P<dim, spacedim>::get_name() const
+  {
+    std::ostringstream namebuf;
+    namebuf << "FE_P<" << dim << ">(" << this->degree << ")";
+
+    return namebuf.str();
+  }
+
+
+  template <int dim, int spacedim>
+  FE_DGP<dim, spacedim>::FE_DGP(const unsigned int degree)
+    : FE_Poly<dim, spacedim>(degree, get_dpo_vector_fe_dgp(dim, degree))
+  {}
+
+
+
+  template <int dim, int spacedim>
+  std::unique_ptr<FiniteElement<dim, spacedim>>
+  FE_DGP<dim, spacedim>::clone() const
+  {
+    return std::make_unique<FE_DGP<dim, spacedim>>(*this);
+  }
+
+
+
+  template <int dim, int spacedim>
+  std::string
+  FE_DGP<dim, spacedim>::get_name() const
+  {
+    std::ostringstream namebuf;
+    namebuf << "FE_DGP<" << dim << ">(" << this->degree << ")";
+
+    return namebuf.str();
+  }
+
+
+
+} // namespace Simplex
+
+// explicit instantiations
+#include "fe_lib.inst"
+
+DEAL_II_NAMESPACE_CLOSE
diff --git a/source/simplex/fe_lib.inst.in b/source/simplex/fe_lib.inst.in
new file mode 100644
index 0000000000..0f29f676d1
--- /dev/null
+++ b/source/simplex/fe_lib.inst.in
@@ -0,0 +1,25 @@
+// ---------------------------------------------------------------------
+//
+// 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.
+//
+// ---------------------------------------------------------------------
+
+
+
+for (deal_II_dimension : DIMENSIONS; deal_II_space_dimension : DIMENSIONS)
+  {
+#if deal_II_dimension <= deal_II_space_dimension
+    template class Simplex::FE_Poly<deal_II_dimension, deal_II_space_dimension>;
+    template class Simplex::FE_P<deal_II_dimension, deal_II_space_dimension>;
+    template class Simplex::FE_DGP<deal_II_dimension, deal_II_space_dimension>;
+#endif
+  }
diff --git a/tests/simplex/fe_lib_01.cc b/tests/simplex/fe_lib_01.cc
new file mode 100644
index 0000000000..e8f01a75bc
--- /dev/null
+++ b/tests/simplex/fe_lib_01.cc
@@ -0,0 +1,56 @@
+// ---------------------------------------------------------------------
+//
+// 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.
+//
+// ---------------------------------------------------------------------
+
+
+// Test n_dofs_per-methods of Simplex::FE_P  and Simplex::FE_DGP.
+
+
+#include <deal.II/simplex/fe_lib.h>
+
+#include "../tests.h"
+
+using namespace dealii;
+
+template <int dim, int spacedim>
+void
+test(const FiniteElement<dim, spacedim> &fe)
+{
+  deallog << fe.get_name() << ": " << std::endl;
+
+  deallog << "  n_dofs_per_vertex(): " << fe.n_dofs_per_vertex() << std::endl;
+  deallog << "  n_dofs_per_line():   " << fe.n_dofs_per_line() << std::endl;
+  deallog << "  n_dofs_per_quad():   " << fe.n_dofs_per_quad() << std::endl;
+  deallog << "  n_dofs_per_hex():    " << fe.n_dofs_per_hex() << std::endl;
+  deallog << "  n_dofs_per_face():   " << fe.n_dofs_per_face() << std::endl;
+  deallog << "  n_dofs_per_cell():   " << fe.n_dofs_per_cell() << std::endl;
+  deallog << "  tensor_degree():     " << fe.tensor_degree() << std::endl;
+
+  deallog << std::endl;
+}
+
+int
+main()
+{
+  initlog();
+
+  test(Simplex::FE_P<2>(1));
+  test(Simplex::FE_P<2>(2));
+  test(Simplex::FE_P<3>(1));
+  test(Simplex::FE_P<3>(2));
+  test(Simplex::FE_DGP<2>(1));
+  test(Simplex::FE_DGP<2>(2));
+  test(Simplex::FE_DGP<3>(1));
+  test(Simplex::FE_DGP<3>(2));
+}
diff --git a/tests/simplex/fe_lib_01.output b/tests/simplex/fe_lib_01.output
new file mode 100644
index 0000000000..4410d62f69
--- /dev/null
+++ b/tests/simplex/fe_lib_01.output
@@ -0,0 +1,73 @@
+
+DEAL::FE_P<2>(1): 
+DEAL::  n_dofs_per_vertex(): 1
+DEAL::  n_dofs_per_line():   0
+DEAL::  n_dofs_per_quad():   0
+DEAL::  n_dofs_per_hex():    0
+DEAL::  n_dofs_per_face():   2
+DEAL::  n_dofs_per_cell():   3
+DEAL::  tensor_degree():     1
+DEAL::
+DEAL::FE_P<2>(2): 
+DEAL::  n_dofs_per_vertex(): 1
+DEAL::  n_dofs_per_line():   1
+DEAL::  n_dofs_per_quad():   0
+DEAL::  n_dofs_per_hex():    0
+DEAL::  n_dofs_per_face():   3
+DEAL::  n_dofs_per_cell():   6
+DEAL::  tensor_degree():     2
+DEAL::
+DEAL::FE_P<3>(1): 
+DEAL::  n_dofs_per_vertex(): 1
+DEAL::  n_dofs_per_line():   0
+DEAL::  n_dofs_per_quad():   0
+DEAL::  n_dofs_per_hex():    0
+DEAL::  n_dofs_per_face():   3
+DEAL::  n_dofs_per_cell():   4
+DEAL::  tensor_degree():     1
+DEAL::
+DEAL::FE_P<3>(2): 
+DEAL::  n_dofs_per_vertex(): 1
+DEAL::  n_dofs_per_line():   1
+DEAL::  n_dofs_per_quad():   0
+DEAL::  n_dofs_per_hex():    0
+DEAL::  n_dofs_per_face():   6
+DEAL::  n_dofs_per_cell():   10
+DEAL::  tensor_degree():     2
+DEAL::
+DEAL::FE_DGP<2>(1): 
+DEAL::  n_dofs_per_vertex(): 0
+DEAL::  n_dofs_per_line():   0
+DEAL::  n_dofs_per_quad():   3
+DEAL::  n_dofs_per_hex():    0
+DEAL::  n_dofs_per_face():   0
+DEAL::  n_dofs_per_cell():   3
+DEAL::  tensor_degree():     1
+DEAL::
+DEAL::FE_DGP<2>(2): 
+DEAL::  n_dofs_per_vertex(): 0
+DEAL::  n_dofs_per_line():   0
+DEAL::  n_dofs_per_quad():   6
+DEAL::  n_dofs_per_hex():    0
+DEAL::  n_dofs_per_face():   0
+DEAL::  n_dofs_per_cell():   6
+DEAL::  tensor_degree():     2
+DEAL::
+DEAL::FE_DGP<3>(1): 
+DEAL::  n_dofs_per_vertex(): 0
+DEAL::  n_dofs_per_line():   0
+DEAL::  n_dofs_per_quad():   0
+DEAL::  n_dofs_per_hex():    4
+DEAL::  n_dofs_per_face():   0
+DEAL::  n_dofs_per_cell():   4
+DEAL::  tensor_degree():     1
+DEAL::
+DEAL::FE_DGP<3>(2): 
+DEAL::  n_dofs_per_vertex(): 0
+DEAL::  n_dofs_per_line():   0
+DEAL::  n_dofs_per_quad():   0
+DEAL::  n_dofs_per_hex():    10
+DEAL::  n_dofs_per_face():   0
+DEAL::  n_dofs_per_cell():   10
+DEAL::  tensor_degree():     2
+DEAL::
diff --git a/tests/simplex/fe_lib_02.cc b/tests/simplex/fe_lib_02.cc
new file mode 100644
index 0000000000..c65e02abf8
--- /dev/null
+++ b/tests/simplex/fe_lib_02.cc
@@ -0,0 +1,60 @@
+// ---------------------------------------------------------------------
+//
+// 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.
+//
+// ---------------------------------------------------------------------
+
+
+// Evaluate Simplex::FE_P  and Simplex::FE_DGP at quadrature points.
+
+
+#include <deal.II/simplex/fe_lib.h>
+#include <deal.II/simplex/quadrature_lib.h>
+
+#include "../tests.h"
+
+using namespace dealii;
+
+template <int dim, int spacedim>
+void
+test(const FiniteElement<dim, spacedim> &fe, const Quadrature<dim> &quad)
+{
+  deallog.push(fe.get_name());
+
+  for (const auto &point : quad.get_points())
+    {
+      deallog << point << " : " << std::endl;
+      for (unsigned int i = 0; i < fe.n_dofs_per_cell(); i++)
+        deallog << fe.shape_value(i, point) << " ";
+      for (unsigned int i = 0; i < fe.n_dofs_per_cell(); i++)
+        deallog << fe.shape_grad(i, point) << " ";
+      deallog << std::endl;
+    }
+  deallog << std::endl;
+
+  deallog.pop();
+}
+
+int
+main()
+{
+  initlog();
+
+  test(Simplex::FE_P<2>(1), Simplex::PGauss<2>(3));
+  test(Simplex::FE_P<2>(2), Simplex::PGauss<2>(7));
+  test(Simplex::FE_P<3>(1), Simplex::PGauss<3>(4));
+  test(Simplex::FE_P<3>(2), Simplex::PGauss<3>(10));
+  test(Simplex::FE_DGP<2>(1), Simplex::PGauss<2>(3));
+  test(Simplex::FE_DGP<2>(2), Simplex::PGauss<2>(7));
+  test(Simplex::FE_DGP<3>(1), Simplex::PGauss<3>(4));
+  test(Simplex::FE_DGP<3>(2), Simplex::PGauss<3>(10));
+}
diff --git a/tests/simplex/fe_lib_02.output b/tests/simplex/fe_lib_02.output
new file mode 100644
index 0000000000..781569e78c
--- /dev/null
+++ b/tests/simplex/fe_lib_02.output
@@ -0,0 +1,105 @@
+
+DEAL:FE_P<2>(1)::0.666667 0.166667 : 
+DEAL:FE_P<2>(1)::0.166667 0.666667 0.166667 -1.00000 -1.00000 1.00000 0.00000 0.00000 1.00000 
+DEAL:FE_P<2>(1)::0.166667 0.666667 : 
+DEAL:FE_P<2>(1)::0.166667 0.166667 0.666667 -1.00000 -1.00000 1.00000 0.00000 0.00000 1.00000 
+DEAL:FE_P<2>(1)::0.166667 0.166667 : 
+DEAL:FE_P<2>(1)::0.666667 0.166667 0.166667 -1.00000 -1.00000 1.00000 0.00000 0.00000 1.00000 
+DEAL:FE_P<2>(1)::
+DEAL:FE_P<2>(2)::0.333333 0.333333 : 
+DEAL:FE_P<2>(2)::-0.111111 -0.111111 -0.111111 0.444444 0.444444 0.444444 -0.333333 -0.333333 0.333333 0.00000 0.00000 0.333333 3.99969e-12 -1.33333 1.33333 1.33333 -1.33333 3.99991e-12 
+DEAL:FE_P<2>(2)::0.797427 0.101287 : 
+DEAL:FE_P<2>(2)::-0.0807686 0.474353 -0.0807686 0.323074 0.323074 0.0410358 0.594854 0.594854 2.18971 0.00000 0.00000 -0.594854 -2.78456 -3.18971 0.405146 3.18971 -0.405146 3.99980e-12 
+DEAL:FE_P<2>(2)::0.101287 0.797427 : 
+DEAL:FE_P<2>(2)::-0.0807686 -0.0807686 0.474353 0.0410358 0.323074 0.323074 0.594854 0.594854 -0.594854 0.00000 0.00000 2.18971 3.99991e-12 -0.405146 3.18971 0.405146 -3.18971 -2.78456 
+DEAL:FE_P<2>(2)::0.101287 0.101287 : 
+DEAL:FE_P<2>(2)::0.474353 -0.0807686 -0.0807686 0.323074 0.0410358 0.323074 -2.18971 -2.18971 -0.594854 0.00000 0.00000 -0.594854 2.78456 -0.405146 0.405146 0.405146 -0.405146 2.78456 
+DEAL:FE_P<2>(2)::0.0597159 0.470142 : 
+DEAL:FE_P<2>(2)::-0.0280749 -0.0525839 -0.0280749 0.112300 0.112300 0.884134 -0.880568 -0.880568 -0.761137 0.00000 0.00000 0.880568 1.64170 -0.238863 1.88057 0.238863 -1.88057 8.00249e-13 
+DEAL:FE_P<2>(2)::0.470142 0.0597159 : 
+DEAL:FE_P<2>(2)::-0.0280749 -0.0280749 -0.0525839 0.884134 0.112300 0.112300 -0.880568 -0.880568 0.880568 0.00000 0.00000 -0.761137 8.00165e-13 -1.88057 0.238863 1.88057 -0.238863 1.64170 
+DEAL:FE_P<2>(2)::0.470142 0.470142 : 
+DEAL:FE_P<2>(2)::-0.0525839 -0.0280749 -0.0280749 0.112300 0.884134 0.112300 0.761137 0.761137 0.880568 0.00000 0.00000 0.880568 -1.64170 -1.88057 1.88057 1.88057 -1.88057 -1.64170 
+DEAL:FE_P<2>(2)::
+DEAL:FE_P<3>(1)::0.138197 0.138197 0.138197 : 
+DEAL:FE_P<3>(1)::0.585410 0.138197 0.138197 0.138197 -1.00000 -1.00000 -1.00000 1.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 1.00000 
+DEAL:FE_P<3>(1)::0.585410 0.138197 0.138197 : 
+DEAL:FE_P<3>(1)::0.138197 0.585410 0.138197 0.138197 -1.00000 -1.00000 -1.00000 1.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 1.00000 
+DEAL:FE_P<3>(1)::0.138197 0.585410 0.138197 : 
+DEAL:FE_P<3>(1)::0.138197 0.138197 0.585410 0.138197 -1.00000 -1.00000 -1.00000 1.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 1.00000 
+DEAL:FE_P<3>(1)::0.138197 0.138197 0.585410 : 
+DEAL:FE_P<3>(1)::0.138197 0.138197 0.138197 0.585410 -1.00000 -1.00000 -1.00000 1.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 1.00000 
+DEAL:FE_P<3>(1)::
+DEAL:FE_P<3>(2)::0.568431 0.143856 0.143856 : 
+DEAL:FE_P<3>(2)::-0.102467 0.0777961 -0.102467 -0.102467 0.327090 0.327090 0.0827787 0.0827787 0.327090 0.0827787 0.424574 0.424574 0.424574 1.27372 0.00000 0.00000 0.00000 -0.424574 0.00000 0.00000 0.00000 -0.424574 -1.69830 -2.27372 -2.27372 0.575426 2.27372 0.00000 -0.575426 -2.22045e-16 -0.575426 -0.575426 -0.575426 -2.22045e-16 0.575426 0.00000 2.27372 0.00000 0.575426 0.575426 
+DEAL:FE_P<3>(2)::0.143856 0.143856 0.143856 : 
+DEAL:FE_P<3>(2)::0.0777961 -0.102467 -0.102467 -0.102467 0.327090 0.0827787 0.327090 0.327090 0.0827787 0.0827787 -1.27372 -1.27372 -1.27372 -0.424574 0.00000 0.00000 0.00000 -0.424574 0.00000 0.00000 0.00000 -0.424574 1.69830 -0.575426 -0.575426 0.575426 0.575426 0.00000 -0.575426 1.69830 -0.575426 -0.575426 -0.575426 1.69830 0.575426 0.00000 0.575426 0.00000 0.575426 0.575426 
+DEAL:FE_P<3>(2)::0.143856 0.143856 0.568431 : 
+DEAL:FE_P<3>(2)::-0.102467 -0.102467 -0.102467 0.0777961 0.0827787 0.0827787 0.0827787 0.327090 0.327090 0.327090 0.424574 0.424574 0.424574 -0.424574 0.00000 0.00000 0.00000 -0.424574 0.00000 0.00000 0.00000 1.27372 -3.33067e-16 -0.575426 -0.575426 0.575426 0.575426 0.00000 -0.575426 -3.33067e-16 -0.575426 -2.27372 -2.27372 -1.69830 2.27372 0.00000 0.575426 0.00000 2.27372 0.575426 
+DEAL:FE_P<3>(2)::0.143856 0.568431 0.143856 : 
+DEAL:FE_P<3>(2)::-0.102467 -0.102467 0.0777961 -0.102467 0.0827787 0.327090 0.327090 0.0827787 0.0827787 0.327090 0.424574 0.424574 0.424574 -0.424574 0.00000 0.00000 0.00000 1.27372 0.00000 0.00000 0.00000 -0.424574 -2.22045e-16 -0.575426 -0.575426 2.27372 0.575426 0.00000 -2.27372 -1.69830 -2.27372 -0.575426 -0.575426 -2.22045e-16 0.575426 0.00000 0.575426 0.00000 0.575426 2.27372 
+DEAL:FE_P<3>(2)::0.00000 0.500000 0.500000 : 
+DEAL:FE_P<3>(2)::0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 2.00000 0.00000 0.00000 -2.00000 -2.00000 -2.00000 -2.00000 -2.00000 -2.00000 2.00000 0.00000 0.00000 0.00000 2.00000 2.00000 
+DEAL:FE_P<3>(2)::0.500000 0.00000 0.500000 : 
+DEAL:FE_P<3>(2)::0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1.00000 0.00000 1.00000 1.00000 1.00000 1.00000 0.00000 0.00000 0.00000 -1.00000 0.00000 0.00000 0.00000 1.00000 -2.00000 -2.00000 -2.00000 0.00000 2.00000 0.00000 0.00000 0.00000 0.00000 -2.00000 -2.00000 -2.00000 2.00000 0.00000 2.00000 0.00000 2.00000 0.00000 
+DEAL:FE_P<3>(2)::0.500000 0.500000 0.00000 : 
+DEAL:FE_P<3>(2)::0.00000 0.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 0.00000 1.00000 1.00000 1.00000 1.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 -1.00000 -2.00000 -2.00000 -2.00000 2.00000 2.00000 0.00000 -2.00000 -2.00000 -2.00000 0.00000 0.00000 0.00000 0.00000 0.00000 2.00000 0.00000 0.00000 2.00000 
+DEAL:FE_P<3>(2)::0.500000 0.00000 0.00000 : 
+DEAL:FE_P<3>(2)::0.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 0.00000 0.00000 -1.00000 -1.00000 -1.00000 1.00000 0.00000 0.00000 0.00000 -1.00000 0.00000 0.00000 0.00000 -1.00000 0.00000 -2.00000 -2.00000 0.00000 2.00000 0.00000 0.00000 2.00000 0.00000 0.00000 0.00000 2.00000 0.00000 0.00000 2.00000 0.00000 0.00000 0.00000 
+DEAL:FE_P<3>(2)::0.00000 0.500000 0.00000 : 
+DEAL:FE_P<3>(2)::0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 -1.00000 -1.00000 -1.00000 -1.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 -1.00000 2.00000 0.00000 0.00000 2.00000 0.00000 0.00000 -2.00000 0.00000 -2.00000 0.00000 0.00000 2.00000 0.00000 0.00000 0.00000 0.00000 0.00000 2.00000 
+DEAL:FE_P<3>(2)::0.00000 0.00000 0.500000 : 
+DEAL:FE_P<3>(2)::0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 -1.00000 -1.00000 -1.00000 -1.00000 0.00000 0.00000 0.00000 -1.00000 0.00000 0.00000 0.00000 1.00000 2.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 2.00000 0.00000 -2.00000 -2.00000 0.00000 2.00000 0.00000 0.00000 0.00000 2.00000 0.00000 
+DEAL:FE_P<3>(2)::
+DEAL:FE_DGP<2>(1)::0.666667 0.166667 : 
+DEAL:FE_DGP<2>(1)::0.166667 0.666667 0.166667 -1.00000 -1.00000 1.00000 0.00000 0.00000 1.00000 
+DEAL:FE_DGP<2>(1)::0.166667 0.666667 : 
+DEAL:FE_DGP<2>(1)::0.166667 0.166667 0.666667 -1.00000 -1.00000 1.00000 0.00000 0.00000 1.00000 
+DEAL:FE_DGP<2>(1)::0.166667 0.166667 : 
+DEAL:FE_DGP<2>(1)::0.666667 0.166667 0.166667 -1.00000 -1.00000 1.00000 0.00000 0.00000 1.00000 
+DEAL:FE_DGP<2>(1)::
+DEAL:FE_DGP<2>(2)::0.333333 0.333333 : 
+DEAL:FE_DGP<2>(2)::-0.111111 -0.111111 -0.111111 0.444444 0.444444 0.444444 -0.333333 -0.333333 0.333333 0.00000 0.00000 0.333333 3.99969e-12 -1.33333 1.33333 1.33333 -1.33333 3.99991e-12 
+DEAL:FE_DGP<2>(2)::0.797427 0.101287 : 
+DEAL:FE_DGP<2>(2)::-0.0807686 0.474353 -0.0807686 0.323074 0.323074 0.0410358 0.594854 0.594854 2.18971 0.00000 0.00000 -0.594854 -2.78456 -3.18971 0.405146 3.18971 -0.405146 3.99980e-12 
+DEAL:FE_DGP<2>(2)::0.101287 0.797427 : 
+DEAL:FE_DGP<2>(2)::-0.0807686 -0.0807686 0.474353 0.0410358 0.323074 0.323074 0.594854 0.594854 -0.594854 0.00000 0.00000 2.18971 3.99991e-12 -0.405146 3.18971 0.405146 -3.18971 -2.78456 
+DEAL:FE_DGP<2>(2)::0.101287 0.101287 : 
+DEAL:FE_DGP<2>(2)::0.474353 -0.0807686 -0.0807686 0.323074 0.0410358 0.323074 -2.18971 -2.18971 -0.594854 0.00000 0.00000 -0.594854 2.78456 -0.405146 0.405146 0.405146 -0.405146 2.78456 
+DEAL:FE_DGP<2>(2)::0.0597159 0.470142 : 
+DEAL:FE_DGP<2>(2)::-0.0280749 -0.0525839 -0.0280749 0.112300 0.112300 0.884134 -0.880568 -0.880568 -0.761137 0.00000 0.00000 0.880568 1.64170 -0.238863 1.88057 0.238863 -1.88057 8.00249e-13 
+DEAL:FE_DGP<2>(2)::0.470142 0.0597159 : 
+DEAL:FE_DGP<2>(2)::-0.0280749 -0.0280749 -0.0525839 0.884134 0.112300 0.112300 -0.880568 -0.880568 0.880568 0.00000 0.00000 -0.761137 8.00165e-13 -1.88057 0.238863 1.88057 -0.238863 1.64170 
+DEAL:FE_DGP<2>(2)::0.470142 0.470142 : 
+DEAL:FE_DGP<2>(2)::-0.0525839 -0.0280749 -0.0280749 0.112300 0.884134 0.112300 0.761137 0.761137 0.880568 0.00000 0.00000 0.880568 -1.64170 -1.88057 1.88057 1.88057 -1.88057 -1.64170 
+DEAL:FE_DGP<2>(2)::
+DEAL:FE_DGP<3>(1)::0.138197 0.138197 0.138197 : 
+DEAL:FE_DGP<3>(1)::0.585410 0.138197 0.138197 0.138197 -1.00000 -1.00000 -1.00000 1.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 1.00000 
+DEAL:FE_DGP<3>(1)::0.585410 0.138197 0.138197 : 
+DEAL:FE_DGP<3>(1)::0.138197 0.585410 0.138197 0.138197 -1.00000 -1.00000 -1.00000 1.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 1.00000 
+DEAL:FE_DGP<3>(1)::0.138197 0.585410 0.138197 : 
+DEAL:FE_DGP<3>(1)::0.138197 0.138197 0.585410 0.138197 -1.00000 -1.00000 -1.00000 1.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 1.00000 
+DEAL:FE_DGP<3>(1)::0.138197 0.138197 0.585410 : 
+DEAL:FE_DGP<3>(1)::0.138197 0.138197 0.138197 0.585410 -1.00000 -1.00000 -1.00000 1.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 1.00000 
+DEAL:FE_DGP<3>(1)::
+DEAL:FE_DGP<3>(2)::0.568431 0.143856 0.143856 : 
+DEAL:FE_DGP<3>(2)::-0.102467 0.0777961 -0.102467 -0.102467 0.327090 0.327090 0.0827787 0.0827787 0.327090 0.0827787 0.424574 0.424574 0.424574 1.27372 0.00000 0.00000 0.00000 -0.424574 0.00000 0.00000 0.00000 -0.424574 -1.69830 -2.27372 -2.27372 0.575426 2.27372 0.00000 -0.575426 -2.22045e-16 -0.575426 -0.575426 -0.575426 -2.22045e-16 0.575426 0.00000 2.27372 0.00000 0.575426 0.575426 
+DEAL:FE_DGP<3>(2)::0.143856 0.143856 0.143856 : 
+DEAL:FE_DGP<3>(2)::0.0777961 -0.102467 -0.102467 -0.102467 0.327090 0.0827787 0.327090 0.327090 0.0827787 0.0827787 -1.27372 -1.27372 -1.27372 -0.424574 0.00000 0.00000 0.00000 -0.424574 0.00000 0.00000 0.00000 -0.424574 1.69830 -0.575426 -0.575426 0.575426 0.575426 0.00000 -0.575426 1.69830 -0.575426 -0.575426 -0.575426 1.69830 0.575426 0.00000 0.575426 0.00000 0.575426 0.575426 
+DEAL:FE_DGP<3>(2)::0.143856 0.143856 0.568431 : 
+DEAL:FE_DGP<3>(2)::-0.102467 -0.102467 -0.102467 0.0777961 0.0827787 0.0827787 0.0827787 0.327090 0.327090 0.327090 0.424574 0.424574 0.424574 -0.424574 0.00000 0.00000 0.00000 -0.424574 0.00000 0.00000 0.00000 1.27372 -3.33067e-16 -0.575426 -0.575426 0.575426 0.575426 0.00000 -0.575426 -3.33067e-16 -0.575426 -2.27372 -2.27372 -1.69830 2.27372 0.00000 0.575426 0.00000 2.27372 0.575426 
+DEAL:FE_DGP<3>(2)::0.143856 0.568431 0.143856 : 
+DEAL:FE_DGP<3>(2)::-0.102467 -0.102467 0.0777961 -0.102467 0.0827787 0.327090 0.327090 0.0827787 0.0827787 0.327090 0.424574 0.424574 0.424574 -0.424574 0.00000 0.00000 0.00000 1.27372 0.00000 0.00000 0.00000 -0.424574 -2.22045e-16 -0.575426 -0.575426 2.27372 0.575426 0.00000 -2.27372 -1.69830 -2.27372 -0.575426 -0.575426 -2.22045e-16 0.575426 0.00000 0.575426 0.00000 0.575426 2.27372 
+DEAL:FE_DGP<3>(2)::0.00000 0.500000 0.500000 : 
+DEAL:FE_DGP<3>(2)::0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 2.00000 0.00000 0.00000 -2.00000 -2.00000 -2.00000 -2.00000 -2.00000 -2.00000 2.00000 0.00000 0.00000 0.00000 2.00000 2.00000 
+DEAL:FE_DGP<3>(2)::0.500000 0.00000 0.500000 : 
+DEAL:FE_DGP<3>(2)::0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1.00000 0.00000 1.00000 1.00000 1.00000 1.00000 0.00000 0.00000 0.00000 -1.00000 0.00000 0.00000 0.00000 1.00000 -2.00000 -2.00000 -2.00000 0.00000 2.00000 0.00000 0.00000 0.00000 0.00000 -2.00000 -2.00000 -2.00000 2.00000 0.00000 2.00000 0.00000 2.00000 0.00000 
+DEAL:FE_DGP<3>(2)::0.500000 0.500000 0.00000 : 
+DEAL:FE_DGP<3>(2)::0.00000 0.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 0.00000 1.00000 1.00000 1.00000 1.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 -1.00000 -2.00000 -2.00000 -2.00000 2.00000 2.00000 0.00000 -2.00000 -2.00000 -2.00000 0.00000 0.00000 0.00000 0.00000 0.00000 2.00000 0.00000 0.00000 2.00000 
+DEAL:FE_DGP<3>(2)::0.500000 0.00000 0.00000 : 
+DEAL:FE_DGP<3>(2)::0.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 0.00000 0.00000 -1.00000 -1.00000 -1.00000 1.00000 0.00000 0.00000 0.00000 -1.00000 0.00000 0.00000 0.00000 -1.00000 0.00000 -2.00000 -2.00000 0.00000 2.00000 0.00000 0.00000 2.00000 0.00000 0.00000 0.00000 2.00000 0.00000 0.00000 2.00000 0.00000 0.00000 0.00000 
+DEAL:FE_DGP<3>(2)::0.00000 0.500000 0.00000 : 
+DEAL:FE_DGP<3>(2)::0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 -1.00000 -1.00000 -1.00000 -1.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 -1.00000 2.00000 0.00000 0.00000 2.00000 0.00000 0.00000 -2.00000 0.00000 -2.00000 0.00000 0.00000 2.00000 0.00000 0.00000 0.00000 0.00000 0.00000 2.00000 
+DEAL:FE_DGP<3>(2)::0.00000 0.00000 0.500000 : 
+DEAL:FE_DGP<3>(2)::0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 -1.00000 -1.00000 -1.00000 -1.00000 0.00000 0.00000 0.00000 -1.00000 0.00000 0.00000 0.00000 1.00000 2.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 2.00000 0.00000 -2.00000 -2.00000 0.00000 2.00000 0.00000 0.00000 0.00000 2.00000 0.00000 
+DEAL:FE_DGP<3>(2)::