Reference documentation for deal.II version Git 5a2787e538 20210921 14:55:10 0600

#include <deal.II/fe/mapping_q_eulerian.h>
Classes  
class  SupportQuadrature 
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 
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 
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 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 
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 
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 
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 ::ExceptionBase &  ExcInactiveCell () 
static ::ExceptionBase &  ExcInUse (int arg1, std::string arg2, std::string arg3) 
static ::ExceptionBase &  ExcNoSubscriber (std::string arg1, std::string arg2) 
Exceptions  
static ::ExceptionBase &  ExcInvalidData () 
static ::ExceptionBase &  ExcTransformationFailed () 
static ::ExceptionBase &  ExcDistortedMappedCell (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 
template<>  
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 
template<>  
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 
template<>  
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 
template<>  
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 
template<>  
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 
template<>  
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 
template<>  
void  add_quad_support_points (const Triangulation< 3, 3 >::cell_iterator &cell, std::vector< Point< 3 >> &a) const 
template<>  
void  add_quad_support_points (const Triangulation< 2, 3 >::cell_iterator &cell, std::vector< Point< 3 >> &a) const 
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, double >  support_point_weights_cell 
Private Attributes  
const unsigned int  level 
const SupportQuadrature  support_quadrature 
FEValues< dim, spacedim >  fe_values 
Threads::Mutex  fe_values_mutex 
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.
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:
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 93 of file mapping_q_eulerian.h.
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.
[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 multigrid 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 49 of file mapping_q_eulerian.cc.

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 106 of file mapping_q_eulerian.cc.

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, spacedim >.
Definition at line 68 of file mapping_q_eulerian.cc.

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, spacedim >.

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 lowerdimensional 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, spacedim >.
Definition at line 124 of file mapping_q_eulerian.cc.

overrideprotectedvirtual 
Compute mappingrelated 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.
Definition at line 197 of file mapping_q_eulerian.cc.

inherited 
Return the degree of the mapping, i.e. the value which was passed to the constructor.
Definition at line 444 of file mapping_q.cc.

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.
[in]  cell  The cell for which you want to compute the bounding box 
Reimplemented from Mapping< dim, spacedim >.
Definition at line 1779 of file mapping_q.cc.

overridevirtualinherited 
Returns if this instance of Mapping is compatible with the type of cell in reference_cell
.
Implements Mapping< dim, spacedim >.
Definition at line 1789 of file mapping_q.cc.

overridevirtualinherited 
Map the point p
on the unit cell to the corresponding point on the real cell cell
.
cell  Iterator to the cell that will be used to define the mapping. 
p  Location of a point on the reference cell. 
Implements Mapping< dim, spacedim >.
Definition at line 453 of file mapping_q.cc.

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
.
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.cell  Iterator to the cell that will be used to define the mapping. 
p  Location of a point on the given cell. 
Implements Mapping< dim, spacedim >.
Definition at line 631 of file mapping_q.cc.

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 752 of file mapping_q.cc.

overridevirtualinherited 
Transform a field of vectors or 1differential forms according to the selected MappingKind.
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 oneforms on the reference cell to a field of oneforms 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 dim1forms on the reference cell is also represented by a vector field, but again transforms differently, namely by the Piola transform \[ \mathbf u(\mathbf x) = \frac{1}{\text{det}\;J(\hat{\mathbf x})} J(\hat{\mathbf x}) \hat{\mathbf u}(\hat{\mathbf x}). \]
[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 1365 of file mapping_q.cc.

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 spacedimvector valued differentiable functions are transformed this way.
In the case when dim=spacedim the previous formula reduces to
\[ \mathbf T(\mathbf x) = \hat{\mathbf u}(\hat{\mathbf x}) J(\hat{\mathbf x})^{1}. \]
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.[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 1381 of file mapping_q.cc.

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 \[ \mathbf T(\mathbf x) = \frac{1}{\text{det}\;J(\hat{\mathbf x})} J(\hat{\mathbf x}) \hat{\mathbf T}(\hat{\mathbf x}) J(\hat{\mathbf x})^{1}. \]
[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 1397 of file mapping_q.cc.

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 spacedimvector 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
\[J^{\dagger} = J^{1}\]
[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 1429 of file mapping_q.cc.

overridevirtualinherited 
Transform a field of 3differential 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 \[ \mathbf T_{ijk}(\mathbf x) = \frac{1}{\text{det}\;J(\hat{\mathbf x})} J_{iI}(\hat{\mathbf x}) \hat{\mathbf T}_{IJK}(\hat{\mathbf x}) J_{jJ}(\hat{\mathbf x})^{1} J_{kK}(\hat{\mathbf x})^{1}. \]
[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 1478 of file mapping_q.cc.

overridevirtualinherited 
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.
Implements Mapping< dim, spacedim >.
Definition at line 834 of file mapping_q.cc.

overridevirtualinherited 
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 precompute 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.
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. 
Implements Mapping< dim, spacedim >.
Definition at line 890 of file mapping_q.cc.

overridevirtualinherited 
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.
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. 
Reimplemented from Mapping< dim, spacedim >.
Definition at line 905 of file mapping_q.cc.

protectedvirtualinherited 

overridevirtualinherited 
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.
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. 
Implements Mapping< dim, spacedim >.
Definition at line 926 of file mapping_q.cc.

overridevirtualinherited 
Compute information about the mapping from the reference cell to the real cell indicated by the first argument to this function. Derived classes will have to implement this function based on the kind of mapping they represent. It is called by FEValues::reinit().
Conceptually, this function's represents the application of the mapping \(\mathbf x=\mathbf F_K(\hat {\mathbf x})\) from reference coordinates \(\mathbf\in [0,1]^d\) to real space coordinates \(\mathbf x\) for a given cell \(K\). Its purpose is to compute the following kinds of data:
The information computed by this function is used to fill the various member variables of the output argument of this function. Which of the member variables of that structure should be filled is determined by the update flags stored in the Mapping::InternalDataBase object passed to this function.
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.
[in]  cell  The cell of the triangulation for which this function is to compute a mapping from the reference cell to. 
[in]  cell_similarity  Whether or not the cell given as first argument is simply a translation, rotation, etc of the cell for which this function was called the most recent time. This information is computed simply by matching the vertices (as stored by the Triangulation) between the previous and the current cell. The value passed here may be modified by implementations of this function and should then be returned (see the discussion of the return value of this function). 
[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. 
cell_similarity
argument to this function. The returned value will be used for the corresponding argument when FEValues::reinit() calls FiniteElement::fill_fe_values(). In most cases, derived classes will simply want to return the value passed for cell_similarity
. However, implementations of this function may downgrade the level of cell similarity. This is, for example, the case for classes that take not only into account the locations of the vertices of a cell (as reported by the Triangulation), but also other information specific to the mapping. The purpose is that FEValues::reinit() can compute whether a cell is similar to the previous one only based on the cell's vertices, whereas the mapping may also consider displacement fields (e.g., in the MappingQ1Eulerian and MappingFEField classes). In such cases, the mapping may conclude that the previously computed cell similarity is too optimistic, and invalidate it for subsequent use in FiniteElement::fill_fe_values() by returning a less optimistic cell similarity value.internal_data
and output_data
objects. In other words, if an implementation of this function knows that it has written a piece of data into the output argument in a previous call, then there is no need to copy it there again in a later call if the implementation knows that this is the same value. Implements Mapping< dim, spacedim >.
Definition at line 945 of file mapping_q.cc.

overridevirtualinherited 
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().
[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 1165 of file mapping_q.cc.

protectedvirtualinherited 

overridevirtualinherited 
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().
[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 1214 of file mapping_q.cc.

inherited 
As opposed to the other fill_fe_values() and fill_fe_face_values() functions that rely on precomputed information of InternalDataBase, this function chooses the flexible evaluation path on the cell and points passed in to the current function.
[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 1264 of file mapping_q.cc.

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 488 of file mapping_q.cc.

protectedinherited 
Definition at line 502 of file mapping_q.cc.

protectedinherited 
Definition at line 522 of file mapping_q.cc.

protectedinherited 
Definition at line 540 of file mapping_q.cc.

protectedinherited 
Definition at line 558 of file mapping_q.cc.

protectedinherited 
Definition at line 589 of file mapping_q.cc.

protectedinherited 
Definition at line 618 of file mapping_q.cc.

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 1503 of file mapping_q.cc.

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 1669 of file mapping_q.cc.

protectedinherited 
Definition at line 1570 of file mapping_q.cc.

protectedinherited 
Definition at line 1639 of file mapping_q.cc.

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.
[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. 
Definition at line 49 of file mapping.cc.

inherited 
Transform the point p
on the real cell
to the corresponding point on the reference cell, and then project this point to a (dim1)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.
Definition at line 111 of file mapping.cc.

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.

inherited 
Unsubscribes a user from the object.
identifier
and the validity
pointer must be the same as the one supplied to subscribe(). Definition at line 156 of file subscriptor.cc.

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 301 of file subscriptor.h.

inlineinherited 
List the subscribers to the input stream
.
Definition at line 318 of file subscriptor.h.

inherited 
List the subscribers to deallog
.
Definition at line 204 of file subscriptor.cc.

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 310 of file subscriptor.h.

protected 
Reference to the vector of shifts.
Definition at line 177 of file mapping_q_eulerian.h.

protected 
Pointer to the DoFHandler to which the mapping vector is associated.
Definition at line 184 of file mapping_q_eulerian.h.

private 
Multigrid level at which the mapping is to be used.
Definition at line 190 of file mapping_q_eulerian.h.

private 
A member variable holding the quadrature points in the right order.
Definition at line 208 of file mapping_q_eulerian.h.

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 218 of file mapping_q_eulerian.h.

mutableprivate 
A variable to guard access to the fe_values variable.
Definition at line 223 of file mapping_q_eulerian.h.

protectedinherited 
The degree of the polynomials used as shape functions for the mapping of cells.
Definition at line 628 of file mapping_q.h.

protectedinherited 
Definition at line 638 of file mapping_q.h.

protectedinherited 
Definition at line 645 of file mapping_q.h.

protectedinherited 
Definition at line 652 of file mapping_q.h.

protectedinherited 
Definition at line 664 of file mapping_q.h.

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 [structdim1], 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*(degree1)
for a quad, 8 + 12*(degree1) + 6*(degree1)*(degree1)
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.

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 subentities 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.