Reference documentation for deal.II version GIT d77e5ebb0a 2023-01-27 22:35:02+00:00
MappingQEulerian< dim, VectorType, spacedim > Class Template Reference

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

Inheritance diagram for MappingQEulerian< dim, VectorType, spacedim >:
[legend]

## Public Member Functions

MappingQEulerian (const unsigned int degree, const DoFHandler< dim, spacedim > &euler_dof_handler, const VectorType &euler_vector, const unsigned int level=numbers::invalid_unsigned_int)

virtual boost::container::small_vector< Point< spacedim >, GeometryInfo< dim >::vertices_per_cell > get_vertices (const typename Triangulation< dim, spacedim >::cell_iterator &cell) const override

virtual std::unique_ptr< Mapping< dim, spacedim > > clone () const override

virtual bool preserves_vertex_locations () const override

virtual std::vector< Point< spacedim > > compute_mapping_support_points (const typename Triangulation< dim, spacedim >::cell_iterator &cell) const override

unsigned int get_degree () const

virtual BoundingBox< spacedim > get_bounding_box (const typename Triangulation< dim, spacedim >::cell_iterator &cell) const override

virtual bool is_compatible_with (const ReferenceCell &reference_cell) const override

void fill_mapping_data_for_generic_points (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const ArrayView< const Point< dim >> &unit_points, const UpdateFlags update_flags, internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &output_data) const

virtual Point< spacedim > get_center (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const bool map_center_of_reference_cell=true) const

template<class Archive >
void serialize (Archive &ar, const unsigned int version)

Mapping points between reference and real cells
virtual Point< spacedim > transform_unit_to_real_cell (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const Point< dim > &p) const override

virtual Point< dim > transform_real_to_unit_cell (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const Point< spacedim > &p) const override

virtual void transform_points_real_to_unit_cell (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const ArrayView< const Point< spacedim >> &real_points, const ArrayView< Point< dim >> &unit_points) const override

Functions to transform tensors from reference to real coordinates
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

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

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

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

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

Mapping points between reference and real cells
Point< dim - 1 > project_real_point_to_unit_point_on_face (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const unsigned int face_no, const Point< spacedim > &p) const

Subscriptor functionality

Classes derived from Subscriptor provide a facility to subscribe to this object. This is mostly used by the SmartPointer class.

void subscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

void unsubscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

unsigned int n_subscriptions () const

template<typename StreamType >
void list_subscribers (StreamType &stream) const

void list_subscribers () const

## Static Public Member Functions

static ::ExceptionBaseExcInactiveCell ()

static ::ExceptionBaseExcInUse (int arg1, std::string arg2, std::string arg3)

static ::ExceptionBaseExcNoSubscriber (std::string arg1, std::string arg2)

Exceptions
static ::ExceptionBaseExcInvalidData ()

static ::ExceptionBaseExcTransformationFailed ()

static ::ExceptionBaseExcDistortedMappedCell (Point< spacedim > arg1, double arg2, int arg3)

## Protected Member Functions

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, internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &output_data) const override

Point< dim > transform_real_to_unit_cell_internal (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const Point< spacedim > &p, const Point< dim > &initial_p_unit) const

Point< 1 > transform_real_to_unit_cell_internal (const Triangulation< 1, 1 >::cell_iterator &cell, const Point< 1 > &p, const Point< 1 > &initial_p_unit) const

Point< 2 > transform_real_to_unit_cell_internal (const Triangulation< 2, 2 >::cell_iterator &cell, const Point< 2 > &p, const Point< 2 > &initial_p_unit) const

Point< 3 > transform_real_to_unit_cell_internal (const Triangulation< 3, 3 >::cell_iterator &cell, const Point< 3 > &p, const Point< 3 > &initial_p_unit) const

Point< 1 > transform_real_to_unit_cell_internal (const Triangulation< 1, 2 >::cell_iterator &cell, const Point< 2 > &p, const Point< 1 > &initial_p_unit) const

Point< 2 > transform_real_to_unit_cell_internal (const Triangulation< 2, 3 >::cell_iterator &cell, const Point< 3 > &p, const Point< 2 > &initial_p_unit) const

Point< 1 > transform_real_to_unit_cell_internal (const Triangulation< 1, 3 >::cell_iterator &, const Point< 3 > &, const Point< 1 > &) const

virtual void add_line_support_points (const typename Triangulation< dim, spacedim >::cell_iterator &cell, std::vector< Point< spacedim >> &a) const

virtual void add_quad_support_points (const typename Triangulation< dim, spacedim >::cell_iterator &cell, std::vector< Point< spacedim >> &a) const

void add_quad_support_points (const Triangulation< 3, 3 >::cell_iterator &cell, std::vector< Point< 3 >> &a) const

void add_quad_support_points (const Triangulation< 2, 3 >::cell_iterator &cell, std::vector< Point< 3 >> &a) const

Interface with FEValues and friends
virtual UpdateFlags requires_update_flags (const UpdateFlags update_flags) const override

virtual std::unique_ptr< typename Mapping< dim, spacedim >::InternalDataBase > get_data (const UpdateFlags, const Quadrature< dim > &quadrature) const override

virtual std::unique_ptr< typename Mapping< dim, spacedim >::InternalDataBase > get_face_data (const UpdateFlags flags, const hp::QCollection< dim - 1 > &quadrature) const override

virtual std::unique_ptr< typename Mapping< dim, spacedim >::InternalDataBase > get_subface_data (const UpdateFlags flags, const Quadrature< dim - 1 > &quadrature) const override

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, internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &output_data) const override

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, internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &output_data) const override

virtual void fill_fe_immersed_surface_values (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const NonMatching::ImmersedSurfaceQuadrature< dim > &quadrature, const typename Mapping< dim, spacedim >::InternalDataBase &internal_data, internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &output_data) const override

Interface with FEValues
virtual std::unique_ptr< InternalDataBase > get_face_data (const UpdateFlags update_flags, const Quadrature< dim - 1 > &quadrature) const

virtual void fill_fe_face_values (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const unsigned int face_no, const Quadrature< dim - 1 > &quadrature, const typename Mapping< dim, spacedim >::InternalDataBase &internal_data, internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &output_data) const

## Protected Attributes

SmartPointer< const VectorType, MappingQEulerian< dim, VectorType, spacedim > > euler_vector

SmartPointer< const DoFHandler< dim, spacedim >, MappingQEulerian< dim, VectorType, spacedim > > euler_dof_handler

const unsigned int polynomial_degree

const std::vector< Point< 1 > > line_support_points

const std::vector< Polynomials::Polynomial< double > > polynomials_1d

const std::vector< unsigned int > renumber_lexicographic_to_hierarchic

const std::vector< Point< dim > > unit_cell_support_points

const std::vector< Table< 2, double > > support_point_weights_perimeter_to_interior

const Table< 2, doublesupport_point_weights_cell

## Private Types

using map_value_type = decltype(counter_map)::value_type

using map_iterator = decltype(counter_map)::iterator

## Private Member Functions

void check_no_subscribers () const noexcept

## Private Attributes

const unsigned int level

FEValues< dim, spacedim > fe_values

std::atomic< unsigned int > counter

std::map< std::string, unsigned int > counter_map

std::vector< std::atomic< bool > * > validity_pointers

const std::type_info * object_info

## Static Private Attributes

static std::mutex mutex

## Detailed Description

### template<int dim, typename VectorType = Vector<double>, int spacedim = dim> class MappingQEulerian< dim, VectorType, spacedim >

This class is an extension of the MappingQ1Eulerian class to higher order $$Q_p$$ mappings. It is useful when one wants to calculate shape function information on a domain that is deforming as the computation proceeds.

### Usage

The constructor of this class takes three arguments: the polynomial degree of the desired Qp mapping, a reference to the vector that defines the mapping from the initial configuration to the current configuration, and a reference to the DoFHandler. The most common case is to use the solution vector for the problem under consideration as the shift vector. The key requirement is that the number of components of the given vector field must be equal to (or possibly greater than) the number of space dimensions. If there are more components than space dimensions (for example, if one is working with a coupled problem where there are additional solution variables), the first dim components are assumed to represent the displacement field, and the remaining components are ignored. If this assumption does not hold one may need to set up a separate DoFHandler on the triangulation and associate the desired shift vector to it.

Typically, the DoFHandler operates on a finite element that is constructed as a system element (FESystem) from continuous FE_Q objects. An example is shown below:

FESystem<dim> fe(FE_Q<dim>(2), dim, FE_Q<dim>(1), 1);
dof_handler.distribute_dofs(fe);
Vector<double> displacement_field(dof_handler.n_dofs());
// ... compute displacement field somehow...
MappingQEulerian<dim> q2_mapping(2, dof_handler, displacement_field);
Definition: fe_q.h:551
const ::parallel::distributed::Triangulation< dim, spacedim > * triangulation

In this example, our element consists of (dim+1) components. Only the first dim components will be used, however, to define the Q2 mapping. The remaining components are ignored.

Note that it is essential to call the distribute_dofs(...) function before constructing a mapping object.

Also note that since the vector of shift values and the dof handler are only associated to this object at construction time, you have to make sure that whenever you use this object, the given objects still represent valid data.

To enable the use of the MappingQEulerian class also in the context of parallel codes using the PETSc or Trilinos wrapper classes, the type of the vector can be specified as template parameter VectorType.

Definition at line 95 of file mapping_q_eulerian.h.

## ◆ map_value_type

 using Subscriptor::map_value_type = decltype(counter_map)::value_type
privateinherited

The data type used in counter_map.

Definition at line 230 of file subscriptor.h.

## ◆ map_iterator

 using Subscriptor::map_iterator = decltype(counter_map)::iterator
privateinherited

The iterator type used in counter_map.

Definition at line 235 of file subscriptor.h.

## ◆ MappingQEulerian()

template<int dim, class VectorType , int spacedim>
 MappingQEulerian< dim, VectorType, spacedim >::MappingQEulerian ( const unsigned int degree, const DoFHandler< dim, spacedim > & euler_dof_handler, const VectorType & euler_vector, const unsigned int level = numbers::invalid_unsigned_int )

Constructor.

Parameters
 [in] degree The polynomial degree of the desired $$Q_p$$ mapping. [in] euler_dof_handler A DoFHandler object that defines a finite element space. This space needs to have at least dim components and the first dim components of the space will be considered displacements relative to the original positions of the cells of the triangulation. [in] euler_vector A finite element function in the space defined by the second argument. The first dim components of this function will be interpreted as the displacement we use in defining the mapping, relative to the location of cells of the underlying triangulation. [in] level The multi-grid level at which the mapping will be used. It is mainly used to check if the size of the euler_vector is consistent with the euler_dof_handler .

Definition at line 48 of file mapping_q_eulerian.cc.

## ◆ get_vertices()

template<int dim, class VectorType , int spacedim>
 boost::container::small_vector< Point< spacedim >, GeometryInfo< dim >::vertices_per_cell > MappingQEulerian< dim, VectorType, spacedim >::get_vertices ( const typename Triangulation< dim, spacedim >::cell_iterator & cell ) const
overridevirtual

Return the mapped vertices of the cell. For the current class, this function does not use the support points from the geometry of the current cell but instead evaluates an externally given displacement field in addition to the geometry of the cell.

Reimplemented from Mapping< dim, spacedim >.

Definition at line 105 of file mapping_q_eulerian.cc.

## ◆ clone()

template<int dim, class VectorType , int spacedim>
 std::unique_ptr< Mapping< dim, spacedim > > MappingQEulerian< dim, VectorType, spacedim >::clone
overridevirtual

Return a pointer to a copy of the present object. The caller of this copy then assumes ownership of it.

Reimplemented from MappingQ< dim, dim >.

Definition at line 67 of file mapping_q_eulerian.cc.

## ◆ preserves_vertex_locations()

template<int dim, typename VectorType = Vector<double>, int spacedim = dim>
 virtual bool MappingQEulerian< dim, VectorType, spacedim >::preserves_vertex_locations ( ) const
overridevirtual

Always return false because MappingQEulerian does not in general preserve vertex locations (unless the translation vector happens to provide zero displacements at vertex locations).

Reimplemented from MappingQ< dim, dim >.

## ◆ compute_mapping_support_points()

template<int dim, class VectorType , int spacedim>
 std::vector< Point< spacedim > > MappingQEulerian< dim, VectorType, spacedim >::compute_mapping_support_points ( const typename Triangulation< dim, spacedim >::cell_iterator & cell ) const
overridevirtual

Return the locations of support points for the mapping. For example, for $$Q_1$$ mappings these are the vertices, and for higher order polynomial mappings they are the vertices plus interior points on edges, faces, and the cell interior that are placed in consultation with the Manifold description of the domain and its boundary. However, other classes may override this function differently. In particular, the MappingQ1Eulerian class does exactly this by not computing the support points from the geometry of the current cell but instead evaluating an externally given displacement field in addition to the geometry of the cell.

The default implementation of this function is appropriate for most cases. It takes the locations of support points on the boundary of the cell from the underlying manifold. Interior support points (ie. support points in quads for 2d, in hexes for 3d) are then computed using an interpolation from the lower-dimensional entities (lines, quads) in order to make the transformation as smooth as possible without introducing additional boundary layers within the cells due to the placement of support points.

The function works its way from the vertices (which it takes from the given cell) via the support points on the line (for which it calls the add_line_support_points() function) and the support points on the quad faces (in 3d, for which it calls the add_quad_support_points() function). It then adds interior support points that are either computed by interpolation from the surrounding points using weights for transfinite interpolation, or if dim<spacedim, it asks the underlying manifold for the locations of interior points.

Reimplemented from MappingQ< dim, dim >.

Definition at line 123 of file mapping_q_eulerian.cc.

## ◆ fill_fe_values()

template<int dim, class VectorType , int spacedim>
 CellSimilarity::Similarity MappingQEulerian< dim, VectorType, spacedim >::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, internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > & output_data ) const
overrideprotectedvirtual

Compute mapping-related information for a cell. See the documentation of Mapping::fill_fe_values() for a discussion of purpose, arguments, and return value of this function.

This function overrides the function in the base class since we cannot use any cell similarity for this class.

Reimplemented from MappingQ< dim, dim >.

Definition at line 196 of file mapping_q_eulerian.cc.

## ◆ get_degree()

 unsigned int MappingQ< dim, spacedim >::get_degree
inherited

Return the degree of the mapping, i.e. the value which was passed to the constructor.

Definition at line 144 of file mapping_q.cc.

## ◆ get_bounding_box()

 BoundingBox< spacedim > MappingQ< dim, spacedim >::get_bounding_box ( const typename Triangulation< dim, spacedim >::cell_iterator & cell ) const
overridevirtualinherited

Return the bounding box of a mapped cell.

If you are using a (bi-,tri-)linear mapping that preserves vertex locations, this function simply returns the value also produced by cell->bounding_box(). However, there are also mappings that add displacements or choose completely different locations, e.g., MappingQEulerian, MappingQ1Eulerian, or MappingFEField.

For linear mappings, this function returns the bounding box containing all the vertices of the cell, as returned by the get_vertices() method. For higher order mappings defined through support points, the bounding box is only guaranteed to contain all the support points, and it is, in general, only an approximation of the true bounding box, which may be larger.

Parameters
 [in] cell The cell for which you want to compute the bounding box

Reimplemented from Mapping< dim, spacedim >.

Definition at line 155 of file mapping_q.cc.

## ◆ is_compatible_with()

 bool MappingQ< dim, spacedim >::is_compatible_with ( const ReferenceCell & reference_cell ) const
overridevirtualinherited

Returns if this instance of Mapping is compatible with the type of cell in reference_cell.

Implements Mapping< dim, spacedim >.

Definition at line 159 of file mapping_q.cc.

## ◆ transform_unit_to_real_cell()

 Point< spacedim > MappingQ< dim, spacedim >::transform_unit_to_real_cell ( const typename Triangulation< dim, spacedim >::cell_iterator & cell, const Point< dim > & p ) const
overridevirtualinherited

Map the point p on the unit cell to the corresponding point on the real cell cell.

Parameters
 cell Iterator to the cell that will be used to define the mapping. p Location of a point on the reference cell.
Returns
The location of the reference point mapped to real space using the mapping defined by the class derived from the current one that implements the mapping, and the coordinates of the cell identified by the first argument.

Implements Mapping< dim, spacedim >.

Definition at line 168 of file mapping_q.cc.

## ◆ transform_real_to_unit_cell()

 Point< dim > MappingQ< dim, spacedim >::transform_real_to_unit_cell ( const typename Triangulation< dim, spacedim >::cell_iterator & cell, const Point< spacedim > & p ) const
overridevirtualinherited

Map the point p on the real cell to the corresponding point on the unit cell, and return its coordinates. This function provides the inverse of the mapping provided by transform_unit_to_real_cell().

In the codimension one case, this function returns the normal projection of the real point p on the curve or surface identified by the cell.

Note
Polynomial mappings from the reference (unit) cell coordinates to the coordinate system of a real cell are not always invertible if the point for which the inverse mapping is to be computed lies outside the cell's boundaries. In such cases, the current function may fail to compute a point on the reference cell whose image under the mapping equals the given point p. If this is the case then this function throws an exception of type Mapping::ExcTransformationFailed . Whether the given point p lies outside the cell can therefore be determined by checking whether the returned reference coordinates lie inside or outside the reference cell (e.g., using GeometryInfo::is_inside_unit_cell()) or whether the exception mentioned above has been thrown.
Parameters
 cell Iterator to the cell that will be used to define the mapping. p Location of a point on the given cell.
Returns
The reference cell location of the point that when mapped to real space equals the coordinates given by the second argument. This mapping uses the mapping defined by the class derived from the current one that implements the mapping, and the coordinates of the cell identified by the first argument.

Implements Mapping< dim, spacedim >.

Definition at line 174 of file mapping_q.cc.

## ◆ transform_points_real_to_unit_cell()

 void MappingQ< dim, spacedim >::transform_points_real_to_unit_cell ( const typename Triangulation< dim, spacedim >::cell_iterator & cell, const ArrayView< const Point< spacedim >> & real_points, const ArrayView< Point< dim >> & unit_points ) const
overridevirtualinherited

Map multiple points from the real point locations to points in reference locations. The functionality is essentially the same as looping over all points and calling the Mapping::transform_real_to_unit_cell() function for each point individually, but it can be much faster for certain mappings that implement a more specialized version such as MappingQ. The only difference in behavior is that this function will never throw an ExcTransformationFailed() exception. If the transformation fails for real_points[i], the returned unit_points[i] contains std::numeric_limits<double>::infinity() as the first entry.

Reimplemented from Mapping< dim, spacedim >.

Definition at line 180 of file mapping_q.cc.

## ◆ transform() [1/5]

 void MappingQ< dim, spacedim >::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
overridevirtualinherited

Transform a field of vectors or 1-differential forms according to the selected MappingKind.

Note
Normally, this function is called by a finite element, filling FEValues objects. For this finite element, there should be an alias MappingKind like mapping_bdm, mapping_nedelec, etc. This alias should be preferred to using the kinds below.

The mapping kinds currently implemented by derived classes are:

• mapping_contravariant: maps a vector field on the reference cell to the physical cell through the Jacobian:

$\mathbf u(\mathbf x) = J(\hat{\mathbf x})\hat{\mathbf u}(\hat{\mathbf x}).$

In physics, this is usually referred to as the contravariant transformation. Mathematically, it is the push forward of a vector field.

• mapping_covariant: maps a field of one-forms on the reference cell to a field of one-forms on the physical cell. (Theoretically this would refer to a DerivativeForm<1,dim,1> but we canonically identify this type with a Tensor<1,dim>). Mathematically, it is the pull back of the differential form

$\mathbf u(\mathbf x) = J(\hat{\mathbf x})(J(\hat{\mathbf x})^{T} J(\hat{\mathbf x}))^{-1}\hat{\mathbf u}(\hat{\mathbf x}).$

Gradients of scalar differentiable functions are transformed this way.

In the case when dim=spacedim the previous formula reduces to

$\mathbf u(\mathbf x) = J(\hat{\mathbf x})^{-T}\hat{\mathbf u}(\hat{\mathbf x})$

because we assume that the mapping $$\mathbf F_K$$ is always invertible, and consequently its Jacobian $$J$$ is an invertible matrix.

• mapping_piola: A field of dim-1-forms on the reference cell is also represented by a vector field, but again transforms differently, namely by the Piola transform

$\mathbf u(\mathbf x) = \frac{1}{\text{det}\;J(\hat{\mathbf x})} J(\hat{\mathbf x}) \hat{\mathbf u}(\hat{\mathbf x}).$

Parameters
 [in] input An array (or part of an array) of input objects that should be mapped. [in] kind The kind of mapping to be applied. [in] internal A pointer to an object of type Mapping::InternalDataBase that contains information previously stored by the mapping. The object pointed to was created by the get_data(), get_face_data(), or get_subface_data() function, and will have been updated as part of a call to fill_fe_values(), fill_fe_face_values(), or fill_fe_subface_values() for the current cell, before calling the current function. In other words, this object also represents with respect to which cell the transformation should be applied to. [out] output An array (or part of an array) into which the transformed objects should be placed. (Note that the array view is const, but the tensors it points to are not.)

Implements Mapping< dim, spacedim >.

Definition at line 196 of file mapping_q.cc.

## ◆ transform() [2/5]

 void MappingQ< dim, spacedim >::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
overridevirtualinherited

Transform a field of differential forms from the reference cell to the physical cell. It is useful to think of $$\mathbf{T} = \nabla \mathbf u$$ and $$\hat{\mathbf T} = \hat \nabla \hat{\mathbf u}$$, with $$\mathbf u$$ a vector field. The mapping kinds currently implemented by derived classes are:

• mapping_covariant: maps a field of forms on the reference cell to a field of forms on the physical cell. Mathematically, it is the pull back of the differential form

$\mathbf T(\mathbf x) = \hat{\mathbf T}(\hat{\mathbf x}) J(\hat{\mathbf x})(J(\hat{\mathbf x})^{T} J(\hat{\mathbf x}))^{-1}.$

Jacobians of spacedim-vector valued differentiable functions are transformed this way.

In the case when dim=spacedim the previous formula reduces to

$\mathbf T(\mathbf x) = \hat{\mathbf u}(\hat{\mathbf x}) J(\hat{\mathbf x})^{-1}.$

Note
It would have been more reasonable to make this transform a template function with the rank in DerivativeForm<1, dim, rank>. Unfortunately C++ does not allow templatized virtual functions. This is why we identify DerivativeForm<1, dim, 1> with a Tensor<1,dim> when using mapping_covariant() in the function transform() above this one.
Parameters
 [in] input An array (or part of an array) of input objects that should be mapped. [in] kind The kind of mapping to be applied. [in] internal A pointer to an object of type Mapping::InternalDataBase that contains information previously stored by the mapping. The object pointed to was created by the get_data(), get_face_data(), or get_subface_data() function, and will have been updated as part of a call to fill_fe_values(), fill_fe_face_values(), or fill_fe_subface_values() for the current cell, before calling the current function. In other words, this object also represents with respect to which cell the transformation should be applied to. [out] output An array (or part of an array) into which the transformed objects should be placed. (Note that the array view is const, but the tensors it points to are not.)

Implements Mapping< dim, spacedim >.

Definition at line 203 of file mapping_q.cc.

## ◆ transform() [3/5]

 void MappingQ< dim, spacedim >::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
overridevirtualinherited

Transform a tensor field from the reference cell to the physical cell. These tensors are usually the Jacobians in the reference cell of vector fields that have been pulled back from the physical cell. The mapping kinds currently implemented by derived classes are:

• mapping_contravariant_gradient: it assumes $$\mathbf u(\mathbf x) = J \hat{\mathbf u}$$ so that

$\mathbf T(\mathbf x) = J(\hat{\mathbf x}) \hat{\mathbf T}(\hat{\mathbf x}) J(\hat{\mathbf x})^{-1}.$

• mapping_covariant_gradient: it assumes $$\mathbf u(\mathbf x) = J^{-T} \hat{\mathbf u}$$ so that

$\mathbf T(\mathbf x) = J(\hat{\mathbf x})^{-T} \hat{\mathbf T}(\hat{\mathbf x}) J(\hat{\mathbf x})^{-1}.$

• mapping_piola_gradient: it assumes $$\mathbf u(\mathbf x) = \frac{1}{\text{det}\;J(\hat{\mathbf x})} J(\hat{\mathbf x}) \hat{\mathbf u}(\hat{\mathbf x})$$ so that

$\mathbf T(\mathbf x) = \frac{1}{\text{det}\;J(\hat{\mathbf x})} J(\hat{\mathbf x}) \hat{\mathbf T}(\hat{\mathbf x}) J(\hat{\mathbf x})^{-1}.$

Todo:
The formulas for mapping_covariant_gradient, mapping_contravariant_gradient and mapping_piola_gradient are only true as stated for linear mappings. If, for example, the mapping is bilinear (or has a higher order polynomial degree) then there is a missing term associated with the derivative of $$J$$.
Parameters
 [in] input An array (or part of an array) of input objects that should be mapped. [in] kind The kind of mapping to be applied. [in] internal A pointer to an object of type Mapping::InternalDataBase that contains information previously stored by the mapping. The object pointed to was created by the get_data(), get_face_data(), or get_subface_data() function, and will have been updated as part of a call to fill_fe_values(), fill_fe_face_values(), or fill_fe_subface_values() for the current cell, before calling the current function. In other words, this object also represents with respect to which cell the transformation should be applied to. [out] output An array (or part of an array) into which the transformed objects should be placed. (Note that the array view is const, but the tensors it points to are not.)

Implements Mapping< dim, spacedim >.

Definition at line 210 of file mapping_q.cc.

## ◆ transform() [4/5]

 void MappingQ< dim, spacedim >::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
overridevirtualinherited

Transform a tensor field from the reference cell to the physical cell. This tensors are most of times the hessians in the reference cell of vector fields that have been pulled back from the physical cell.

The mapping kinds currently implemented by derived classes are:

• mapping_covariant_gradient: maps a field of forms on the reference cell to a field of forms on the physical cell. Mathematically, it is the pull back of the differential form

$\mathbf T_{ijk}(\mathbf x) = \hat{\mathbf T}_{iJK}(\hat{\mathbf x}) J_{jJ}^{\dagger} J_{kK}^{\dagger}$

,

where

$J^{\dagger} = J(\hat{\mathbf x})(J(\hat{\mathbf x})^{T} J(\hat{\mathbf x}))^{-1}.$

Hessians of spacedim-vector valued differentiable functions are transformed this way (After subtraction of the product of the derivative with the Jacobian gradient).

In the case when dim=spacedim the previous formula reduces to

$J^{\dagger} = J^{-1}$

Parameters
 [in] input An array (or part of an array) of input objects that should be mapped. [in] kind The kind of mapping to be applied. [in] internal A pointer to an object of type Mapping::InternalDataBase that contains information previously stored by the mapping. The object pointed to was created by the get_data(), get_face_data(), or get_subface_data() function, and will have been updated as part of a call to fill_fe_values(), fill_fe_face_values(), or fill_fe_subface_values() for the current cell, before calling the current function. In other words, this object also represents with respect to which cell the transformation should be applied to. [out] output An array (or part of an array) into which the transformed objects should be placed. (Note that the array view is const, but the tensors it points to are not.)

Implements Mapping< dim, spacedim >.

Definition at line 217 of file mapping_q.cc.

## ◆ transform() [5/5]

 void MappingQ< dim, spacedim >::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
overridevirtualinherited

Transform a field of 3-differential forms from the reference cell to the physical cell. It is useful to think of $$\mathbf{T}_{ijk} = D^2_{jk} \mathbf u_i$$ and $$\mathbf{\hat T}_{IJK} = \hat D^2_{JK} \mathbf{\hat u}_I$$, with $$\mathbf u_i$$ a vector field.

The mapping kinds currently implemented by derived classes are:

• mapping_contravariant_hessian: it assumes $$\mathbf u_i(\mathbf x) = J_{iI} \hat{\mathbf u}_I$$ so that

$\mathbf T_{ijk}(\mathbf x) = J_{iI}(\hat{\mathbf x}) \hat{\mathbf T}_{IJK}(\hat{\mathbf x}) J_{jJ}(\hat{\mathbf x})^{-1} J_{kK}(\hat{\mathbf x})^{-1}.$

• mapping_covariant_hessian: it assumes $$\mathbf u_i(\mathbf x) = J_{iI}^{-T} \hat{\mathbf u}_I$$ so that

$\mathbf T_{ijk}(\mathbf x) = J_iI(\hat{\mathbf x})^{-1} \hat{\mathbf T}_{IJK}(\hat{\mathbf x}) J_{jJ}(\hat{\mathbf x})^{-1} J_{kK}(\hat{\mathbf x})^{-1}.$

• mapping_piola_hessian: it assumes $$\mathbf u_i(\mathbf x) = \frac{1}{\text{det}\;J(\hat{\mathbf x})} J_{iI}(\hat{\mathbf x}) \hat{\mathbf u}(\hat{\mathbf x})$$ so that

$\mathbf T_{ijk}(\mathbf x) = \frac{1}{\text{det}\;J(\hat{\mathbf x})} J_{iI}(\hat{\mathbf x}) \hat{\mathbf T}_{IJK}(\hat{\mathbf x}) J_{jJ}(\hat{\mathbf x})^{-1} J_{kK}(\hat{\mathbf x})^{-1}.$

Parameters
 [in] input An array (or part of an array) of input objects that should be mapped. [in] kind The kind of mapping to be applied. [in] internal A pointer to an object of type Mapping::InternalDataBase that contains information previously stored by the mapping. The object pointed to was created by the get_data(), get_face_data(), or get_subface_data() function, and will have been updated as part of a call to fill_fe_values(), fill_fe_face_values(), or fill_fe_subface_values() for the current cell, before calling the current function. In other words, this object also represents with respect to which cell the transformation should be applied to. [out] output An array (or part of an array) into which the transformed objects should be placed.

Implements Mapping< dim, spacedim >.

Definition at line 224 of file mapping_q.cc.

## ◆ fill_mapping_data_for_generic_points()

 void MappingQ< dim, spacedim >::fill_mapping_data_for_generic_points ( const typename Triangulation< dim, spacedim >::cell_iterator & cell, const ArrayView< const Point< dim >> & unit_points, const UpdateFlags update_flags, internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > & output_data ) const
inherited

As opposed to the other fill_fe_values() and fill_fe_face_values() functions that rely on pre-computed information of InternalDataBase, this function chooses the flexible evaluation path on the cell and points passed in to the current function.

Parameters
 [in] cell The cell where to evaluate the mapping [in] unit_points The points in reference coordinates where the transformation (Jacobians, positions) should be computed. [in] update_flags The kind of information that should be computed. [out] output_data A struct containing the evaluated quantities such as the Jacobian resulting from application of the mapping on the given cell with its underlying manifolds.

Definition at line 251 of file mapping_q.cc.

## ◆ requires_update_flags()

 UpdateFlags MappingQ< dim, spacedim >::requires_update_flags ( const UpdateFlags update_flags ) const
overrideprotectedvirtualinherited

Given a set of update flags, compute which other quantities also need to be computed in order to satisfy the request by the given flags. Then return the combination of the original set of flags and those just computed.

As an example, if update_flags contains update_JxW_values (i.e., the product of the determinant of the Jacobian and the weights provided by the quadrature formula), a mapping may require the computation of the full Jacobian matrix in order to compute its determinant. They would then return not just update_JxW_values, but also update_jacobians. (This is not how it is actually done internally in the derived classes that compute the JxW values – they set update_contravariant_transformation instead, from which the determinant can also be computed – but this does not take away from the instructiveness of the example.)

An extensive discussion of the interaction between this function and FEValues can be found in the How Mapping, FiniteElement, and FEValues work together documentation module.

UpdateFlags

Implements Mapping< dim, spacedim >.

Definition at line 560 of file mapping_q.cc.

## ◆ get_data()

 std::unique_ptr< typename Mapping< dim, spacedim >::InternalDataBase > MappingQ< dim, spacedim >::get_data ( const UpdateFlags update_flags, const Quadrature< dim > & quadrature ) const
overrideprotectedvirtualinherited

Create and return a pointer to an object into which mappings can store data that only needs to be computed once but that can then be used whenever the mapping is applied to a concrete cell (e.g., in the various transform() functions, as well as in the fill_fe_values(), fill_fe_face_values() and fill_fe_subface_values() that form the interface of mappings with the FEValues class).

Derived classes will return pointers to objects of a type derived from Mapping::InternalDataBase (see there for more information) and may pre- compute some information already (in accordance with what will be asked of the mapping in the future, as specified by the update flags) and for the given quadrature object. Subsequent calls to transform() or fill_fe_values() and friends will then receive back the object created here (with the same set of update flags and for the same quadrature object). Derived classes can therefore pre-compute some information in their get_data() function and store it in the internal data object.

The mapping classes do not keep track of the objects created by this function. Ownership will therefore rest with the caller.

An extensive discussion of the interaction between this function and FEValues can be found in the How Mapping, FiniteElement, and FEValues work together documentation module.

Parameters
 update_flags A set of flags that define what is expected of the mapping class in future calls to transform() or the fill_fe_values() group of functions. This set of flags may contain flags that mappings do not know how to deal with (e.g., for information that is in fact computed by the finite element classes, such as UpdateFlags::update_values). Derived classes will need to store these flags, or at least that subset of flags that will require the mapping to perform any actions in fill_fe_values(), in InternalDataBase::update_each. quadrature The quadrature object for which mapping information will have to be computed. This includes the locations and weights of quadrature points.
Returns
A pointer to a newly created object of type InternalDataBase (or a derived class). Ownership of this object passes to the calling function.
Note
C++ allows that virtual functions in derived classes may return pointers to objects not of type InternalDataBase but in fact pointers to objects of classes derived from InternalDataBase. (This feature is called "covariant return types".) This is useful in some contexts where the calling is within the derived class and will immediately make use of the returned object, knowing its real (derived) type.

Implements Mapping< dim, spacedim >.

Definition at line 564 of file mapping_q.cc.

## ◆ get_face_data() [1/2]

 std::unique_ptr< typename Mapping< dim, spacedim >::InternalDataBase > MappingQ< dim, spacedim >::get_face_data ( const UpdateFlags update_flags, const hp::QCollection< dim - 1 > & quadrature ) const
overrideprotectedvirtualinherited

Like get_data(), but in preparation for later calls to transform() or fill_fe_face_values() that will need information about mappings from the reference face to a face of a concrete cell.

Parameters
 update_flags A set of flags that define what is expected of the mapping class in future calls to transform() or the fill_fe_values() group of functions. This set of flags may contain flags that mappings do not know how to deal with (e.g., for information that is in fact computed by the finite element classes, such as UpdateFlags::update_values). Derived classes will need to store these flags, or at least that subset of flags that will require the mapping to perform any actions in fill_fe_values(), in InternalDataBase::update_each. quadrature The quadrature object for which mapping information will have to be computed. This includes the locations and weights of quadrature points.
Returns
A pointer to a newly created object of type InternalDataBase (or a derived class). Ownership of this object passes to the calling function.
Note
C++ allows that virtual functions in derived classes may return pointers to objects not of type InternalDataBase but in fact pointers to objects of classes derived from InternalDataBase. (This feature is called "covariant return types".) This is useful in some contexts where the calling is within the derived class and will immediately make use of the returned object, knowing its real (derived) type.

Reimplemented from Mapping< dim, spacedim >.

Definition at line 570 of file mapping_q.cc.

## ◆ get_face_data() [2/2]

 virtual std::unique_ptr Mapping< dim, spacedim >::get_face_data ( const UpdateFlags update_flags, const Quadrature< dim - 1 > & quadrature ) const
protectedvirtualinherited
Deprecated:
Use the version taking a hp::QCollection argument.

## ◆ get_subface_data()

 std::unique_ptr< typename Mapping< dim, spacedim >::InternalDataBase > MappingQ< dim, spacedim >::get_subface_data ( const UpdateFlags update_flags, const Quadrature< dim - 1 > & quadrature ) const
overrideprotectedvirtualinherited

Like get_data() and get_face_data(), but in preparation for later calls to transform() or fill_fe_subface_values() that will need information about mappings from the reference face to a child of a face (i.e., subface) of a concrete cell.

Parameters
 update_flags A set of flags that define what is expected of the mapping class in future calls to transform() or the fill_fe_values() group of functions. This set of flags may contain flags that mappings do not know how to deal with (e.g., for information that is in fact computed by the finite element classes, such as UpdateFlags::update_values). Derived classes will need to store these flags, or at least that subset of flags that will require the mapping to perform any actions in fill_fe_values(), in InternalDataBase::update_each. quadrature The quadrature object for which mapping information will have to be computed. This includes the locations and weights of quadrature points.
Returns
A pointer to a newly created object of type InternalDataBase (or a derived class). Ownership of this object passes to the calling function.
Note
C++ allows that virtual functions in derived classes may return pointers to objects not of type InternalDataBase but in fact pointers to objects of classes derived from InternalDataBase. (This feature is called "covariant return types".) This is useful in some contexts where the calling is within the derived class and will immediately make use of the returned object, knowing its real (derived) type.

Implements Mapping< dim, spacedim >.

Definition at line 575 of file mapping_q.cc.

## ◆ fill_fe_face_values() [1/2]

 void MappingQ< 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 typename Mapping< dim, spacedim >::InternalDataBase & internal_data, internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > & output_data ) const
overrideprotectedvirtualinherited

This function is the equivalent to Mapping::fill_fe_values(), but for faces of cells. See there for an extensive discussion of its purpose. It is called by FEFaceValues::reinit().

Parameters
 [in] cell The cell of the triangulation for which this function is to compute a mapping from the reference cell to. [in] face_no The number of the face of the given cell for which information is requested. [in] quadrature A reference to the quadrature formula in use for the current evaluation. This quadrature object is the same as the one used when creating the internal_data object. The object is used both to map the location of quadrature points, as well as to compute the JxW values for each quadrature point (which involves the quadrature weights). [in] internal_data A reference to an object previously created by get_data() and that may be used to store information the mapping can compute once on the reference cell. See the documentation of the Mapping::InternalDataBase class for an extensive description of the purpose of these objects. [out] output_data A reference to an object whose member variables should be computed. Not all of the members of this argument need to be filled; which ones need to be filled is determined by the update flags stored inside the internal_data object.

Reimplemented from Mapping< dim, spacedim >.

Definition at line 592 of file mapping_q.cc.

## ◆ fill_fe_face_values() [2/2]

 virtual void Mapping< dim, spacedim >::fill_fe_face_values ( const typename Triangulation< dim, spacedim >::cell_iterator & cell, const unsigned int face_no, const Quadrature< dim - 1 > & quadrature, const typename Mapping< dim, spacedim >::InternalDataBase & internal_data, internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > & output_data ) const
protectedvirtualinherited
Deprecated:
Use the version taking a hp::QCollection argument.

## ◆ fill_fe_subface_values()

 void MappingQ< dim, spacedim >::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, internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > & output_data ) const
overrideprotectedvirtualinherited

This function is the equivalent to Mapping::fill_fe_values(), but for subfaces (i.e., children of faces) of cells. See there for an extensive discussion of its purpose. It is called by FESubfaceValues::reinit().

Parameters
 [in] cell The cell of the triangulation for which this function is to compute a mapping from the reference cell to. [in] face_no The number of the face of the given cell for which information is requested. [in] subface_no The number of the child of a face of the given cell for which information is requested. [in] quadrature A reference to the quadrature formula in use for the current evaluation. This quadrature object is the same as the one used when creating the internal_data object. The object is used both to map the location of quadrature points, as well as to compute the JxW values for each quadrature point (which involves the quadrature weights). [in] internal_data A reference to an object previously created by get_data() and that may be used to store information the mapping can compute once on the reference cell. See the documentation of the Mapping::InternalDataBase class for an extensive description of the purpose of these objects. [out] output_data A reference to an object whose member variables should be computed. Not all of the members of this argument need to be filled; which ones need to be filled is determined by the update flags stored inside the internal_data object.

Implements Mapping< dim, spacedim >.

Definition at line 602 of file mapping_q.cc.

## ◆ fill_fe_immersed_surface_values()

 void MappingQ< dim, spacedim >::fill_fe_immersed_surface_values ( const typename Triangulation< dim, spacedim >::cell_iterator & cell, const NonMatching::ImmersedSurfaceQuadrature< dim > & quadrature, const typename Mapping< dim, spacedim >::InternalDataBase & internal_data, internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > & output_data ) const
overrideprotectedvirtualinherited

The equivalent of Mapping::fill_fe_values(), but for the case that the quadrature is an ImmersedSurfaceQuadrature. See there for a comprehensive description of the input parameters. This function is called by FEImmersedSurfaceValues::reinit().

Reimplemented from Mapping< dim, spacedim >.

Definition at line 613 of file mapping_q.cc.

## ◆ transform_real_to_unit_cell_internal() [1/7]

 Point< dim > MappingQ< dim, spacedim >::transform_real_to_unit_cell_internal ( const typename Triangulation< dim, spacedim >::cell_iterator & cell, const Point< spacedim > & p, const Point< dim > & initial_p_unit ) const
protectedinherited

Transform the point p on the real cell to the corresponding point on the unit cell cell by a Newton iteration.

Definition at line 740 of file mapping_q.cc.

## ◆ transform_real_to_unit_cell_internal() [2/7]

 Point< 1 > MappingQ< 1, 1 >::transform_real_to_unit_cell_internal ( const Triangulation< 1, 1 >::cell_iterator & cell, const Point< 1 > & p, const Point< 1 > & initial_p_unit ) const
protectedinherited

Definition at line 503 of file mapping_q.cc.

## ◆ transform_real_to_unit_cell_internal() [3/7]

 Point< 2 > MappingQ< 2, 2 >::transform_real_to_unit_cell_internal ( const Triangulation< 2, 2 >::cell_iterator & cell, const Point< 2 > & p, const Point< 2 > & initial_p_unit ) const
protectedinherited

Definition at line 523 of file mapping_q.cc.

## ◆ transform_real_to_unit_cell_internal() [4/7]

 Point< 3 > MappingQ< 3, 3 >::transform_real_to_unit_cell_internal ( const Triangulation< 3, 3 >::cell_iterator & cell, const Point< 3 > & p, const Point< 3 > & initial_p_unit ) const
protectedinherited

Definition at line 541 of file mapping_q.cc.

## ◆ transform_real_to_unit_cell_internal() [5/7]

 Point< 1 > MappingQ< 1, 2 >::transform_real_to_unit_cell_internal ( const Triangulation< 1, 2 >::cell_iterator & cell, const Point< 2 > & p, const Point< 1 > & initial_p_unit ) const
protectedinherited

Definition at line 559 of file mapping_q.cc.

## ◆ transform_real_to_unit_cell_internal() [6/7]

 Point< 2 > MappingQ< 2, 3 >::transform_real_to_unit_cell_internal ( const Triangulation< 2, 3 >::cell_iterator & cell, const Point< 3 > & p, const Point< 2 > & initial_p_unit ) const
protectedinherited

Definition at line 590 of file mapping_q.cc.

## ◆ transform_real_to_unit_cell_internal() [7/7]

 Point< 1 > MappingQ< 1, 3 >::transform_real_to_unit_cell_internal ( const Triangulation< 1, 3 >::cell_iterator & , const Point< 3 > & , const Point< 1 > & ) const
protectedinherited

Definition at line 619 of file mapping_q.cc.

 void MappingQ< dim, spacedim >::add_line_support_points ( const typename Triangulation< dim, spacedim >::cell_iterator & cell, std::vector< Point< spacedim >> & a ) const
protectedvirtualinherited

Append the support points of all shape functions located on bounding lines of the given cell to the vector a. Points located on the vertices of a line are not included.

This function uses the underlying manifold object of the line (or, if none is set, of the cell) for the location of the requested points. This function is usually called by compute_mapping_support_points() function.

This function is made virtual in order to allow derived classes to choose shape function support points differently than the present class, which chooses the points as interpolation points on the boundary.

Definition at line 759 of file mapping_q.cc.

 void MappingQ< dim, spacedim >::add_quad_support_points ( const typename Triangulation< dim, spacedim >::cell_iterator & cell, std::vector< Point< spacedim >> & a ) const
protectedvirtualinherited

Append the support points of all shape functions located on bounding faces (quads in 3d) of the given cell to the vector a. This function is only defined for dim=3. Points located on the vertices or lines of a quad are not included.

This function uses the underlying manifold object of the quad (or, if none is set, of the cell) for the location of the requested points. This function is usually called by compute_mapping_support_points().

This function is made virtual in order to allow derived classes to choose shape function support points differently than the present class, which chooses the points as interpolation points on the boundary.

Definition at line 778 of file mapping_q.cc.

 void MappingQ< 3, 3 >::add_quad_support_points ( const Triangulation< 3, 3 >::cell_iterator & cell, std::vector< Point< 3 >> & a ) const
protectedinherited

Definition at line 1704 of file mapping_q.cc.

 void MappingQ< 2, 3 >::add_quad_support_points ( const Triangulation< 2, 3 >::cell_iterator & cell, std::vector< Point< 3 >> & a ) const
protectedinherited

Definition at line 1773 of file mapping_q.cc.

## ◆ get_center()

 virtual Point Mapping< dim, spacedim >::get_center ( const typename Triangulation< dim, spacedim >::cell_iterator & cell, const bool map_center_of_reference_cell = true ) const
virtualinherited

Return the mapped center of a cell.

If you are using a (bi-,tri-)linear mapping that preserves vertex locations, this function simply returns the value also produced by cell->center(). However, there are also mappings that add displacements or choose completely different locations, e.g., MappingQEulerian, MappingQ1Eulerian, or MappingFEField, and mappings based on high order polynomials, for which the center may not coincide with the average of the vertex locations.

By default, this function returns the push forward of the center of the reference cell. If the parameter map_center_of_reference_cell is set to false, than the return value will be the average of the vertex locations, as returned by the get_vertices() method.

Parameters
 [in] cell The cell for which you want to compute the center [in] map_center_of_reference_cell A flag that switches the algorithm for the computation of the cell center from transform_unit_to_real_cell() applied to the center of the reference cell to computing the vertex averages.

## ◆ project_real_point_to_unit_point_on_face()

 Point Mapping< dim, spacedim >::project_real_point_to_unit_point_on_face ( const typename Triangulation< dim, spacedim >::cell_iterator & cell, const unsigned int face_no, const Point< spacedim > & p ) const
inherited

Transform the point p on the real cell to the corresponding point on the reference cell, and then project this point to a (dim-1)-dimensional point in the coordinate system of the face with the given face number face_no. Ideally the point p is near the face face_no, but any point in the cell can technically be projected.

This function does not make physical sense when dim=1, so it throws an exception in this case.

## ◆ ExcInvalidData()

 static ::ExceptionBase& Mapping< dim, spacedim >::ExcInvalidData ( )
staticinherited

Exception

## ◆ ExcTransformationFailed()

 static ::ExceptionBase& Mapping< dim, spacedim >::ExcTransformationFailed ( )
staticinherited

Computing the mapping between a real space point and a point in reference space failed, typically because the given point lies outside the cell where the inverse mapping is not unique.

Note
The message that will be printed by this exception reads:
"Computing the mapping between a real space point and a point in reference " "space failed, typically because the given point lies outside the cell " "where the inverse mapping is not unique."

## ◆ ExcDistortedMappedCell()

 static ::ExceptionBase& Mapping< dim, spacedim >::ExcDistortedMappedCell ( Point< spacedim > arg1, double arg2, int arg3 )
staticinherited

deal.II assumes the Jacobian determinant to be positive. When the cell geometry is distorted under the image of the mapping, the mapping becomes invalid and this exception is thrown.

Note
The message that will be printed by this exception reads:
<< "The image of the mapping applied to cell with center [" << arg1 << "] is distorted. The cell geometry or the " << "mapping are invalid, giving a non-positive volume " << "fraction of " << arg2 << " in quadrature point " << arg3 << '.'

## ◆ subscribe()

 void Subscriptor::subscribe ( std::atomic< bool > *const validity, const std::string & identifier = "" ) const
inherited

Subscribes a user of the object by storing the pointer validity. The subscriber may be identified by text supplied as identifier.

Definition at line 136 of file subscriptor.cc.

## ◆ unsubscribe()

 void Subscriptor::unsubscribe ( std::atomic< bool > *const validity, const std::string & identifier = "" ) const
inherited

Unsubscribes a user from the object.

Note
The identifier and the validity pointer must be the same as the one supplied to subscribe().

Definition at line 156 of file subscriptor.cc.

## ◆ n_subscriptions()

 unsigned int Subscriptor::n_subscriptions ( ) const
inlineinherited

Return the present number of subscriptions to this object. This allows to use this class for reference counted lifetime determination where the last one to unsubscribe also deletes the object.

Definition at line 300 of file subscriptor.h.

## ◆ list_subscribers() [1/2]

template<typename StreamType >
 void Subscriptor::list_subscribers ( StreamType & stream ) const
inlineinherited

List the subscribers to the input stream.

Definition at line 317 of file subscriptor.h.

## ◆ list_subscribers() [2/2]

 void Subscriptor::list_subscribers ( ) const
inherited

List the subscribers to deallog.

Definition at line 204 of file subscriptor.cc.

## ◆ serialize()

template<class Archive >
 void Subscriptor::serialize ( Archive & ar, const unsigned int version )
inlineinherited

Read or write the data of this object to or from a stream for the purpose of serialization using the BOOST serialization library.

This function does not actually serialize any of the member variables of this class. The reason is that what this class stores is only who subscribes to this object, but who does so at the time of storing the contents of this object does not necessarily have anything to do with who subscribes to the object when it is restored. Consequently, we do not want to overwrite the subscribers at the time of restoring, and then there is no reason to write the subscribers out in the first place.

Definition at line 309 of file subscriptor.h.

## ◆ check_no_subscribers()

 void Subscriptor::check_no_subscribers ( ) const
privatenoexceptinherited

Check that there are no objects subscribing to this object. If this check passes then it is safe to destroy the current object. It this check fails then this function will either abort or print an error message to deallog (by using the AssertNothrow mechanism), but will not throw an exception.

Note
Since this function is just a consistency check it does nothing in release mode.
If this function is called when there is an uncaught exception then, rather than aborting, this function prints an error message to the standard error stream and returns.

Definition at line 53 of file subscriptor.cc.

## ◆ euler_vector

template<int dim, typename VectorType = Vector<double>, int spacedim = dim>
 SmartPointer > MappingQEulerian< dim, VectorType, spacedim >::euler_vector
protected

Reference to the vector of shifts.

Definition at line 179 of file mapping_q_eulerian.h.

## ◆ euler_dof_handler

template<int dim, typename VectorType = Vector<double>, int spacedim = dim>
 SmartPointer, MappingQEulerian > MappingQEulerian< dim, VectorType, spacedim >::euler_dof_handler
protected

Pointer to the DoFHandler to which the mapping vector is associated.

Definition at line 186 of file mapping_q_eulerian.h.

## ◆ level

template<int dim, typename VectorType = Vector<double>, int spacedim = dim>
 const unsigned int MappingQEulerian< dim, VectorType, spacedim >::level
private

Multigrid level at which the mapping is to be used.

Definition at line 192 of file mapping_q_eulerian.h.

template<int dim, typename VectorType = Vector<double>, int spacedim = dim>
private

A member variable holding the quadrature points in the right order.

Definition at line 210 of file mapping_q_eulerian.h.

## ◆ fe_values

template<int dim, typename VectorType = Vector<double>, int spacedim = dim>
 FEValues MappingQEulerian< dim, VectorType, spacedim >::fe_values
mutableprivate

FEValues object used to query the given finite element field at the support points in the reference configuration.

The variable is marked as mutable since we have to call FEValues::reinit from compute_mapping_support_points, a function that is 'const'.

Definition at line 220 of file mapping_q_eulerian.h.

## ◆ fe_values_mutex

template<int dim, typename VectorType = Vector<double>, int spacedim = dim>
 Threads::Mutex MappingQEulerian< dim, VectorType, spacedim >::fe_values_mutex
mutableprivate

Definition at line 225 of file mapping_q_eulerian.h.

## ◆ polynomial_degree

 const unsigned int MappingQ< dim, spacedim >::polynomial_degree
protectedinherited

The degree of the polynomials used as shape functions for the mapping of cells.

Definition at line 628 of file mapping_q.h.

## ◆ line_support_points

 const std::vector > MappingQ< dim, spacedim >::line_support_points
protectedinherited

Definition at line 638 of file mapping_q.h.

## ◆ polynomials_1d

 const std::vector > MappingQ< dim, spacedim >::polynomials_1d
protectedinherited

Definition at line 645 of file mapping_q.h.

## ◆ renumber_lexicographic_to_hierarchic

 const std::vector MappingQ< dim, spacedim >::renumber_lexicographic_to_hierarchic
protectedinherited

Definition at line 652 of file mapping_q.h.

## ◆ unit_cell_support_points

 const std::vector > MappingQ< dim, spacedim >::unit_cell_support_points
protectedinherited

Definition at line 664 of file mapping_q.h.

## ◆ support_point_weights_perimeter_to_interior

 const std::vector > MappingQ< dim, spacedim >::support_point_weights_perimeter_to_interior
protectedinherited

A vector of tables of weights by which we multiply the locations of the support points on the perimeter of an object (line, quad, hex) to get the location of interior support points.

Access into this table is by [structdim-1], i.e., use 0 to access the support point weights on a line (i.e., the interior points of the GaussLobatto quadrature), use 1 to access the support point weights from to perimeter to the interior of a quad, and use 2 to access the support point weights from the perimeter to the interior of a hex.

The table itself contains as many columns as there are surrounding points to a particular object (2 for a line, 4 + 4*(degree-1) for a quad, 8 + 12*(degree-1) + 6*(degree-1)*(degree-1) for a hex) and as many rows as there are strictly interior points.

For the definition of this table see equation (8) of the ‘mapping’ report.

Definition at line 686 of file mapping_q.h.

## ◆ support_point_weights_cell

 const Table<2, double> MappingQ< dim, spacedim >::support_point_weights_cell
protectedinherited

A table of weights by which we multiply the locations of the vertex points of the cell to get the location of all additional support points, both on lines, quads, and hexes (as appropriate). This data structure is used when we fill all support points at once, which is the case if the same manifold is attached to all sub-entities of a cell. This way, we can avoid some of the overhead in transforming data for mappings.

The table has as many rows as there are vertices to the cell (2 in 1D, 4 in 2D, 8 in 3D), and as many rows as there are additional support points in the mapping, i.e., (degree+1)^dim - 2^dim.

Definition at line 700 of file mapping_q.h.

## ◆ counter

 std::atomic Subscriptor::counter
mutableprivateinherited

Store the number of objects which subscribed to this object. Initially, this number is zero, and upon destruction it shall be zero again (i.e. all objects which subscribed should have unsubscribed again).

The creator (and owner) of an object is counted in the map below if HE manages to supply identification.

We use the mutable keyword in order to allow subscription to constant objects also.

This counter may be read from and written to concurrently in multithreaded code: hence we use the std::atomic class template.

Definition at line 219 of file subscriptor.h.

## ◆ counter_map

 std::map Subscriptor::counter_map
mutableprivateinherited

In this map, we count subscriptions for each different identification string supplied to subscribe().

Definition at line 225 of file subscriptor.h.

## ◆ validity_pointers

 std::vector *> Subscriptor::validity_pointers
mutableprivateinherited

In this vector, we store pointers to the validity bool in the SmartPointer objects that subscribe to this class.

Definition at line 241 of file subscriptor.h.

## ◆ object_info

 const std::type_info* Subscriptor::object_info
mutableprivateinherited

Pointer to the typeinfo object of this object, from which we can later deduce the class name. Since this information on the derived class is neither available in the destructor, nor in the constructor, we obtain it in between and store it here.

Definition at line 249 of file subscriptor.h.

## ◆ mutex

 std::mutex Subscriptor::mutex
staticprivateinherited

A mutex used to ensure data consistency when printing out the list of subscribers.

Definition at line 271 of file subscriptor.h.

The documentation for this class was generated from the following files: