Reference documentation for deal.II version Git 5ba5df0 20150903 22:21:05 0500

#include <deal.II/fe/mapping_q1.h>
Classes  
class  InternalData 
Public Member Functions  
MappingQ1 ()  
virtual Mapping< dim, spacedim > *  clone () const 
virtual bool  preserves_vertex_locations () const 
template<int dim_>  
Point< dim_ >  transform_real_to_unit_cell_internal_codim1 (const typename Triangulation< dim_, dim_+1 >::cell_iterator &cell, const Point< dim_+1 > &p, const Point< dim_ > &initial_p_unit, typename MappingQ1< dim, spacedim >::InternalData &mdata) 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 VectorSlice< const std::vector< Tensor< 1, dim > > > input, const MappingType type, const typename Mapping< dim, spacedim >::InternalDataBase &internal, VectorSlice< std::vector< Tensor< 1, spacedim > > > output) const 
virtual void  transform (const VectorSlice< const std::vector< DerivativeForm< 1, dim, spacedim > > > input, const MappingType type, const typename Mapping< dim, spacedim >::InternalDataBase &internal, VectorSlice< std::vector< Tensor< 2, spacedim > > > output) const 
virtual void  transform (const VectorSlice< const std::vector< Tensor< 2, dim > > > input, const MappingType type, const typename Mapping< dim, spacedim >::InternalDataBase &internal, VectorSlice< std::vector< Tensor< 2, spacedim > > > output) const 
virtual void  transform (const VectorSlice< const std::vector< DerivativeForm< 2, dim, spacedim > > > input, const MappingType type, const typename Mapping< dim, spacedim >::InternalDataBase &internal, VectorSlice< std::vector< Tensor< 3, spacedim > > > output) const 
virtual void  transform (const VectorSlice< const std::vector< Tensor< 3, dim > > > input, const MappingType type, const typename Mapping< dim, spacedim >::InternalDataBase &internal, VectorSlice< std::vector< Tensor< 3, spacedim > > > output) 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< dim1 >  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 
DeclException0 (ExcInvalidData)  
DeclExceptionMsg (ExcTransformationFailed,"Computing the mapping between a real space point and a point in reference ""space failed, typically because the given point lies outside the cell ""where the inverse mapping is not unique.")  
DeclException3 (ExcDistortedMappedCell, Point< spacedim >, double, int,<< "The image of the mapping applied to cell with center ["<< arg1<< "] is distorted. The cell geometry or the "<< "mapping are invalid, giving a nonpositive volume "<< "fraction of "<< arg2<< " in quadrature point "<< arg3<< ".")  
Public Member Functions inherited from Subscriptor  
Subscriptor ()  
Subscriptor (const Subscriptor &)  
virtual  ~Subscriptor () 
Subscriptor &  operator= (const Subscriptor &) 
void  subscribe (const char *identifier=0) const 
void  unsubscribe (const char *identifier=0) const 
unsigned int  n_subscriptions () const 
void  list_subscribers () const 
DeclException3 (ExcInUse, int, char *, std::string &,<< "Object of class "<< arg2<< " is still used by "<< arg1<< " other objects."<< "\n\n"<< "(Additional information: "<< arg3<< ")\n\n"<< "See the entry in the Frequently Asked Questions of "<< "deal.II (linked to from http://www.dealii.org/) for "<< "a lot more information on what this error means and "<< "how to fix programs in which it happens.")  
DeclException2 (ExcNoSubscriber, char *, char *,<< "No subscriber with identifier <"<< arg2<< "> subscribes to this object of class "<< arg1<< ". Consequently, it cannot be unsubscribed.")  
template<class Archive >  
void  serialize (Archive &ar, const unsigned int version) 
Protected Types  
typedef QProjector< dim >::DataSetDescriptor  DataSetDescriptor 
Protected Member Functions  
template<int dim_>  
Point< dim_ >  transform_real_to_unit_cell_internal_codim1 (const typename Triangulation< dim_, dim_+1 >::cell_iterator &cell, const Point< dim_+1 > &p, const Point< dim_ > &initial_p_unit, InternalData &mdata) const 
Point< dim >  transform_real_to_unit_cell_initial_guess (const std::vector< Point< spacedim > > &vertex, const Point< spacedim > &p) const 
void  compute_shapes (const std::vector< Point< dim > > &unit_points, InternalData &data) const 
virtual void  compute_shapes_virtual (const std::vector< Point< dim > > &unit_points, InternalData &data) const 
Point< spacedim >  transform_unit_to_real_cell_internal (const InternalData &mdata) const 
Point< dim >  transform_real_to_unit_cell_internal (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const Point< spacedim > &p, const Point< dim > &initial_p_unit, InternalData &mdata) const 
virtual void  compute_mapping_support_points (const typename Triangulation< dim, spacedim >::cell_iterator &cell, std::vector< Point< spacedim > > &a) const 
template<>  
Point< 2 >  transform_real_to_unit_cell_internal (const Triangulation< 2, 3 >::cell_iterator &cell, const Point< 3 > &p, const Point< 2 > &initial_p_unit, InternalData &mdata) const 
template<>  
Point< 1 >  transform_real_to_unit_cell_internal (const Triangulation< 1, 2 >::cell_iterator &cell, const Point< 2 > &p, const Point< 1 > &initial_p_unit, InternalData &mdata) const 
template<>  
Point< 1 >  transform_real_to_unit_cell_internal (const Triangulation< 1, 3 >::cell_iterator &, const Point< 3 > &, const Point< 1 > &, InternalData &) const 
Interface with FEValues  
virtual UpdateFlags  requires_update_flags (const UpdateFlags update_flags) const 
virtual InternalData *  get_data (const UpdateFlags, const Quadrature< dim > &quadrature) const 
virtual Mapping< dim, spacedim >::InternalDataBase *  get_face_data (const UpdateFlags flags, const Quadrature< dim1 > &quadrature) const 
virtual Mapping< dim, spacedim >::InternalDataBase *  get_subface_data (const UpdateFlags flags, const Quadrature< dim1 > &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< dim1 > &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< dim1 > &quadrature, const typename Mapping< dim, spacedim >::InternalDataBase &internal_data,::internal::FEValues::MappingRelatedData< dim, spacedim > &output_data) const 
Interface with FEValues 
Static Protected Attributes  
static const unsigned int  n_shape_functions = GeometryInfo<dim>::vertices_per_cell 
Mapping of the reference to cell to a general quadrilateral/hexahedra by \(d\)linear shape functions.
This function maps the reference (unit) cell to a general grid cell with straight lines in \(d\) dimensions. (Note, however, that in 3D the faces of a general, trilinearly mapped cell may be curved, even if the edges are not). This is the standard mapping used for polyhedral domains. It is also the mapping used throughout deal.II for many functions that two variants, one that allows to pass a mapping argument explicitly and one that simply falls back to the MappingQ1 class declared here.
The shape functions for this mapping are the same as for the finite element FE_Q of order 1. Therefore, coupling these two yields an isoparametric element.
Definition at line 54 of file mapping_q1.h.

protected 
Declare a convenience typedef for the class that describes offsets into quadrature formulas projected onto faces and subfaces.
Definition at line 462 of file mapping_q1.h.
Default constructor.
Definition at line 197 of file mapping_q1.cc.

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 >.
Reimplemented in MappingQ< dim, spacedim >, MappingQEulerian< dim, VECTOR, spacedim >, MappingQ1Eulerian< dim, VECTOR, spacedim >, and MappingC1< dim, spacedim >.
Definition at line 2861 of file mapping_q1.cc.

virtual 
Always returns true
because MappingQ1 preserves vertex locations.
Implements Mapping< dim, spacedim >.
Reimplemented in MappingQEulerian< dim, VECTOR, spacedim >, and MappingQ1Eulerian< dim, VECTOR, spacedim >.

virtual 
Maps the point p
on the unit cell to the corresponding point on the real cell cell
.
cell  Iterator to the cell that will be used to define the mapping. 
p  Location of a point on the reference cell. 
Implements Mapping< dim, spacedim >.
Reimplemented in MappingQ< dim, spacedim >.
Definition at line 2182 of file mapping_q1.cc.

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
.
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 or outside the reference cell (e.g., using GeometryInfo::is_inside_unit_cell()) or whether the exception mentioned above has been thrown.cell  Iterator to the cell that will be used to define the mapping. 
p  Location of a point on the given cell. 
Implements Mapping< dim, spacedim >.
Reimplemented in MappingQ< dim, spacedim >.
Definition at line 2372 of file mapping_q1.cc.

virtual 
Transform a field of vectors or 1differential forms according to the selected MappingType.
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 is to the physical cell through the Jacobian:
\[ \mathbf u(\mathbf x) = J(\hat{\mathbf x})\hat{\mathbf u}(\hat{\mathbf x}). \]
In physics, this is usually referred to as the contravariant transformation. Mathematically, it is the push forward of a vector field.
mapping_covariant:
maps a field of oneforms on the reference cell to a field of oneforms on the physical cell. (Theoretically this would refer to a DerivativeForm<1,dim,1> but we canonically identify this type with a Tensor<1,dim>). Mathematically, it is the pull back of the differential form
\[ \mathbf u(\mathbf x) = J(\hat{\mathbf x})(J(\hat{\mathbf x})^{T} J(\hat{\mathbf x}))^{1}\hat{\mathbf u}(\hat{\mathbf x}). \]
Gradients of scalar differentiable functions are transformed this way.
In the case when dim=spacedim the previous formula reduces to
\[ \mathbf u(\mathbf x) = J(\hat{\mathbf x})^{T}\hat{\mathbf u}(\hat{\mathbf x}) \]
because we assume that the mapping \(\mathbf F_K\) is always invertible, and consequently its Jacobian \(J\) is an invertible matrix.
mapping_piola:
A field of dim1forms on the reference cell is also represented by a vector field, but again transforms differently, namely by the Piola transform \[ \mathbf u(\mathbf x) = \frac{1}{\text{det}\;J(\mathbf x)} J(\mathbf x) \hat{\mathbf u}(\mathbf x). \]
[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 >.
Reimplemented in MappingQ< dim, spacedim >.
Definition at line 2058 of file mapping_q1.cc.

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 spacedimvector valued differentiable functions are transformed this way.
In the case when dim=spacedim the previous formula reduces to
\[ \mathbf T(\mathbf x) = \hat{\mathbf u}(\hat{\mathbf x}) J(\hat{\mathbf x})^{1}. \]
DerivativeForm<1, dim, rank>
. Unfortunately C++ does not allow templatized virtual functions. This is why we identify DerivativeForm<1, dim, 1>
with a Tensor<1,dim>
when using mapping_covariant() in the function transform() above this one.[in]  input  An array (or part of an array) of input objects that should be mapped. 
[in]  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 >.
Reimplemented in MappingQ< dim, spacedim >.
Definition at line 2071 of file mapping_q1.cc.

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 \[ \mathbf T(\mathbf x) = \frac{1}{\text{det}\;J(\mathbf x)} J(\hat{\mathbf x}) \hat{\mathbf T}(\hat{\mathbf x}) J(\hat{\mathbf x})^{1}. \]
[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 >.
Reimplemented in MappingQ< dim, spacedim >.
Definition at line 2084 of file mapping_q1.cc.

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 spacedimvector valued differentiable functions are transformed this way (After subtraction of the product of the derivative with the Jacobian gradient).
In the case when dim=spacedim the previous formula reduces to
\[J^{\dagger} = J^{1}\]
[in]  input  An array (or part of an array) of input objects that should be mapped. 
[in]  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 >.
Reimplemented in MappingQ< dim, spacedim >.
Definition at line 2110 of file mapping_q1.cc.

virtual 
Transform a field of 3differential forms from the reference cell to the physical cell. It is useful to think of \(\mathbf{T}_{ijk} = D^2_{jk} \mathbf u_i\) and \(\mathbf{\hat T}_{IJK} = \hat D^2_{JK} \mathbf{\hat u}_I\), with \(\mathbf u_i\) a vector field.
The mapping 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 \[ \mathbf T_{ijk}(\mathbf x) = \frac{1}{\text{det}\;J(\mathbf x)} J_{iI}(\hat{\mathbf x}) \hat{\mathbf T}_{IJK}(\hat{\mathbf x}) J_{jJ}(\hat{\mathbf x})^{1} J_{kK}(\hat{\mathbf x})^{1}. \]
[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 >.
Reimplemented in MappingQ< dim, spacedim >.
Definition at line 2159 of file mapping_q1.cc.

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.
Implements Mapping< dim, spacedim >.
Definition at line 685 of file mapping_q1.cc.

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 precompute some information already (in accordance with what will be asked of the mapping in the future, as specified by the update flags) and for the given quadrature object. Subsequent calls to transform() or fill_fe_values() and friends will then receive back the object created here (with the same set of update flags and for the same quadrature object). Derived classes can therefore precompute some information in their get_data() function and store it in the internal data object.
The mapping classes do not keep track of the objects created by this function. Ownership will therefore rest with the caller.
An extensive discussion of the interaction between this function and FEValues can be found in the How Mapping, FiniteElement, and FEValues work together documentation module.
update_flags  A set of flags that define what is expected of the mapping class in future calls to transform() or the fill_fe_values() group of functions. This set of flags may contain flags that mappings do not know how to deal with (e.g., for information that is in fact computed by the finite element classes, such as UpdateFlags::update_values). Derived classes will need to store these flags, or at least that subset of flags that will require the mapping to perform any actions in fill_fe_values(), in InternalDataBase::update_each. 
quadrature  The quadrature object for which mapping information will have to be computed. This includes the locations and weights of quadrature points. 
Implements Mapping< dim, spacedim >.
Reimplemented in MappingQ< dim, spacedim >.
Definition at line 745 of file mapping_q1.cc.

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.
update_flags  A set of flags that define what is expected of the mapping class in future calls to transform() or the fill_fe_values() group of functions. This set of flags may contain flags that mappings do not know how to deal with (e.g., for information that is in fact computed by the finite element classes, such as UpdateFlags::update_values). Derived classes will need to store these flags, or at least that subset of flags that will require the mapping to perform any actions in fill_fe_values(), in InternalDataBase::update_each. 
quadrature  The quadrature object for which mapping information will have to be computed. This includes the locations and weights of quadrature points. 
Implements Mapping< dim, spacedim >.
Reimplemented in MappingQ< dim, spacedim >.
Definition at line 759 of file mapping_q1.cc.

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.
update_flags  A set of flags that define what is expected of the mapping class in future calls to transform() or the fill_fe_values() group of functions. This set of flags may contain flags that mappings do not know how to deal with (e.g., for information that is in fact computed by the finite element classes, such as UpdateFlags::update_values). Derived classes will need to store these flags, or at least that subset of flags that will require the mapping to perform any actions in fill_fe_values(), in InternalDataBase::update_each. 
quadrature  The quadrature object for which mapping information will have to be computed. This includes the locations and weights of quadrature points. 
Implements Mapping< dim, spacedim >.
Reimplemented in MappingQ< dim, spacedim >.
Definition at line 776 of file mapping_q1.cc.

protectedvirtual 
Compute information about the mapping from the reference cell to the real cell indicated by the first argument to this function. Derived classes will have to implement this function based on the kind of mapping they represent. It is called by FEValues::reinit().
Conceptually, this function's represents the application of the mapping \(\mathbf x=\mathbf F_K(\hat {\mathbf x})\) from reference coordinates \(\mathbf\in [0,1]^d\) to real space coordinates \(\mathbf x\) for a given cell \(K\). Its purpose is to compute the following kinds of data:
The information computed by this function is used to fill the various member variables of the output argument of this function. Which of the member variables of that structure should be filled is determined by the update flags stored in the Mapping::InternalDataBase object passed to this function.
An extensive discussion of the interaction between this function and FEValues can be found in the How Mapping, FiniteElement, and FEValues work together documentation module.
[in]  cell  The cell of the triangulation for which this function is to compute a mapping from the reference cell to. 
[in]  cell_similarity  Whether or not the cell given as first argument is simply a translation, rotation, etc of the cell for which this function was called the most recent time. This information is computed simply by matching the vertices (as stored by the Triangulation) between the previous and the current cell. The value passed here may be modified by implementations of this function and should then be returned (see the discussion of the return value of this function). 
[in]  quadrature  A reference to the quadrature formula in use for the current evaluation. This quadrature object is the same as the one used when creating the internal_data object. The object is used both to map the location of quadrature points, as well as to compute the JxW values for each quadrature point (which involves the quadrature weights). 
[in]  internal_data  A reference to an object previously created by get_data() and that may be used to store information the mapping can compute once on the reference cell. See the documentation of the Mapping::InternalDataBase class for an extensive description of the purpose of these objects. 
[out]  output_data  A reference to an object whose member variables should be computed. Not all of the members of this argument need to be filled; which ones need to be filled is determined by the update flags stored inside the internal_data object. 
cell_similarity
argument to this function. The returned value will be used for the corresponding argument when FEValues::reinit() calls FiniteElement::fill_fe_values(). In most cases, derived classes will simply want to return the value passed for cell_similarity
. However, implementations of this function may downgrade the level of cell similarity. This is, for example, the case for classes that take not only into account the locations of the vertices of a cell (as reported by the Triangulation), but also other information specific to the mapping. The purpose is that FEValues::reinit() can compute whether a cell is similar to the previous one only based on the cell's vertices, whereas the mapping may also consider displacement fields (e.g., in the MappingQ1Eulerian and MappingFEField classes). In such cases, the mapping may conclude that the previously computed cell similarity is too optimistic, and invalidate it for subsequent use in FiniteElement::fill_fe_values() by returning a less optimistic cell similarity value.internal_data
and 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 1284 of file mapping_q1.cc.

protectedvirtual 
This function is the equivalent to Mapping::fill_fe_values(), but for faces of cells. See there for an extensive discussion of its purpose. It is called by FEFaceValues::reinit().
[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 1669 of file mapping_q1.cc.

protectedvirtual 
This function is the equivalent to Mapping::fill_fe_values(), but for subfaces (i.e., children of faces) of cells. See there for an extensive discussion of its purpose. It is called by FESubfaceValues::reinit().
[in]  cell  The cell of the triangulation for which this function is to compute a mapping from the reference cell to. 
[in]  face_no  The number of the face of the given cell for which information is requested. 
[in]  subface_no  The number of the child of a face of the given cell for which information is requested. 
[in]  quadrature  A reference to the quadrature formula in use for the current evaluation. This quadrature object is the same as the one used when creating the internal_data object. The object is used both to map the location of quadrature points, as well as to compute the JxW values for each quadrature point (which involves the quadrature weights). 
[in]  internal_data  A reference to an object previously created by get_data() and that may be used to store information the mapping can compute once on the reference cell. See the documentation of the Mapping::InternalDataBase class for an extensive description of the purpose of these objects. 
[out]  output_data  A reference to an object whose member variables should be computed. Not all of the members of this argument need to be filled; which ones need to be filled is determined by the update flags stored inside the internal_data object. 
Implements Mapping< dim, spacedim >.
Definition at line 1713 of file mapping_q1.cc.

protected 
Compute an initial guess to pass to the Newton method in transform_real_to_unit_cell. For the initial guess we proceed in the following way:
Definition at line 2325 of file mapping_q1.cc.

protected 
Compute shape values and/or derivatives.
Calls either the compute_shapes_virtual
of this class or that of the derived class, depending on whether data.is_mapping_q1_data
equals true
or false
.
Definition at line 204 of file mapping_q1.cc.

protectedvirtual 
Compute shape values and/or derivatives.
Definition at line 673 of file mapping_q1.cc.

protected 
Transforms 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 2215 of file mapping_q1.cc.

protected 
Transforms 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 2505 of file mapping_q1.cc.

protectedvirtual 
Computes the support points of the mapping. For MappingQ1
these are the vertices. However, other classes may override this function. In particular, the MappingQ1Eulerian class does exactly this by not computing the support points from the geometry of the current cell but instead evaluating an externally given displacement field in addition to the geometry of the cell.
Reimplemented in MappingQ< dim, spacedim >, and MappingQEulerian< dim, VECTOR, spacedim >.
Definition at line 1267 of file mapping_q1.cc.

staticprotected 
Number of shape functions. Is simply the number of vertices per cell for the Q1 mapping.
Definition at line 535 of file mapping_q1.h.