Reference documentation for deal.II version Git 1dafe5d 2017-11-23 14:01:03 +0100
MappingManifold< dim, spacedim > Class Template Reference

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

Inheritance diagram for MappingManifold< dim, spacedim >:
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## Classes

class  InternalData

## Public Member Functions

MappingManifold ()=default

MappingManifold (const MappingManifold< dim, spacedim > &mapping)

virtual Mapping< dim, spacedim > * clone () const

virtual bool preserves_vertex_locations () const

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

virtual Point< dim > transform_real_to_unit_cell (const typename Triangulation< dim, spacedim >::cell_iterator &cell, 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 MappingType type, const typename Mapping< dim, spacedim >::InternalDataBase &internal, const ArrayView< Tensor< 1, spacedim > > &output) const

virtual void transform (const ArrayView< const DerivativeForm< 1, dim, spacedim > > &input, const MappingType type, const typename Mapping< dim, spacedim >::InternalDataBase &internal, const ArrayView< Tensor< 2, spacedim > > &output) const

virtual void transform (const ArrayView< const Tensor< 2, dim > > &input, const MappingType type, const typename Mapping< dim, spacedim >::InternalDataBase &internal, const ArrayView< Tensor< 2, spacedim > > &output) const

virtual void transform (const ArrayView< const DerivativeForm< 2, dim, spacedim > > &input, const MappingType type, const typename Mapping< dim, spacedim >::InternalDataBase &internal, const ArrayView< Tensor< 3, spacedim > > &output) const

virtual void transform (const ArrayView< const Tensor< 3, dim > > &input, const MappingType type, const typename Mapping< dim, spacedim >::InternalDataBase &internal, const ArrayView< Tensor< 3, spacedim > > &output) const

Interface with FEValues
virtual UpdateFlags requires_update_flags (const UpdateFlags update_flags) const

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::FEValues::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::FEValues::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::FEValues::MappingRelatedData< dim, spacedim > &output_data) const

Public Member Functions inherited from Mapping< dim, spacedim >
virtual ~Mapping ()=default

virtual std::array< Point< spacedim >, GeometryInfo< dim >::vertices_per_cell > get_vertices (const typename Triangulation< dim, spacedim >::cell_iterator &cell) 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

Public Member Functions inherited from Subscriptor
Subscriptor ()

Subscriptor (const Subscriptor &)

Subscriptor (Subscriptor &&)

virtual ~Subscriptor ()

Subscriptoroperator= (const Subscriptor &)

Subscriptoroperator= (Subscriptor &&)

void subscribe (const char *identifier=nullptr) const

void unsubscribe (const char *identifier=nullptr) const

unsigned int n_subscriptions () const

void list_subscribers () const

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

Static Public Member Functions inherited from Mapping< dim, spacedim >
static::ExceptionBase & ExcInvalidData ()

static::ExceptionBase & ExcTransformationFailed ()

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

Static Public Member Functions inherited from Subscriptor
static::ExceptionBase & ExcInUse (int arg1, char *arg2, std::string &arg3)

static::ExceptionBase & ExcNoSubscriber (char *arg1, char *arg2)

Interface with FEValues

## Detailed Description

### template<int dim, int spacedim = dim> class MappingManifold< dim, spacedim >

This class implements the functionality for Manifold conforming mappings. This Mapping computes the transformation between the reference and real cell by exploiting the geometrical information coming from the underlying Manifold object.

Quadrature points computed using this mapping lie on the exact geometrical objects, and tangent and normal vectors computed using this class are tangent and normal to the underlying geometry. This is in constrast with the MappingQ class, which approximates the geometry using a polynomial of some order, and then computes the normals and tangents using the approximated surface.

Warning
It is not possible, for mathematical reasons, for one to use this class with a geometry described by a SphericalManifold: see the note in that class for more information.

Definition at line 60 of file mapping_manifold.h.

## Constructor & Destructor Documentation

template<int dim, int spacedim = dim>
 MappingManifold< dim, spacedim >::MappingManifold ( )
default

Constructor.

template<int dim, int spacedim>
 MappingManifold< dim, spacedim >::MappingManifold ( const MappingManifold< dim, spacedim > & mapping )

Copy constructor.

Definition at line 173 of file mapping_manifold.cc.

## Member Function Documentation

template<int dim, int spacedim>
 Mapping< dim, spacedim > * MappingManifold< dim, spacedim >::clone ( ) const
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.

Implements Mapping< dim, spacedim >.

Definition at line 180 of file mapping_manifold.cc.

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

Always returns true because this class assumes that the vertices always lies on the underlying Manifold.

Implements Mapping< dim, spacedim >.

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

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

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

Implements Mapping< dim, spacedim >.

Definition at line 202 of file mapping_manifold.cc.

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

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

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

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

Implements Mapping< dim, spacedim >.

Definition at line 190 of file mapping_manifold.cc.

template<int dim, int spacedim>
 void MappingManifold< dim, spacedim >::transform ( const ArrayView< const Tensor< 1, dim > > & input, const MappingType type, const typename Mapping< dim, spacedim >::InternalDataBase & internal, const ArrayView< Tensor< 1, spacedim > > & output ) const
virtual

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

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

The mapping types 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

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

Parameters
 [in] input An array (or part of an array) of input objects that should be mapped. [in] type 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 1199 of file mapping_manifold.cc.

template<int dim, int spacedim>
 void MappingManifold< dim, spacedim >::transform ( const ArrayView< const DerivativeForm< 1, dim, spacedim > > & input, const MappingType type, const typename Mapping< dim, spacedim >::InternalDataBase & internal, const ArrayView< Tensor< 2, spacedim > > & output ) const
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 types currently implemented by derived classes are:

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

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

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

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

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

Note
It would have been more reasonable to make this transform a template function with the rank in DerivativeForm<1, dim, rank>. Unfortunately C++ does not allow templatized virtual functions. This is why we identify DerivativeForm<1, dim, 1> with a Tensor<1,dim> when using mapping_covariant() in the function transform() above this one.
Parameters
 [in] input An array (or part of an array) of input objects that should be mapped. [in] type 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 1212 of file mapping_manifold.cc.

template<int dim, int spacedim>
 void MappingManifold< dim, spacedim >::transform ( const ArrayView< const Tensor< 2, dim > > & input, const MappingType type, const typename Mapping< dim, spacedim >::InternalDataBase & internal, const ArrayView< Tensor< 2, spacedim > > & output ) const
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 types 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 $$\mathbf u(\mathbf x) = \frac{1}{\text{det}\;J(\mathbf x)} J(\mathbf x) \hat{\mathbf u}(\mathbf x)$$ so that

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

Todo:
The formulas for mapping_covariant_gradient, mapping_contravariant_gradient and mapping_piola_gradient are only true as stated for linear mappings. If, for example, the mapping is bilinear (or has a higher order polynomial degree) then there is a missing term associated with the derivative of $$J$$.
Parameters
 [in] input An array (or part of an array) of input objects that should be mapped. [in] type 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 1225 of file mapping_manifold.cc.

template<int dim, int spacedim>
 void MappingManifold< dim, spacedim >::transform ( const ArrayView< const DerivativeForm< 2, dim, spacedim > > & input, const MappingType type, const typename Mapping< dim, spacedim >::InternalDataBase & internal, const ArrayView< Tensor< 3, spacedim > > & output ) const
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 types currently implemented by derived classes are:

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

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

,

where

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

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

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

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

Parameters
 [in] input An array (or part of an array) of input objects that should be mapped. [in] type 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 1251 of file mapping_manifold.cc.

template<int dim, int spacedim>
 void MappingManifold< dim, spacedim >::transform ( const ArrayView< const Tensor< 3, dim > > & input, const MappingType type, const typename Mapping< dim, spacedim >::InternalDataBase & internal, const ArrayView< Tensor< 3, spacedim > > & output ) const
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 types 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 $$\mathbf u_i(\mathbf x) = \frac{1}{\text{det}\;J(\mathbf x)} J_{iI}(\mathbf x) \hat{\mathbf u}(\mathbf x)$$ so that

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

Parameters
 [in] input An array (or part of an array) of input objects that should be mapped. [in] type 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 1300 of file mapping_manifold.cc.

template<int dim, int spacedim>
 UpdateFlags MappingManifold< dim, spacedim >::requires_update_flags ( const UpdateFlags update_flags ) const
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.

UpdateFlags

Implements Mapping< dim, spacedim >.

Definition at line 236 of file mapping_manifold.cc.

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

Implements Mapping< dim, spacedim >.

Definition at line 303 of file mapping_manifold.cc.

template<int dim, int spacedim>
 MappingManifold< dim, spacedim >::InternalData * MappingManifold< dim, spacedim >::get_face_data ( const UpdateFlags update_flags, const Quadrature< dim-1 > & quadrature ) const
virtual

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

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

Implements Mapping< dim, spacedim >.

Definition at line 316 of file mapping_manifold.cc.

template<int dim, int spacedim>
 MappingManifold< dim, spacedim >::InternalData * MappingManifold< dim, spacedim >::get_subface_data ( const UpdateFlags update_flags, const Quadrature< dim-1 > & quadrature ) const
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_flags A set of flags that define what is expected of the mapping class in future calls to transform() or the fill_fe_values() group of functions. This set of flags may contain flags that mappings do not know how to deal with (e.g., for information that is in fact computed by the finite element classes, such as UpdateFlags::update_values). Derived classes will need to store these flags, or at least that subset of flags that will require the mapping to perform any actions in fill_fe_values(), in InternalDataBase::update_each. quadrature The quadrature object for which mapping information will have to be computed. This includes the locations and weights of quadrature points.
Returns
A pointer to a newly created object of type InternalDataBase (or a derived class). Ownership of this object passes to the calling function.
Note
C++ allows that virtual functions in derived classes may return pointers to objects not of type InternalDataBase but in fact pointers to objects of classes derived from InternalDataBase. (This feature is called "covariant return types".) This is useful in some contexts where the calling is within the derived class and will immediately make use of the returned object, knowing its real (derived) type.

Implements Mapping< dim, spacedim >.

Definition at line 331 of file mapping_manifold.cc.

template<int dim, int spacedim = dim>
 CellSimilarity::Similarity MappingManifold< 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::FEValues::MappingRelatedData< dim, spacedim > & output_data ) const
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] 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.
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.

Implements Mapping< dim, spacedim >.

Definition at line 475 of file mapping_manifold.cc.

template<int dim, int spacedim = dim>
 void MappingManifold< 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::FEValues::MappingRelatedData< dim, spacedim > & output_data ) const
virtual

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

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

Implements Mapping< dim, spacedim >.

Definition at line 1137 of file mapping_manifold.cc.

template<int dim, int spacedim = dim>
 void MappingManifold< 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::FEValues::MappingRelatedData< dim, spacedim > & output_data ) const
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] 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 1167 of file mapping_manifold.cc.

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