mapping_manifold.h

// ---------------------------------------------------------------------
//
// Copyright (C) 2016 - 2021 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.
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// ---------------------------------------------------------------------

#ifndef dealii_mapping_manifold_h
#define dealii_mapping_manifold_h


#include <deal.II/base/config.h>

#include <deal.II/base/array_view.h>
#include <deal.II/base/derivative_form.h>
#include <deal.II/base/quadrature.h>

#include <deal.II/fe/mapping.h>

#include <cmath>

DEAL_II_NAMESPACE_OPEN

template <int, int>
class MappingQ;


/*!@addtogroup mapping */
/*@{*/


/**
 * This class implements the functionality for Manifold conforming
 * mappings. This Mapping computes the transformation between the
 * reference and real cell by exploiting the geometrical information
 * coming from the underlying Manifold object.
 *
 * Quadrature points computed using this mapping lie on the exact
 * geometrical objects, and tangent and normal vectors computed using
 * this class are tangent and normal to the underlying geometry. This
 * is in contrast with the MappingQ class, which approximates the
 * geometry using a polynomial of some order, and then computes the
 * normals and tangents using the approximated surface.
 *
 * @warning It is not possible, for mathematical reasons, for one to use this
 * class with a geometry described by a SphericalManifold: see the note in
 * that class for more information.
 */
template <int dim, int spacedim = dim>
class MappingManifold : public Mapping<dim, spacedim>
{
public:
  /**
   * Constructor.
   */
  MappingManifold() = default;

  /**
   * Copy constructor.
   */
  MappingManifold(const MappingManifold<dim, spacedim> &mapping);

  // for documentation, see the Mapping base class
  virtual std::unique_ptr<Mapping<dim, spacedim>>
  clone() const override;

  /**
   * Always returns @p true because this class assumes that the
   * vertices always lies on the underlying Manifold.
   */
  virtual bool
  preserves_vertex_locations() const override;

  virtual bool
  is_compatible_with(const ReferenceCell &cell_type) const override;

  /**
   * @name Mapping points between reference and real cells
   * @{
   */

  // for documentation, see the Mapping base class
  virtual Point<spacedim>
  transform_unit_to_real_cell(
    const typename Triangulation<dim, spacedim>::cell_iterator &cell,
    const Point<dim> &p) const override;

  // for documentation, see the Mapping base class
  virtual Point<dim>
  transform_real_to_unit_cell(
    const typename Triangulation<dim, spacedim>::cell_iterator &cell,
    const Point<spacedim> &p) const override;

  /**
   * @}
   */

  /**
   * @name Functions to transform tensors from reference to real coordinates
   * @{
   */

  // for documentation, see the Mapping base class
  virtual void
  transform(const ArrayView<const Tensor<1, dim>> &                  input,
            const MappingKind                                        kind,
            const typename Mapping<dim, spacedim>::InternalDataBase &internal,
            const ArrayView<Tensor<1, spacedim>> &output) const override;

  // for documentation, see the Mapping base class
  virtual void
  transform(const ArrayView<const DerivativeForm<1, dim, spacedim>> &input,
            const MappingKind                                        kind,
            const typename Mapping<dim, spacedim>::InternalDataBase &internal,
            const ArrayView<Tensor<2, spacedim>> &output) const override;

  // for documentation, see the Mapping base class
  virtual void
  transform(const ArrayView<const Tensor<2, dim>> &                  input,
            const MappingKind                                        kind,
            const typename Mapping<dim, spacedim>::InternalDataBase &internal,
            const ArrayView<Tensor<2, spacedim>> &output) const override;

  // for documentation, see the Mapping base class
  virtual void
  transform(const ArrayView<const DerivativeForm<2, dim, spacedim>> &input,
            const MappingKind                                        kind,
            const typename Mapping<dim, spacedim>::InternalDataBase &internal,
            const ArrayView<Tensor<3, spacedim>> &output) const override;

  // for documentation, see the Mapping base class
  virtual void
  transform(const ArrayView<const Tensor<3, dim>> &                  input,
            const MappingKind                                        kind,
            const typename Mapping<dim, spacedim>::InternalDataBase &internal,
            const ArrayView<Tensor<3, spacedim>> &output) const override;

  /**
   * @}
   */

  /**
   * @name Interface with FEValues
   * @{
   */

  /**
   * Storage for internal data of polynomial mappings. See
   * Mapping::InternalDataBase for an extensive description.
   *
   * For the current class, the InternalData class stores data that is
   * computed once when the object is created (in get_data()) as well as data
   * the class wants to store from between the call to fill_fe_values(),
   * fill_fe_face_values(), or fill_fe_subface_values() until possible later
   * calls from the finite element to functions such as transform(). The
   * latter class of member variables are marked as 'mutable'.
   */
  class InternalData : public Mapping<dim, spacedim>::InternalDataBase
  {
  public:
    /**
     * Constructor.
     */
    InternalData() = default;

    /**
     * Initialize the object's member variables related to cell data based on
     * the given arguments.
     *
     * The function also calls compute_shape_function_values() to actually set
     * the member variables related to the values and derivatives of the
     * mapping shape functions.
     */
    void
    initialize(const UpdateFlags      update_flags,
               const Quadrature<dim> &quadrature,
               const unsigned int     n_original_q_points);

    /**
     * Initialize the object's member variables related to cell and face data
     * based on the given arguments. In order to initialize cell data, this
     * function calls initialize().
     */
    void
    initialize_face(const UpdateFlags      update_flags,
                    const Quadrature<dim> &quadrature,
                    const unsigned int     n_original_q_points);

    /**
     * Compute the weights associated to the Manifold object, that
     * need to be passed when computing the location of the quadrature
     * points.
     */
    void
    compute_manifold_quadrature_weights(const Quadrature<dim> &quadrature);

    /**
     * Store vertices internally.
     */
    void
    store_vertices(
      const typename Triangulation<dim, spacedim>::cell_iterator &cell) const;

    /**
     * Return an estimate (in bytes) for the memory consumption of this object.
     */
    virtual std::size_t
    memory_consumption() const override;

    /**
     * The current cell vertices.
     *
     * Computed each.
     */
    mutable std::vector<Point<spacedim>> vertices;

    /**
     * The current cell.
     *
     * Computed each.
     */
    mutable typename Triangulation<dim, spacedim>::cell_iterator cell;

    /**
     * The actual quadrature on the reference cell.
     *
     * Computed once.
     */
    Quadrature<dim> quad;

    /**
     * Values of quadrature weights for manifold quadrature
     * formulas.
     *
     * The Manifold class has a function (Manifold::get_new_point())
     * that returns new points according to a weighted average of some
     * surrounding points on the Manifold. For each quadrature point,
     * we call this function with a Quadrature formula constructed
     * using the vertices of the current cell, and the values of the
     * basis functions of an FE_Q(1) finite element evaluated at the
     * quadrature point itself. While the vertices of the cell change
     * for every cell, the weights can be computed once for each
     * quadrature point. We store this information in the following
     * variable, where the first index runs through the quadrature
     * points, and the second index runs through the vertex indices.
     *
     * Computed once.
     */
    std::vector<std::vector<double>> cell_manifold_quadrature_weights;

    /**
     * A vector of weights for use in Manifold::get_new_point(). For
     * each point (interior to a cell), we compute the weight each
     * vertex has for this point. If the point lies at a vertex, then
     * this vertex has weight one and all others have weight zero. If
     * the point lies interior to a cell, then the weight every vertex
     * has is just the $d$-linear shape functions associated with each
     * vertex evaluated at that point.
     *
     * This array has size GeometryInfo<dim>::vertices_per_cell, but it
     * can't be converted into a fixed size array because it is used
     * as input for Manifold::get_new_point() which wants to see a
     * std::vector<double> for the weights.
     */
    mutable std::vector<double> vertex_weights;

    /**
     * Unit tangential vectors. Used for the computation of boundary forms and
     * normal vectors.
     *
     * This array has `(dim-1) * GeometryInfo::faces_per_cell` entries. The
     * first GeometryInfo::faces_per_cell contain the vectors in the first
     * tangential direction for each face; the second set of
     * GeometryInfo<dim>::faces_per_cell entries contain the vectors in the
     * second tangential direction (only in 3d, since there we have 2 tangential
     * directions per face), etc.
     *
     * Filled once.
     */
    std::array<std::vector<Tensor<1, dim>>,
               GeometryInfo<dim>::faces_per_cell *(dim - 1)>
      unit_tangentials;

    /**
     * Tensors of covariant transformation at each of the quadrature points.
     * The matrix stored is the Jacobian * G^{-1}, where G = Jacobian^{t} *
     * Jacobian, is the first fundamental form of the map; if dim=spacedim
     * then it reduces to the transpose of the inverse of the Jacobian matrix,
     * which itself is stored in the @p contravariant field of this structure.
     *
     * Computed on each cell.
     */
    mutable std::vector<DerivativeForm<1, dim, spacedim>> covariant;

    /**
     * Tensors of contravariant transformation at each of the quadrature
     * points. The contravariant matrix is the Jacobian of the transformation,
     * i.e. $J_{ij}=dx_i/d\hat x_j$.
     *
     * Computed on each cell.
     */
    mutable std::vector<DerivativeForm<1, dim, spacedim>> contravariant;

    /**
     * Auxiliary vectors for internal use.
     */
    mutable std::vector<std::vector<Tensor<1, spacedim>>> aux;

    /**
     * The determinant of the Jacobian in each quadrature point. Filled if
     * #update_volume_elements.
     */
    mutable std::vector<double> volume_elements;

    /**
     * A pointer to the Manifold in use.
     *
     * Updated each.
     */
    mutable SmartPointer<const Manifold<dim, spacedim>> manifold;
  };

private:
  // documentation can be found in Mapping::requires_update_flags()
  virtual UpdateFlags
  requires_update_flags(const UpdateFlags update_flags) const override;

  // documentation can be found in Mapping::get_data()
  virtual std::unique_ptr<typename Mapping<dim, spacedim>::InternalDataBase>
  get_data(const UpdateFlags, const Quadrature<dim> &quadrature) const override;

  using Mapping<dim, spacedim>::get_face_data;

  // documentation can be found in Mapping::get_face_data()
  virtual std::unique_ptr<typename Mapping<dim, spacedim>::InternalDataBase>
  get_face_data(const UpdateFlags               flags,
                const hp::QCollection<dim - 1> &quadrature) const override;

  // documentation can be found in Mapping::get_subface_data()
  virtual std::unique_ptr<typename Mapping<dim, spacedim>::InternalDataBase>
  get_subface_data(const UpdateFlags          flags,
                   const Quadrature<dim - 1> &quadrature) const override;

  // documentation can be found in Mapping::fill_fe_values()
  virtual CellSimilarity::Similarity
  fill_fe_values(
    const typename Triangulation<dim, spacedim>::cell_iterator &cell,
    const CellSimilarity::Similarity                            cell_similarity,
    const Quadrature<dim> &                                     quadrature,
    const typename Mapping<dim, spacedim>::InternalDataBase &   internal_data,
    dealii::internal::FEValuesImplementation::MappingRelatedData<dim, spacedim>
      &output_data) const override;

  using Mapping<dim, spacedim>::fill_fe_face_values;

  // documentation can be found in Mapping::fill_fe_face_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 typename Mapping<dim, spacedim>::InternalDataBase &   internal_data,
    dealii::internal::FEValuesImplementation::MappingRelatedData<dim, spacedim>
      &output_data) const override;

  // documentation can be found in Mapping::fill_fe_subface_values()
  virtual void
  fill_fe_subface_values(
    const typename Triangulation<dim, spacedim>::cell_iterator &cell,
    const unsigned int                                          face_no,
    const unsigned int                                          subface_no,
    const Quadrature<dim - 1> &                                 quadrature,
    const typename Mapping<dim, spacedim>::InternalDataBase &   internal_data,
    dealii::internal::FEValuesImplementation::MappingRelatedData<dim, spacedim>
      &output_data) const override;

  /**
   * @}
   */
};



/*@}*/

/*----------------------------------------------------------------------*/

#ifndef DOXYGEN

template <int dim, int spacedim>
inline void
MappingManifold<dim, spacedim>::InternalData::store_vertices(
  const typename Triangulation<dim, spacedim>::cell_iterator &cell) const
{
  vertices.resize(GeometryInfo<dim>::vertices_per_cell);
  for (const unsigned int i : GeometryInfo<dim>::vertex_indices())
    vertices[i] = cell->vertex(i);
  this->cell = cell;
}


template <int dim, int spacedim>
inline void
MappingManifold<dim, spacedim>::InternalData::
  compute_manifold_quadrature_weights(const Quadrature<dim> &quad)
{
  cell_manifold_quadrature_weights.resize(
    quad.size(), std::vector<double>(GeometryInfo<dim>::vertices_per_cell));
  for (unsigned int q = 0; q < quad.size(); ++q)
    {
      for (const unsigned int i : GeometryInfo<dim>::vertex_indices())
        {
          cell_manifold_quadrature_weights[q][i] =
            GeometryInfo<dim>::d_linear_shape_function(quad.point(q), i);
        }
    }
}



template <int dim, int spacedim>
inline bool
MappingManifold<dim, spacedim>::preserves_vertex_locations() const
{
  return true;
}


template <int dim, int spacedim>
bool
MappingManifold<dim, spacedim>::is_compatible_with(
  const ReferenceCell &cell_type) const
{
  if (cell_type.get_dimension() != dim)
    return false; // TODO: or is this an error?

  if (cell_type.is_hyper_cube())
    return true;

  return false;
}



#endif // DOXYGEN

/* -------------- declaration of explicit specializations ------------- */


DEAL_II_NAMESPACE_CLOSE

#endif