Reference documentation for deal.II version Git b304a00 2017-03-22 15:49:17 -0600
Classes | Public Member Functions | Protected Attributes | List of all members
MappingQ< dim, spacedim > Class Template Reference

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

Inheritance diagram for MappingQ< dim, spacedim >:
[legend]

Classes

class  InternalData
 

Public Member Functions

 MappingQ (const unsigned int polynomial_degree, const bool use_mapping_q_on_all_cells=false)
 
 MappingQ (const MappingQ< dim, spacedim > &mapping)
 
unsigned int get_degree () const
 
virtual bool preserves_vertex_locations () const
 
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
 
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
 
virtual Mapping< dim, spacedim > * clone () const
 
- Public Member Functions inherited from Mapping< dim, spacedim >
virtual ~Mapping ()
 
virtual std_cxx11::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=0) const
 
void unsubscribe (const char *identifier=0) const
 
unsigned int n_subscriptions () const
 
void list_subscribers () const
 
template<class Archive >
void serialize (Archive &ar, const unsigned int version)
 

Protected Member Functions

Interface with FEValues
virtual UpdateFlags requires_update_flags (const UpdateFlags update_flags) const
 
virtual InternalDataget_data (const UpdateFlags, const Quadrature< dim > &quadrature) const
 
virtual InternalDataget_face_data (const UpdateFlags flags, const Quadrature< dim-1 > &quadrature) const
 
virtual InternalDataget_subface_data (const UpdateFlags flags, const Quadrature< dim-1 > &quadrature) 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
 
- 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::FEValues::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::FEValues::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::FEValues::MappingRelatedData< dim, spacedim > &output_data) const =0
 

Protected Attributes

const unsigned int polynomial_degree
 
const bool use_mapping_q_on_all_cells
 
std_cxx11::shared_ptr< const MappingQGeneric< dim, spacedim > > q1_mapping
 
std_cxx11::shared_ptr< const MappingQGeneric< dim, spacedim > > qp_mapping
 

Additional Inherited Members

- 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)
 

Detailed Description

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

A class that implements a polynomial mapping \(Q_p\) of degree \(p\) on cells at the boundary of the domain (or, if requested in the constructor, for all cells) and linear mappings for interior cells.

The class is in fact poorly named since (unless explicitly specified during the construction of the object, see below), it does not actually use mappings of degree \(p\) everywhere, but only on cells at the boundary. This is in contrast to the MappingQGeneric class which indeed does use a polynomial mapping \(Q_p\) of degree \(p\) everywhere. The point of the current class is that in many situations, curved domains are only provided with information about how exactly edges at the boundary are shaped, but we do not know anything about internal edges. Thus, in the absence of other information, we can only assume that internal edges are straight lines, and in that case internal cells may as well be treated is bilinear quadrilaterals or trilinear hexahedra. (An example of how such meshes look is shown in step-1 already, but it is also discussed in the "Results" section of step-6.) Because bi-/trilinear mappings are significantly cheaper to compute than higher order mappings, it is advantageous in such situations to use the higher order mapping only on cells at the boundary of the domain. This class implements exactly this behavior.

There are a number of special cases worth considering:

Behavior along curved boundaries and with different manifolds

For the behavior of the mapping and convergence rates in case of mixing different manifolds, please consult the respective section of MappingQGeneric.

Author
Ralf Hartmann, 2000, 2001, 2005; Guido Kanschat 2000, 2001, Wolfgang Bangerth, 2015

Definition at line 26 of file fe_dgq.h.

Constructor & Destructor Documentation

template<int dim, int spacedim>
MappingQ< dim, spacedim >::MappingQ ( const unsigned int  polynomial_degree,
const bool  use_mapping_q_on_all_cells = false 
)

Constructor. polynomial_degree denotes the polynomial degree of the polynomials that are used to map cells boundary.

The second argument determines whether the higher order mapping should also be used on interior cells. If its value is false (the default), then a lower order mapping is used in the interior. This is sufficient for most cases where higher order mappings are only used to better approximate the boundary. In that case, cells bounded by straight lines are acceptable in the interior. However, there are cases where one would also like to use a higher order mapping in the interior. The MappingQEulerian class is one such case.

The value of use_mapping_q_on_all_cells is ignored if dim is not equal to spacedim, i.e., if we are considering meshes on surfaces embedded into higher dimensional spaces.

Definition at line 56 of file mapping_q.cc.

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

Copy constructor.

Definition at line 87 of file mapping_q.cc.

Member Function Documentation

template<int dim, int spacedim>
unsigned int MappingQ< dim, spacedim >::get_degree ( ) const

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

Definition at line 107 of file mapping_q.cc.

template<int dim, int spacedim>
bool MappingQ< dim, spacedim >::preserves_vertex_locations ( ) const
inlinevirtual

Always returns true because the default implementation of functions in this class preserves vertex locations.

Implements Mapping< dim, spacedim >.

Reimplemented in MappingQEulerian< dim, VectorType, spacedim >.

Definition at line 117 of file mapping_q.cc.

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

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

Implements Mapping< dim, spacedim >.

Definition at line 471 of file mapping_q.cc.

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

Transforms the point p on the real cell to the point p_unit on the unit cell cell and returns p_unit.

Uses Newton iteration and the transform_unit_to_real_cell function.

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 return reference coordinates lie inside of outside the reference cell (e.g., using GeometryInfo::is_inside_unit_cell) or whether the exception mentioned above has been thrown.

Implements Mapping< dim, spacedim >.

Definition at line 488 of file mapping_q.cc.

template<int dim, int spacedim>
void MappingQ< 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]inputAn array (or part of an array) of input objects that should be mapped.
[in]typeThe 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.)

Implements Mapping< dim, spacedim >.

Definition at line 352 of file mapping_q.cc.

template<int dim, int spacedim>
void MappingQ< 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]inputAn array (or part of an array) of input objects that should be mapped.
[in]typeThe 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.)

Implements Mapping< dim, spacedim >.

Definition at line 376 of file mapping_q.cc.

template<int dim, int spacedim>
void MappingQ< 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]inputAn array (or part of an array) of input objects that should be mapped.
[in]typeThe 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.)

Implements Mapping< dim, spacedim >.

Definition at line 400 of file mapping_q.cc.

template<int dim, int spacedim>
void MappingQ< 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]inputAn array (or part of an array) of input objects that should be mapped.
[in]typeThe 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.)

Implements Mapping< dim, spacedim >.

Definition at line 424 of file mapping_q.cc.

template<int dim, int spacedim>
void MappingQ< 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]inputAn array (or part of an array) of input objects that should be mapped.
[in]typeThe 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.

Implements Mapping< dim, spacedim >.

Definition at line 447 of file mapping_q.cc.

template<int dim, int spacedim>
Mapping< dim, spacedim > * MappingQ< 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.

Implements Mapping< dim, spacedim >.

Reimplemented in MappingQEulerian< dim, VectorType, spacedim >, and MappingC1< dim, spacedim >.

Definition at line 505 of file mapping_q.cc.

template<int dim, int spacedim>
UpdateFlags MappingQ< dim, spacedim >::requires_update_flags ( const UpdateFlags  update_flags) const
protectedvirtual

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
UpdateFlags

Implements Mapping< dim, spacedim >.

Definition at line 126 of file mapping_q.cc.

template<int dim, int spacedim>
MappingQ< dim, spacedim >::InternalData * MappingQ< dim, spacedim >::get_data ( const UpdateFlags  update_flags,
const Quadrature< dim > &  quadrature 
) const
protectedvirtual

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.

Implements Mapping< dim, spacedim >.

Definition at line 137 of file mapping_q.cc.

template<int dim, int spacedim>
MappingQ< dim, spacedim >::InternalData * MappingQ< dim, spacedim >::get_face_data ( const UpdateFlags  update_flags,
const Quadrature< 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.

Implements Mapping< dim, spacedim >.

Definition at line 161 of file mapping_q.cc.

template<int dim, int spacedim>
MappingQ< dim, spacedim >::InternalData * MappingQ< dim, spacedim >::get_subface_data ( const UpdateFlags  update_flags,
const Quadrature< dim-1 > &  quadrature 
) const
protectedvirtual

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.

Implements Mapping< dim, spacedim >.

Definition at line 185 of file mapping_q.cc.

Member Data Documentation

template<int dim, int spacedim = dim>
const unsigned int MappingQ< dim, spacedim >::polynomial_degree
protected

The polynomial degree of the cells to be used on all cells at the boundary of the domain, or everywhere if so specified.

Definition at line 330 of file mapping_q.h.

template<int dim, int spacedim = dim>
const bool MappingQ< dim, spacedim >::use_mapping_q_on_all_cells
protected

If this flag is set true then MappingQ is used on all cells, not only on boundary cells.

Definition at line 336 of file mapping_q.h.

template<int dim, int spacedim = dim>
std_cxx11::shared_ptr<const MappingQGeneric<dim,spacedim> > MappingQ< dim, spacedim >::q1_mapping
protected

Pointer to a Q1 mapping. This mapping is used on interior cells unless use_mapping_q_on_all_cells was set in the call to the constructor. The mapping is also used on any cell in the transform_real_to_unit_cell() to compute a cheap initial guess for the position of the point before we employ the more expensive Newton iteration using the full mapping.

Note
MappingQEulerian resets this pointer to an object of type MappingQ1Eulerian to ensure that the Q1 mapping also knows about the proper shifts and transformations of the Eulerian displacements. This also means that we really need to store our own Q1 mapping here, rather than simply resorting to StaticMappingQ1::mapping.
If the polynomial degree used for the current object is one, then the qp_mapping and q1_mapping variables point to the same underlying object.

Definition at line 355 of file mapping_q.h.

template<int dim, int spacedim = dim>
std_cxx11::shared_ptr<const MappingQGeneric<dim,spacedim> > MappingQ< dim, spacedim >::qp_mapping
protected

Pointer to a Q_p mapping. This mapping is used on boundary cells unless use_mapping_q_on_all_cells was set in the call to the constructor (in which case it is used for all cells).

Note
MappingQEulerian and MappingC1 reset this pointer to an object of their own implementation to ensure that the Q_p mapping also knows about the proper shifts and transformations of the Eulerian displacements (Eulerian case) and proper choice of support points (C1 case).
If the polynomial degree used for the current object is one, then the qp_mapping and q1_mapping variables point to the same underlying object.

Definition at line 371 of file mapping_q.h.


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