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Reference documentation for deal.II version 9.3.3
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Reference documentation for deal.II version 9.3.3
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#include <deal.II/fe/mapping_q.h>
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 override |
| virtual BoundingBox< spacedim > | get_bounding_box (const typename Triangulation< dim, spacedim >::cell_iterator &cell) const override |
| virtual bool | is_compatible_with (const ReferenceCell &reference_cell) const override |
| 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 |
| virtual void | transform (const ArrayView< const Tensor< 1, dim > > &input, const MappingKind kind, const typename Mapping< dim, spacedim >::InternalDataBase &internal, const ArrayView< Tensor< 1, spacedim > > &output) const override |
| virtual void | transform (const ArrayView< const DerivativeForm< 1, dim, spacedim > > &input, const MappingKind kind, const typename Mapping< dim, spacedim >::InternalDataBase &internal, const ArrayView< Tensor< 2, spacedim > > &output) const override |
| virtual void | transform (const ArrayView< const Tensor< 2, dim > > &input, const MappingKind kind, const typename Mapping< dim, spacedim >::InternalDataBase &internal, const ArrayView< Tensor< 2, spacedim > > &output) const override |
| virtual void | transform (const ArrayView< const DerivativeForm< 2, dim, spacedim > > &input, const MappingKind kind, const typename Mapping< dim, spacedim >::InternalDataBase &internal, const ArrayView< Tensor< 3, spacedim > > &output) const override |
| virtual void | transform (const ArrayView< const Tensor< 3, dim > > &input, const MappingKind kind, const typename Mapping< dim, spacedim >::InternalDataBase &internal, const ArrayView< Tensor< 3, spacedim > > &output) const override |
| virtual std::unique_ptr< Mapping< dim, spacedim > > | clone () const override |
| virtual boost::container::small_vector< Point< spacedim >, GeometryInfo< dim >::vertices_per_cell > | get_vertices (const typename Triangulation< dim, spacedim >::cell_iterator &cell) const |
| virtual Point< spacedim > | get_center (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const bool map_center_of_reference_cell=true) const |
Mapping points between reference and real cells | |
| virtual void | transform_points_real_to_unit_cell (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const ArrayView< const Point< spacedim > > &real_points, const ArrayView< Point< dim > > &unit_points) const |
| Point< dim - 1 > | project_real_point_to_unit_point_on_face (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const unsigned int face_no, const Point< spacedim > &p) const |
Static Public Member Functions | |
Exceptions | |
| static ::ExceptionBase & | ExcInvalidData () |
| static ::ExceptionBase & | ExcTransformationFailed () |
| static ::ExceptionBase & | ExcDistortedMappedCell (Point< spacedim > arg1, double arg2, int arg3) |
Protected Member Functions | |
Interface with FEValues | |
| virtual UpdateFlags | requires_update_flags (const UpdateFlags update_flags) const override |
| virtual std::unique_ptr< typename Mapping< dim, spacedim >::InternalDataBase > | get_data (const UpdateFlags, const Quadrature< dim > &quadrature) const override |
| virtual std::unique_ptr< typename Mapping< dim, spacedim >::InternalDataBase > | get_face_data (const UpdateFlags flags, const hp::QCollection< dim - 1 > &quadrature) const override |
| virtual std::unique_ptr< typename Mapping< dim, spacedim >::InternalDataBase > | get_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 hp::QCollection< 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 |
Interface with FEValues | |
| virtual std::unique_ptr< InternalDataBase > | get_face_data (const UpdateFlags update_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::FEValuesImplementation::MappingRelatedData< dim, spacedim > &output_data) const=0 |
| virtual void | fill_fe_face_values (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const unsigned int face_no, const hp::QCollection< dim - 1 > &quadrature, const typename Mapping< dim, spacedim >::InternalDataBase &internal_data, ::internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &output_data) const |
| virtual void | fill_fe_face_values (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const unsigned int face_no, const Quadrature< dim - 1 > &quadrature, const typename Mapping< dim, spacedim >::InternalDataBase &internal_data, internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &output_data) const |
| virtual void | fill_fe_subface_values (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const unsigned int face_no, const unsigned int subface_no, const Quadrature< dim - 1 > &quadrature, const typename Mapping< dim, spacedim >::InternalDataBase &internal_data, ::internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &output_data) const=0 |
Protected Attributes | |
| const unsigned int | polynomial_degree |
| const bool | use_mapping_q_on_all_cells |
| std::shared_ptr< const MappingQGeneric< dim, spacedim > > | q1_mapping |
| std::shared_ptr< const MappingQGeneric< dim, spacedim > > | qp_mapping |
Subscriptor functionality | |
Classes derived from Subscriptor provide a facility to subscribe to this object. This is mostly used by the SmartPointer class. | |
| void | subscribe (std::atomic< bool > *const validity, const std::string &identifier="") const |
| void | unsubscribe (std::atomic< bool > *const validity, const std::string &identifier="") const |
| unsigned int | n_subscriptions () const |
| template<typename StreamType > | |
| void | list_subscribers (StreamType &stream) const |
| void | list_subscribers () const |
| template<class Archive > | |
| void | serialize (Archive &ar, const unsigned int version) |
| static ::ExceptionBase & | ExcInUse (int arg1, std::string arg2, std::string arg3) |
| static ::ExceptionBase & | ExcNoSubscriber (std::string arg1, std::string arg2) |
| using | map_value_type = decltype(counter_map)::value_type |
| using | map_iterator = decltype(counter_map)::iterator |
| std::atomic< unsigned int > | counter |
| std::map< std::string, unsigned int > | counter_map |
| std::vector< std::atomic< bool > * > | validity_pointers |
| const std::type_info * | object_info |
| static std::mutex | mutex |
| void | check_no_subscribers () const noexcept |
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:
For the behavior of the mapping and convergence rates in case of mixing different manifolds, please consult the respective section of MappingQGeneric.
Definition at line 80 of file mapping_q.h.
| MappingQ< dim, spacedim >::MappingQ | ( | const unsigned int | polynomial_degree, |
| const bool | use_mapping_q_on_all_cells = false |
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| ) |
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.
| MappingQ< dim, spacedim >::MappingQ | ( | const MappingQ< dim, spacedim > & | mapping | ) |
Copy constructor.
Return the degree of the mapping, i.e. the value which was passed to the constructor.
Always returns true because the default implementation of functions in this class preserves vertex locations.
Implements Mapping< dim, dim >.
Reimplemented in MappingQEulerian< dim, VectorType, spacedim >, and MappingQEulerian< dim, VectorType, spacedim >.
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overridevirtual |
Return the bounding box of a mapped cell.
If you are using a (bi-,tri-)linear mapping that preserves vertex locations, this function simply returns the value also produced by cell->bounding_box(). However, there are also mappings that add displacements or choose completely different locations, e.g., MappingQEulerian, MappingQ1Eulerian, or MappingFEField.
For linear mappings, this function returns the bounding box containing all the vertices of the cell, as returned by the get_vertices() method. For higher order mappings defined through support points, the bounding box is only guaranteed to contain all the support points, and it is, in general, only an approximation of the true bounding box, which may be larger.
| [in] | cell | The cell for which you want to compute the bounding box |
Reimplemented from Mapping< dim, dim >.
Reimplemented in MappingQEulerian< dim, VectorType, spacedim >, and MappingQEulerian< dim, VectorType, spacedim >.
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overridevirtual |
Returns if this instance of Mapping is compatible with the type of cell in reference_cell.
Implements Mapping< dim, dim >.
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overridevirtual |
Transform the point p on the unit cell to the point p_real on the real cell cell and returns p_real.
Implements Mapping< dim, dim >.
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overridevirtual |
Transform 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.
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, dim >.
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Transform a field of vectors or 1-differential forms according to the selected MappingKind.
mapping_bdm, mapping_nedelec, etc. This alias should be preferred to using the kinds below.The mapping kinds currently implemented by derived classes are:
mapping_contravariant: maps a vector field on the reference cell to the physical cell through the Jacobian:
\[ \mathbf u(\mathbf x) = J(\hat{\mathbf x})\hat{\mathbf u}(\hat{\mathbf x}). \]
In physics, this is usually referred to as the contravariant transformation. Mathematically, it is the push forward of a vector field.
mapping_covariant: maps a field of one-forms on the reference cell to a field of one-forms on the physical cell. (Theoretically this would refer to a DerivativeForm<1,dim,1> but we canonically identify this type with a Tensor<1,dim>). Mathematically, it is the pull back of the differential form
\[ \mathbf u(\mathbf x) = J(\hat{\mathbf x})(J(\hat{\mathbf x})^{T} J(\hat{\mathbf x}))^{-1}\hat{\mathbf u}(\hat{\mathbf x}). \]
Gradients of scalar differentiable functions are transformed this way.
In the case when dim=spacedim the previous formula reduces to
\[ \mathbf u(\mathbf x) = J(\hat{\mathbf x})^{-T}\hat{\mathbf u}(\hat{\mathbf x}) \]
because we assume that the mapping \(\mathbf F_K\) is always invertible, and consequently its Jacobian \(J\) is an invertible matrix.
mapping_piola: A field of dim-1-forms on the reference cell is also represented by a vector field, but again transforms differently, namely by the Piola transform \[ \mathbf u(\mathbf x) = \frac{1}{\text{det}\;J(\hat{\mathbf x})} J(\hat{\mathbf x}) \hat{\mathbf u}(\hat{\mathbf x}). \]
| [in] | input | An array (or part of an array) of input objects that should be mapped. |
| [in] | kind | 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, dim >.
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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 kinds currently implemented by derived classes are:
mapping_covariant: maps a field of forms on the reference cell to a field of forms on the physical cell. Mathematically, it is the pull back of the differential form
\[ \mathbf T(\mathbf x) = \hat{\mathbf T}(\hat{\mathbf x}) J(\hat{\mathbf x})(J(\hat{\mathbf x})^{T} J(\hat{\mathbf x}))^{-1}. \]
Jacobians of spacedim-vector valued differentiable functions are transformed this way.
In the case when dim=spacedim the previous formula reduces to
\[ \mathbf T(\mathbf x) = \hat{\mathbf u}(\hat{\mathbf x}) J(\hat{\mathbf x})^{-1}. \]
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] | kind | 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, dim >.
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Transform a tensor field from the reference cell to the physical cell. These tensors are usually the Jacobians in the reference cell of vector fields that have been pulled back from the physical cell. The mapping kinds currently implemented by derived classes are:
mapping_contravariant_gradient: it assumes \(\mathbf u(\mathbf x)
= J \hat{\mathbf u}\) so that \[ \mathbf T(\mathbf x) = J(\hat{\mathbf x}) \hat{\mathbf T}(\hat{\mathbf x}) J(\hat{\mathbf x})^{-1}. \]
mapping_covariant_gradient: it assumes \(\mathbf u(\mathbf x) =
J^{-T} \hat{\mathbf u}\) so that \[ \mathbf T(\mathbf x) = J(\hat{\mathbf x})^{-T} \hat{\mathbf T}(\hat{\mathbf x}) J(\hat{\mathbf x})^{-1}. \]
mapping_piola_gradient: it assumes \(\mathbf u(\mathbf x) =
\frac{1}{\text{det}\;J(\hat{\mathbf x})} J(\hat{\mathbf x}) \hat{\mathbf
u}(\hat{\mathbf x})\) so that \[ \mathbf T(\mathbf x) = \frac{1}{\text{det}\;J(\hat{\mathbf x})} J(\hat{\mathbf x}) \hat{\mathbf T}(\hat{\mathbf x}) J(\hat{\mathbf x})^{-1}. \]
| [in] | input | An array (or part of an array) of input objects that should be mapped. |
| [in] | kind | 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, dim >.
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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 kinds currently implemented by derived classes are:
mapping_covariant_gradient: maps a field of forms on the reference cell to a field of forms on the physical cell. Mathematically, it is the pull back of the differential form
\[ \mathbf T_{ijk}(\mathbf x) = \hat{\mathbf T}_{iJK}(\hat{\mathbf x}) J_{jJ}^{\dagger} J_{kK}^{\dagger}\]
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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
\[J^{\dagger} = J^{-1}\]
| [in] | input | An array (or part of an array) of input objects that should be mapped. |
| [in] | kind | 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, dim >.
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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 kinds currently implemented by derived classes are:
mapping_contravariant_hessian: it assumes \(\mathbf u_i(\mathbf x)
= J_{iI} \hat{\mathbf u}_I\) so that \[ \mathbf T_{ijk}(\mathbf x) = J_{iI}(\hat{\mathbf x}) \hat{\mathbf T}_{IJK}(\hat{\mathbf x}) J_{jJ}(\hat{\mathbf x})^{-1} J_{kK}(\hat{\mathbf x})^{-1}. \]
mapping_covariant_hessian: it assumes \(\mathbf u_i(\mathbf x) =
J_{iI}^{-T} \hat{\mathbf u}_I\) so that \[ \mathbf T_{ijk}(\mathbf x) = J_iI(\hat{\mathbf x})^{-1} \hat{\mathbf T}_{IJK}(\hat{\mathbf x}) J_{jJ}(\hat{\mathbf x})^{-1} J_{kK}(\hat{\mathbf x})^{-1}. \]
mapping_piola_hessian: it assumes \(\mathbf u_i(\mathbf x) =
\frac{1}{\text{det}\;J(\hat{\mathbf x})} J_{iI}(\hat{\mathbf x})
\hat{\mathbf u}(\hat{\mathbf x})\) so that \[ \mathbf T_{ijk}(\mathbf x) = \frac{1}{\text{det}\;J(\hat{\mathbf x})} J_{iI}(\hat{\mathbf x}) \hat{\mathbf T}_{IJK}(\hat{\mathbf x}) J_{jJ}(\hat{\mathbf x})^{-1} J_{kK}(\hat{\mathbf x})^{-1}. \]
| [in] | input | An array (or part of an array) of input objects that should be mapped. |
| [in] | kind | 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, dim >.
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overridevirtual |
Return a pointer to a copy of the present object. The caller of this copy then assumes ownership of it.
Implements Mapping< dim, dim >.
Reimplemented in MappingC1< dim, spacedim >, MappingQEulerian< dim, VectorType, spacedim >, and MappingQEulerian< dim, VectorType, spacedim >.
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overrideprotectedvirtual |
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, dim >.
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overrideprotectedvirtual |
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.
| 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, dim >.
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overrideprotectedvirtual |
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. |
Reimplemented from Mapping< dim, dim >.
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overrideprotectedvirtual |
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, dim >.
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overrideprotectedvirtual |
Reimplemented in MappingQEulerian< dim, VectorType, spacedim >, and MappingQEulerian< dim, VectorType, spacedim >.
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overrideprotectedvirtual |
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overrideprotectedvirtual |
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virtualinherited |
Return the mapped vertices of a cell.
Most of the time, these values will simply be the coordinates of the vertices of a cell as returned by cell->vertex(v) for vertex v, i.e., information stored by the triangulation. However, there are also mappings that add displacements or choose completely different locations, e.g., MappingQEulerian, MappingQ1Eulerian, or MappingFEField.
The default implementation of this function simply returns the information stored by the triangulation, i.e., cell->vertex(v).
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virtualinherited |
Return the mapped center of a cell.
If you are using a (bi-,tri-)linear mapping that preserves vertex locations, this function simply returns the value also produced by cell->center(). However, there are also mappings that add displacements or choose completely different locations, e.g., MappingQEulerian, MappingQ1Eulerian, or MappingFEField, and mappings based on high order polynomials, for which the center may not coincide with the average of the vertex locations.
By default, this function returns the push forward of the center of the reference cell. If the parameter map_center_of_reference_cell is set to false, than the return value will be the average of the vertex locations, as returned by the get_vertices() method.
| [in] | cell | The cell for which you want to compute the center |
| [in] | map_center_of_reference_cell | A flag that switches the algorithm for the computation of the cell center from transform_unit_to_real_cell() applied to the center of the reference cell to computing the vertex averages. |
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Map multiple points from the real point locations to points in reference locations. The functionality is essentially the same as looping over all points and calling the Mapping::transform_real_to_unit_cell() function for each point individually, but it can be much faster for certain mappings that implement a more specialized version such as MappingQGeneric. The only difference in behavior is that this function will never throw an ExcTransformationFailed() exception. If the transformation fails for real_points[i], the returned unit_points[i] contains std::numeric_limits<double>::infinity() as the first entry.
Reimplemented in MappingQGeneric< dim, spacedim >.
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inherited |
Transform the point p on the real cell to the corresponding point on the reference cell, and then project this point to a (dim-1)-dimensional point in the coordinate system of the face with the given face number face_no. Ideally the point p is near the face face_no, but any point in the cell can technically be projected.
This function does not make physical sense when dim=1, so it throws an exception in this case.
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Exception
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staticinherited |
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.
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staticinherited |
deal.II assumes the Jacobian determinant to be positive. When the cell geometry is distorted under the image of the mapping, the mapping becomes invalid and this exception is thrown.
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protectedvirtualinherited |
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protectedpure virtualinherited |
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 output_data objects. In other words, if an implementation of this function knows that it has written a piece of data into the output argument in a previous call, then there is no need to copy it there again in a later call if the implementation knows that this is the same value. Implemented in MappingFE< dim, spacedim >, MappingManifold< dim, spacedim >, and MappingQGeneric< dim, spacedim >.
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protectedvirtualinherited |
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. |
Reimplemented in MappingFE< dim, spacedim >, MappingManifold< dim, spacedim >, and MappingQGeneric< dim, spacedim >.
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protectedvirtualinherited |
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protectedpure virtualinherited |
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. |
Implemented in MappingFE< dim, spacedim >, MappingManifold< dim, spacedim >, and MappingQGeneric< dim, spacedim >.
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 336 of file mapping_q.h.
If this flag is set true then MappingQ is used on all cells, not only on boundary cells.
Definition at line 342 of file mapping_q.h.
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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.
Definition at line 361 of file mapping_q.h.
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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).
Definition at line 377 of file mapping_q.h.
1.9.4