Reference documentation for deal.II version GIT relicensing-233-g802318d791 2024-03-28 20:20:02+00:00
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Classes | Public Member Functions | Static Public Member Functions | Private Types | Private Member Functions | Private Attributes | Static Private Attributes | Friends | List of all members
Mapping< dim, spacedim > Class Template Referenceabstract

Abstract base class for mapping classes. More...

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

Inheritance diagram for Mapping< dim, spacedim >:
Inheritance graph
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Classes

class  InternalDataBase
 

Public Member Functions

virtual ~Mapping () override=default
 
virtual std::unique_ptr< Mapping< dim, spacedim > > clone () const =0
 
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
 
virtual BoundingBox< spacedim > get_bounding_box (const typename Triangulation< dim, spacedim >::cell_iterator &cell) const
 
virtual bool preserves_vertex_locations () const =0
 
virtual bool is_compatible_with (const ReferenceCell &reference_cell) const =0
 
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 =0
 
virtual Point< dim > transform_real_to_unit_cell (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const Point< spacedim > &p) const =0
 
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
 
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
 
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 =0
 
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 =0
 
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 =0
 
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 =0
 
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 =0
 
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 ::ExceptionBaseExcInUse (int arg1, std::string arg2, std::string arg3)
 
static ::ExceptionBaseExcNoSubscriber (std::string arg1, std::string arg2)
 
Exceptions
static ::ExceptionBaseExcInvalidData ()
 
static ::ExceptionBaseExcTransformationFailed ()
 
static ::ExceptionBaseExcDistortedMappedCell (Point< spacedim > arg1, double arg2, int arg3)
 

Protected Member Functions

Interface with FEValues
virtual UpdateFlags requires_update_flags (const UpdateFlags update_flags) const =0
 
virtual std::unique_ptr< InternalDataBaseget_data (const UpdateFlags update_flags, const Quadrature< dim > &quadrature) const =0
 
virtual std::unique_ptr< InternalDataBaseget_face_data (const UpdateFlags update_flags, const hp::QCollection< dim - 1 > &quadrature) const
 
virtual std::unique_ptr< InternalDataBaseget_face_data (const UpdateFlags update_flags, const Quadrature< dim - 1 > &quadrature) const
 
virtual std::unique_ptr< InternalDataBaseget_subface_data (const UpdateFlags update_flags, const Quadrature< dim - 1 > &quadrature) const =0
 
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 =0
 
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
 
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
 
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 =0
 
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
 

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 intcounter
 
std::map< std::string, unsigned intcounter_map
 
std::vector< std::atomic< bool > * > validity_pointers
 
const std::type_info * object_info
 

Static Private Attributes

static std::mutex mutex
 

Friends

class FEValuesBase< dim, spacedim >
 
class FEValues< dim, spacedim >
 
class FEFaceValues< dim, spacedim >
 
class FESubfaceValues< dim, spacedim >
 
class NonMatching::FEImmersedSurfaceValues< dim >
 
class NonMatching::internal::ComputeMappingDataHelper< dim, spacedim >
 

Detailed Description

template<int dim, int spacedim = dim>
class Mapping< dim, spacedim >

Abstract base class for mapping classes.

This class declares the interface for the functionality to describe mappings from the reference (unit) cell to a cell in real space, as well as for filling the information necessary to use the FEValues, FEFaceValues, and FESubfaceValues classes. Concrete implementations of these interfaces are provided in derived classes.

Mathematics of the mapping

The mapping is a transformation \(\mathbf x = \mathbf F_K(\hat{\mathbf x})\) which maps points \(\hat{\mathbf x}\) in the reference cell \([0,1]^\text{dim}\) to points \(\mathbf x\) in the actual grid cell \(K\subset{\mathbb R}^\text{spacedim}\). Many of the applications of such mappings require the Jacobian of this mapping, \(J(\hat{\mathbf x}) = \hat\nabla {\mathbf F}_K(\hat{\mathbf x})\). For instance, if dim=spacedim=2, we have

\[ J(\hat{\mathbf x}) = \left(\begin{matrix} \frac{\partial x}{\partial \hat x} & \frac{\partial x}{\partial \hat y} \\ \frac{\partial y}{\partial \hat x} & \frac{\partial y}{\partial \hat y} \end{matrix}\right) \]

Mapping of scalar functions

The shape functions of scalar finite elements are typically defined on a reference cell and are then simply mapped according to the rule

\[ \varphi(\mathbf x) = \varphi\bigl(\mathbf F_K(\hat{\mathbf x})\bigr) = \hat \varphi(\hat{\mathbf x}). \]

Mapping of integrals

Using simply a change of variables, integrals of scalar functions over a cell \(K\) can be expressed as an integral over the reference cell \(\hat K\). Specifically, The volume form \(d\hat x\) is transformed so that

\[ \int_K u(\mathbf x)\,dx = \int_{\hat K} \hat u(\hat{\mathbf x}) \left|\text{det}J(\hat{\mathbf x})\right| \,d\hat x. \]

In expressions where such integrals are approximated by quadrature, this then leads to terms of the form

\[ \int_K u(\mathbf x)\,dx \approx \sum_{q} \hat u(\hat{\mathbf x}_q) \underbrace{\left|\text{det}J(\hat{\mathbf x}_q)\right| w_q}_{=: \text{JxW}_q}. \]

Here, the weights \(\text{JxW}_q\) of each quadrature point (where JxW mnemonically stands for Jacobian times Quadrature Weights) take the role of the \(dx\) in the original integral. Consequently, they appear in all code that computes integrals approximated by quadrature, and are accessed by FEValues::JxW().

Todo:
Document what happens in the codimension-1 case.

Mapping of vector fields, differential forms and gradients of vector fields

The transformation of vector fields or differential forms (gradients of scalar functions) \(\mathbf v\), and gradients of vector fields \(\mathbf T\) follows the general form

\[ \mathbf v(\mathbf x) = \mathbf A(\hat{\mathbf x}) \hat{\mathbf v}(\hat{\mathbf x}), \qquad \mathbf T(\mathbf x) = \mathbf A(\hat{\mathbf x}) \hat{\mathbf T}(\hat{\mathbf x}) \mathbf B(\hat{\mathbf x}). \]

The differential forms A and B are determined by the kind of object being transformed. These transformations are performed through the transform() functions, and the type of object being transformed is specified by their MappingKind argument. See the documentation there for possible choices.

Derivatives of the mapping

Some applications require the derivatives of the mapping, of which the first order derivative is the mapping Jacobian, \(J_{iJ}(\hat{\mathbf x})=\frac{\partial x_i}{\partial \hat x_J}\), described above. Higher order derivatives of the mapping are similarly defined, for example the Jacobian derivative, \(\hat H_{iJK}(\hat{\mathbf x}) = \frac{\partial^2 x_i}{\partial \hat x_J \partial \hat x_K}\), and the Jacobian second derivative, \(\hat K_{iJKL}(\hat{\mathbf x}) = \frac{\partial^3 x_i}{\partial \hat x_J \partial \hat x_K \partial \hat x_L}\). It is also useful to define the "pushed-forward" versions of the higher order derivatives: the Jacobian pushed-forward derivative, \(H_{ijk}(\hat{\mathbf x}) = \frac{\partial^2 x_i}{\partial \hat x_J \partial \hat x_K}(J_{jJ})^{-1}(J_{kK})^{-1}\), and the Jacobian pushed-forward second derivative, \(K_{ijkl}(\hat{\mathbf x}) = \frac{\partial^3 x_i}{\partial \hat x_J \partial \hat x_K \partial \hat x_L}(J_{jJ})^{-1}(J_{kK})^{-1}(J_{lL})^{-1}\). These pushed-forward versions can be used to compute the higher order derivatives of functions defined on the reference cell with respect to the real cell coordinates. For instance, the Jacobian derivative with respect to the real cell coordinates is given by:

\[ \frac{\partial}{\partial x_j}\left[J_{iJ}(\hat{\mathbf x})\right] = H_{ikn}(\hat{\mathbf x})J_{nJ}(\hat{\mathbf x}), \]

and the derivative of the Jacobian inverse with respect to the real cell coordinates is similarly given by:

\[ \frac{\partial}{\partial x_j}\left[\left(J_{iJ}(\hat{\mathbf x})\right)^{-1}\right] = -H_{nik}(\hat{\mathbf x})\left(J_{nJ}(\hat{\mathbf x})\right)^{-1}. \]

In a similar fashion, higher order derivatives, with respect to the real cell coordinates, of functions defined on the reference cell can be defined using the Jacobian pushed-forward higher-order derivatives. For example, the derivative, with respect to the real cell coordinates, of the Jacobian pushed-forward derivative is given by:

\[ \frac{\partial}{\partial x_l}\left[H_{ijk}(\hat{\mathbf x})\right] = K_{ijkl}(\hat{\mathbf x}) -H_{mjl}(\hat{\mathbf x})H_{imk}(\hat{\mathbf x})-H_{mkl}(\hat{\mathbf x})H_{imj}(\hat{\mathbf x}). \]

References

A general publication on differential geometry and finite elements is the survey

The description of the Piola transform has been taken from the lecture notes by Ronald H. W. Hoppe, University of Houston, Chapter 7.

Definition at line 315 of file mapping.h.

Member Typedef Documentation

◆ map_value_type

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

The data type used in counter_map.

Definition at line 229 of file subscriptor.h.

◆ map_iterator

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

The iterator type used in counter_map.

Definition at line 234 of file subscriptor.h.

Constructor & Destructor Documentation

◆ ~Mapping()

template<int dim, int spacedim = dim>
virtual Mapping< dim, spacedim >::~Mapping ( )
overridevirtualdefault

Virtual destructor.

Member Function Documentation

◆ clone()

template<int dim, int spacedim = dim>
virtual std::unique_ptr< Mapping< dim, spacedim > > Mapping< dim, spacedim >::clone ( ) const
pure virtual

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.

Implemented in MappingC1< dim, spacedim >, MappingCartesian< dim, spacedim >, MappingFE< dim, spacedim >, MappingFEField< dim, spacedim, VectorType >, MappingManifold< dim, spacedim >, MappingQ< dim, spacedim >, MappingQ< dim, dim >, MappingQ1< dim, spacedim >, MappingQ1Eulerian< dim, VectorType, spacedim >, MappingQCache< dim, spacedim >, and MappingQEulerian< dim, VectorType, spacedim >.

◆ get_vertices() [1/2]

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

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 >.

◆ get_vertices() [2/2]

template<int dim, int spacedim = dim>
boost::container::small_vector< Point< spacedim >, GeometryInfo< dim >::vertices_per_face > Mapping< dim, spacedim >::get_vertices ( const typename Triangulation< dim, spacedim >::cell_iterator &  cell,
const unsigned int  face_no 
) const

Return the mapped vertices of a face.

Same as above but working on a given face of a cell.

Parameters
[in]cellThe cell containing the face.
[in]face_noThe number of the face within the cell.

◆ get_center()

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

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.

Parameters
[in]cellThe cell for which you want to compute the center
[in]map_barycenter_of_reference_cellA 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.

◆ get_bounding_box()

template<int dim, int spacedim = dim>
virtual BoundingBox< spacedim > Mapping< dim, spacedim >::get_bounding_box ( const typename Triangulation< dim, spacedim >::cell_iterator &  cell) const
virtual

Return the bounding box of a mapped cell.

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

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

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

Reimplemented in MappingFE< dim, spacedim >, MappingQ< dim, spacedim >, and MappingQ< dim, dim >.

◆ preserves_vertex_locations()

template<int dim, int spacedim = dim>
virtual bool Mapping< dim, spacedim >::preserves_vertex_locations ( ) const
pure virtual

Return whether the mapping preserves vertex locations. In other words, this function returns whether the mapped location of the reference cell vertices (given by GeometryInfo::unit_cell_vertex()) equals the result of cell->vertex() (i.e., information stored by the triangulation).

For example, implementations in derived classes return true for MappingQ, MappingCartesian, but false for MappingQEulerian, MappingQ1Eulerian, and MappingFEField.

Implemented in MappingCartesian< dim, spacedim >, MappingFE< dim, spacedim >, MappingFEField< dim, spacedim, VectorType >, MappingManifold< dim, spacedim >, MappingQ< dim, spacedim >, MappingQ< dim, dim >, MappingQ1Eulerian< dim, VectorType, spacedim >, MappingQCache< dim, spacedim >, and MappingQEulerian< dim, VectorType, spacedim >.

◆ is_compatible_with()

template<int dim, int spacedim = dim>
virtual bool Mapping< dim, spacedim >::is_compatible_with ( const ReferenceCell reference_cell) const
pure virtual

◆ transform_unit_to_real_cell()

template<int dim, int spacedim = dim>
virtual Point< spacedim > Mapping< dim, spacedim >::transform_unit_to_real_cell ( const typename Triangulation< dim, spacedim >::cell_iterator &  cell,
const Point< dim > &  p 
) const
pure virtual

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

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

Implemented in MappingCartesian< dim, spacedim >, MappingFE< dim, spacedim >, MappingFEField< dim, spacedim, VectorType >, MappingManifold< dim, spacedim >, MappingQ< dim, spacedim >, and MappingQ< dim, dim >.

◆ transform_real_to_unit_cell()

template<int dim, int spacedim = dim>
virtual Point< dim > Mapping< dim, spacedim >::transform_real_to_unit_cell ( const typename Triangulation< dim, spacedim >::cell_iterator &  cell,
const Point< spacedim > &  p 
) const
pure virtual

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

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

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

Implemented in MappingCartesian< dim, spacedim >, MappingFE< dim, spacedim >, MappingFEField< dim, spacedim, VectorType >, MappingManifold< dim, spacedim >, MappingQ< dim, spacedim >, and MappingQ< dim, dim >.

◆ transform_points_real_to_unit_cell()

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

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 in MappingQ< dim, spacedim >, and MappingQ< dim, dim >.

◆ project_real_point_to_unit_point_on_face()

template<int dim, int spacedim = dim>
Point< dim - 1 > Mapping< dim, spacedim >::project_real_point_to_unit_point_on_face ( const typename Triangulation< dim, spacedim >::cell_iterator &  cell,
const unsigned int  face_no,
const Point< spacedim > &  p 
) const

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

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

◆ requires_update_flags()

template<int dim, int spacedim = dim>
virtual UpdateFlags Mapping< dim, spacedim >::requires_update_flags ( const UpdateFlags  update_flags) const
protectedpure virtual

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.

See also
The interplay of UpdateFlags, Mapping, and FiniteElement in FEValues

Implemented in MappingCartesian< dim, spacedim >, MappingFE< dim, spacedim >, MappingFEField< dim, spacedim, VectorType >, MappingManifold< dim, spacedim >, MappingQ< dim, spacedim >, and MappingQ< dim, dim >.

◆ get_data()

template<int dim, int spacedim = dim>
virtual std::unique_ptr< InternalDataBase > Mapping< dim, spacedim >::get_data ( const UpdateFlags  update_flags,
const Quadrature< dim > &  quadrature 
) const
protectedpure virtual

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

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

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

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

Parameters
update_flagsA 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.
quadratureThe quadrature object for which mapping information will have to be computed. This includes the locations and weights of quadrature points.
Returns
A pointer to a newly created object of type InternalDataBase (or a derived class). Ownership of this object passes to the calling function.
Note
C++ allows that virtual functions in derived classes may return pointers to objects not of type InternalDataBase but in fact pointers to objects of classes derived from InternalDataBase. (This feature is called "covariant return types".) This is useful in some contexts where the calling is within the derived class and will immediately make use of the returned object, knowing its real (derived) type.

Implemented in MappingCartesian< dim, spacedim >, MappingFE< dim, spacedim >, MappingFEField< dim, spacedim, VectorType >, MappingManifold< dim, spacedim >, MappingQ< dim, spacedim >, and MappingQ< dim, dim >.

◆ get_face_data() [1/2]

template<int dim, int spacedim = dim>
virtual std::unique_ptr< InternalDataBase > Mapping< dim, spacedim >::get_face_data ( const UpdateFlags  update_flags,
const hp::QCollection< dim - 1 > &  quadrature 
) const
protectedvirtual

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

Parameters
update_flagsA 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.
quadratureThe quadrature object for which mapping information will have to be computed. This includes the locations and weights of quadrature points.
Returns
A pointer to a newly created object of type InternalDataBase (or a derived class). Ownership of this object passes to the calling function.
Note
C++ allows that virtual functions in derived classes may return pointers to objects not of type InternalDataBase but in fact pointers to objects of classes derived from InternalDataBase. (This feature is called "covariant return types".) This is useful in some contexts where the calling is within the derived class and will immediately make use of the returned object, knowing its real (derived) type.

Reimplemented in MappingCartesian< dim, spacedim >, MappingFE< dim, spacedim >, MappingFEField< dim, spacedim, VectorType >, MappingManifold< dim, spacedim >, MappingQ< dim, spacedim >, and MappingQ< dim, dim >.

◆ get_face_data() [2/2]

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

◆ get_subface_data()

template<int dim, int spacedim = dim>
virtual std::unique_ptr< InternalDataBase > Mapping< dim, spacedim >::get_subface_data ( const UpdateFlags  update_flags,
const Quadrature< dim - 1 > &  quadrature 
) const
protectedpure virtual

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

Parameters
update_flagsA 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.
quadratureThe quadrature object for which mapping information will have to be computed. This includes the locations and weights of quadrature points.
Returns
A pointer to a newly created object of type InternalDataBase (or a derived class). Ownership of this object passes to the calling function.
Note
C++ allows that virtual functions in derived classes may return pointers to objects not of type InternalDataBase but in fact pointers to objects of classes derived from InternalDataBase. (This feature is called "covariant return types".) This is useful in some contexts where the calling is within the derived class and will immediately make use of the returned object, knowing its real (derived) type.

Implemented in MappingCartesian< dim, spacedim >, MappingFE< dim, spacedim >, MappingFEField< dim, spacedim, VectorType >, MappingManifold< dim, spacedim >, MappingQ< dim, spacedim >, and MappingQ< dim, dim >.

◆ fill_fe_values()

template<int dim, int spacedim = dim>
virtual CellSimilarity::Similarity Mapping< dim, spacedim >::fill_fe_values ( const typename Triangulation< dim, spacedim >::cell_iterator &  cell,
const CellSimilarity::Similarity  cell_similarity,
const Quadrature< dim > &  quadrature,
const typename Mapping< dim, spacedim >::InternalDataBase &  internal_data,
internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &  output_data 
) const
protectedpure virtual

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:

  • Data that results from the application of the mapping itself, e.g., computing the location \(\mathbf x_q = \mathbf F_K(\hat{\mathbf x}_q)\) of quadrature points on the real cell, and that is directly useful to users of FEValues, for example during assembly.
  • Data that is necessary for finite element implementations to compute their shape functions on the real cell. To this end, the FEValues::reinit() function calls FiniteElement::fill_fe_values() after the current function, and the output of this function serves as input to FiniteElement::fill_fe_values(). Examples of information that needs to be computed here for use by the finite element classes is the Jacobian of the mapping, \(\hat\nabla \mathbf F_K(\hat{\mathbf x})\) or its inverse, for example to transform the gradients of shape functions on the reference cell to the gradients of shape functions on the real cell.

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.

Parameters
[in]cellThe cell of the triangulation for which this function is to compute a mapping from the reference cell to.
[in]cell_similarityWhether 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]quadratureA 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_dataA 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_dataA 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.
Returns
An updated value of the 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.
Note
FEValues ensures that this function is always called with the same pair of 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.

Implemented in MappingCartesian< dim, spacedim >, MappingFE< dim, spacedim >, MappingFEField< dim, spacedim, VectorType >, MappingManifold< dim, spacedim >, MappingQ< dim, spacedim >, MappingQ< dim, dim >, MappingQ1Eulerian< dim, VectorType, spacedim >, and MappingQEulerian< dim, VectorType, spacedim >.

◆ fill_fe_face_values() [1/2]

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

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

Parameters
[in]cellThe cell of the triangulation for which this function is to compute a mapping from the reference cell to.
[in]face_noThe number of the face of the given cell for which information is requested.
[in]quadratureA 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_dataA 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_dataA 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 in MappingCartesian< dim, spacedim >, MappingFE< dim, spacedim >, MappingFEField< dim, spacedim, VectorType >, MappingManifold< dim, spacedim >, MappingQ< dim, spacedim >, and MappingQ< dim, dim >.

◆ fill_fe_face_values() [2/2]

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

◆ fill_fe_subface_values()

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

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

Parameters
[in]cellThe cell of the triangulation for which this function is to compute a mapping from the reference cell to.
[in]face_noThe number of the face of the given cell for which information is requested.
[in]subface_noThe number of the child of a face of the given cell for which information is requested.
[in]quadratureA 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_dataA 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_dataA 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.

Implemented in MappingCartesian< dim, spacedim >, MappingFE< dim, spacedim >, MappingFEField< dim, spacedim, VectorType >, MappingManifold< dim, spacedim >, MappingQ< dim, spacedim >, and MappingQ< dim, dim >.

◆ fill_fe_immersed_surface_values()

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

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 in MappingCartesian< dim, spacedim >, MappingFEField< dim, spacedim, VectorType >, MappingQ< dim, spacedim >, and MappingQ< dim, dim >.

◆ transform() [1/5]

template<int dim, int spacedim = dim>
virtual void Mapping< dim, spacedim >::transform ( const ArrayView< const Tensor< 1, dim > > &  input,
const MappingKind  kind,
const typename Mapping< dim, spacedim >::InternalDataBase &  internal,
const ArrayView< Tensor< 1, spacedim > > &  output 
) const
pure virtual

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

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

The mapping kinds currently implemented by derived classes are:

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

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

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

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

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

    Gradients of scalar differentiable functions are transformed this way.

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

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

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

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

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

Parameters
[in]inputAn array (or part of an array) of input objects that should be mapped.
[in]kindThe kind of mapping to be applied.
[in]internalA 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]outputAn 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.)

Implemented in MappingCartesian< dim, spacedim >, MappingFE< dim, spacedim >, MappingFEField< dim, spacedim, VectorType >, MappingManifold< dim, spacedim >, MappingQ< dim, spacedim >, and MappingQ< dim, dim >.

◆ transform() [2/5]

template<int dim, int spacedim = dim>
virtual void Mapping< dim, spacedim >::transform ( const ArrayView< const DerivativeForm< 1, dim, spacedim > > &  input,
const MappingKind  kind,
const typename Mapping< dim, spacedim >::InternalDataBase &  internal,
const ArrayView< Tensor< 2, spacedim > > &  output 
) const
pure virtual

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

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

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

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

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

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

Note
It would have been more reasonable to make this transform a template function with the rank in DerivativeForm<1, dim, rank>. Unfortunately C++ does not allow templatized virtual functions. This is why we identify DerivativeForm<1, dim, 1> with a Tensor<1,dim> when using mapping_covariant() in the function transform() above this one.
Parameters
[in]inputAn array (or part of an array) of input objects that should be mapped.
[in]kindThe kind of mapping to be applied.
[in]internalA 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]outputAn 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.)

Implemented in MappingCartesian< dim, spacedim >, MappingFE< dim, spacedim >, MappingFEField< dim, spacedim, VectorType >, MappingManifold< dim, spacedim >, MappingQ< dim, spacedim >, and MappingQ< dim, dim >.

◆ transform() [3/5]

template<int dim, int spacedim = dim>
virtual void Mapping< dim, spacedim >::transform ( const ArrayView< const Tensor< 2, dim > > &  input,
const MappingKind  kind,
const typename Mapping< dim, spacedim >::InternalDataBase &  internal,
const ArrayView< Tensor< 2, spacedim > > &  output 
) const
pure virtual

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

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

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

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

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

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

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

Todo:
The formulas for mapping_covariant_gradient, mapping_contravariant_gradient and mapping_piola_gradient are only true as stated for linear mappings. If, for example, the mapping is bilinear (or has a higher order polynomial degree) then there is a missing term associated with the derivative of \(J\).
Parameters
[in]inputAn array (or part of an array) of input objects that should be mapped.
[in]kindThe kind of mapping to be applied.
[in]internalA 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]outputAn 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.)

Implemented in MappingCartesian< dim, spacedim >, MappingFE< dim, spacedim >, MappingFEField< dim, spacedim, VectorType >, MappingManifold< dim, spacedim >, MappingQ< dim, spacedim >, and MappingQ< dim, dim >.

◆ transform() [4/5]

template<int dim, int spacedim = dim>
virtual void Mapping< dim, spacedim >::transform ( const ArrayView< const DerivativeForm< 2, dim, spacedim > > &  input,
const MappingKind  kind,
const typename Mapping< dim, spacedim >::InternalDataBase &  internal,
const ArrayView< Tensor< 3, spacedim > > &  output 
) const
pure virtual

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

The mapping kinds currently implemented by derived classes are:

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

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

    ,

    where

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

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

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

\[J^{\dagger} = J^{-1}\]

Parameters
[in]inputAn array (or part of an array) of input objects that should be mapped.
[in]kindThe kind of mapping to be applied.
[in]internalA 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]outputAn 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.)

Implemented in MappingCartesian< dim, spacedim >, MappingFE< dim, spacedim >, MappingFEField< dim, spacedim, VectorType >, MappingManifold< dim, spacedim >, MappingQ< dim, spacedim >, and MappingQ< dim, dim >.

◆ transform() [5/5]

template<int dim, int spacedim = dim>
virtual void Mapping< dim, spacedim >::transform ( const ArrayView< const Tensor< 3, dim > > &  input,
const MappingKind  kind,
const typename Mapping< dim, spacedim >::InternalDataBase &  internal,
const ArrayView< Tensor< 3, spacedim > > &  output 
) const
pure virtual

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

The mapping kinds currently implemented by derived classes are:

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

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

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

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

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

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

Parameters
[in]inputAn array (or part of an array) of input objects that should be mapped.
[in]kindThe kind of mapping to be applied.
[in]internalA 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]outputAn array (or part of an array) into which the transformed objects should be placed.

Implemented in MappingCartesian< dim, spacedim >, MappingFE< dim, spacedim >, MappingFEField< dim, spacedim, VectorType >, MappingManifold< dim, spacedim >, MappingQ< dim, spacedim >, and MappingQ< dim, dim >.

◆ subscribe()

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

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

Definition at line 135 of file subscriptor.cc.

◆ unsubscribe()

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

Unsubscribes a user from the object.

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

Definition at line 155 of file subscriptor.cc.

◆ n_subscriptions()

unsigned int Subscriptor::n_subscriptions ( ) const
inlineinherited

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

Definition at line 300 of file subscriptor.h.

◆ list_subscribers() [1/2]

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

List the subscribers to the input stream.

Definition at line 317 of file subscriptor.h.

◆ list_subscribers() [2/2]

void Subscriptor::list_subscribers ( ) const
inherited

List the subscribers to deallog.

Definition at line 203 of file subscriptor.cc.

◆ serialize()

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

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

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

Definition at line 309 of file subscriptor.h.

◆ check_no_subscribers()

void Subscriptor::check_no_subscribers ( ) const
privatenoexceptinherited

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

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

Definition at line 52 of file subscriptor.cc.

Friends And Related Symbol Documentation

◆ FEValuesBase< dim, spacedim >

template<int dim, int spacedim = dim>
friend class FEValuesBase< dim, spacedim >
friend

Definition at line 1308 of file mapping.h.

◆ FEValues< dim, spacedim >

template<int dim, int spacedim = dim>
friend class FEValues< dim, spacedim >
friend

Definition at line 1308 of file mapping.h.

◆ FEFaceValues< dim, spacedim >

template<int dim, int spacedim = dim>
friend class FEFaceValues< dim, spacedim >
friend

Definition at line 1308 of file mapping.h.

◆ FESubfaceValues< dim, spacedim >

template<int dim, int spacedim = dim>
friend class FESubfaceValues< dim, spacedim >
friend

Definition at line 1308 of file mapping.h.

◆ NonMatching::FEImmersedSurfaceValues< dim >

template<int dim, int spacedim = dim>
friend class NonMatching::FEImmersedSurfaceValues< dim >
friend

Definition at line 1308 of file mapping.h.

◆ NonMatching::internal::ComputeMappingDataHelper< dim, spacedim >

template<int dim, int spacedim = dim>
friend class NonMatching::internal::ComputeMappingDataHelper< dim, spacedim >
friend

Definition at line 1308 of file mapping.h.

Member Data Documentation

◆ counter

std::atomic<unsigned int> Subscriptor::counter
mutableprivateinherited

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

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

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

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

Definition at line 218 of file subscriptor.h.

◆ counter_map

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

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

Definition at line 224 of file subscriptor.h.

◆ validity_pointers

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

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

Definition at line 240 of file subscriptor.h.

◆ object_info

const std::type_info* Subscriptor::object_info
mutableprivateinherited

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

Definition at line 248 of file subscriptor.h.

◆ mutex

std::mutex Subscriptor::mutex
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.


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