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//
// Copyright (C) 2001 - 2016 by the deal.II authors
//
// This file is part of the deal.II library.
//
// The deal.II library is free software; you can use it, redistribute
// it, and/or modify it under the terms of the GNU Lesser General
// Public License as published by the Free Software Foundation; either
// version 2.1 of the License, or (at your option) any later version.
// The full text of the license can be found in the file LICENSE at
// the top level of the deal.II distribution.
//
// ---------------------------------------------------------------------
#ifndef dealii__mapping_fe_h
#define dealii__mapping_fe_h
#include <deal.II/base/config.h>
#include <deal.II/base/table.h>
#include <deal.II/fe/mapping.h>
#include <deal.II/fe/fe.h>
#include <deal.II/base/qprojector.h>
#include <deal.II/base/thread_management.h>
DEAL_II_NAMESPACE_OPEN
/*!@addtogroup mapping */
/*@{*/
/**
* The MappingFEField is a generalization of the MappingQEulerian class, for
* arbitrary vector finite elements. The two main differences are that this
* class uses a vector of absolute positions instead of a vector of
* displacements, and it allows for arbitrary FiniteElement types, instead of
* only FE_Q.
*
* This class effectively decouples the topology from the geometry, by
* relegating all geometrical information to some components of a
* FiniteElement vector field. The components that are used for the geometry
* can be arbitrarily selected at construction time.
*
* The idea is to consider the Triangulation as a parameter configuration
* space, on which we construct an arbitrary geometrical mapping, using the
* instruments of the deal.II library: a vector of degrees of freedom, a
* DoFHandler associated to the geometry of the problem and a ComponentMask
* that tells us which components of the FiniteElement to use for the mapping.
*
* Typically, the DoFHandler operates on a finite element that is constructed
* as a system element (FESystem()) from continuous FE_Q() (for iso-parametric
* discretizations) or FE_Bernstein() (for iso-geometric discretizations)
* objects. An example is shown below:
*
* @code
* const FE_Q<dim,spacedim> feq(1);
* const FESystem<dim,spacedim> fesystem(feq, spacedim);
* DoFHandler<dim,spacedim> dhq(triangulation);
* dhq.distribute_dofs(fesystem);
* const ComponentMask mask(spacedim, true);
* Vector<double> eulerq(dhq.n_dofs());
* // Fills the euler vector with information from the Triangulation
* VectorTools::get_position_vector(dhq, eulerq, mask);
* MappingFEField<dim,spacedim> map(dhq, eulerq, mask);
* @endcode
*
* @author Luca Heltai, Marco Tezzele 2013, 2015
*/
template <int dim, int spacedim=dim,
typename VectorType=Vector<double>,
typename DoFHandlerType=DoFHandler<dim,spacedim> >
class MappingFEField : public Mapping<dim,spacedim>
{
public:
/**
* Constructor. The first argument is a VectorType that specifies the
* transformation of the domain from the reference to the current
* configuration.
*
* In general this class decouples geometry from topology, allowing users to
* define geometries which are only topologically equivalent to the
* underlying Triangulation, but which may otherwise be arbitrary.
* Differently from what happens in MappingQEulerian, the FiniteElement
* field which is passed to the constructor is interpreted as an absolute
* geometrical configuration, therefore one has to make sure that the
* euler_vector actually represents a valid geometry (i.e., one with no
* inverted cells, or with no zero-volume cells).
*
* If the underlying FiniteElement is a system of FE_Q(), and euler_vector
* is initialized using VectorTools::get_position_vector(), then this class
* is in all respects identical to MappingQ().
*
* The optional ComponentMask argument can be used to specify what
* components of the FiniteElement to use for the geometrical
* transformation. If no mask is specified at construction time, then a
* default one is used, which makes this class works in the same way of
* MappingQEulerian(), i.e., the first spacedim components of the
* FiniteElement are assumed to represent the geometry of the problem.
*
* Notice that if a mask is specified, it has to match in size the
* underlying FiniteElement, and it has to have exactly spacedim non-zero
* elements, indicating the components (in order) of the FiniteElement which
* will be used for the geometry.
*
* If an incompatible mask is passed, an exception is thrown.
*/
MappingFEField (const DoFHandlerType &euler_dof_handler,
const VectorType &euler_vector,
const ComponentMask mask = ComponentMask());
/**
* Copy constructor.
*/
MappingFEField (const MappingFEField<dim,spacedim,VectorType,DoFHandlerType> &mapping);
/**
* Return a pointer to a copy of the present object. The caller of this copy
* then assumes ownership of it.
*/
virtual
Mapping<dim,spacedim> *clone () const;
/**
* Always returns @p false.
*/
virtual
bool preserves_vertex_locations () const;
/**
* @name Mapping points between reference and real cells
* @{
*/
// for documentation, see the Mapping base class
virtual
Point<spacedim>
transform_unit_to_real_cell (const typename Triangulation<dim,spacedim>::cell_iterator &cell,
const Point<dim> &p) const;
// for documentation, see the Mapping base class
virtual
Point<dim>
transform_real_to_unit_cell (const typename Triangulation<dim,spacedim>::cell_iterator &cell,
const Point<spacedim> &p) const;
/**
* @}
*/
/**
* @name Functions to transform tensors from reference to real coordinates
* @{
*/
// for documentation, see the Mapping base class
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;
// for documentation, see the Mapping base class
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;
// for documentation, see the Mapping base class
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;
// for documentation, see the Mapping base class
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;
// for documentation, see the Mapping base class
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;
/**
* @}
*/
/**
* Return the degree of the mapping, i.e. the value which was passed to the
* constructor.
*/
unsigned int get_degree () const;
/**
* Return the ComponentMask of the mapping, i.e. which components to use for
* the mapping.
*/
ComponentMask get_component_mask () const;
/**
* Exception
*/
DeclException0(ExcInactiveCell);
private:
/**
* @name Interface with FEValues
* @{
*/
// documentation can be found in Mapping::requires_update_flags()
virtual
UpdateFlags
requires_update_flags (const UpdateFlags update_flags) const;
public:
/**
* Storage for internal data of this mapping. See Mapping::InternalDataBase
* for an extensive description.
*
* This includes data that is computed once when the object is created (in
* get_data()) as well as data the class wants to store from between the
* call to fill_fe_values(), fill_fe_face_values(), or
* fill_fe_subface_values() until possible later calls from the finite
* element to functions such as transform(). The latter class of member
* variables are marked as 'mutable', along with scratch arrays.
*/
class InternalData : public Mapping<dim,spacedim>::InternalDataBase
{
public:
/**
* Constructor.
*/
InternalData(const FiniteElement<dim,spacedim> &fe,
const ComponentMask mask);
/**
* Shape function at quadrature point. Shape functions are in tensor
* product order, so vertices must be reordered to obtain transformation.
*/
const double &shape (const unsigned int qpoint,
const unsigned int shape_nr) const;
/**
* Shape function at quadrature point. See above.
*/
double &shape (const unsigned int qpoint,
const unsigned int shape_nr);
/**
* Gradient of shape function in quadrature point. See above.
*/
const Tensor<1,dim> &derivative (const unsigned int qpoint,
const unsigned int shape_nr) const;
/**
* Gradient of shape function in quadrature point. See above.
*/
Tensor<1,dim> &derivative (const unsigned int qpoint,
const unsigned int shape_nr);
/**
* Second derivative of shape function in quadrature point. See above.
*/
const Tensor<2,dim> &second_derivative (const unsigned int qpoint,
const unsigned int shape_nr) const;
/**
* Second derivative of shape function in quadrature point. See above.
*/
Tensor<2,dim> &second_derivative (const unsigned int qpoint,
const unsigned int shape_nr);
/**
* Third derivative of shape function in quadrature point. See above.
*/
const Tensor<3,dim> &third_derivative (const unsigned int qpoint,
const unsigned int shape_nr) const;
/**
* Fourth derivative of shape function in quadrature point. See above.
*/
Tensor<3,dim> &third_derivative (const unsigned int qpoint,
const unsigned int shape_nr);
/**
* Fourth derivative of shape function in quadrature point. See above.
*/
const Tensor<4,dim> &fourth_derivative (const unsigned int qpoint,
const unsigned int shape_nr) const;
/**
* Third derivative of shape function in quadrature point. See above.
*/
Tensor<4,dim> &fourth_derivative (const unsigned int qpoint,
const unsigned int shape_nr);
/**
* Return an estimate (in bytes) or the memory consumption of this object.
*/
virtual std::size_t memory_consumption () const;
/**
* Values of shape functions. Access by function @p shape.
*
* Computed once.
*/
std::vector<double> shape_values;
/**
* Values of shape function derivatives. Access by function @p derivative.
*
* Computed once.
*/
std::vector<Tensor<1,dim> > shape_derivatives;
/**
* Values of shape function second derivatives. Access by function @p
* second_derivative.
*
* Computed once.
*/
std::vector<Tensor<2,dim> > shape_second_derivatives;
/**
* Values of shape function third derivatives. Access by function @p
* third_derivative.
*
* Computed once.
*/
std::vector<Tensor<3,dim> > shape_third_derivatives;
/**
* Values of shape function fourth derivatives. Access by function @p
* fourth_derivative.
*
* Computed once.
*/
std::vector<Tensor<4,dim> > shape_fourth_derivatives;
/**
* Unit tangential vectors. Used for the computation of boundary forms and
* normal vectors.
*
* This vector has (dim-1)GeometryInfo::faces_per_cell entries. The first
* GeometryInfo::faces_per_cell contain the vectors in the first
* tangential direction for each face; the second set of
* GeometryInfo::faces_per_cell entries contain the vectors in the second
* tangential direction (only in 3d, since there we have 2 tangential
* directions per face), etc.
*
* Filled once.
*/
std::vector<std::vector<Tensor<1,dim> > > unit_tangentials;
/**
* Number of shape functions. If this is a Q1 mapping, then it is simply
* the number of vertices per cell. However, since also derived classes
* use this class (e.g. the Mapping_Q() class), the number of shape
* functions may also be different.
*/
unsigned int n_shape_functions;
/**
* Stores the mask given at construction time. If no mask was specified at
* construction time, then a default one is used, which makes this class
* works in the same way of MappingQEulerian(), i.e., the first spacedim
* components of the FiniteElement are used for the euler_vector and the
* euler_dh.
*
* If a mask is specified, then it has to match the underlying
* FiniteElement, and it has to have exactly spacedim non-zero elements,
* indicating the components (in order) of the FiniteElement which will be
* used for the euler vector and the euler dof handler.
*/
ComponentMask mask;
/**
* Tensors of covariant transformation at each of the quadrature points.
* The matrix stored is the Jacobian * G^{-1}, where G = Jacobian^{t} *
* Jacobian, is the first fundamental form of the map; if dim=spacedim
* then it reduces to the transpose of the inverse of the Jacobian matrix,
* which itself is stored in the @p contravariant field of this structure.
*
* Computed on each cell.
*/
mutable std::vector<DerivativeForm<1,dim, spacedim > > covariant;
/**
* Tensors of contravariant transformation at each of the quadrature
* points. The contravariant matrix is the Jacobian of the transformation,
* i.e. $J_{ij}=dx_i/d\hat x_j$.
*
* Computed on each cell.
*/
mutable std::vector< DerivativeForm<1,dim,spacedim> > contravariant;
/**
* The determinant of the Jacobian in each quadrature point. Filled if
* #update_volume_elements.
*/
mutable std::vector<double> volume_elements;
/**
* Auxiliary vectors for internal use.
*/
mutable std::vector<std::vector<Tensor<1,spacedim> > > aux;
/**
* Storage for the indices of the local degrees of freedom.
*/
mutable std::vector<types::global_dof_index> local_dof_indices;
/**
* Storage for local degrees of freedom.
*/
mutable std::vector<double> local_dof_values;
};
private:
// documentation can be found in Mapping::get_data()
virtual
InternalData *
get_data (const UpdateFlags,
const Quadrature<dim> &quadrature) const;
// documentation can be found in Mapping::get_face_data()
virtual
typename Mapping<dim,spacedim>::InternalDataBase *
get_face_data (const UpdateFlags flags,
const Quadrature<dim-1>& quadrature) const;
// documentation can be found in Mapping::get_subface_data()
virtual
typename Mapping<dim,spacedim>::InternalDataBase *
get_subface_data (const UpdateFlags flags,
const Quadrature<dim-1>& quadrature) const;
// documentation can be found in Mapping::fill_fe_values()
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;
// documentation can be found in Mapping::fill_fe_face_values()
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;
// documentation can be found in Mapping::fill_fe_subface_values()
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;
/**
* @}
*/
/**
* Reference to the vector of shifts.
*/
SmartPointer<const VectorType, MappingFEField<dim,spacedim,VectorType,DoFHandlerType> > euler_vector;
/**
* A FiniteElement object which is only needed in 3D, since it knows how to
* reorder shape functions/DoFs on non-standard faces. This is used to
* reorder support points in the same way. We could make this a pointer to
* prevent construction in 1D and 2D, but since memory and time requirements
* are not particularly high this seems unnecessary at the moment.
*/
SmartPointer<const FiniteElement<dim,spacedim>, MappingFEField<dim,spacedim,VectorType,DoFHandlerType> > fe;
/**
* Pointer to the DoFHandler to which the mapping vector is associated.
*/
SmartPointer<const DoFHandlerType,MappingFEField<dim,spacedim,VectorType,DoFHandlerType> > euler_dof_handler;
private:
/**
* Transforms a point @p p on the unit cell to the point @p p_real on the
* real cell @p cell and returns @p p_real.
*
* This function is called by @p transform_unit_to_real_cell and multiple
* times (through the Newton iteration) by @p
* transform_real_to_unit_cell_internal.
*
* Takes a reference to an @p InternalData that must already include the
* shape values at point @p p and the mapping support points of the cell.
*
* This @p InternalData argument avoids multiple computations of the shape
* values at point @p p and especially multiple computations of the mapping
* support points.
*/
Point<spacedim>
do_transform_unit_to_real_cell (const InternalData &mdata) const;
/**
* Transforms the point @p p on the real cell to the corresponding point on
* the unit cell @p cell by a Newton iteration.
*
* Takes a reference to an @p InternalData that is assumed to be previously
* created by the @p get_data function with @p UpdateFlags including @p
* update_transformation_values and @p update_transformation_gradients and a
* one point Quadrature that includes the given initial guess for the
* transformation @p initial_p_unit. Hence this function assumes that @p
* mdata already includes the transformation shape values and gradients
* computed at @p initial_p_unit.
*
* @p mdata will be changed by this function.
*/
Point<dim>
do_transform_real_to_unit_cell (const typename Triangulation<dim,spacedim>::cell_iterator &cell,
const Point<spacedim> &p,
const Point<dim> &initial_p_unit,
InternalData &mdata) const;
/**
* Update internal degrees of freedom.
*/
void update_internal_dofs(const typename Triangulation<dim,spacedim>::cell_iterator &cell,
const typename MappingFEField<dim, spacedim>::InternalData &data) const;
/**
* See the documentation of the base class for detailed information.
*/
virtual void
compute_shapes_virtual (const std::vector<Point<dim> > &unit_points,
typename MappingFEField<dim, spacedim>::InternalData &data) const;
/*
* Which components to use for the mapping.
*/
const ComponentMask fe_mask;
/**
* Mapping between indices in the FE space and the real space. This vector
* contains one index for each component of the finite element space. If the
* index is one for which the ComponentMask which is used to construct this
* element is false, then numbers::invalid_unsigned_int is returned,
* otherwise the component in real space is returned. For example, if we
* construct the mapping using ComponentMask(spacedim, true), then this
* vector contains {0,1,2} in spacedim = 3.
*/
std::vector<unsigned int> fe_to_real;
void
compute_data (const UpdateFlags update_flags,
const Quadrature<dim> &q,
const unsigned int n_original_q_points,
InternalData &data) const;
void
compute_face_data (const UpdateFlags update_flags,
const Quadrature<dim> &q,
const unsigned int n_original_q_points,
InternalData &data) const;
/**
* Declare other MappingFEField classes friends.
*/
template <int,int,class,class> friend class MappingFEField;
};
/*@}*/
/* -------------- declaration of explicit specializations ------------- */
#ifndef DOXYGEN
#endif // DOXYGEN
DEAL_II_NAMESPACE_CLOSE
#endif
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