Reference documentation for deal.II version Git 3fe3ec1 2018-08-21 21:05:51 -0400
MappingFEField< dim, spacedim, VectorType, DoFHandlerType > Class Template Reference

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

Inheritance diagram for MappingFEField< dim, spacedim, VectorType, DoFHandlerType >:
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## Classes

class  InternalData

## Public Member Functions

MappingFEField (const MappingFEField< dim, spacedim, VectorType, DoFHandlerType > &mapping)

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

virtual bool preserves_vertex_locations () const override

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

unsigned int get_degree () 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 override

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

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 override

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 override

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 override

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 override

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 override

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

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 &&) noexcept

virtual ~Subscriptor ()

Subscriptoroperator= (const Subscriptor &)

Subscriptoroperator= (Subscriptor &&) noexcept

void subscribe (const char *identifier=nullptr) const

void unsubscribe (const char *identifier=nullptr) const

unsigned int n_subscriptions () const

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

void list_subscribers () const

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

## Static Public Member Functions

static::ExceptionBase & ExcInactiveCell ()

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, std::string arg2, std::string arg3)

static::ExceptionBase & ExcNoSubscriber (std::string arg1, std::string arg2)

## Private Member Functions

Point< spacedim > do_transform_unit_to_real_cell (const InternalData &mdata) const

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

void update_internal_dofs (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const typename MappingFEField< dim, spacedim, VectorType, DoFHandlerType >::InternalData &data) const

virtual void compute_shapes_virtual (const std::vector< Point< dim >> &unit_points, typename MappingFEField< dim, spacedim, VectorType, DoFHandlerType >::InternalData &data) const

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

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

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

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

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

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

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

## Private Attributes

SmartPointer< const VectorType, MappingFEField< dim, spacedim, VectorType, DoFHandlerType > > euler_vector

SmartPointer< const DoFHandlerType, MappingFEField< dim, spacedim, VectorType, DoFHandlerType > > euler_dof_handler

std::vector< unsigned int > fe_to_real

FEValues< dim, spacedim > fe_values

## Friends

template<int , int , class , class >
class MappingFEField

Protected Member Functions inherited from Mapping< dim, spacedim >
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 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_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

## Detailed Description

### template<int dim, int spacedim = dim, typename VectorType = Vector<double>, typename DoFHandlerType = DoFHandler<dim, spacedim>> class MappingFEField< dim, spacedim, VectorType, DoFHandlerType >

The MappingFEField is a generalization of the MappingQEulerian class, for arbitrary vector finite elements. The two main differences are that this class uses a vector of absolute positions instead of a vector of displacements, and it allows for arbitrary FiniteElement types, instead of only FE_Q.

This class effectively decouples the topology from the geometry, by relegating all geometrical information to some components of a FiniteElement vector field. The components that are used for the geometry can be arbitrarily selected at construction time.

The idea is to consider the Triangulation as a parameter configuration space, on which we construct an arbitrary geometrical mapping, using the instruments of the deal.II library: a vector of degrees of freedom, a DoFHandler associated to the geometry of the problem and a ComponentMask that tells us which components of the FiniteElement to use for the mapping.

Typically, the DoFHandler operates on a finite element that is constructed as a system element (FESystem()) from continuous FE_Q() (for iso-parametric discretizations) or FE_Bernstein() (for iso-geometric discretizations) objects. An example is shown below:

const FE_Q<dim,spacedim> feq(1);
const FESystem<dim,spacedim> fesystem(feq, spacedim);
DoFHandler<dim,spacedim> dhq(triangulation);
dhq.distribute_dofs(fesystem);
Vector<double> eulerq(dhq.n_dofs());
// Fills the euler vector with information from the Triangulation

Definition at line 83 of file mapping_fe_field.h.

## Constructor & Destructor Documentation

template<int dim, int spacedim, typename VectorType , typename DoFHandlerType >
 MappingFEField< dim, spacedim, VectorType, DoFHandlerType >::MappingFEField ( const DoFHandlerType & euler_dof_handler, const VectorType & euler_vector, const ComponentMask & mask = ComponentMask() )

Constructor. The first argument is a VectorType that specifies the transformation of the domain from the reference to the current configuration.

In general this class decouples geometry from topology, allowing users to define geometries which are only topologically equivalent to the underlying Triangulation, but which may otherwise be arbitrary. Differently from what happens in MappingQEulerian, the FiniteElement field which is passed to the constructor is interpreted as an absolute geometrical configuration, therefore one has to make sure that the euler_vector actually represents a valid geometry (i.e., one with no inverted cells, or with no zero-volume cells).

If the underlying FiniteElement is a system of FE_Q(), and euler_vector is initialized using VectorTools::get_position_vector(), then this class is in all respects identical to MappingQ().

The optional ComponentMask argument can be used to specify what components of the FiniteElement to use for the geometrical transformation. If no mask is specified at construction time, then a default one is used, which makes this class works in the same way of MappingQEulerian(), i.e., the first spacedim components of the FiniteElement are assumed to represent the geometry of the problem.

Notice that if a mask is specified, it has to match in size the underlying FiniteElement, and it has to have exactly spacedim non-zero elements, indicating the components (in order) of the FiniteElement which will be used for the geometry.

If an incompatible mask is passed, an exception is thrown.

Definition at line 237 of file mapping_fe_field.cc.

template<int dim, int spacedim, typename VectorType , typename DoFHandlerType >
 MappingFEField< dim, spacedim, VectorType, DoFHandlerType >::MappingFEField ( const MappingFEField< dim, spacedim, VectorType, DoFHandlerType > & mapping )

Copy constructor.

Definition at line 264 of file mapping_fe_field.cc.

## Member Function Documentation

template<int dim, int spacedim, typename VectorType , typename DoFHandlerType >
 std::unique_ptr< Mapping< dim, spacedim > > MappingFEField< dim, spacedim, VectorType, DoFHandlerType >::clone ( ) const
overridevirtual

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

Implements Mapping< dim, spacedim >.

Definition at line 2328 of file mapping_fe_field.cc.

template<int dim, int spacedim, typename VectorType , typename DoFHandlerType >
 bool MappingFEField< dim, spacedim, VectorType, DoFHandlerType >::preserves_vertex_locations ( ) const
overridevirtual

See the documentation of Mapping::preserves_vertex_locations() for the purpose of this function. The implementation in this class always returns false.

Implements Mapping< dim, spacedim >.

Definition at line 295 of file mapping_fe_field.cc.

template<int dim, int spacedim, typename VectorType , typename DoFHandlerType >
 std::array< Point< spacedim >, GeometryInfo< dim >::vertices_per_cell > MappingFEField< dim, spacedim, VectorType, DoFHandlerType >::get_vertices ( const typename Triangulation< dim, spacedim >::cell_iterator & cell ) const
overridevirtual

Return the mapped vertices of a cell.

This mapping ignores the vertices of the Triangulation it is associated to, and constructs the position of the vertices according to the euler_vector that was passed at construction time.

Reimplemented from Mapping< dim, spacedim >.

Definition at line 304 of file mapping_fe_field.cc.

template<int dim, int spacedim, typename VectorType , typename DoFHandlerType >
 Point< spacedim > MappingFEField< dim, spacedim, VectorType, DoFHandlerType >::transform_unit_to_real_cell ( const typename Triangulation< dim, spacedim >::cell_iterator & cell, const Point< dim > & p ) const
overridevirtual

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

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

Implements Mapping< dim, spacedim >.

Definition at line 2108 of file mapping_fe_field.cc.

template<int dim, int spacedim, typename VectorType , typename DoFHandlerType >
 Point< dim > MappingFEField< dim, spacedim, VectorType, DoFHandlerType >::transform_real_to_unit_cell ( const typename Triangulation< dim, spacedim >::cell_iterator & cell, const Point< spacedim > & p ) const
overridevirtual

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

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

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 2151 of file mapping_fe_field.cc.

template<int dim, int spacedim, typename VectorType , typename DoFHandlerType >
 void MappingFEField< dim, spacedim, VectorType, DoFHandlerType >::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
overridevirtual

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

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

Parameters
 [in] input An array (or part of an array) of input objects that should be mapped. [in] 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 1987 of file mapping_fe_field.cc.

template<int dim, int spacedim, typename VectorType , typename DoFHandlerType >
 void MappingFEField< dim, spacedim, VectorType, DoFHandlerType >::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
overridevirtual

Transform a field of differential forms from the reference cell to the physical cell. It is useful to think of $$\mathbf{T} = \nabla \mathbf u$$ and $$\hat{\mathbf T} = \hat \nabla \hat{\mathbf u}$$, with $$\mathbf u$$ a vector field. The mapping 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 2006 of file mapping_fe_field.cc.

template<int dim, int spacedim, typename VectorType , typename DoFHandlerType >
 void MappingFEField< dim, spacedim, VectorType, DoFHandlerType >::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
overridevirtual

Transform a tensor field from the reference cell to the physical cell. These tensors are usually the Jacobians in the reference cell of vector fields that have been pulled back from the physical cell. The mapping 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 $$\mathbf u(\mathbf x) = \frac{1}{\text{det}\;J(\hat{\mathbf x})} J(\hat{\mathbf x}) \hat{\mathbf u}(\hat{\mathbf x})$$ so that

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

Todo:
The formulas for mapping_covariant_gradient, mapping_contravariant_gradient and mapping_piola_gradient are only true as stated for linear mappings. If, for example, the mapping is bilinear (or has a higher order polynomial degree) then there is a missing term associated with the derivative of $$J$$.
Parameters
 [in] input An array (or part of an array) of input objects that should be mapped. [in] 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 2023 of file mapping_fe_field.cc.

template<int dim, int spacedim, typename VectorType , typename DoFHandlerType >
 void MappingFEField< dim, spacedim, VectorType, DoFHandlerType >::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
overridevirtual

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

The mapping 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 2041 of file mapping_fe_field.cc.

template<int dim, int spacedim, typename VectorType , typename DoFHandlerType >
 void MappingFEField< dim, spacedim, VectorType, DoFHandlerType >::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
overridevirtual

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

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

Parameters
 [in] input An array (or part of an array) of input objects that should be mapped. [in] 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 2089 of file mapping_fe_field.cc.

template<int dim, int spacedim, typename VectorType , typename DoFHandlerType >
 unsigned int MappingFEField< dim, spacedim, VectorType, DoFHandlerType >::get_degree ( ) const

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

Definition at line 2311 of file mapping_fe_field.cc.

template<int dim, int spacedim, typename VectorType , typename DoFHandlerType >

Return the ComponentMask of the mapping, i.e. which components to use for the mapping.

Definition at line 2319 of file mapping_fe_field.cc.

template<int dim, int spacedim, typename VectorType , typename DoFHandlerType >
 UpdateFlags MappingFEField< dim, spacedim, VectorType, DoFHandlerType >::requires_update_flags ( const UpdateFlags update_flags ) const
overrideprivatevirtual

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 382 of file mapping_fe_field.cc.

template<int dim, int spacedim, typename VectorType , typename DoFHandlerType >
 std::unique_ptr< typename Mapping< dim, spacedim >::InternalDataBase > MappingFEField< dim, spacedim, VectorType, DoFHandlerType >::get_data ( const UpdateFlags update_flags, const Quadrature< dim > & quadrature ) const
overrideprivatevirtual

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

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

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

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

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

Implements Mapping< dim, spacedim >.

Definition at line 564 of file mapping_fe_field.cc.

template<int dim, int spacedim, typename VectorType , typename DoFHandlerType >
 std::unique_ptr< typename Mapping< dim, spacedim >::InternalDataBase > MappingFEField< dim, spacedim, VectorType, DoFHandlerType >::get_face_data ( const UpdateFlags update_flags, const Quadrature< dim-1 > & quadrature ) const
overrideprivatevirtual

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 578 of file mapping_fe_field.cc.

template<int dim, int spacedim, typename VectorType , typename DoFHandlerType >
 std::unique_ptr< typename Mapping< dim, spacedim >::InternalDataBase > MappingFEField< dim, spacedim, VectorType, DoFHandlerType >::get_subface_data ( const UpdateFlags update_flags, const Quadrature< dim-1 > & quadrature ) const
overrideprivatevirtual

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 593 of file mapping_fe_field.cc.

template<int dim, int spacedim, typename VectorType , typename DoFHandlerType >
 Point< spacedim > MappingFEField< dim, spacedim, VectorType, DoFHandlerType >::do_transform_unit_to_real_cell ( const InternalData & mdata ) const
private

Transform a point p on the unit cell to the point p_real on the real cell cell and returns p_real.

This function is called by transform_unit_to_real_cell and multiple times (through the Newton iteration) by transform_real_to_unit_cell_internal.

Takes a reference to an InternalData that must already include the shape values at point p and the mapping support points of the cell.

This InternalData argument avoids multiple computations of the shape values at point p and especially multiple computations of the mapping support points.

Definition at line 2130 of file mapping_fe_field.cc.

template<int dim, int spacedim, typename VectorType , typename DoFHandlerType >
 Point< dim > MappingFEField< dim, spacedim, VectorType, DoFHandlerType >::do_transform_real_to_unit_cell ( const typename Triangulation< dim, spacedim >::cell_iterator & cell, const Point< spacedim > & p, const Point< dim > & initial_p_unit, InternalData & mdata ) const
private

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

Takes a reference to an InternalData that is assumed to be previously created by the get_data function with UpdateFlags including update_transformation_values and update_transformation_gradients and a one point Quadrature that includes the given initial guess for the transformation initial_p_unit. Hence this function assumes that mdata already includes the transformation shape values and gradients computed at initial_p_unit.

mdata will be changed by this function.

Definition at line 2200 of file mapping_fe_field.cc.

template<int dim, int spacedim, typename VectorType , typename DoFHandlerType >
 void MappingFEField< dim, spacedim, VectorType, DoFHandlerType >::update_internal_dofs ( const typename Triangulation< dim, spacedim >::cell_iterator & cell, const typename MappingFEField< dim, spacedim, VectorType, DoFHandlerType >::InternalData & data ) const
private

Update internal degrees of freedom.

Definition at line 2337 of file mapping_fe_field.cc.

template<int dim, int spacedim, typename VectorType , typename DoFHandlerType >
 void MappingFEField< dim, spacedim, VectorType, DoFHandlerType >::compute_shapes_virtual ( const std::vector< Point< dim >> & unit_points, typename MappingFEField< dim, spacedim, VectorType, DoFHandlerType >::InternalData & data ) const
privatevirtual

See the documentation of the base class for detailed information.

Definition at line 343 of file mapping_fe_field.cc.

## Friends And Related Function Documentation

template<int dim, int spacedim = dim, typename VectorType = Vector<double>, typename DoFHandlerType = DoFHandler<dim, spacedim>>
template<int , int , class , class >
 friend class MappingFEField
friend

Declare other MappingFEField classes friends.

Definition at line 626 of file mapping_fe_field.h.

## Member Data Documentation

template<int dim, int spacedim = dim, typename VectorType = Vector<double>, typename DoFHandlerType = DoFHandler<dim, spacedim>>
 SmartPointer > MappingFEField< dim, spacedim, VectorType, DoFHandlerType >::euler_vector
private

Reference to the vector of shifts.

Definition at line 514 of file mapping_fe_field.h.

template<int dim, int spacedim = dim, typename VectorType = Vector<double>, typename DoFHandlerType = DoFHandler<dim, spacedim>>
 SmartPointer > MappingFEField< dim, spacedim, VectorType, DoFHandlerType >::euler_dof_handler
private

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

Definition at line 521 of file mapping_fe_field.h.

template<int dim, int spacedim = dim, typename VectorType = Vector<double>, typename DoFHandlerType = DoFHandler<dim, spacedim>>
 std::vector MappingFEField< dim, spacedim, VectorType, DoFHandlerType >::fe_to_real
private

Mapping between indices in the FE space and the real space. This vector contains one index for each component of the finite element space. If the index is one for which the ComponentMask which is used to construct this element is false, then numbers::invalid_unsigned_int is returned, otherwise the component in real space is returned. For example, if we construct the mapping using ComponentMask(spacedim, true), then this vector contains {0,1,2} in spacedim = 3.

Definition at line 596 of file mapping_fe_field.h.

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

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

Definition at line 602 of file mapping_fe_field.h.

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