Reference documentation for deal.II version GIT relicensing1304g0b6601e2da 20240724 19:10:02+00:00

#include <deal.II/fe/mapping_q.h>
Classes  
class  InternalData 
Public Member Functions  
MappingQ (const unsigned int polynomial_degree)  
MappingQ (const unsigned int polynomial_degree, const bool use_mapping_q_on_all_cells)  
MappingQ (const MappingQ< dim, spacedim > &mapping)  
virtual std::unique_ptr< Mapping< dim, spacedim > >  clone () const override 
unsigned int  get_degree () const 
virtual bool  preserves_vertex_locations () const override 
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 
void  fill_mapping_data_for_face_quadrature (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const unsigned int face_number, const Quadrature< dim  1 > &face_quadrature, const typename Mapping< dim, spacedim >::InternalDataBase &internal_data, internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &output_data) const 
virtual boost::container::small_vector< Point< spacedim >, GeometryInfo< dim >::vertices_per_cell >  get_vertices (const typename Triangulation< dim, spacedim >::cell_iterator &cell) const 
boost::container::small_vector< Point< spacedim >, GeometryInfo< dim >::vertices_per_face >  get_vertices (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const unsigned int face_no) const 
virtual Point< spacedim >  get_center (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const bool map_barycenter_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 ::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 std::vector< Point< spacedim > >  compute_mapping_support_points (const typename Triangulation< dim, spacedim >::cell_iterator &cell) const 
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 
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 
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 
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 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 
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  
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 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  
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 
Friends  
template<int , int >  
class  MappingQCache 
This class implements the functionality for polynomial mappings \(Q_p\) of polynomial degree \(p\) that will be used on all cells of the mesh. In order to get a genuine higherorder mapping for all cells, it is important to provide information about how interior edges and faces of the mesh should be curved. This is typically done by associating a Manifold with interior cells and edges. A simple example of this is discussed in the "Results" section of step6; a full discussion of manifolds is provided in step53. If manifolds are only attached to the boundaries of a domain, the current class with higher polynomial degrees will provide the same information as a mere MappingQ1 object. If you are working on meshes that describe a (curved) manifold embedded in higher space dimensions, i.e., if dim!=spacedim, then every cell is at the boundary of the domain you will likely already have attached a manifold object to all cells that can then also be used by the mapping classes for higher order mappings.
For a number of applications, one only knows a manifold description of a surface but not the interior of the computational domain. In such a case, a FlatManifold object will be assigned to the interior entities that describes a usual planar coordinate system where the additional points for the higher order mapping are placed exactly according to a bi/trilinear mapping. When combined with a nonflat manifold on the boundary, for example a circle bulging into the interior of a square cell, the two manifold descriptions are in general incompatible. For example, a FlatManifold defined solely through the cell's vertices would put an interior point located at some small distance epsilon away from the boundary along a straight line and thus in general outside the concave part of a circle. If the polynomial degree of MappingQ is sufficiently high, the transformation from the reference cell to such a cell would in general contain inverted regions close to the boundary.
In order to avoid this situation, this class applies an algorithm for making this transition smooth using a socalled transfinite interpolation that is essentially a linear blend between the descriptions along the surrounding entities. In the algorithm that computes additional points, the compute_mapping_support_points() method, all the entities of the cells are passed through hierarchically, starting from the lines to the quads and finally hexes. Points on objects higher up in the hierarchy are obtained from the manifold associated with that object, taking into account all the points previously computed by the manifolds associated with the lowerdimensional objects, not just the vertices. If only a line is assigned a curved boundary but the adjacent quad is on a flat manifold, the flat manifold on the quad will take the points on the deformed line into account when interpolating the position of the additional points inside the quad and thus always result in a welldefined transformation.
The interpolation scheme used in this class makes sure that curved descriptions can go over to flat descriptions within a single layer of elements, maintaining the overall optimal convergence rates of the finite element interpolation. However, this only helps as long as opposite faces of a cell are far enough away from each other: If a curved part is indeed curved to the extent that it would come close or even intersect some of the other faces, as is often the case with long and sliver cells, the current approach still leads to bad mesh quality. Therefore, the recommended way is to spread the transition between curved boundaries and flat interior domains over a larger range as the mesh is refined. This is provided by the special manifold TransfiniteInterpolationManifold.
Definition at line 109 of file mapping_q.h.

privateinherited 
The data type used in counter_map.
Definition at line 229 of file subscriptor.h.

privateinherited 
The iterator type used in counter_map.
Definition at line 234 of file subscriptor.h.
MappingQ< dim, spacedim >::MappingQ  (  const unsigned int  polynomial_degree  ) 
Constructor. polynomial_degree
denotes the polynomial degree of the polynomials that are used to map cells from the reference to the real cell.
Definition at line 217 of file mapping_q.cc.
MappingQ< dim, spacedim >::MappingQ  (  const unsigned int  polynomial_degree, 
const bool  use_mapping_q_on_all_cells  
) 
The second argument is here for backward compatibility with previous versions of deal.II, but it does not have any effect on the workings of this class.
Definition at line 246 of file mapping_q.cc.
MappingQ< dim, spacedim >::MappingQ  (  const MappingQ< dim, spacedim > &  mapping  ) 
Copy constructor.
Definition at line 253 of file mapping_q.cc.

overridevirtual 
Return a pointer to a copy of the present object. The caller of this copy then assumes ownership of it.
The function is declared abstract virtual in this base class, and derived classes will have to implement it.
This function is mainly used by the hp::MappingCollection class.
Implements Mapping< dim, spacedim >.
Reimplemented in MappingC1< dim, spacedim >, MappingQ1< dim, spacedim >, MappingQ1Eulerian< dim, VectorType, spacedim >, MappingQCache< dim, spacedim >, and MappingQEulerian< dim, VectorType, spacedim >.
Definition at line 269 of file mapping_q.cc.
Return the degree of the mapping, i.e. the value which was passed to the constructor.
Definition at line 278 of file mapping_q.cc.
Always returns true
because the default implementation of functions in this class preserves vertex locations.
Implements Mapping< dim, spacedim >.
Reimplemented in MappingQ1Eulerian< dim, VectorType, spacedim >, MappingQCache< dim, spacedim >, and MappingQEulerian< dim, VectorType, spacedim >.

overridevirtual 
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 1835 of file mapping_q.cc.

overridevirtual 
Returns if this instance of Mapping is compatible with the type of cell in reference_cell
.
Implements Mapping< dim, spacedim >.
Definition at line 1845 of file mapping_q.cc.

overridevirtual 
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 287 of file mapping_q.cc.

overridevirtual 
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 513 of file mapping_q.cc.

overridevirtual 
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 633 of file mapping_q.cc.

overridevirtual 
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 1418 of file mapping_q.cc.

overridevirtual 
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 1434 of file mapping_q.cc.

overridevirtual 
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 1450 of file mapping_q.cc.

overridevirtual 
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 1482 of file mapping_q.cc.

overridevirtual 
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 1534 of file mapping_q.cc.
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 
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 1326 of file mapping_q.cc.
void MappingQ< dim, spacedim >::fill_mapping_data_for_face_quadrature  (  const typename Triangulation< dim, spacedim >::cell_iterator &  cell, 
const unsigned int  face_number,  
const Quadrature< dim  1 > &  face_quadrature,  
const typename Mapping< dim, spacedim >::InternalDataBase &  internal_data,  
internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &  output_data  
)  const 
As opposed to the fill_fe_face_values() function that relies 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]  face_number  The face number where to evaluate the mapping. 
[in]  face_quadrature  The quadrature points where the transformation (Jacobians, positions) should be computed. 
[in]  internal_data  A reference to an object previously created 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 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 1374 of file mapping_q.cc.

overrideprotectedvirtual 
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 topic.
Implements Mapping< dim, spacedim >.
Definition at line 725 of file mapping_q.cc.

overrideprotectedvirtual 
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 precompute 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 topic.
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 781 of file mapping_q.cc.

overrideprotectedvirtual 
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 794 of file mapping_q.cc.

overrideprotectedvirtual 
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 815 of file mapping_q.cc.

overrideprotectedvirtual 
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 topic.
[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 >.
Reimplemented in MappingQ1Eulerian< dim, VectorType, spacedim >, and MappingQEulerian< dim, VectorType, spacedim >.
Definition at line 834 of file mapping_q.cc.

overrideprotectedvirtual 
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 1062 of file mapping_q.cc.

overrideprotectedvirtual 
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 1125 of file mapping_q.cc.

overrideprotectedvirtual 
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 1187 of file mapping_q.cc.

protectedvirtual 
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 in MappingQ1Eulerian< dim, VectorType, spacedim >, MappingQCache< dim, spacedim >, and MappingQEulerian< dim, VectorType, spacedim >.
Definition at line 1736 of file mapping_q.cc.

protected 
Transform the point p
on the real cell to the corresponding point on the unit cell cell
by a Newton iteration.
Definition at line 328 of file mapping_q.cc.

protectedvirtual 
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.
Reimplemented in MappingC1< dim, spacedim >.
Definition at line 1559 of file mapping_q.cc.

protectedvirtual 
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.
Reimplemented in MappingC1< dim, spacedim >.
Definition at line 1725 of file mapping_q.cc.

protected 
Definition at line 342 of file mapping_q.cc.

protected 
Definition at line 374 of file mapping_q.cc.

protected 
Definition at line 404 of file mapping_q.cc.

protected 
Definition at line 434 of file mapping_q.cc.

protected 
Definition at line 467 of file mapping_q.cc.

protected 
Definition at line 500 of file mapping_q.cc.

protected 
Definition at line 1626 of file mapping_q.cc.

protected 
Definition at line 1695 of file mapping_q.cc.

virtualinherited 
Return the mapped vertices of a cell.
Most of the time, these values will simply be the coordinates of the vertices of a cell as returned by cell>vertex(v)
for vertex v
, i.e., information stored by the triangulation. However, there are also mappings that add displacements or choose completely different locations, e.g., MappingQEulerian, MappingQ1Eulerian, or MappingFEField.
The default implementation of this function simply returns the information stored by the triangulation, i.e., cell>vertex(v)
.
Reimplemented in MappingFEField< dim, spacedim, VectorType >, MappingQ1Eulerian< dim, VectorType, spacedim >, MappingQCache< dim, spacedim >, and MappingQEulerian< dim, VectorType, spacedim >.

inherited 
Return the mapped vertices of a face.
Same as above but working on a given face of a cell.
[in]  cell  The cell containing the face. 
[in]  face_no  The number of the face within the cell. 

virtualinherited 
Return one of two possible mapped centers 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 barycenter of the reference cell. If the parameter map_barycenter_of_reference_cell
is set to false, then the returned 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_barycenter_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. 

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.

protectedvirtualinherited 

protectedvirtualinherited 

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 135 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 155 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 300 of file subscriptor.h.

inlineinherited 
List the subscribers to the input stream
.
Definition at line 317 of file subscriptor.h.

inherited 
List the subscribers to deallog
.
Definition at line 203 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 309 of file subscriptor.h.

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.
Definition at line 52 of file subscriptor.cc.
Definition at line 681 of file mapping_q.h.
The degree of the polynomials used as shape functions for the mapping of cells.
Definition at line 524 of file mapping_q.h.
Definition at line 534 of file mapping_q.h.

protected 
Definition at line 541 of file mapping_q.h.

protected 
Definition at line 548 of file mapping_q.h.
Definition at line 560 of file mapping_q.h.

protected 
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 582 of file mapping_q.h.
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 596 of file mapping_q.h.

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 218 of file subscriptor.h.

mutableprivateinherited 
In this map, we count subscriptions for each different identification string supplied to subscribe().
Definition at line 224 of file subscriptor.h.

mutableprivateinherited 
In this vector, we store pointers to the validity bool in the SmartPointer objects that subscribe to this class.
Definition at line 240 of file subscriptor.h.

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 248 of file subscriptor.h.

staticprivateinherited 
A mutex used to ensure data consistency when accessing the mutable
members of this class. This lock is used in the subscribe() and unsubscribe() functions, as well as in list_subscribers()
.
Definition at line 271 of file subscriptor.h.