--- /dev/null
+New: Added classes for conforming Hermite interpolation polynomials, allowing for higher levels of regularity between elements to be directly enforced.
+<br>
+(Ivy Weber, 2023/06/21)
--- /dev/null
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2023 - 2023 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_fe_hermite_h
+#define dealii_fe_hermite_h
+
+#include <deal.II/base/config.h>
+
+#include <deal.II/base/polynomials_hermite.h>
+
+#include <deal.II/fe/fe_poly.h>
+
+#include <string>
+#include <vector>
+
+DEAL_II_NAMESPACE_OPEN
+
+/**
+ * @addtogroup fe
+ * @{
+ */
+
+
+
+/**
+ * This class implements a Hermite interpolation basis of maximum regularity
+ * elements (see @cite CiarletRiavart1972interpolation). These bases are always
+ * of odd polynomial degree, have regularity $r=\frac{p-1}{2}$ and are defined
+ * up to polynomial degree $p=13$, with larger degrees currently being
+ * ill-conditioned.
+ *
+ * Each node has $(r+1)^{d}$ degrees of freedom (DoFs) assigned to it,
+ * corresponding to various derivatives up to order $r$ in each direction.
+ * DoFs at each node are not consecutive in lexicographic ordering for
+ * $d>1$ due to the tensor product construction of the basis. The
+ * ordering is determined by the direction of the derivative each function
+ * corresponds to; first by $x$-derivatives, then $y$, then $z$. Locally over
+ * each element the DoFs are ordered similarly. See below for the local ordering
+ * for $r=1$, where DoFs are ordered from 0 to $(2r+2)^{d}-1$:
+ *
+ * <code>FE_Hermite<1>(3)</code>
+ *
+ * @verbatim
+ * (0)________________(2)
+ * (1) (3)
+ * @endverbatim
+ *
+ * <code>FE_Hermite<2>(3)</code>:
+ *
+ * @verbatim
+ * ( 8, 9)__________(10,11)
+ * (12,13) (14,15)
+ * | |
+ * | |
+ * | |
+ * | |
+ * | |
+ * | |
+ * ( 0, 1)__________( 2, 3)
+ * ( 4, 5) ( 6, 7)
+ * @endverbatim
+ *
+ * <code>FE_Hermite<3>(3)</code>:
+ *
+ * @verbatim
+ * (40,41,44,45)__(42,43,46,47) (40,41,44,45)__(42,43,46,47)
+ * (56,57,60,61) (58,59,62,63) (56,57,60,61) (58,59,62,63)
+ * / | | / / |
+ * / | | / / |
+ * / | | / / |
+ *(32,33,36,37)| | (32,33,36,37)__(34,35,38,39)
+ *(48,49,52,53)| | (48,49,52,53) (50,51,54,55)
+ * | | | | | |
+ * |( 8,9,12,13 )__(10,11,14,15) | |(10,11,14,15)
+ * |(24,25,28,29) (26,27,30,31) | |(26,27,30,31)
+ * | / / | | /
+ * | / / | | /
+ * | / / | | /
+ * ( 0,1,4,5 )___( 2,3,6,7 ) ( 0,1,4,5 )___( 2,3,6,7 )
+ * (16,17,20,21) (18,19,22,23) (16,17,20,21) (18,19,22,23)
+ * @endverbatim
+ *
+ * Due to the increased regularity constraints many of these functions are
+ * shared between elements, leading to fewer DoFs overall.
+ */
+template <int dim, int spacedim = dim>
+class FE_Hermite : public FE_Poly<dim, spacedim>
+{
+public:
+ /**
+ * Constructor that creates an Hermite finite element object with
+ * continuity in derivatives up to and including order
+ * <code>(fe_degree - 1)/2</code>.
+ * Throws an exception if @p fe_degree is not an odd positive integer.
+ */
+ FE_Hermite(const unsigned int fe_degree);
+
+ // Other functions
+ /**
+ * Returns <code>FE_Hermite<dim,spacedim>(fe_degree)</code> as a
+ * <code>std::string</code> with @p dim, @p spacedim and @p fe_degree
+ * replaced with the correct values.
+ */
+ virtual std::string
+ get_name() const override;
+
+ /**
+ * @copydoc dealii::FiniteElement::clone()
+ */
+ virtual std::unique_ptr<FiniteElement<dim, spacedim>>
+ clone() const override;
+
+ /**
+ * @copydoc dealii::FiniteElement::requires_update_flags()
+ */
+ virtual UpdateFlags
+ requires_update_flags(const UpdateFlags update_flags) const override;
+
+ /**
+ * @copydoc dealii::FiniteElement::hp_vertex_dof_identities()
+ */
+ virtual std::vector<std::pair<unsigned int, unsigned int>>
+ hp_vertex_dof_identities(
+ const FiniteElement<dim, spacedim> &fe_other) const override;
+
+ /**
+ * @copydoc dealii::FiniteElement::hp_line_dof_identities()
+ */
+ virtual std::vector<std::pair<unsigned int, unsigned int>>
+ hp_line_dof_identities(
+ const FiniteElement<dim, spacedim> &fe_other) const override;
+
+ /**
+ * @copydoc dealii::FiniteElement::hp_quad_dof_identities()
+ */
+ virtual std::vector<std::pair<unsigned int, unsigned int>>
+ hp_quad_dof_identities(const FiniteElement<dim, spacedim> &fe_other,
+ const unsigned int face_no = 0) const override;
+
+ /**
+ * @copydoc FiniteElement::compare_for_domination()
+ */
+ virtual FiniteElementDomination::Domination
+ compare_for_domination(const FiniteElement<dim, spacedim> &other_fe,
+ const unsigned int codim) const override;
+
+ /**
+ * Returns the mapping between lexicographic and hierarchic numbering
+ * schemes for Hermite. See the class documentation for diagrams of
+ * examples of lexicographic numbering for Hermite elements.
+ */
+ std::vector<unsigned int>
+ get_lexicographic_to_hierarchic_numbering() const;
+
+ /**
+ * This re-implements FiniteElement::fill_fe_values() for a Hermite
+ * polynomial basis, to account for the more complicated shape function
+ * re-scaling that a Hermite basis requires. On a non-uniform grid,
+ * failing to rescale function values will lead to discontinuities across
+ * element boundaries in derivatives if elements have different sizes.
+ *
+ * At present this is done by only allowing certain cell mappings
+ * (currently only MappingCartesian) which guarantee
+ * rectangular cells, which ensures that the directions of derivatives
+ * does not change. The derivative continuity can then be enforced by
+ * re-scaling shape functions so that the magnitude of each derivative is
+ * equivalent to the derivative on the reference interval.
+ */
+ virtual void
+ fill_fe_values(
+ const typename Triangulation<dim, spacedim>::cell_iterator &cell,
+ const CellSimilarity::Similarity cell_similarity,
+ const Quadrature<dim> &quadrature,
+ const Mapping<dim, spacedim> &mapping,
+ const typename Mapping<dim, spacedim>::InternalDataBase &mapping_internal,
+ const dealii::internal::FEValuesImplementation::MappingRelatedData<dim,
+ spacedim>
+ &mapping_data,
+ const typename FiniteElement<dim, spacedim>::InternalDataBase &fe_internal,
+ dealii::internal::FEValuesImplementation::FiniteElementRelatedData<dim,
+ spacedim>
+ &output_data) const override;
+
+ using FiniteElement<dim, spacedim>::fill_fe_face_values;
+
+ /**
+ * This re-implements FiniteElement::fill_fe_face_values() for
+ * Hermite polynomial bases, for the same reasons as described for
+ * FEHermite::fill_fe_values().
+ */
+ virtual void
+ fill_fe_face_values(
+ const typename Triangulation<dim, spacedim>::cell_iterator &cell,
+ const unsigned int face_no,
+ const hp::QCollection<dim - 1> &quadrature,
+ const Mapping<dim, spacedim> &mapping,
+ const typename Mapping<dim, spacedim>::InternalDataBase &mapping_internal,
+ const dealii::internal::FEValuesImplementation::MappingRelatedData<dim,
+ spacedim>
+ &mapping_data,
+ const typename FiniteElement<dim, spacedim>::InternalDataBase &fe_internal,
+ dealii::internal::FEValuesImplementation::FiniteElementRelatedData<dim,
+ spacedim>
+ &output_data) const override;
+
+
+
+protected:
+ /**
+ * Wrapper function for FE_Poly::get_data() that ensures relevant
+ * shape value data is copied to the new data object as well.
+ */
+ virtual std::unique_ptr<
+ typename FiniteElement<dim, spacedim>::InternalDataBase>
+ get_data(
+ const UpdateFlags update_flags,
+ const Mapping<dim, spacedim> &mapping,
+ const Quadrature<dim> &quadrature,
+ dealii::internal::FEValuesImplementation::FiniteElementRelatedData<dim,
+ spacedim>
+ &output_data) const override
+ {
+ std::unique_ptr<typename FiniteElement<dim, spacedim>::InternalDataBase>
+ data_ptr = FE_Poly<dim, spacedim>::get_data(update_flags,
+ mapping,
+ quadrature,
+ output_data);
+ auto &data =
+ dynamic_cast<typename FE_Poly<dim, spacedim>::InternalData &>(*data_ptr);
+
+ const unsigned int n_q_points = quadrature.size();
+
+ if ((update_flags & update_values) &&
+ ((output_data.shape_values.n_rows() > 0) &&
+ (output_data.shape_values.n_cols() == n_q_points)))
+ data.shape_values = output_data.shape_values;
+
+ return data_ptr;
+ }
+
+
+
+private:
+ /**
+ * Variable storing the order of the highest derivative that the current
+ * FE_Hermite object can enforce continuity for. Here the order
+ * of derivative only counts in one spatial direction, so the derivative
+ * $\frac{d^{2}f}{dx \; dy}$ would be counted as a first order derivative
+ * of $f$, as an example.
+ */
+ unsigned int regularity;
+
+
+
+public:
+ /**
+ * Returns the regularity of the Hermite FE basis.
+ */
+ inline unsigned int
+ get_regularity() const
+ {
+ return this->regularity;
+ };
+};
+
+/** @} */
+
+
+
+DEAL_II_NAMESPACE_CLOSE
+
+#endif
* pushed forward to the real cell coordinates.
*/
update_jacobian_pushed_forward_3rd_derivatives = 0x1000000,
+ /**
+ * Update rescaling for Hermite elements.
+ */
+ update_rescale = 0x2000000,
//! Values needed for Piola transform
/**
* Combination of the flags needed for Piola transform of Hdiv elements.
update_covariant_transformation | update_contravariant_transformation |
update_transformation_values | update_transformation_gradients |
// Volume data
- update_volume_elements
+ update_volume_elements |
+ // Hermite needs several DOFs to be rescaled each time
+ update_rescale
};
#include <deal.II/dofs/dof_tools.h>
#include <deal.II/fe/fe.h>
+#include <deal.II/fe/fe_hermite.h>
#include <deal.II/fe/fe_values.h>
#include <deal.II/fe/mapping.h>
const unsigned int n_components = fe.n_components();
const unsigned int n_function_components =
boundary_functions.begin()->second->n_components;
- const bool fe_is_system = (n_components != 1);
+ const bool fe_is_system = (n_components != 1);
+ const bool fe_is_hermite =
+ (dynamic_cast<const FE_Hermite<dim, spacedim> *>(&fe) != nullptr);
const bool fe_is_primitive = fe.is_primitive();
copy_data.cell = cell;
UpdateFlags update_flags =
UpdateFlags(update_values | update_JxW_values | update_normal_vectors |
update_quadrature_points);
+ if (fe_is_hermite)
+ update_flags = update_flags | update_mapping;
FEFaceValues<dim, spacedim> fe_values(mapping, fe, q, update_flags);
// two variables for the coefficient, one for the two cases
const FiniteElement<dim, spacedim> &fe = dof.get_fe();
const unsigned int n_components = fe.n_components();
+ const bool fe_is_hermite =
+ (dynamic_cast<const FE_Hermite<dim, spacedim> *>(&fe) != nullptr);
- Assert(matrix.n() == dof.n_boundary_dofs(boundary_functions),
+ Assert(fe_is_hermite ||
+ matrix.n() == dof.n_boundary_dofs(boundary_functions),
ExcInternalError());
+ (void)fe_is_hermite;
+
Assert(matrix.n() == matrix.m(), ExcInternalError());
Assert(matrix.n() == rhs_vector.size(), ExcInternalError());
Assert(boundary_functions.size() != 0, ExcInternalError());
#include <deal.II/dofs/dof_tools.h>
#include <deal.II/fe/fe.h>
+#include <deal.II/fe/fe_hermite.h>
#include <deal.II/fe/fe_tools.h>
#include <deal.II/fe/fe_values.h>
Assert(boundary_ids.find(numbers::internal_face_boundary_id) ==
boundary_ids.end(),
(typename DoFHandler<dim, spacedim>::ExcInvalidBoundaryIndicator()));
- Assert(sparsity.n_rows() == dof.n_boundary_dofs(boundary_ids),
+
+ const bool fe_is_hermite = (dynamic_cast<const FE_Hermite<dim, spacedim> *>(
+ &(dof.get_fe())) != nullptr);
+
+ Assert(fe_is_hermite ||
+ sparsity.n_rows() == dof.n_boundary_dofs(boundary_ids),
ExcDimensionMismatch(sparsity.n_rows(),
dof.n_boundary_dofs(boundary_ids)));
- Assert(sparsity.n_cols() == dof.n_boundary_dofs(boundary_ids),
+ Assert(fe_is_hermite ||
+ sparsity.n_cols() == dof.n_boundary_dofs(boundary_ids),
ExcDimensionMismatch(sparsity.n_cols(),
dof.n_boundary_dofs(boundary_ids)));
+ (void)fe_is_hermite;
+
#ifdef DEBUG
if (sparsity.n_rows() != 0)
{
fe_dg_vector.cc
fe_enriched.cc
fe_face.cc
+ fe_hermite.cc
fe_nedelec.cc
fe_nedelec_sz.cc
fe_nothing.cc
fe_dg_vector.inst.in
fe_face.inst.in
fe.inst.in
+ fe_hermite.inst.in
fe_nedelec.inst.in
fe_nedelec_sz.inst.in
fe_nothing.inst.in
--- /dev/null
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2023 - 2023 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/base/quadrature.h>
+#include <deal.II/base/quadrature_lib.h>
+#include <deal.II/base/table.h>
+#include <deal.II/base/template_constraints.h>
+
+#include <deal.II/dofs/dof_tools.h>
+
+#include <deal.II/fe/fe_dgq.h>
+#include <deal.II/fe/fe_face.h>
+#include <deal.II/fe/fe_hermite.h>
+#include <deal.II/fe/fe_nothing.h>
+#include <deal.II/fe/fe_pyramid_p.h>
+#include <deal.II/fe/fe_q.h>
+#include <deal.II/fe/fe_simplex_p.h>
+#include <deal.II/fe/fe_simplex_p_bubbles.h>
+#include <deal.II/fe/fe_tools.h>
+#include <deal.II/fe/fe_values.h>
+#include <deal.II/fe/fe_wedge_p.h>
+#include <deal.II/fe/mapping_cartesian.h>
+
+#include <deal.II/grid/reference_cell.h>
+
+#include <deal.II/lac/dynamic_sparsity_pattern.h>
+#include <deal.II/lac/precondition.h>
+#include <deal.II/lac/solver_cg.h>
+#include <deal.II/lac/solver_control.h>
+#include <deal.II/lac/sparse_direct.h>
+#include <deal.II/lac/sparse_matrix.h>
+#include <deal.II/lac/sparsity_pattern.h>
+#include <deal.II/lac/sparsity_tools.h>
+#include <deal.II/lac/vector.h>
+
+#include <deal.II/numerics/matrix_tools.h>
+
+#include <algorithm>
+#include <cmath>
+#include <iostream>
+#include <iterator>
+#include <memory>
+#include <sstream>
+
+DEAL_II_NAMESPACE_OPEN
+
+
+
+namespace internal
+{
+ inline unsigned int
+ get_regularity_from_degree(const unsigned int fe_degree)
+ {
+ return (fe_degree == 0) ? 0 : (fe_degree - 1) / 2;
+ }
+
+
+
+ inline std::vector<unsigned int>
+ get_hermite_dpo_vector(const unsigned int dim, const unsigned int regularity)
+ {
+ std::vector<unsigned int> result(dim + 1, 0);
+ result[0] = Utilities::pow(regularity + 1, dim);
+
+ return result;
+ }
+
+
+
+ /*
+ * Renumbering function. Function needs different levels of for loop nesting
+ * for different values of dim, so different definitions are used for
+ * simplicity.
+ */
+ template <int dim>
+ void
+ hermite_hierarchic_to_lexicographic_numbering(const unsigned int regularity,
+ std::vector<unsigned int> &h2l);
+
+
+
+ template <>
+ void
+ hermite_hierarchic_to_lexicographic_numbering<1>(
+ const unsigned int regularity,
+ std::vector<unsigned int> &h2l)
+ {
+ const unsigned int node_dofs_1d = regularity + 1;
+
+ AssertDimension(h2l.size(), 2 * node_dofs_1d);
+
+ // Assign DOFs at vertices
+ for (unsigned int di = 0; di < 2; ++di)
+ for (unsigned int i = 0; i < node_dofs_1d; ++i)
+ h2l[i + di * node_dofs_1d] = i + di * node_dofs_1d;
+ }
+
+
+
+ template <>
+ void
+ hermite_hierarchic_to_lexicographic_numbering<2>(
+ const unsigned int regularity,
+ std::vector<unsigned int> &h2l)
+ {
+ const unsigned int node_dofs_1d = regularity + 1;
+ const unsigned int dim_dofs_1d = 2 * node_dofs_1d;
+ unsigned int offset = 0;
+
+ AssertDimension(h2l.size(), dim_dofs_1d * dim_dofs_1d);
+
+ // Assign DOFs at vertices
+ for (unsigned int di = 0; di < 2; ++di)
+ for (unsigned int dj = 0; dj < 2; ++dj)
+ {
+ for (unsigned int i = 0; i < node_dofs_1d; ++i)
+ for (unsigned int j = 0; j < node_dofs_1d; ++j)
+ h2l[j + i * node_dofs_1d + offset] =
+ j + i * dim_dofs_1d + (dj + di * dim_dofs_1d) * node_dofs_1d;
+
+ offset += node_dofs_1d * node_dofs_1d;
+ }
+ }
+
+
+
+ template <>
+ void
+ hermite_hierarchic_to_lexicographic_numbering<3>(
+ const unsigned int regularity,
+ std::vector<unsigned int> &h2l)
+ {
+ const unsigned int node_dofs_1d = regularity + 1;
+ const unsigned int node_dofs_2d = node_dofs_1d * node_dofs_1d;
+
+ const unsigned int dim_dofs_1d = 2 * node_dofs_1d;
+ const unsigned int dim_dofs_2d = dim_dofs_1d * dim_dofs_1d;
+
+ unsigned int offset = 0;
+
+ AssertDimension(h2l.size(), dim_dofs_2d * dim_dofs_1d);
+
+ // Assign DOFs at nodes
+ for (unsigned int di = 0; di < 2; ++di)
+ for (unsigned int dj = 0; dj < 2; ++dj)
+ for (unsigned int dk = 0; dk < 2; ++dk)
+ {
+ for (unsigned int i = 0; i < node_dofs_1d; ++i)
+ for (unsigned int j = 0; j < node_dofs_1d; ++j)
+ for (unsigned int k = 0; k < node_dofs_1d; ++k)
+ h2l[k + j * node_dofs_1d + i * node_dofs_2d + offset] =
+ k + j * dim_dofs_1d + i * dim_dofs_2d +
+ node_dofs_1d * (dk + dj * dim_dofs_1d + di * dim_dofs_2d);
+
+ offset += node_dofs_1d * node_dofs_2d;
+ }
+ }
+
+
+
+ template <int dim>
+ inline std::vector<unsigned int>
+ hermite_hierarchic_to_lexicographic_numbering(const unsigned int regularity)
+ {
+ const std::vector<unsigned int> dpo =
+ get_hermite_dpo_vector(dim, regularity);
+ const dealii::FiniteElementData<dim> face_data(dpo, 1, 2 * regularity + 1);
+ std::vector<unsigned int> renumbering(face_data.dofs_per_cell);
+
+ hermite_hierarchic_to_lexicographic_numbering<dim>(regularity, renumbering);
+
+ return renumbering;
+ }
+
+
+
+ template <int dim>
+ std::vector<unsigned int>
+ hermite_lexicographic_to_hierarchic_numbering(const unsigned int regularity)
+ {
+ return Utilities::invert_permutation(
+ hermite_hierarchic_to_lexicographic_numbering<dim>(regularity));
+ }
+
+
+
+ template <int dim>
+ inline std::vector<unsigned int>
+ hermite_face_lexicographic_to_hierarchic_numbering(
+ const unsigned int regularity)
+ {
+ return (dim > 1) ? hermite_lexicographic_to_hierarchic_numbering<dim - 1>(
+ regularity) :
+ std::vector<unsigned int>();
+ }
+
+
+
+ template <int dim>
+ TensorProductPolynomials<dim>
+ get_hermite_polynomials(const unsigned int fe_degree)
+ {
+ const unsigned int regularity = get_regularity_from_degree(fe_degree);
+
+ TensorProductPolynomials<dim> polynomial_basis(
+ Polynomials::PolynomialsHermite::generate_complete_basis(regularity));
+
+ std::vector<unsigned int> renumber =
+ internal::hermite_hierarchic_to_lexicographic_numbering<dim>(regularity);
+ polynomial_basis.set_numbering(renumber);
+
+ return polynomial_basis;
+ }
+
+
+
+ /**
+ * The @p Rescaler class implements the re-scaling of individual shape
+ * functions required by Hermite bases on non-uniform meshes. The three
+ * cases for different element dimensions are all defined separately
+ * due to the requirement for different levels of nesting of for loops.
+ */
+ template <int xdim, int xspacedim = xdim, typename xNumber = double>
+ class Rescaler
+ {
+ public:
+ template <int spacedim, typename Number>
+ void
+ rescale_fe_hermite_values(
+ Rescaler<1, spacedim, Number> & /*rescaler*/,
+ const FE_Hermite<1, spacedim> &fe_herm,
+ const typename Mapping<1, spacedim>::InternalDataBase &mapping_data,
+ Table<2, Number> &value_list)
+ {
+ unsigned int n_q_points;
+ double cell_extent = 1.0;
+
+ // Check mapping_data is associated with a compatible mapping class
+ if (dynamic_cast<const typename MappingCartesian<1>::InternalData *>(
+ &mapping_data) != nullptr)
+ {
+ const typename MappingCartesian<1>::InternalData *data =
+ dynamic_cast<const typename MappingCartesian<1>::InternalData *>(
+ &mapping_data);
+ n_q_points = data->quadrature_points.size();
+ cell_extent = data->cell_extents[0];
+ }
+ else
+ Assert(false, ExcInternalError());
+
+ const unsigned int regularity = fe_herm.get_regularity();
+ const unsigned int n_dofs_per_cell = fe_herm.n_dofs_per_cell();
+ const unsigned int n_q_points_out = value_list.size(1);
+ (void)n_dofs_per_cell;
+
+ AssertDimension(value_list.size(0), n_dofs_per_cell);
+ AssertDimension(n_dofs_per_cell, 2 * regularity + 2);
+ AssertIndexRange(n_q_points_out, n_q_points + 1);
+ (void)n_q_points;
+
+ std::vector<unsigned int> l2h =
+ hermite_lexicographic_to_hierarchic_numbering<1>(regularity);
+
+ for (unsigned int q = 0; q < n_q_points_out; ++q)
+ {
+ double factor_1 = 1.0;
+
+ for (unsigned int d1 = 0, d2 = regularity + 1; d2 < n_dofs_per_cell;
+ ++d1, ++d2)
+ {
+ /*
+ * d1 is used to count over indices on the left and d2 counts
+ * over indices on the right. These variables are used
+ * to avoid the need to loop over vertices.
+ */
+ value_list(l2h[d1], q) *= factor_1;
+ value_list(l2h[d2], q) *= factor_1;
+
+ factor_1 *= cell_extent;
+ }
+ }
+ }
+
+
+
+ template <int spacedim, typename Number>
+ void
+ rescale_fe_hermite_values(
+ Rescaler<2, spacedim, Number> & /*rescaler*/,
+ const FE_Hermite<2, spacedim> &fe_herm,
+ const typename Mapping<2, spacedim>::InternalDataBase &mapping_data,
+ Table<2, Number> &value_list)
+ {
+ unsigned int n_q_points;
+ Tensor<1, 2> cell_extents;
+
+ // Check mapping_data is associated with a compatible mapping class
+ if (dynamic_cast<const typename MappingCartesian<2>::InternalData *>(
+ &mapping_data) != nullptr)
+ {
+ const typename MappingCartesian<2>::InternalData *data =
+ dynamic_cast<const typename MappingCartesian<2>::InternalData *>(
+ &mapping_data);
+ n_q_points = data->quadrature_points.size();
+ cell_extents = data->cell_extents;
+ }
+ else
+ Assert(false, ExcInternalError());
+
+ const unsigned int regularity = fe_herm.get_regularity();
+ const unsigned int n_dofs_per_cell = fe_herm.n_dofs_per_cell();
+ const unsigned int n_dofs_per_dim = 2 * regularity + 2;
+ const unsigned int n_q_points_out = value_list.size(1);
+ (void)n_dofs_per_cell;
+
+ AssertDimension(value_list.size(0), n_dofs_per_cell);
+ AssertDimension(n_dofs_per_dim * n_dofs_per_dim, n_dofs_per_cell);
+ AssertIndexRange(n_q_points_out, n_q_points + 1);
+ (void)n_q_points;
+
+ std::vector<unsigned int> l2h =
+ hermite_lexicographic_to_hierarchic_numbering<2>(regularity);
+
+ AssertDimension(l2h.size(), n_dofs_per_cell);
+
+ for (unsigned int q = 0; q < n_q_points_out; ++q)
+ {
+ double factor_2 = 1.0;
+
+ for (unsigned int d3 = 0, d4 = regularity + 1; d4 < n_dofs_per_dim;
+ ++d3, ++d4)
+ {
+ double factor_1 = factor_2;
+
+ for (unsigned int d1 = 0, d2 = regularity + 1;
+ d2 < n_dofs_per_dim;
+ ++d1, ++d2)
+ {
+ /*
+ * d1 and d2 represent "left" and "right" in the
+ * x-direction, d3 and d4 represent "bottom" and "top"
+ * in the y-direction. As before, this is to avoid looping
+ * over vertices.
+ */
+ value_list(l2h[d1 + d3 * n_dofs_per_dim], q) *= factor_1;
+ value_list(l2h[d2 + d3 * n_dofs_per_dim], q) *= factor_1;
+ value_list(l2h[d1 + d4 * n_dofs_per_dim], q) *= factor_1;
+ value_list(l2h[d2 + d4 * n_dofs_per_dim], q) *= factor_1;
+
+ factor_1 *= cell_extents[0];
+ }
+
+ factor_2 *= cell_extents[1];
+ }
+ }
+ }
+
+
+
+ template <int spacedim, typename Number>
+ void
+ rescale_fe_hermite_values(
+ Rescaler<3, spacedim, Number> & /*rescaler*/,
+ const FE_Hermite<3, spacedim> &fe_herm,
+ const typename Mapping<3, spacedim>::InternalDataBase &mapping_data,
+ Table<2, Number> &value_list)
+ {
+ unsigned int n_q_points;
+ Tensor<1, 3> cell_extents;
+
+ // Check mapping_data is associated with a compatible mapping class
+ if (dynamic_cast<const typename MappingCartesian<3>::InternalData *>(
+ &mapping_data) != nullptr)
+ {
+ const typename MappingCartesian<3>::InternalData *data =
+ dynamic_cast<const typename MappingCartesian<3>::InternalData *>(
+ &mapping_data);
+ n_q_points = data->quadrature_points.size();
+ cell_extents = data->cell_extents;
+ }
+ else
+ Assert(false, ExcInternalError());
+
+ const unsigned int regularity = fe_herm.get_regularity();
+ const unsigned int n_dofs_per_cell = fe_herm.n_dofs_per_cell();
+ const unsigned int n_dofs_per_dim = 2 * regularity + 2;
+ const unsigned int n_dofs_per_quad = n_dofs_per_dim * n_dofs_per_dim;
+ const unsigned int n_q_points_out = value_list.size(1);
+ (void)n_dofs_per_cell;
+
+ AssertDimension(value_list.size(0), n_dofs_per_cell);
+ AssertDimension(Utilities::pow(n_dofs_per_dim, 3), n_dofs_per_cell);
+ AssertIndexRange(n_q_points_out, n_q_points + 1);
+ (void)n_q_points;
+
+ std::vector<unsigned int> l2h =
+ hermite_lexicographic_to_hierarchic_numbering<3>(regularity);
+
+ for (unsigned int q = 0; q < n_q_points_out; ++q)
+ {
+ double factor_3 = 1.0;
+
+ for (unsigned int d5 = 0, d6 = regularity + 1; d6 < n_dofs_per_dim;
+ ++d5, ++d6)
+ {
+ double factor_2 = factor_3;
+
+ for (unsigned int d3 = 0, d4 = regularity + 1;
+ d4 < n_dofs_per_dim;
+ ++d3, ++d4)
+ {
+ double factor_1 = factor_2;
+
+ for (unsigned int d1 = 0, d2 = regularity + 1;
+ d2 < n_dofs_per_dim;
+ ++d1, ++d2)
+ {
+ /*
+ * d1, d2: "left" and "right" (x-direction)
+ * d3, d4: "bottom" and "top" (y-direction)
+ * d5, d6: "down" and "up" (z-direction)
+ * This avoids looping over vertices
+ */
+ value_list(
+ l2h[d1 + d3 * n_dofs_per_dim + d5 * n_dofs_per_quad],
+ q) *= factor_1;
+ value_list(
+ l2h[d2 + d3 * n_dofs_per_dim + d5 * n_dofs_per_quad],
+ q) *= factor_1;
+ value_list(
+ l2h[d1 + d4 * n_dofs_per_dim + d5 * n_dofs_per_quad],
+ q) *= factor_1;
+ value_list(
+ l2h[d2 + d4 * n_dofs_per_dim + d5 * n_dofs_per_quad],
+ q) *= factor_1;
+ value_list(
+ l2h[d1 + d3 * n_dofs_per_dim + d6 * n_dofs_per_quad],
+ q) *= factor_1;
+ value_list(
+ l2h[d2 + d3 * n_dofs_per_dim + d6 * n_dofs_per_quad],
+ q) *= factor_1;
+ value_list(
+ l2h[d1 + d4 * n_dofs_per_dim + d6 * n_dofs_per_quad],
+ q) *= factor_1;
+ value_list(
+ l2h[d2 + d4 * n_dofs_per_dim + d6 * n_dofs_per_quad],
+ q) *= factor_1;
+
+ factor_1 *= cell_extents[0];
+ }
+
+ factor_2 *= cell_extents[1];
+ }
+
+ factor_3 *= cell_extents[2];
+ }
+ }
+ }
+ }; // class Rescaler
+} // namespace internal
+
+
+
+// Constructors
+template <int dim, int spacedim>
+FE_Hermite<dim, spacedim>::FE_Hermite(const unsigned int fe_degree)
+ : FE_Poly<dim, spacedim>(
+ internal::get_hermite_polynomials<dim>(fe_degree),
+ FiniteElementData<dim>(internal::get_hermite_dpo_vector(
+ dim,
+ internal::get_regularity_from_degree(fe_degree)),
+ 1,
+ std::max(1U, fe_degree),
+ ((fe_degree > 2) ? FiniteElementData<dim>::H2 :
+ FiniteElementData<dim>::H1)),
+ std::vector<bool>(Utilities::pow(std::max(2U, fe_degree + 1), dim),
+ false),
+ std::vector<ComponentMask>(Utilities::pow(std::max(2U, fe_degree + 1),
+ dim),
+ std::vector<bool>(1, true)))
+ , regularity(internal::get_regularity_from_degree(fe_degree))
+{
+ Assert((fe_degree % 2 == 1),
+ ExcMessage(
+ "ERROR: The current implementation of Hermite interpolation "
+ "polynomials is only defined for odd polynomial degrees. Running "
+ "in release mode will use a polynomial degree of max(1,fe_degree-1) "
+ "to protect against unexpected internal bugs."));
+}
+
+
+
+template <int dim, int spacedim>
+std::string
+FE_Hermite<dim, spacedim>::get_name() const
+{
+ std::ostringstream name_buffer;
+ name_buffer << "FE_Hermite<" << dim << "," << spacedim << ">(" << this->degree
+ << ")";
+ return name_buffer.str();
+}
+
+
+
+template <int dim, int spacedim>
+std::unique_ptr<FiniteElement<dim, spacedim>>
+FE_Hermite<dim, spacedim>::clone() const
+{
+ return std::make_unique<FE_Hermite<dim, spacedim>>(*this);
+}
+
+
+
+template <int dim, int spacedim>
+UpdateFlags
+FE_Hermite<dim, spacedim>::requires_update_flags(const UpdateFlags flags) const
+{
+ UpdateFlags out = FE_Poly<dim, spacedim>::requires_update_flags(flags);
+ if (flags & (update_values | update_gradients | update_hessians |
+ update_3rd_derivatives))
+ out |= update_rescale; // since we need to rescale values, gradients, ...
+ return out;
+}
+
+
+
+/**
+ * A large part of the following function is copied from FE_Q_Base. At present
+ * the case of two Hermite bases meeting is not implemented as it is unlikely
+ * that two different Hermite bases would be used in an hp method. This may be
+ * implemented in a later update.
+ */
+template <int dim, int spacedim>
+std::vector<std::pair<unsigned int, unsigned int>>
+FE_Hermite<dim, spacedim>::hp_vertex_dof_identities(
+ const FiniteElement<dim, spacedim> &fe_other) const
+{
+ if (dynamic_cast<const FE_Q_Base<dim, spacedim> *>(&fe_other) != nullptr)
+ {
+ // there should be exactly one single DoF of FE_Q_Base at a vertex, and it
+ // should have an identical value to the first Hermite DoF
+ return {{0U, 0U}};
+ }
+ else if (dynamic_cast<const FE_SimplexP<dim, spacedim> *>(&fe_other) !=
+ nullptr)
+ {
+ // there should be exactly one single DoF of FE_Q_Base at a vertex, and it
+ // should have an identical value to the first Hermite DoF
+ return {{0U, 0U}};
+ }
+ else if (dynamic_cast<const FE_Nothing<dim> *>(&fe_other) != nullptr)
+ {
+ // the FE_Nothing has no degrees of freedom, so there are no
+ // equivalencies to be recorded
+ return {};
+ }
+ else if (fe_other.n_unique_faces() == 1 && fe_other.n_dofs_per_face(0) == 0)
+ {
+ // if the other element has no elements on faces at all,
+ // then it would be impossible to enforce any kind of
+ // continuity even if we knew exactly what kind of element
+ // we have -- simply because the other element declares
+ // that it is discontinuous because it has no DoFs on
+ // its faces. in that case, just state that we have no
+ // constraints to declare
+ return {};
+ }
+ else if (const FE_Hermite<dim, spacedim> *fe_herm_other =
+ dynamic_cast<const FE_Hermite<dim, spacedim> *>(&fe_other))
+ {
+ Assert(false, ExcNotImplemented());
+ return {};
+ }
+ else
+ {
+ Assert(false, ExcNotImplemented());
+ return {};
+ }
+}
+
+
+
+/**
+ * This function only supplies empty lists of pairs, since Hermite
+ * stores all DoFs on vertices meaning there is no continuity that
+ * could be enforced.
+ */
+template <int dim, int spacedim>
+std::vector<std::pair<unsigned int, unsigned int>>
+FE_Hermite<dim, spacedim>::hp_line_dof_identities(
+ const FiniteElement<dim, spacedim> &fe_other) const
+{
+ (void)fe_other;
+ return {};
+}
+
+
+
+/**
+ * Similar to above, no continuity can be enforced on quads
+ * for Hermite.
+ */
+template <int dim, int spacedim>
+std::vector<std::pair<unsigned int, unsigned int>>
+FE_Hermite<dim, spacedim>::hp_quad_dof_identities(
+ const FiniteElement<dim, spacedim> &fe_other,
+ const unsigned int face_no) const
+{
+ (void)fe_other;
+ (void)face_no;
+ return {};
+}
+
+
+
+/*
+ * The layout of this function is largely copied directly from FE_Q,
+ * however FE_Hermite can behave significantly differently in terms
+ * of domination due to how the function space is defined */
+template <int dim, int spacedim>
+FiniteElementDomination::Domination
+FE_Hermite<dim, spacedim>::compare_for_domination(
+ const FiniteElement<dim, spacedim> &fe_other,
+ const unsigned int codim) const
+{
+ Assert(codim <= dim, ExcImpossibleInDim(dim));
+
+ if (codim > 0)
+ if (dynamic_cast<const FE_DGQ<dim, spacedim> *>(&fe_other) != nullptr)
+ // there are no requirements between continuous and discontinuous elements
+ return FiniteElementDomination::no_requirements;
+
+
+ // vertex/line/face domination
+ // (if fe_other is not derived from FE_DGQ)
+ // & cell domination
+ // ----------------------------------------
+ if (const FE_Hermite<dim, spacedim> *fe_hermite_other =
+ dynamic_cast<const FE_Hermite<dim, spacedim> *>(&fe_other))
+ {
+ if (this->degree < fe_hermite_other->degree)
+ return FiniteElementDomination::this_element_dominates;
+ else if (this->degree == fe_hermite_other->degree)
+ return FiniteElementDomination::either_element_can_dominate;
+ else
+ return FiniteElementDomination::other_element_dominates;
+ }
+ if (const FE_Q<dim, spacedim> *fe_q_other =
+ dynamic_cast<const FE_Q<dim, spacedim> *>(&fe_other))
+ {
+ if (fe_q_other->degree == 1)
+ {
+ if (this->degree == 1)
+ return FiniteElementDomination::either_element_can_dominate;
+ else
+ return FiniteElementDomination::other_element_dominates;
+ }
+ else if (this->degree <= fe_q_other->degree)
+ return FiniteElementDomination::this_element_dominates;
+ else
+ return FiniteElementDomination::neither_element_dominates;
+ }
+ else if (const FE_SimplexP<dim, spacedim> *fe_p_other =
+ dynamic_cast<const FE_SimplexP<dim, spacedim> *>(&fe_other))
+ {
+ if (fe_p_other->degree == 1)
+ {
+ if (this->degree == 1)
+ return FiniteElementDomination::either_element_can_dominate;
+ else
+ return FiniteElementDomination::other_element_dominates;
+ }
+ else if (this->degree <= fe_p_other->degree)
+ return FiniteElementDomination::this_element_dominates;
+ else
+ return FiniteElementDomination::neither_element_dominates;
+ }
+ else if (const FE_WedgeP<dim, spacedim> *fe_wp_other =
+ dynamic_cast<const FE_WedgeP<dim, spacedim> *>(&fe_other))
+ {
+ if (fe_wp_other->degree == 1)
+ {
+ if (this->degree == 1)
+ return FiniteElementDomination::either_element_can_dominate;
+ else
+ return FiniteElementDomination::other_element_dominates;
+ }
+ else if (this->degree <= fe_wp_other->degree)
+ return FiniteElementDomination::this_element_dominates;
+ else
+ return FiniteElementDomination::neither_element_dominates;
+ }
+ else if (const FE_PyramidP<dim, spacedim> *fe_pp_other =
+ dynamic_cast<const FE_PyramidP<dim, spacedim> *>(&fe_other))
+ {
+ if (fe_pp_other->degree == 1)
+ {
+ if (this->degree == 1)
+ return FiniteElementDomination::either_element_can_dominate;
+ else
+ return FiniteElementDomination::other_element_dominates;
+ }
+ else if (this->degree <= fe_pp_other->degree)
+ return FiniteElementDomination::this_element_dominates;
+ else
+ return FiniteElementDomination::neither_element_dominates;
+ }
+ else if (const FE_Nothing<dim, spacedim> *fe_nothing =
+ dynamic_cast<const FE_Nothing<dim, spacedim> *>(&fe_other))
+ {
+ if (fe_nothing->is_dominating())
+ return FiniteElementDomination::other_element_dominates;
+ else
+ // the FE_Nothing has no degrees of freedom and it is typically used
+ // in a context where we don't require any continuity along the
+ // interface
+ return FiniteElementDomination::no_requirements;
+ }
+
+ Assert(false, ExcNotImplemented());
+ return FiniteElementDomination::neither_element_dominates;
+}
+
+
+
+template <int dim, int spacedim>
+std::vector<unsigned int>
+FE_Hermite<dim, spacedim>::get_lexicographic_to_hierarchic_numbering() const
+{
+ return internal::hermite_lexicographic_to_hierarchic_numbering<dim>(
+ this->regularity);
+}
+
+
+
+template <int dim, int spacedim>
+void
+FE_Hermite<dim, spacedim>::fill_fe_values(
+ const typename Triangulation<dim, spacedim>::cell_iterator &,
+ const CellSimilarity::Similarity cell_similarity,
+ const Quadrature<dim> & /*quadrature*/,
+ const Mapping<dim, spacedim> &mapping,
+ const typename Mapping<dim, spacedim>::InternalDataBase &mapping_internal,
+ const dealii::internal::FEValuesImplementation::MappingRelatedData<dim,
+ spacedim>
+ & /*mapping_data*/,
+ const typename FiniteElement<dim, spacedim>::InternalDataBase &fe_internal,
+ dealii::internal::FEValuesImplementation::FiniteElementRelatedData<dim,
+ spacedim>
+ &output_data) const
+{
+ // Convert data object to internal data for this class.
+ // Fails with an exception if that is not possible.
+ Assert(
+ (dynamic_cast<const typename FE_Hermite<dim, spacedim>::InternalData *>(
+ &fe_internal) != nullptr),
+ ExcInternalError());
+ const typename FE_Hermite<dim, spacedim>::InternalData &fe_data =
+ static_cast<const typename FE_Hermite<dim, spacedim>::InternalData &>(
+ fe_internal);
+
+ const UpdateFlags flags(fe_data.update_each);
+
+ // Transform values, gradients and higher derivatives. Values also need to
+ // be rescaled according the the nodal derivative they correspond to.
+ if ((flags & update_values) &&
+ (cell_similarity != CellSimilarity::translation))
+ {
+ internal::Rescaler<dim, spacedim, double> shape_fix;
+ for (unsigned int i = 0; i < output_data.shape_values.size(0); ++i)
+ for (unsigned int q = 0; q < output_data.shape_values.size(1); ++q)
+ output_data.shape_values(i, q) = fe_data.shape_values(i, q);
+ shape_fix.rescale_fe_hermite_values(shape_fix,
+ *this,
+ mapping_internal,
+ output_data.shape_values);
+ }
+
+ if ((flags & update_gradients) &&
+ (cell_similarity != CellSimilarity::translation))
+ {
+ for (unsigned int k = 0; k < this->n_dofs_per_cell(); ++k)
+ mapping.transform(make_array_view(fe_data.shape_gradients, k),
+ mapping_covariant,
+ mapping_internal,
+ make_array_view(output_data.shape_gradients, k));
+
+ internal::Rescaler<dim, spacedim, Tensor<1, spacedim>> grad_fix;
+ grad_fix.rescale_fe_hermite_values(grad_fix,
+ *this,
+ mapping_internal,
+ output_data.shape_gradients);
+ }
+
+ if ((flags & update_hessians) &&
+ (cell_similarity != CellSimilarity::translation))
+ {
+ for (unsigned int k = 0; k < this->n_dofs_per_cell(); ++k)
+ mapping.transform(make_array_view(fe_data.shape_hessians, k),
+ mapping_covariant_gradient,
+ mapping_internal,
+ make_array_view(output_data.shape_hessians, k));
+
+ internal::Rescaler<dim, spacedim, Tensor<2, spacedim>> hessian_fix;
+ hessian_fix.rescale_fe_hermite_values(hessian_fix,
+ *this,
+ mapping_internal,
+ output_data.shape_hessians);
+ }
+
+ if ((flags & update_3rd_derivatives) &&
+ (cell_similarity != CellSimilarity::translation))
+ {
+ for (unsigned int k = 0; k < this->n_dofs_per_cell(); ++k)
+ mapping.transform(make_array_view(fe_data.shape_3rd_derivatives, k),
+ mapping_covariant_hessian,
+ mapping_internal,
+ make_array_view(output_data.shape_3rd_derivatives,
+ k));
+
+ internal::Rescaler<dim, spacedim, Tensor<3, spacedim>> third_dev_fix;
+ third_dev_fix.rescale_fe_hermite_values(
+ third_dev_fix,
+ *this,
+ mapping_internal,
+ output_data.shape_3rd_derivatives);
+ }
+}
+
+
+
+template <int dim, int spacedim>
+void
+FE_Hermite<dim, spacedim>::fill_fe_face_values(
+ const typename Triangulation<dim, spacedim>::cell_iterator &cell,
+ const unsigned int face_no,
+ const hp::QCollection<dim - 1> &quadrature,
+ const Mapping<dim, spacedim> &mapping,
+ const typename Mapping<dim, spacedim>::InternalDataBase &mapping_internal,
+ const dealii::internal::FEValuesImplementation::MappingRelatedData<dim,
+ spacedim>
+ &,
+ const typename FiniteElement<dim, spacedim>::InternalDataBase &fe_internal,
+ dealii::internal::FEValuesImplementation::FiniteElementRelatedData<dim,
+ spacedim>
+ &output_data) const
+{
+ /*
+ * Convert data object to internal data for this class. Fails with
+ * an exception if that is not possible.
+ */
+ Assert(
+ (dynamic_cast<const typename FE_Hermite<dim, spacedim>::InternalData *>(
+ &fe_internal) != nullptr),
+ ExcInternalError());
+ const typename FE_Hermite<dim, spacedim>::InternalData &fe_data =
+ static_cast<const typename FE_Hermite<dim, spacedim>::InternalData &>(
+ fe_internal);
+
+ Assert((dynamic_cast<
+ const typename MappingCartesian<dim, spacedim>::InternalData *>(
+ &mapping_internal) != nullptr),
+ ExcInternalError());
+
+ AssertDimension(quadrature.size(), 1U);
+
+ /*
+ * offset determines which data set to take (all data sets for all
+ * faces are stored contiguously)
+ */
+ const typename QProjector<dim>::DataSetDescriptor offset =
+ QProjector<dim>::DataSetDescriptor::face(
+ ReferenceCells::get_hypercube<dim>(),
+ face_no,
+ cell->face_orientation(face_no),
+ cell->face_flip(face_no),
+ cell->face_rotation(face_no),
+ quadrature[0].size());
+
+ const UpdateFlags flags(fe_data.update_each);
+
+ // Transform values, gradients and higher derivatives.
+ if (flags & update_values)
+ {
+ for (unsigned int k = 0; k < this->dofs_per_cell; ++k)
+ for (unsigned int i = 0; i < quadrature[0].size(); ++i)
+ output_data.shape_values(k, i) = fe_data.shape_values[k][i + offset];
+
+ internal::Rescaler<dim, spacedim, double> shape_face_fix;
+ shape_face_fix.rescale_fe_hermite_values(shape_face_fix,
+ *this,
+ mapping_internal,
+ output_data.shape_values);
+ }
+
+ if (flags & update_gradients)
+ {
+ for (unsigned int k = 0; k < this->dofs_per_cell; ++k)
+ mapping.transform(make_array_view(fe_data.shape_gradients,
+ k,
+ offset,
+ quadrature[0].size()),
+ mapping_covariant,
+ mapping_internal,
+ make_array_view(output_data.shape_gradients, k));
+
+ internal::Rescaler<dim, spacedim, Tensor<1, spacedim>> grad_face_fix;
+ grad_face_fix.rescale_fe_hermite_values(grad_face_fix,
+ *this,
+ mapping_internal,
+ output_data.shape_gradients);
+ }
+
+ if (flags & update_hessians)
+ {
+ for (unsigned int k = 0; k < this->dofs_per_cell; ++k)
+ mapping.transform(make_array_view(fe_data.shape_hessians,
+ k,
+ offset,
+ quadrature[0].size()),
+ mapping_covariant_gradient,
+ mapping_internal,
+ make_array_view(output_data.shape_hessians, k));
+
+ internal::Rescaler<dim, spacedim, Tensor<2, spacedim>> hessian_face_fix;
+ hessian_face_fix.rescale_fe_hermite_values(hessian_face_fix,
+ *this,
+ mapping_internal,
+ output_data.shape_hessians);
+ }
+
+ if (flags & update_3rd_derivatives)
+ {
+ for (unsigned int k = 0; k < this->dofs_per_cell; ++k)
+ mapping.transform(make_array_view(fe_data.shape_3rd_derivatives,
+ k,
+ offset,
+ quadrature[0].size()),
+ mapping_covariant_hessian,
+ mapping_internal,
+ make_array_view(output_data.shape_3rd_derivatives,
+ k));
+
+ internal::Rescaler<dim, spacedim, Tensor<3, spacedim>> shape_3rd_face_fix;
+ shape_3rd_face_fix.rescale_fe_hermite_values(
+ shape_3rd_face_fix,
+ *this,
+ mapping_internal,
+ output_data.shape_3rd_derivatives);
+ }
+}
+
+
+
+// Explicit instantiations
+#include "fe_hermite.inst"
+
+DEAL_II_NAMESPACE_CLOSE
--- /dev/null
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2023 - 2023 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)
+ {
+ template std::vector<unsigned int>
+ internal::hermite_lexicographic_to_hierarchic_numbering<deal_II_dimension>(
+ const unsigned int regularity);
+
+ template std::vector<unsigned int> internal::
+ hermite_face_lexicographic_to_hierarchic_numbering<deal_II_dimension>(
+ const unsigned int regularity);
+ }
+
+
+
+for (deal_II_dimension : DIMENSIONS; deal_II_space_dimension : SPACE_DIMENSIONS;
+ tensor_rank : RANKS)
+ {
+#if deal_II_dimension <= deal_II_space_dimension
+ template class internal::Rescaler<
+ deal_II_dimension,
+ deal_II_space_dimension,
+ Tensor<tensor_rank, deal_II_space_dimension>>;
+#endif
+ }
+
+
+
+for (deal_II_dimension : DIMENSIONS; deal_II_space_dimension : SPACE_DIMENSIONS)
+ {
+#if deal_II_dimension <= deal_II_space_dimension
+ template class internal::Rescaler<deal_II_dimension,
+ deal_II_space_dimension>;
+ template class FE_Hermite<deal_II_dimension, deal_II_space_dimension>;
+#endif
+ }
// ---------------------------------------------------------------------
//
-// Copyright (C) 2000 - 2021 by the deal.II authors
+// Copyright (C) 2000 - 2023 by the deal.II authors
//
// This file is part of the deal.II library.
//
#include <deal.II/base/quadrature_lib.h>
#include <deal.II/fe/fe_dgq.h>
+#include <deal.II/fe/fe_hermite.h>
#include <deal.II/fe/fe_nothing.h>
#include <deal.II/fe/fe_pyramid_p.h>
#include <deal.II/fe/fe_q.h>
// interface
return FiniteElementDomination::no_requirements;
}
+ else if (const FE_Hermite<dim, spacedim> *fe_hermite_other =
+ dynamic_cast<const FE_Hermite<dim, spacedim> *>(&fe_other))
+ {
+ if (this->degree == 1)
+ {
+ if (fe_hermite_other->degree > 1)
+ return FiniteElementDomination::this_element_dominates;
+ else
+ return FiniteElementDomination::either_element_can_dominate;
+ }
+ else if (this->degree >= fe_hermite_other->degree)
+ return FiniteElementDomination::other_element_dominates;
+ else
+ return FiniteElementDomination::neither_element_dominates;
+ }
Assert(false, ExcNotImplemented());
return FiniteElementDomination::neither_element_dominates;
#include <deal.II/fe/fe_dgp.h>
#include <deal.II/fe/fe_dgq.h>
+#include <deal.II/fe/fe_hermite.h>
#include <deal.II/fe/fe_nothing.h>
#include <deal.II/fe/fe_pyramid_p.h>
#include <deal.II/fe/fe_q_base.h>
// should have identical value
return {{0U, 0U}};
}
+ else if (dynamic_cast<const FE_Hermite<dim, spacedim> *>(&fe_other) !=
+ nullptr)
+ {
+ // FE_Hermite will usually have several degrees of freedom on
+ // each vertex, however only the first one will actually
+ // correspond to the shape value at the vertex, meaning it's
+ // the only one of interest for FE_Q_Base
+ return {{0U, 0U}};
+ }
else if (dynamic_cast<const FE_Nothing<dim> *>(&fe_other) != nullptr)
{
// the FE_Nothing has no degrees of freedom, so there are no
--- /dev/null
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2023 - 2023 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/base/function.h>
+#include <deal.II/base/multithread_info.h>
+#include <deal.II/base/quadrature_lib.h>
+#include <deal.II/base/utilities.h>
+
+#include <deal.II/dofs/dof_accessor.h>
+#include <deal.II/dofs/dof_handler.h>
+#include <deal.II/dofs/dof_tools.h>
+
+#include <deal.II/fe/fe_hermite.h>
+#include <deal.II/fe/fe_values.h>
+#include <deal.II/fe/mapping_cartesian.h>
+
+#include <deal.II/grid/grid_generator.h>
+#include <deal.II/grid/tria.h>
+#include <deal.II/grid/tria_accessor.h>
+#include <deal.II/grid/tria_iterator.h>
+
+#include <deal.II/lac/affine_constraints.h>
+#include <deal.II/lac/dynamic_sparsity_pattern.h>
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/precondition.h>
+#include <deal.II/lac/solver_cg.h>
+#include <deal.II/lac/sparse_matrix.h>
+#include <deal.II/lac/vector.h>
+
+#include <deal.II/numerics/data_out.h>
+#include <deal.II/numerics/matrix_tools.h>
+#include <deal.II/numerics/vector_tools.h>
+
+#include <cmath>
+#include <cstdlib>
+#include <fstream>
+#include <iterator>
+#include <sstream>
+#include <vector>
+
+#define PRECISION 8
+
+/**
+ * Test case for Hermite interpolation elements on an regular square grid,
+ * to check the assembly of the mass matrix works as intended and
+ * representation of polynomials up to $1D$ degree @p poly_degree by
+ * <code>FE_Hermite<dim>(poly_degree)<\code> is exact.
+ */
+
+using namespace dealii;
+
+// Define a function to project onto the domain [-1,1]^2
+class Solution : public Function<2>
+{
+public:
+ virtual double
+ value(const Point<2> &p, unsigned int c = 0) const override
+ {
+ (void)c;
+ return p(0) * p(1);
+ }
+
+
+ std::string
+ get_function_string()
+ {
+ return "XY";
+ }
+};
+
+
+void
+test_fe_on_domain(const unsigned int regularity)
+{
+ deallog << std::endl;
+ char fname[50];
+ sprintf(fname, "Cell-%dd-Hermite-%d", 2, 2 * regularity + 1);
+ deallog.push(fname);
+ deallog.depth_file(2);
+
+ Triangulation<2> tr;
+ DoFHandler<2> dof(tr);
+
+ double left = -1.0, right = 1.0;
+ Point<2> left_point, right_point;
+ for (unsigned int i = 0; i < 2; ++i)
+ left_point(i) = left, right_point(i) = right;
+ GridGenerator::subdivided_hyper_cube(tr, 4, left, right);
+
+ FE_Hermite<2> herm(2 * regularity + 1);
+ dof.distribute_dofs(herm);
+
+ MappingCartesian<2> mapping;
+
+ QGauss<2> quadr(2 * regularity + 2);
+
+ Vector<double> sol(dof.n_dofs());
+ Vector<double> rhs(dof.n_dofs());
+
+ Solution sol_object;
+
+ AffineConstraints<double> constraints;
+ constraints.close();
+
+ DynamicSparsityPattern dsp(dof.n_dofs());
+ DoFTools::make_sparsity_pattern(dof, dsp);
+ SparsityPattern sp;
+ sp.copy_from(dsp);
+
+ SparseMatrix<double> mass_matrix;
+ mass_matrix.reinit(sp);
+ MatrixCreator::create_mass_matrix(mapping, dof, quadr, mass_matrix);
+
+ FEValues<2> fe_herm(mapping,
+ herm,
+ quadr,
+ update_values | update_quadrature_points |
+ update_JxW_values);
+
+ std::vector<types::global_dof_index> local_to_global(herm.n_dofs_per_cell());
+
+ for (const auto &cell : dof.active_cell_iterators())
+ {
+ fe_herm.reinit(cell);
+ cell->get_dof_indices(local_to_global);
+ for (const unsigned int i : fe_herm.dof_indices())
+ {
+ double rhs_temp = 0;
+ for (const unsigned int q : fe_herm.quadrature_point_indices())
+ rhs_temp += fe_herm.shape_value(i, q) *
+ sol_object.value(fe_herm.quadrature_point(q)) *
+ fe_herm.JxW(q);
+ rhs(local_to_global[i]) += rhs_temp;
+ }
+ }
+
+ IterationNumberControl solver_control_values(8000, 1e-11);
+ SolverCG<> solver(solver_control_values);
+
+ solver.solve(mass_matrix, sol, rhs, PreconditionIdentity());
+
+ double err_sq = 0;
+
+ for (auto &cell : dof.active_cell_iterators())
+ {
+ fe_herm.reinit(cell);
+ cell->get_dof_indices(local_to_global);
+ for (const unsigned int q : fe_herm.quadrature_point_indices())
+ {
+ double sol_at_point = 0;
+ for (const unsigned int i : fe_herm.dof_indices())
+ sol_at_point += fe_herm.shape_value(i, q) * sol(local_to_global[i]);
+
+ sol_at_point -= sol_object.value(fe_herm.quadrature_point(q));
+ err_sq += sol_at_point * sol_at_point * fe_herm.JxW(q);
+ }
+ }
+
+ err_sq = std::sqrt(err_sq);
+
+ deallog << "Test polynomial:" << std::endl;
+ deallog << sol_object.get_function_string() << std::endl;
+ deallog << std::endl;
+
+ deallog << "Interpolation error:" << std::endl;
+ deallog << err_sq << "\n\n" << std::endl;
+ deallog.pop();
+}
+
+
+
+int
+main()
+{
+ std::ofstream logfile("output");
+ deallog << std::setprecision(PRECISION) << std::fixed;
+ deallog.attach(logfile);
+ MultithreadInfo::set_thread_limit(1);
+
+ test_fe_on_domain(0);
+ test_fe_on_domain(1);
+ test_fe_on_domain(2);
+
+ return 0;
+}
--- /dev/null
+
+DEAL::
+DEAL:Cell-2d-Hermite-1::Test polynomial:
+DEAL:Cell-2d-Hermite-1::XY
+DEAL:Cell-2d-Hermite-1::
+DEAL:Cell-2d-Hermite-1::Interpolation error:
+DEAL:Cell-2d-Hermite-1::0.00000000
+
+
+DEAL::
+DEAL:Cell-2d-Hermite-3::Test polynomial:
+DEAL:Cell-2d-Hermite-3::XY
+DEAL:Cell-2d-Hermite-3::
+DEAL:Cell-2d-Hermite-3::Interpolation error:
+DEAL:Cell-2d-Hermite-3::0.00000000
+
+
+DEAL::
+DEAL:Cell-2d-Hermite-5::Test polynomial:
+DEAL:Cell-2d-Hermite-5::XY
+DEAL:Cell-2d-Hermite-5::
+DEAL:Cell-2d-Hermite-5::Interpolation error:
+DEAL:Cell-2d-Hermite-5::0.00000000
+
+
--- /dev/null
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2023 - 2023 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_tests_derivatives_h
+#define dealii_tests_derivatives_h
+
+#include <deal.II/base/quadrature_lib.h>
+
+#include <deal.II/dofs/dof_accessor.h>
+
+#include <deal.II/fe/fe_values.h>
+#include <deal.II/fe/mapping_q1.h>
+
+#include <deal.II/grid/grid_generator.h>
+#include <deal.II/grid/tria.h>
+#include <deal.II/grid/tria_iterator.h>
+
+#include <deal.II/lac/vector.h>
+
+#include <cmath>
+#include <fstream>
+#include <string>
+#include <vector>
+
+#include "../tests.h"
+
+char fname[50];
+
+
+
+//////////////////////////////////////////////////////////////////////////////////////////////////////
+// Plot derivatives of the shape functions at the corner nodes of [0,1]^d
+//
+// Output values on each line take the form
+//
+// x (y) (z) xderiv_index (yderiv_index) (zderiv_index) value[0]+1.
+// value[1]+1. ...
+//////////////////////////////////////////////////////////////////////////////////////////////////////
+
+template <int dim>
+inline void
+plot_function_derivatives(Mapping<dim> &mapping,
+ FiniteElement<dim> &finel,
+ const char *name)
+{
+ Triangulation<dim> tr;
+ DoFHandler<dim> dof(tr);
+ GridGenerator::hyper_cube(tr, 0., 1.);
+ dof.distribute_dofs(finel);
+
+ typename DoFHandler<dim>::cell_iterator c = dof.begin();
+
+ const unsigned int sz = finel.n_dofs_per_cell();
+ const unsigned int degree = finel.tensor_degree();
+ const unsigned int max_test_deriv =
+ std::min(((dim == 1) ? 3 : 1), static_cast<int>(degree - 1) / 2);
+ const unsigned int div = 1;
+
+ QTrapezoid<dim> q;
+ FEValues<dim> fe(mapping,
+ finel,
+ q,
+ UpdateFlags(update_values | update_gradients |
+ update_hessians | update_3rd_derivatives));
+
+ sprintf(fname, "Cell-%dd-%s", dim, name);
+ deallog.push(fname);
+
+ fe.reinit(c);
+
+ unsigned int k = 0;
+
+ // iterate over vertices
+ for (unsigned int mz = 0; mz <= ((dim > 2) ? div : 0); ++mz)
+ {
+ for (unsigned int my = 0; my <= ((dim > 1) ? div : 0); ++my)
+ {
+ for (unsigned int mx = 0; mx <= div; ++mx)
+ {
+ // iterate over derivatives
+ for (unsigned int dz = 0; dz <= ((dim > 2) ? max_test_deriv : 0);
+ ++dz)
+ {
+ for (unsigned int dy = 0;
+ dy <= ((dim > 1) ? max_test_deriv : 0);
+ ++dy)
+ {
+ for (unsigned int dx = 0; dx <= max_test_deriv; ++dx)
+ {
+ deallog << q.point(k) << " " << dx;
+ if (dim > 1)
+ deallog << " " << dy;
+ if (dim > 2)
+ deallog << " " << dz;
+
+ for (unsigned int i = 0; i < sz; ++i)
+ {
+ unsigned int deriv_level = dx + dy + dz;
+ unsigned int entry_1, entry_2, entry_3;
+
+ switch (deriv_level)
+ {
+ case 0:
+ deallog << " " << fe.shape_value(i, k) + 1.;
+ break;
+
+ case 1:
+ entry_1 = dy + 2 * dz;
+ deallog
+ << " "
+ << fe.shape_grad(i, k)[entry_1] + 1.;
+ break;
+
+ case 2:
+ entry_1 = (dx == 0) ? 1 : 0;
+ entry_2 = 2 * dz - dy * dz + dy;
+ deallog
+ << " "
+ << fe.shape_hessian(i,
+ k)[entry_1][entry_2] +
+ 1.;
+ break;
+
+ case 3:
+ if (dz == 1)
+ entry_1 = 0, entry_2 = 1,
+ entry_3 = 2; // only occurs for dx=dy=dz=1
+ else if (dx * dy == 0)
+ entry_1 = entry_2 = entry_3 = dy / 3;
+ else
+ entry_1 = 0, entry_2 = dy - 1,
+ entry_3 = 1;
+ deallog
+ << " "
+ << fe.shape_3rd_derivative(
+ i, k)[entry_1][entry_2][entry_3] +
+ 1.;
+ break;
+
+ default:
+ ExcMessage(
+ "Requested order of derivative is too high for current implementation.");
+ break;
+ }
+ }
+ deallog << std::endl;
+ }
+ }
+ }
+ deallog << std::endl;
+ ++k;
+ }
+ deallog << std::endl;
+ }
+ deallog << std::endl;
+ }
+ deallog.pop();
+}
+
+#endif
--- /dev/null
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2023 - 2023 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 "../tests.h"
+
+#define PRECISION 8
+
+
+#include <deal.II/fe/fe_hermite.h>
+#include <deal.II/fe/mapping_cartesian.h>
+
+#include <string>
+
+#include "derivatives.h"
+
+
+
+/**
+ * Test case to check the derivative values up to @p regularity
+ * of <code>FE_Hermite<dim>(2 \times regularity + 1)<\code> at the vertices
+ * of a hypercube reference element are calculated correctly.
+ * This checks that the basis is assembled correctly and will not
+ * accidentally enforce discontinuities rather than continuity
+ * in derivatives across element boundaries.
+ */
+
+
+template <int dim>
+void
+print_hermite_endpoint_derivatives()
+{
+ MappingCartesian<dim> m;
+
+ FE_Hermite<dim> herm_1(1);
+ plot_function_derivatives<dim>(m, herm_1, "Hermite-1");
+
+ FE_Hermite<dim> herm_3(3);
+ plot_function_derivatives<dim>(m, herm_3, "Hermite-3");
+
+ // Skip the following for dim 3 or greater
+ if (dim < 3)
+ {
+ FE_Hermite<dim> herm_5(5);
+ plot_function_derivatives<dim>(m, herm_5, "Hermite-5");
+ }
+ if (dim == 1)
+ {
+ FE_Hermite<dim> herm_7(7);
+ plot_function_derivatives<dim>(m, herm_7, "Hermite-7");
+
+ FE_Hermite<dim> herm_9(9);
+ plot_function_derivatives<dim>(m, herm_9, "Hermite-9");
+ }
+}
+
+
+
+int
+main()
+{
+ std::ofstream logfile("output");
+
+ deallog << std::setprecision(PRECISION) << std::fixed;
+ deallog.attach(logfile);
+
+ print_hermite_endpoint_derivatives<1>();
+ print_hermite_endpoint_derivatives<2>();
+ print_hermite_endpoint_derivatives<3>();
+
+ return 0;
+}
--- /dev/null
+
+DEAL:Cell-1d-Hermite-1::0.00000000 0 2.00000000 1.00000000
+DEAL:Cell-1d-Hermite-1::
+DEAL:Cell-1d-Hermite-1::1.00000000 0 1.00000000 2.00000000
+DEAL:Cell-1d-Hermite-1::
+DEAL:Cell-1d-Hermite-1::
+DEAL:Cell-1d-Hermite-1::
+DEAL:Cell-1d-Hermite-3::0.00000000 0 2.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-3::0.00000000 1 1.00000000 5.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-3::
+DEAL:Cell-1d-Hermite-3::1.00000000 0 1.00000000 1.00000000 2.00000000 1.00000000
+DEAL:Cell-1d-Hermite-3::1.00000000 1 1.00000000 1.00000000 1.00000000 5.00000000
+DEAL:Cell-1d-Hermite-3::
+DEAL:Cell-1d-Hermite-3::
+DEAL:Cell-1d-Hermite-3::
+DEAL:Cell-1d-Hermite-5::0.00000000 0 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-5::0.00000000 1 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-5::0.00000000 2 1.00000000 1.00000000 33.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-5::
+DEAL:Cell-1d-Hermite-5::1.00000000 0 1.00000000 1.00000000 1.00000000 2.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-5::1.00000000 1 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000
+DEAL:Cell-1d-Hermite-5::1.00000000 2 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 33.00000000
+DEAL:Cell-1d-Hermite-5::
+DEAL:Cell-1d-Hermite-5::
+DEAL:Cell-1d-Hermite-5::
+DEAL:Cell-1d-Hermite-7::0.00000000 0 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-7::0.00000000 1 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-7::0.00000000 2 1.00000000 1.00000000 33.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-7::0.00000000 3 1.00000000 1.00000000 1.00000000 385.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-7::
+DEAL:Cell-1d-Hermite-7::1.00000000 0 1.00000000 1.00000000 1.00000000 1.00000000 2.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-7::1.00000000 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-7::1.00000000 2 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 33.00000000 1.00000000
+DEAL:Cell-1d-Hermite-7::1.00000000 3 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 385.00000000
+DEAL:Cell-1d-Hermite-7::
+DEAL:Cell-1d-Hermite-7::
+DEAL:Cell-1d-Hermite-7::
+DEAL:Cell-1d-Hermite-9::0.00000000 0 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-9::0.00000000 1 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-9::0.00000000 2 1.00000000 1.00000000 33.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-9::0.00000000 3 1.00000000 1.00000000 1.00000000 385.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-9::
+DEAL:Cell-1d-Hermite-9::1.00000000 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-9::1.00000000 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-9::1.00000000 2 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 33.00000000 1.00000000 1.00000000
+DEAL:Cell-1d-Hermite-9::1.00000000 3 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 385.00000000 1.00000000
+DEAL:Cell-1d-Hermite-9::
+DEAL:Cell-1d-Hermite-9::
+DEAL:Cell-1d-Hermite-9::
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+DEAL:Cell-2d-Hermite-1::
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+DEAL:Cell-2d-Hermite-1::
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+DEAL:Cell-2d-Hermite-1::
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+DEAL:Cell-2d-Hermite-1::
+DEAL:Cell-2d-Hermite-1::
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+DEAL:Cell-2d-Hermite-3::
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+DEAL:Cell-2d-Hermite-3::
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+DEAL:Cell-2d-Hermite-5::
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+DEAL:Cell-2d-Hermite-5::
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+DEAL:Cell-2d-Hermite-5::
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+DEAL:Cell-2d-Hermite-5::
+DEAL:Cell-2d-Hermite-5::
+DEAL:Cell-2d-Hermite-5::
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+DEAL:Cell-3d-Hermite-1::
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+DEAL:Cell-3d-Hermite-1::
+DEAL:Cell-3d-Hermite-1::
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+DEAL:Cell-3d-Hermite-1::
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+DEAL:Cell-3d-Hermite-1::
+DEAL:Cell-3d-Hermite-1::
+DEAL:Cell-3d-Hermite-1::
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+DEAL:Cell-3d-Hermite-1::
+DEAL:Cell-3d-Hermite-1::
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+DEAL:Cell-3d-Hermite-1::
+DEAL:Cell-3d-Hermite-1::
+DEAL:Cell-3d-Hermite-1::
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+DEAL:Cell-3d-Hermite-3::0.00000000 0.00000000 0.00000000 1 0 0 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-3::0.00000000 0.00000000 0.00000000 0 1 0 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-3::0.00000000 0.00000000 0.00000000 1 1 0 1.00000000 1.00000000 1.00000000 17.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-3::0.00000000 0.00000000 0.00000000 0 0 1 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-3::0.00000000 0.00000000 0.00000000 1 0 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 17.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-3::0.00000000 0.00000000 0.00000000 0 1 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 17.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-3::0.00000000 0.00000000 0.00000000 1 1 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 65.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-3::
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+DEAL:Cell-3d-Hermite-3::1.00000000 0.00000000 0.00000000 1 0 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-3::1.00000000 0.00000000 0.00000000 0 1 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-3::1.00000000 0.00000000 0.00000000 1 1 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 17.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-3::1.00000000 0.00000000 0.00000000 0 0 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-3::1.00000000 0.00000000 0.00000000 1 0 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 17.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-3::1.00000000 0.00000000 0.00000000 0 1 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 17.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
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+DEAL:Cell-3d-Hermite-3::
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+DEAL:Cell-3d-Hermite-3::
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+DEAL:Cell-3d-Hermite-3::
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+DEAL:Cell-3d-Hermite-3::
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+DEAL:Cell-3d-Hermite-3::0.00000000 1.00000000 1.00000000 1 1 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 17.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-3::0.00000000 1.00000000 1.00000000 0 0 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-3::0.00000000 1.00000000 1.00000000 1 0 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 17.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-3::0.00000000 1.00000000 1.00000000 0 1 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 17.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-3::0.00000000 1.00000000 1.00000000 1 1 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 65.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-3::
+DEAL:Cell-3d-Hermite-3::1.00000000 1.00000000 1.00000000 0 0 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-3::1.00000000 1.00000000 1.00000000 1 0 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-3::1.00000000 1.00000000 1.00000000 0 1 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-3::1.00000000 1.00000000 1.00000000 1 1 0 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 17.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-3::1.00000000 1.00000000 1.00000000 0 0 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 5.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-3::1.00000000 1.00000000 1.00000000 1 0 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 17.00000000 1.00000000 1.00000000
+DEAL:Cell-3d-Hermite-3::1.00000000 1.00000000 1.00000000 0 1 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 17.00000000 1.00000000
+DEAL:Cell-3d-Hermite-3::1.00000000 1.00000000 1.00000000 1 1 1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 65.00000000
+DEAL:Cell-3d-Hermite-3::
+DEAL:Cell-3d-Hermite-3::
+DEAL:Cell-3d-Hermite-3::
// ---------------------------------------------------------------------
//
-// Copyright (C) 2003 - 2021 by the deal.II authors
+// Copyright (C) 2003 - 2023 by the deal.II authors
//
// This file is part of the deal.II library.
//
//
// ---------------------------------------------------------------------
+#ifndef dealii_tests_shapes_h
+#define dealii_tests_shapes_h
// Show the shape functions implemented.
check_values_and_derivatives(fe, fe_values, q);
};
}
+
+#endif
--- /dev/null
+// ---------------------------------------------------------------------
+//
+// Copyright (C) 2023 - 2023 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 "../tests.h"
+
+#define PRECISION 8
+
+
+#include <deal.II/fe/fe_hermite.h>
+#include <deal.II/fe/mapping_cartesian.h>
+
+#include <string>
+
+#include "shapes.h"
+
+
+
+/*
+ * Test case on various hypercube reference cells to check that the shape
+ * values of <code>FE_Hermite<dim>(reg)<\code> are correct. Values of
+ * derivatives at the vertices are checked in the test case
+ * derivatives_hermite.cc.
+ */
+
+template <int dim>
+void
+plot_FE_Hermite_shape_functions()
+{
+ MappingCartesian<dim> m;
+
+ FE_Hermite<dim> herm_1(1);
+ plot_shape_functions(m, herm_1, "Hermite-1");
+ plot_face_shape_functions(m, herm_1, "Hermite-1");
+ test_compute_functions(m, herm_1, "Hermite-1");
+
+ FE_Hermite<dim> herm_3(3);
+ plot_shape_functions(m, herm_3, "Hermite-3");
+ plot_face_shape_functions(m, herm_3, "Hermite-3");
+ test_compute_functions(m, herm_3, "Hermite-3");
+
+ // skip the following tests to
+ // reduce run-time
+ if (dim < 3)
+ {
+ FE_Hermite<dim> herm_5(5);
+ plot_shape_functions(m, herm_5, "Hermite-5");
+ plot_face_shape_functions(m, herm_5, "Hermite-5");
+ test_compute_functions(m, herm_5, "Hermite-5");
+ }
+
+ if (dim == 1)
+ {
+ FE_Hermite<dim> herm_7(7);
+ plot_shape_functions(m, herm_7, "Hermite-7");
+ plot_face_shape_functions(m, herm_7, "Hermite-7");
+ test_compute_functions(m, herm_7, "Hermite-7");
+
+ FE_Hermite<dim> herm_9(9);
+ plot_shape_functions(m, herm_9, "Hermite-9");
+ plot_face_shape_functions(m, herm_9, "Hermite-9");
+ test_compute_functions(m, herm_9, "Hermite-9");
+ };
+}
+
+
+
+int
+main()
+{
+ std::ofstream logfile("output");
+ deallog << std::setprecision(PRECISION) << std::fixed;
+ deallog.attach(logfile);
+
+ plot_FE_Hermite_shape_functions<1>();
+ plot_FE_Hermite_shape_functions<2>();
+ plot_FE_Hermite_shape_functions<3>();
+
+ return 0;
+}
--- /dev/null
+
+DEAL:Cell1d-Hermite-1::0.00000000 2.00000000 1.00000000
+DEAL:Cell1d-Hermite-1::0.25000000 1.75000000 1.25000000
+DEAL:Cell1d-Hermite-1::0.50000000 1.50000000 1.50000000
+DEAL:Cell1d-Hermite-1::0.75000000 1.25000000 1.75000000
+DEAL:Cell1d-Hermite-1::1.00000000 1.00000000 2.00000000
+DEAL:Cell1d-Hermite-1::
+DEAL:Cell1d-Hermite-1::
+DEAL:Cell1d-Hermite-3::0.00000000 2.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell1d-Hermite-3::0.25000000 1.84375000 1.56250000 1.15625000 0.81250000
+DEAL:Cell1d-Hermite-3::0.50000000 1.50000000 1.50000000 1.50000000 0.50000000
+DEAL:Cell1d-Hermite-3::0.75000000 1.15625000 1.18750000 1.84375000 0.43750000
+DEAL:Cell1d-Hermite-3::1.00000000 1.00000000 1.00000000 2.00000000 1.00000000
+DEAL:Cell1d-Hermite-3::
+DEAL:Cell1d-Hermite-3::
+DEAL:Cell1d-Hermite-5::0.00000000 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell1d-Hermite-5::0.25000000 1.89648438 1.73828125 1.42187500 1.10351562 0.84765625 1.14062500
+DEAL:Cell1d-Hermite-5::0.50000000 1.50000000 1.62500000 1.50000000 1.50000000 0.37500000 1.50000000
+DEAL:Cell1d-Hermite-5::0.75000000 1.10351562 1.15234375 1.14062500 1.89648438 0.26171875 1.42187500
+DEAL:Cell1d-Hermite-5::1.00000000 1.00000000 1.00000000 1.00000000 2.00000000 1.00000000 1.00000000
+DEAL:Cell1d-Hermite-5::
+DEAL:Cell1d-Hermite-5::
+DEAL:Cell1d-Hermite-7::0.00000000 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell1d-Hermite-7::0.25000000 1.92944336 1.83056641 1.63281250 1.31640625 1.07055664 0.88720703 1.14062500 0.89453125
+DEAL:Cell1d-Hermite-7::0.50000000 1.50000000 1.68750000 1.75000000 1.50000000 1.50000000 0.31250000 1.75000000 0.50000000
+DEAL:Cell1d-Hermite-7::0.75000000 1.07055664 1.11279297 1.14062500 1.10546875 1.92944336 0.16943359 1.63281250 0.68359375
+DEAL:Cell1d-Hermite-7::1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 2.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell1d-Hermite-7::
+DEAL:Cell1d-Hermite-7::
+DEAL:Cell1d-Hermite-9::0.00000000 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell1d-Hermite-9::0.25000000 1.95107269 1.88618469 1.75640869 1.53393555 1.23730469 1.04892731 0.91810608 1.11590576 0.87475586 1.07910156
+DEAL:Cell1d-Hermite-9::0.50000000 1.50000000 1.72656250 1.90625000 1.87500000 1.50000000 1.50000000 0.27343750 1.90625000 0.12500000 1.50000000
+DEAL:Cell1d-Hermite-9::0.75000000 1.04892731 1.08189392 1.11590576 1.12524414 1.07910156 1.95107269 0.11381531 1.75640869 0.46606445 1.23730469
+DEAL:Cell1d-Hermite-9::1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell1d-Hermite-9::
+DEAL:Cell1d-Hermite-9::
+DEAL:Cell2d-Hermite-1::0.00000000 0.00000000 2.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell2d-Hermite-1::0.25000000 0.00000000 1.75000000 1.25000000 1.00000000 1.00000000
+DEAL:Cell2d-Hermite-1::0.50000000 0.00000000 1.50000000 1.50000000 1.00000000 1.00000000
+DEAL:Cell2d-Hermite-1::0.75000000 0.00000000 1.25000000 1.75000000 1.00000000 1.00000000
+DEAL:Cell2d-Hermite-1::1.00000000 0.00000000 1.00000000 2.00000000 1.00000000 1.00000000
+DEAL:Cell2d-Hermite-1::
+DEAL:Cell2d-Hermite-1::0.00000000 0.25000000 1.75000000 1.00000000 1.25000000 1.00000000
+DEAL:Cell2d-Hermite-1::0.25000000 0.25000000 1.56250000 1.18750000 1.18750000 1.06250000
+DEAL:Cell2d-Hermite-1::0.50000000 0.25000000 1.37500000 1.37500000 1.12500000 1.12500000
+DEAL:Cell2d-Hermite-1::0.75000000 0.25000000 1.18750000 1.56250000 1.06250000 1.18750000
+DEAL:Cell2d-Hermite-1::1.00000000 0.25000000 1.00000000 1.75000000 1.00000000 1.25000000
+DEAL:Cell2d-Hermite-1::
+DEAL:Cell2d-Hermite-1::0.00000000 0.50000000 1.50000000 1.00000000 1.50000000 1.00000000
+DEAL:Cell2d-Hermite-1::0.25000000 0.50000000 1.37500000 1.12500000 1.37500000 1.12500000
+DEAL:Cell2d-Hermite-1::0.50000000 0.50000000 1.25000000 1.25000000 1.25000000 1.25000000
+DEAL:Cell2d-Hermite-1::0.75000000 0.50000000 1.12500000 1.37500000 1.12500000 1.37500000
+DEAL:Cell2d-Hermite-1::1.00000000 0.50000000 1.00000000 1.50000000 1.00000000 1.50000000
+DEAL:Cell2d-Hermite-1::
+DEAL:Cell2d-Hermite-1::0.00000000 0.75000000 1.25000000 1.00000000 1.75000000 1.00000000
+DEAL:Cell2d-Hermite-1::0.25000000 0.75000000 1.18750000 1.06250000 1.56250000 1.18750000
+DEAL:Cell2d-Hermite-1::0.50000000 0.75000000 1.12500000 1.12500000 1.37500000 1.37500000
+DEAL:Cell2d-Hermite-1::0.75000000 0.75000000 1.06250000 1.18750000 1.18750000 1.56250000
+DEAL:Cell2d-Hermite-1::1.00000000 0.75000000 1.00000000 1.25000000 1.00000000 1.75000000
+DEAL:Cell2d-Hermite-1::
+DEAL:Cell2d-Hermite-1::0.00000000 1.00000000 1.00000000 1.00000000 2.00000000 1.00000000
+DEAL:Cell2d-Hermite-1::0.25000000 1.00000000 1.00000000 1.00000000 1.75000000 1.25000000
+DEAL:Cell2d-Hermite-1::0.50000000 1.00000000 1.00000000 1.00000000 1.50000000 1.50000000
+DEAL:Cell2d-Hermite-1::0.75000000 1.00000000 1.00000000 1.00000000 1.25000000 1.75000000
+DEAL:Cell2d-Hermite-1::1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 2.00000000
+DEAL:Cell2d-Hermite-1::
+DEAL:Cell2d-Hermite-1::
+DEAL:Face2d-Hermite-1::0.00000000 0.00000000 2.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Face2d-Hermite-1::0.00000000 0.12500000 1.75000000 1.00000000 1.25000000 1.00000000
+DEAL:Face2d-Hermite-1::0.00000000 0.25000000 1.50000000 1.00000000 1.50000000 1.00000000
+DEAL:Face2d-Hermite-1::0.00000000 0.37500000 1.25000000 1.00000000 1.75000000 1.00000000
+DEAL:Face2d-Hermite-1::0.00000000 0.50000000 1.00000000 1.00000000 2.00000000 1.00000000
+DEAL:Face2d-Hermite-1::
+DEAL:Face2d-Hermite-1::
+DEAL:Face2d-Hermite-1::0.50000000 0.00000000 1.00000000 2.00000000 1.00000000 1.00000000
+DEAL:Face2d-Hermite-1::0.50000000 0.06250000 1.00000000 1.87500000 1.00000000 1.12500000
+DEAL:Face2d-Hermite-1::0.50000000 0.12500000 1.00000000 1.75000000 1.00000000 1.25000000
+DEAL:Face2d-Hermite-1::0.50000000 0.18750000 1.00000000 1.62500000 1.00000000 1.37500000
+DEAL:Face2d-Hermite-1::0.50000000 0.25000000 1.00000000 1.50000000 1.00000000 1.50000000
+DEAL:Face2d-Hermite-1::
+DEAL:Face2d-Hermite-1::
+DEAL:Face2d-Hermite-1::0.50000000 0.25000000 1.00000000 1.50000000 1.00000000 1.50000000
+DEAL:Face2d-Hermite-1::0.50000000 0.31250000 1.00000000 1.37500000 1.00000000 1.62500000
+DEAL:Face2d-Hermite-1::0.50000000 0.37500000 1.00000000 1.25000000 1.00000000 1.75000000
+DEAL:Face2d-Hermite-1::0.50000000 0.43750000 1.00000000 1.12500000 1.00000000 1.87500000
+DEAL:Face2d-Hermite-1::0.50000000 0.50000000 1.00000000 1.00000000 1.00000000 2.00000000
+DEAL:Face2d-Hermite-1::
+DEAL:Face2d-Hermite-1::
+DEAL:Face2d-Hermite-1::0.00000000 0.00000000 2.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Face2d-Hermite-1::0.12500000 0.00000000 1.75000000 1.25000000 1.00000000 1.00000000
+DEAL:Face2d-Hermite-1::0.25000000 0.00000000 1.50000000 1.50000000 1.00000000 1.00000000
+DEAL:Face2d-Hermite-1::0.37500000 0.00000000 1.25000000 1.75000000 1.00000000 1.00000000
+DEAL:Face2d-Hermite-1::0.50000000 0.00000000 1.00000000 2.00000000 1.00000000 1.00000000
+DEAL:Face2d-Hermite-1::
+DEAL:Face2d-Hermite-1::
+DEAL:Face2d-Hermite-1::0.00000000 0.50000000 1.00000000 1.00000000 2.00000000 1.00000000
+DEAL:Face2d-Hermite-1::0.12500000 0.50000000 1.00000000 1.00000000 1.75000000 1.25000000
+DEAL:Face2d-Hermite-1::0.25000000 0.50000000 1.00000000 1.00000000 1.50000000 1.50000000
+DEAL:Face2d-Hermite-1::0.37500000 0.50000000 1.00000000 1.00000000 1.25000000 1.75000000
+DEAL:Face2d-Hermite-1::0.50000000 0.50000000 1.00000000 1.00000000 1.00000000 2.00000000
+DEAL:Face2d-Hermite-1::
+DEAL:Face2d-Hermite-1::
+DEAL:Cell2d-Hermite-3::0.00000000 0.00000000 2.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
+DEAL:Cell2d-Hermite-3::0.25000000 0.00000000 1.84375000 1.56250000 1.00000000 1.00000000 1.15625000 0.81250000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
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projects / dealii.git / commitdiff