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//
// Copyright (C) 1999 - 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__fe_system_h
#define dealii__fe_system_h
/*---------------------------- fe_system.h ---------------------------*/
#include <deal.II/base/config.h>
#include <deal.II/base/thread_management.h>
#include <deal.II/fe/fe.h>
#include <vector>
#include <utility>
DEAL_II_NAMESPACE_OPEN
/**
* This class provides an interface to group several elements together into
* one. To the outside world, the resulting object looks just like a usual
* finite element object, which is composed of several other finite elements
* that are possibly of different type. The result is then a vector-valued
* finite element. %Vector valued elements are discussed in a number of
* tutorial programs, for example step-8, step-20, step-21, and in particular
* in the
* @ref vector_valued
* module.
*
* @dealiiVideoLecture{19,20}
*
* <h3>FESystem, components and blocks</h3>
*
* An FESystem, except in the most trivial case, produces a vector-valued
* finite element with several components. The number of components
* n_components() corresponds to the dimension of the solution function in the
* PDE system, and correspondingly also to the number of equations your PDE
* system has. For example, the mixed Laplace system covered in step-20 has
* $d+1$ components in $d$ space dimensions: the scalar pressure and the $d$
* components of the velocity vector. Similarly, the elasticity equation
* covered in step-8 has $d$ components in $d$ space dimensions. In general,
* the number of components of a FESystem element is the accumulated number of
* components of all base elements times their multiplicities. A bit more on
* components is also given in the
* @ref GlossComponent "glossary entry on components".
*
* While the concept of components is important from the viewpoint of a
* partial differential equation, the finite element side looks a bit
* different Since not only FESystem, but also vector-valued elements like
* FE_RaviartThomas, have several components. The concept needed here is a
* @ref GlossBlock "block".
* Each block encompasses the set of degrees of freedom associated with a
* single base element of an FESystem, where base elements with multiplicities
* count multiple times. These blocks are usually addressed using the
* information in DoFHandler::block_info(). The number of blocks of a FESystem
* object is simply the sum of all multiplicities of base elements and is
* given by n_blocks().
*
* For example, the FESystem for the Taylor-Hood element for the three-
* dimensional Stokes problem can be built using the code
*
* @code
* FE_Q<3> u(2);
* FE_Q<3> p(1);
* FESystem<3> sys1(u,3, p,1);
* @endcode
*
* This example creates an FESystem @p sys1 with four components, three for
* the velocity components and one for the pressure, and also four blocks with
* the degrees of freedom of each of the velocity components and the pressure
* in a separate block each. The number of blocks is four since the first base
* element is repeated three times.
*
* On the other hand, a Taylor-Hood element can also be constructed using
*
* @code
* FESystem<3> U(u,3);
* FESystem<3> sys2(U,1, p,1);
* @endcode
*
* The FESystem @p sys2 created here has the same four components, but the
* degrees of freedom are distributed into only two blocks. The first block
* has all velocity degrees of freedom from @p U, while the second block
* contains the pressure degrees of freedom. Note that while @p U itself has 3
* blocks, the FESystem @p sys2 does not attempt to split @p U into its base
* elements but considers it a block of its own. By blocking all velocities
* into one system first as in @p sys2, we achieve the same block structure
* that would be generated if instead of using a $Q_2^3$ element for the
* velocities we had used vector-valued base elements, for instance like using
* a mixed discretization of Darcy's law using
*
* @code
* FE_RaviartThomas<3> u(1);
* FE_DGQ<3> p(1);
* FESystem<3> sys3(u,1, p,1);
* @endcode
*
* This example also produces a system with four components, but only two
* blocks.
*
* In most cases, the composed element behaves as if it were a usual element.
* It just has more degrees of freedom than most of the "common" elements.
* However the underlying structure is visible in the restriction,
* prolongation and interface constraint matrices, which do not couple the
* degrees of freedom of the base elements. E.g. the continuity requirement is
* imposed for the shape functions of the subobjects separately; no
* requirement exist between shape functions of different subobjects, i.e. in
* the above example: on a hanging node, the respective value of the @p u
* velocity is only coupled to @p u at the vertices and the line on the larger
* cell next to this vertex, but there is no interaction with @p v and @p w of
* this or the other cell.
*
*
* <h3>Internal information on numbering of degrees of freedom</h3>
*
* The overall numbering of degrees of freedom is as follows: for each
* subobject (vertex, line, quad, or hex), the degrees of freedom are numbered
* such that we run over all subelements first, before turning for the next
* dof on this subobject or for the next subobject. For example, for an
* element of three components in one space dimension, the first two
* components being cubic lagrange elements and the third being a quadratic
* lagrange element, the ordering for the system <tt>s=(u,v,p)</tt> is:
*
* <ul>
* <li> First vertex: <tt>u0, v0, p0 = s0, s1, s2</tt>
* <li> Second vertex: <tt>u1, v1, p1 = s3, s4, s5</tt>
* <li> First component on the line: <tt>u2, u3 = s4, s5</tt>
* <li> Second component on the line: <tt>v2, v3 = s6, s7</tt>.
* <li> Third component on the line: <tt>p2 = s8</tt>.
* </ul>
* That said, you should not rely on this numbering in your application as
* these %internals might change in future. Rather use the functions
* system_to_component_index() and component_to_system_index().
*
* For more information on the template parameter <tt>spacedim</tt> see the
* documentation of Triangulation.
*
* @ingroup febase fe vector_valued
*
* @author Wolfgang Bangerth, Guido Kanschat, 1999, 2002, 2003, 2006, Ralf
* Hartmann 2001.
*/
template <int dim, int spacedim=dim>
class FESystem : public FiniteElement<dim,spacedim>
{
public:
/**
* Constructor. Take a finite element and the number of elements you want to
* group together using this class.
*
* The object @p fe is not actually used for anything other than creating a
* copy that will then be owned by the current object. In other words, it is
* completely fine to call this constructor with a temporary object for the
* finite element, as in this code snippet:
* @code
* FESystem<dim> fe (FE_Q<dim>(2), 2);
* @endcode
* Here, <code>FE_Q@<dim@>(2)</code> constructs an unnamed, temporary object
* that is passed to the FESystem constructor to create a finite element
* that consists of two components, both of which are quadratic FE_Q
* elements. The temporary is destroyed again at the end of the code that
* corresponds to this line, but this does not matter because FESystem
* creates its own copy of the FE_Q object.
*
* This constructor (or its variants below) is used in essentially all
* tutorial programs that deal with vector valued problems. See step-8,
* step-20, step-22 and others for use cases. Also see the module on
* @ref vector_valued "Handling vector valued problems".
*
* @dealiiVideoLecture{19,20}
*
* @param[in] fe The finite element that will be used to represent the
* components of this composed element.
* @param[in] n_elements An integer denoting how many copies of @p fe this
* element should consist of.
*/
FESystem (const FiniteElement<dim,spacedim> &fe,
const unsigned int n_elements);
/**
* Constructor for mixed discretizations with two base elements.
*
* See the other constructor above for an explanation of the general idea of
* composing elements.
*/
FESystem (const FiniteElement<dim,spacedim> &fe1, const unsigned int n1,
const FiniteElement<dim,spacedim> &fe2, const unsigned int n2);
/**
* Constructor for mixed discretizations with three base elements.
*
* See the other constructor above for an explanation of the general idea of
* composing elements.
*/
FESystem (const FiniteElement<dim,spacedim> &fe1, const unsigned int n1,
const FiniteElement<dim,spacedim> &fe2, const unsigned int n2,
const FiniteElement<dim,spacedim> &fe3, const unsigned int n3);
/**
* Constructor for mixed discretizations with four base elements.
*
* See the first of the other constructors above for an explanation of the
* general idea of composing elements.
*/
FESystem (const FiniteElement<dim,spacedim> &fe1, const unsigned int n1,
const FiniteElement<dim,spacedim> &fe2, const unsigned int n2,
const FiniteElement<dim,spacedim> &fe3, const unsigned int n3,
const FiniteElement<dim,spacedim> &fe4, const unsigned int n4);
/**
* Constructor for mixed discretizations with five base elements.
*
* See the first of the other constructors above for an explanation of the
* general idea of composing elements.
*/
FESystem (const FiniteElement<dim,spacedim> &fe1, const unsigned int n1,
const FiniteElement<dim,spacedim> &fe2, const unsigned int n2,
const FiniteElement<dim,spacedim> &fe3, const unsigned int n3,
const FiniteElement<dim,spacedim> &fe4, const unsigned int n4,
const FiniteElement<dim,spacedim> &fe5, const unsigned int n5);
/**
* Same as above but for any number of base elements. Pointers to the base
* elements and their multiplicities are passed as vectors to this
* constructor. The lengths of these vectors are assumed to be equal.
*
* As above, the finite element objects pointed to by the first argument are
* not actually used other than to create copies internally. Consequently,
* you can delete these pointers immediately again after calling this
* constructor.
*
* <h4>How to use this constructor</h4>
*
* Using this constructor is a bit awkward at times because you need to pass
* two vectors in a place where it may not be straightforward to construct
* such a vector -- for example, in the member initializer list of a class
* with an FESystem member variable. For example, if your main class looks
* like this:
* @code
* template <int dim>
* class MySimulator {
* public:
* MySimulator (const unsigned int polynomial_degree);
* private:
* FESystem<dim> fe;
* };
*
* template <int dim>
* MySimulator<dim>::MySimulator (const unsigned int polynomial_degree)
* :
* fe (...) // what to pass here???
* {}
* @endcode
*
* If your compiler supports the C++11 language standard (or later) and
* deal.II has been configured to use it, then you could do something like
* this to create an element with four base elements and multiplicities 1,
* 2, 3 and 4:
* @code
* template <int dim>
* MySimulator<dim>::MySimulator (const unsigned int polynomial_degree)
* :
* fe (std::vector<const FiniteElement<dim>*> { new FE_Q<dim>(1),
* new FE_Q<dim>(2),
* new FE_Q<dim>(3),
* new FE_Q<dim>(4) },
* std::vector<unsigned int> { 1, 2, 3, 4 })
* {}
* @endcode
* This creates two vectors in place and initializes them using the
* initializer list enclosed in braces <code>{ ... }</code>.
*
* This code has a problem: it creates four memory leaks because the first
* vector above is created with pointers to elements that are allocated with
* <code>new</code> but never destroyed. Without C++11, you have another
* problem: brace-initializer don't exist in earlier C++ standards.
*
* The solution to the second of these problems is to create two static
* member functions that can create vectors. Here is an example:
* @code
* template <int dim>
* class MySimulator {
* public:
* MySimulator (const unsigned int polynomial_degree);
*
* private:
* FESystem<dim> fe;
*
* static std::vector<const FiniteElement<dim>*>
* create_fe_list (const unsigned int polynomial_degree);
*
* static std::vector<unsigned int>
* create_fe_multiplicities ();
* };
*
* template <int dim>
* std::vector<const FiniteElement<dim>*>
* MySimulator<dim>::create_fe_list (const unsigned int polynomial_degree)
* {
* std::vector<const FiniteElement<dim>*> fe_list;
* fe_list.push_back (new FE_Q<dim>(1));
* fe_list.push_back (new FE_Q<dim>(2));
* fe_list.push_back (new FE_Q<dim>(3));
* fe_list.push_back (new FE_Q<dim>(4));
* return fe_list;
* }
*
* template <int dim>
* std::vector<unsigned int>
* MySimulator<dim>::create_fe_multiplicities ()
* {
* std::vector<unsigned int> multiplicities;
* multiplicities.push_back (1);
* multiplicities.push_back (2);
* multiplicities.push_back (3);
* multiplicities.push_back (4);
* return multiplicities;
* }
*
* template <int dim>
* MySimulator<dim>::MySimulator (const unsigned int polynomial_degree)
* :
* fe (create_fe_list (polynomial_degree),
* create_fe_multiplicities ())
* {}
* @endcode
*
* The way this works is that we have two static member functions that
* create the necessary vectors to pass to the constructor of the member
* variable <code>fe</code>. They need to be static because they are called
* during the constructor of <code>MySimulator</code> at a time when the
* <code>*this</code> object isn't fully constructed and, consequently,
* regular member functions cannot be called yet.
*
* The code above does not solve the problem with the memory leak yet,
* though: the <code>create_fe_list()</code> function creates a vector of
* pointers, but nothing destroys these. This is the solution:
* @code
* template <int dim>
* class MySimulator {
* public:
* MySimulator (const unsigned int polynomial_degree);
*
* private:
* FESystem<dim> fe;
*
* struct VectorElementDestroyer {
* const std::vector<const FiniteElement<dim>*> data;
* VectorElementDestroyer (const std::vector<const FiniteElement<dim>*> &pointers);
* ~VectorElementDestroyer (); // destructor to delete the pointers
* const std::vector<const FiniteElement<dim>*> & get_data () const;
* };
*
* static std::vector<const FiniteElement<dim>*>
* create_fe_list (const unsigned int polynomial_degree);
*
* static std::vector<unsigned int>
* create_fe_multiplicities ();
* };
*
* template <int dim>
* MySimulator<dim>::VectorElementDestroyer::
* VectorElementDestroyer (const std::vector<const FiniteElement<dim>*> &pointers)
* : data(pointers)
* {}
*
* template <int dim>
* MySimulator<dim>::VectorElementDestroyer::
* ~VectorElementDestroyer ()
* {
* for (unsigned int i=0; i<data.size(); ++i)
* delete data[i];
* }
*
* template <int dim>
* const std::vector<const FiniteElement<dim>*> &
* MySimulator<dim>::VectorElementDestroyer::
* get_data () const
* {
* return data;
* }
*
*
* template <int dim>
* MySimulator<dim>::MySimulator (const unsigned int polynomial_degree)
* :
* fe (VectorElementDestroyer(create_fe_list (polynomial_degree)).get_data(),
* create_fe_multiplicities ())
* {}
* @endcode
*
* In other words, the vector we receive from the
* <code>create_fe_list()</code> is packed into a temporary object of type
* <code>VectorElementDestroyer</code>; we then get the vector from this
* temporary object immediately to pass it to the constructor of
* <code>fe</code>; and finally, the <code>VectorElementDestroyer</code>
* destructor is called at the end of the entire expression (after the
* constructor of <code>fe</code> has finished) and destroys the elements of
* the temporary vector. Voila: not short nor elegant, but it works!
*/
FESystem (const std::vector<const FiniteElement<dim,spacedim>*> &fes,
const std::vector<unsigned int> &multiplicities);
/**
* Destructor.
*/
virtual ~FESystem ();
/**
* Return a string that uniquely identifies a finite element. This element
* returns a string that is composed of the strings @p name1...@p nameN
* returned by the basis elements. From these, we create a sequence
* <tt>FESystem<dim>[name1^m1-name2^m2-...-nameN^mN]</tt>, where @p mi are
* the multiplicities of the basis elements. If a multiplicity is equal to
* one, then the superscript is omitted.
*/
virtual std::string get_name () const;
// for documentation, see the FiniteElement base class
virtual
UpdateFlags
requires_update_flags (const UpdateFlags update_flags) const;
/**
* Return the value of the @p ith shape function at the point @p p. @p p is
* a point on the reference element. Since this finite element is always
* vector-valued, we return the value of the only non-zero component of the
* vector value of this shape function. If the shape function has more than
* one non-zero component (which we refer to with the term non-primitive),
* then throw an exception of type @p ExcShapeFunctionNotPrimitive.
*
* An @p ExcUnitShapeValuesDoNotExist is thrown if the shape values of the
* @p FiniteElement (corresponding to the @p ith shape function) depend on
* the shape of the cell in real space.
*/
virtual double shape_value (const unsigned int i,
const Point<dim> &p) const;
/**
* Return the value of the @p componentth vector component of the @p ith
* shape function at the point @p p. See the FiniteElement base class for
* more information about the semantics of this function.
*
* Since this element is vector valued in general, it relays the computation
* of these values to the base elements.
*/
virtual double shape_value_component (const unsigned int i,
const Point<dim> &p,
const unsigned int component) const;
/**
* Return the gradient of the @p ith shape function at the point @p p. @p p
* is a point on the reference element, and likewise the gradient is the
* gradient on the unit cell with respect to unit cell coordinates. Since
* this finite element is always vector-valued, we return the value of the
* only non-zero component of the vector value of this shape function. If
* the shape function has more than one non-zero component (which we refer
* to with the term non-primitive), then throw an exception of type @p
* ExcShapeFunctionNotPrimitive.
*
* An @p ExcUnitShapeValuesDoNotExist is thrown if the shape values of the
* @p FiniteElement (corresponding to the @p ith shape function) depend on
* the shape of the cell in real space.
*/
virtual Tensor<1,dim> shape_grad (const unsigned int i,
const Point<dim> &p) const;
/**
* Return the gradient of the @p componentth vector component of the @p ith
* shape function at the point @p p. See the FiniteElement base class for
* more information about the semantics of this function.
*
* Since this element is vector valued in general, it relays the computation
* of these values to the base elements.
*/
virtual Tensor<1,dim> shape_grad_component (const unsigned int i,
const Point<dim> &p,
const unsigned int component) const;
/**
* Return the tensor of second derivatives of the @p ith shape function at
* point @p p on the unit cell. The derivatives are derivatives on the unit
* cell with respect to unit cell coordinates. Since this finite element is
* always vector-valued, we return the value of the only non-zero component
* of the vector value of this shape function. If the shape function has
* more than one non-zero component (which we refer to with the term non-
* primitive), then throw an exception of type @p
* ExcShapeFunctionNotPrimitive.
*
* An @p ExcUnitShapeValuesDoNotExist is thrown if the shape values of the
* @p FiniteElement (corresponding to the @p ith shape function) depend on
* the shape of the cell in real space.
*/
virtual Tensor<2,dim> shape_grad_grad (const unsigned int i,
const Point<dim> &p) const;
/**
* Return the second derivatives of the @p componentth vector component of
* the @p ith shape function at the point @p p. See the FiniteElement base
* class for more information about the semantics of this function.
*
* Since this element is vector valued in general, it relays the computation
* of these values to the base elements.
*/
virtual
Tensor<2,dim>
shape_grad_grad_component (const unsigned int i,
const Point<dim> &p,
const unsigned int component) const;
/**
* Return the tensor of third derivatives of the @p ith shape function at
* point @p p on the unit cell. The derivatives are derivatives on the unit
* cell with respect to unit cell coordinates. Since this finite element is
* always vector-valued, we return the value of the only non-zero component
* of the vector value of this shape function. If the shape function has
* more than one non-zero component (which we refer to with the term non-
* primitive), then throw an exception of type @p
* ExcShapeFunctionNotPrimitive.
*
* An @p ExcUnitShapeValuesDoNotExist is thrown if the shape values of the
* @p FiniteElement (corresponding to the @p ith shape function) depend on
* the shape of the cell in real space.
*/
virtual Tensor<3,dim> shape_3rd_derivative (const unsigned int i,
const Point<dim> &p) const;
/**
* Return the third derivatives of the @p componentth vector component of
* the @p ith shape function at the point @p p. See the FiniteElement base
* class for more information about the semantics of this function.
*
* Since this element is vector valued in general, it relays the computation
* of these values to the base elements.
*/
virtual Tensor<3,dim> shape_3rd_derivative_component (const unsigned int i,
const Point<dim> &p,
const unsigned int component) const;
/**
* Return the tensor of fourth derivatives of the @p ith shape function at
* point @p p on the unit cell. The derivatives are derivatives on the unit
* cell with respect to unit cell coordinates. Since this finite element is
* always vector-valued, we return the value of the only non-zero component
* of the vector value of this shape function. If the shape function has
* more than one non-zero component (which we refer to with the term non-
* primitive), then throw an exception of type @p
* ExcShapeFunctionNotPrimitive.
*
* An @p ExcUnitShapeValuesDoNotExist is thrown if the shape values of the
* @p FiniteElement (corresponding to the @p ith shape function) depend on
* the shape of the cell in real space.
*/
virtual Tensor<4,dim> shape_4th_derivative (const unsigned int i,
const Point<dim> &p) const;
/**
* Return the fourth derivatives of the @p componentth vector component of
* the @p ith shape function at the point @p p. See the FiniteElement base
* class for more information about the semantics of this function.
*
* Since this element is vector valued in general, it relays the computation
* of these values to the base elements.
*/
virtual Tensor<4,dim> shape_4th_derivative_component (const unsigned int i,
const Point<dim> &p,
const unsigned int component) const;
/**
* Return the matrix interpolating from the given finite element to the
* present one. The size of the matrix is then @p dofs_per_cell times
* <tt>source.dofs_per_cell</tt>.
*
* These matrices are available if source and destination element are both
* @p FESystem elements, have the same number of base elements with same
* element multiplicity, and if these base elements also implement their @p
* get_interpolation_matrix functions. Otherwise, an exception of type
* FiniteElement<dim,spacedim>::ExcInterpolationNotImplemented is thrown.
*/
virtual void
get_interpolation_matrix (const FiniteElement<dim,spacedim> &source,
FullMatrix<double> &matrix) const;
/**
* Access to a composing element. The index needs to be smaller than the
* number of base elements. Note that the number of base elements may in
* turn be smaller than the number of components of the system element, if
* the multiplicities are greater than one.
*/
virtual const FiniteElement<dim,spacedim> &
base_element (const unsigned int index) const;
/**
* This function returns @p true, if the shape function @p shape_index has
* non-zero function values somewhere on the face @p face_index.
*/
virtual bool has_support_on_face (const unsigned int shape_index,
const unsigned int face_index) const;
/**
* Projection from a fine grid space onto a coarse grid space. Overrides the
* respective method in FiniteElement, implementing lazy evaluation
* (initialize when requested).
*
* If this projection operator is associated with a matrix @p P, then the
* restriction of this matrix @p P_i to a single child cell is returned
* here.
*
* The matrix @p P is the concatenation or the sum of the cell matrices @p
* P_i, depending on the #restriction_is_additive_flags. This distinguishes
* interpolation (concatenation) and projection with respect to scalar
* products (summation).
*
* Row and column indices are related to coarse grid and fine grid spaces,
* respectively, consistent with the definition of the associated operator.
*
* If projection matrices are not implemented in the derived finite element
* class, this function aborts with an exception of type
* FiniteElement::ExcProjectionVoid. You can check whether this would happen
* by first calling the restriction_is_implemented() or the
* isotropic_restriction_is_implemented() function.
*/
virtual const FullMatrix<double> &
get_restriction_matrix (const unsigned int child,
const RefinementCase<dim> &refinement_case=RefinementCase<dim>::isotropic_refinement) const;
/**
* Embedding matrix between grids. Overrides the respective method in
* FiniteElement, implementing lazy evaluation (initialize when queried).
*
* The identity operator from a coarse grid space into a fine grid space is
* associated with a matrix @p P. The restriction of this matrix @p P_i to a
* single child cell is returned here.
*
* The matrix @p P is the concatenation, not the sum of the cell matrices @p
* P_i. That is, if the same non-zero entry <tt>j,k</tt> exists in in two
* different child matrices @p P_i, the value should be the same in both
* matrices and it is copied into the matrix @p P only once.
*
* Row and column indices are related to fine grid and coarse grid spaces,
* respectively, consistent with the definition of the associated operator.
*
* These matrices are used by routines assembling the prolongation matrix
* for multi-level methods. Upon assembling the transfer matrix between
* cells using this matrix array, zero elements in the prolongation matrix
* are discarded and will not fill up the transfer matrix.
*
* If prolongation matrices are not implemented in one of the base finite
* element classes, this function aborts with an exception of type
* FiniteElement::ExcEmbeddingVoid. You can check whether this would happen
* by first calling the prolongation_is_implemented() or the
* isotropic_prolongation_is_implemented() function.
*/
virtual const FullMatrix<double> &
get_prolongation_matrix (const unsigned int child,
const RefinementCase<dim> &refinement_case=RefinementCase<dim>::isotropic_refinement) const;
/**
* Given an index in the natural ordering of indices on a face, return the
* index of the same degree of freedom on the cell.
*
* To explain the concept, consider the case where we would like to know
* whether a degree of freedom on a face, for example as part of an FESystem
* element, is primitive. Unfortunately, the is_primitive() function in the
* FiniteElement class takes a cell index, so we would need to find the cell
* index of the shape function that corresponds to the present face index.
* This function does that.
*
* Code implementing this would then look like this:
* @code
* for (i=0; i<dofs_per_face; ++i)
* if (fe.is_primitive(fe.face_to_equivalent_cell_index(i, some_face_no)))
* ... do whatever
* @endcode
* The function takes additional arguments that account for the fact that
* actual faces can be in their standard ordering with respect to the cell
* under consideration, or can be flipped, oriented, etc.
*
* @param face_dof_index The index of the degree of freedom on a face. This
* index must be between zero and dofs_per_face.
* @param face The number of the face this degree of freedom lives on. This
* number must be between zero and GeometryInfo::faces_per_cell.
* @param face_orientation One part of the description of the orientation of
* the face. See
* @ref GlossFaceOrientation.
* @param face_flip One part of the description of the orientation of the
* face. See
* @ref GlossFaceOrientation.
* @param face_rotation One part of the description of the orientation of
* the face. See
* @ref GlossFaceOrientation.
* @return The index of this degree of freedom within the set of degrees of
* freedom on the entire cell. The returned value will be between zero and
* dofs_per_cell.
*/
virtual
unsigned int face_to_cell_index (const unsigned int face_dof_index,
const unsigned int face,
const bool face_orientation = true,
const bool face_flip = false,
const bool face_rotation = false) const;
/**
* Implementation of the respective function in the base class.
*/
virtual
Point<dim>
unit_support_point (const unsigned int index) const;
/**
* Implementation of the respective function in the base class.
*/
virtual
Point<dim-1>
unit_face_support_point (const unsigned int index) const;
/**
* Returns a list of constant modes of the element. The returns table has as
* many rows as there are components in the element and dofs_per_cell
* columns. To each component of the finite element, the row in the returned
* table contains a basis representation of the constant function 1 on the
* element. Concatenates the constant modes of each base element.
*/
virtual std::pair<Table<2,bool>, std::vector<unsigned int> >
get_constant_modes () const;
/**
* @name Functions to support hp
* @{
*/
/**
* Return whether this element implements its hanging node constraints in
* the new way, which has to be used to make elements "hp compatible".
*
* This function returns @p true iff all its base elements return @p true
* for this function.
*/
virtual bool hp_constraints_are_implemented () const;
/**
* Return the matrix interpolating from a face of of one element to the face
* of the neighboring element. The size of the matrix is then
* <tt>source.dofs_per_face</tt> times <tt>this->dofs_per_face</tt>.
*
* Base elements of this element will have to implement this function. They
* may only provide interpolation matrices for certain source finite
* elements, for example those from the same family. If they don't implement
* interpolation from a given element, then they must throw an exception of
* type FiniteElement<dim,spacedim>::ExcInterpolationNotImplemented, which
* will get propagated out from this element.
*/
virtual void
get_face_interpolation_matrix (const FiniteElement<dim,spacedim> &source,
FullMatrix<double> &matrix) const;
/**
* Return the matrix interpolating from a face of of one element to the
* subface of the neighboring element. The size of the matrix is then
* <tt>source.dofs_per_face</tt> times <tt>this->dofs_per_face</tt>.
*
* Base elements of this element will have to implement this function. They
* may only provide interpolation matrices for certain source finite
* elements, for example those from the same family. If they don't implement
* interpolation from a given element, then they must throw an exception of
* type FiniteElement<dim,spacedim>::ExcInterpolationNotImplemented, which
* will get propagated out from this element.
*/
virtual void
get_subface_interpolation_matrix (const FiniteElement<dim,spacedim> &source,
const unsigned int subface,
FullMatrix<double> &matrix) const;
/**
* If, on a vertex, several finite elements are active, the hp code first
* assigns the degrees of freedom of each of these FEs different global
* indices. It then calls this function to find out which of them should get
* identical values, and consequently can receive the same global DoF index.
* This function therefore returns a list of identities between DoFs of the
* present finite element object with the DoFs of @p fe_other, which is a
* reference to a finite element object representing one of the other finite
* elements active on this particular vertex. The function computes which of
* the degrees of freedom of the two finite element objects are equivalent,
* both numbered between zero and the corresponding value of dofs_per_vertex
* of the two finite elements. The first index of each pair denotes one of
* the vertex dofs of the present element, whereas the second is the
* corresponding index of the other finite element.
*/
virtual
std::vector<std::pair<unsigned int, unsigned int> >
hp_vertex_dof_identities (const FiniteElement<dim,spacedim> &fe_other) const;
/**
* Same as hp_vertex_dof_indices(), except that the function treats degrees
* of freedom on lines.
*/
virtual
std::vector<std::pair<unsigned int, unsigned int> >
hp_line_dof_identities (const FiniteElement<dim,spacedim> &fe_other) const;
/**
* Same as hp_vertex_dof_indices(), except that the function treats degrees
* of freedom on quads.
*/
virtual
std::vector<std::pair<unsigned int, unsigned int> >
hp_quad_dof_identities (const FiniteElement<dim,spacedim> &fe_other) const;
/**
* Return whether this element dominates the one given as argument when they
* meet at a common face, whether it is the other way around, whether
* neither dominates, or if either could dominate.
*
* For a definition of domination, see FiniteElementBase::Domination and in
* particular the
* @ref hp_paper "hp paper".
*/
virtual
FiniteElementDomination::Domination
compare_for_face_domination (const FiniteElement<dim,spacedim> &fe_other) const;
//@}
/**
* Determine an estimate for the memory consumption (in bytes) of this
* object.
*
* This function is made virtual, since finite element objects are usually
* accessed through pointers to their base class, rather than the class
* itself.
*/
virtual std::size_t memory_consumption () const;
protected:
/**
* @p clone function instead of a copy constructor.
*
* This function is needed by the constructors of @p FESystem.
*/
virtual FiniteElement<dim,spacedim> *clone() const;
virtual typename FiniteElement<dim,spacedim>::InternalDataBase *
get_data (const UpdateFlags update_flags,
const Mapping<dim,spacedim> &mapping,
const Quadrature<dim> &quadrature,
dealii::internal::FEValues::FiniteElementRelatedData<dim, spacedim> &output_data) const;
virtual
typename FiniteElement<dim,spacedim>::InternalDataBase *
get_face_data (const UpdateFlags update_flags,
const Mapping<dim,spacedim> &mapping,
const Quadrature<dim-1> &quadrature,
dealii::internal::FEValues::FiniteElementRelatedData<dim, spacedim> &output_data) const;
virtual
typename FiniteElement<dim,spacedim>::InternalDataBase *
get_subface_data (const UpdateFlags update_flags,
const Mapping<dim,spacedim> &mapping,
const Quadrature<dim-1> &quadrature,
dealii::internal::FEValues::FiniteElementRelatedData<dim, spacedim> &output_data) const;
virtual
void
fill_fe_values (const typename Triangulation<dim,spacedim>::cell_iterator &cell,
const CellSimilarity::Similarity cell_similarity,
const Quadrature<dim> &quadrature,
const Mapping<dim,spacedim> &mapping,
const typename Mapping<dim,spacedim>::InternalDataBase &mapping_internal,
const dealii::internal::FEValues::MappingRelatedData<dim, spacedim> &mapping_data,
const typename FiniteElement<dim,spacedim>::InternalDataBase &fe_internal,
dealii::internal::FEValues::FiniteElementRelatedData<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 Mapping<dim,spacedim> &mapping,
const typename Mapping<dim,spacedim>::InternalDataBase &mapping_internal,
const dealii::internal::FEValues::MappingRelatedData<dim, spacedim> &mapping_data,
const typename FiniteElement<dim,spacedim>::InternalDataBase &fe_internal,
dealii::internal::FEValues::FiniteElementRelatedData<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 sub_no,
const Quadrature<dim-1> &quadrature,
const Mapping<dim,spacedim> &mapping,
const typename Mapping<dim,spacedim>::InternalDataBase &mapping_internal,
const dealii::internal::FEValues::MappingRelatedData<dim, spacedim> &mapping_data,
const typename FiniteElement<dim,spacedim>::InternalDataBase &fe_internal,
dealii::internal::FEValues::FiniteElementRelatedData<dim, spacedim> &output_data) const;
/**
* Do the work for the three <tt>fill_fe*_values</tt> functions.
*
* Calls (among other things) <tt>fill_fe_([sub]face)_values</tt> of the
* base elements. Calls @p fill_fe_values if
* <tt>face_no==invalid_face_no</tt> and <tt>sub_no==invalid_face_no</tt>;
* calls @p fill_fe_face_values if <tt>face_no==invalid_face_no</tt> and
* <tt>sub_no!=invalid_face_no</tt>; and calls @p fill_fe_subface_values if
* <tt>face_no!=invalid_face_no</tt> and <tt>sub_no!=invalid_face_no</tt>.
*/
template <int dim_1>
void compute_fill (const Mapping<dim,spacedim> &mapping,
const typename Triangulation<dim,spacedim>::cell_iterator &cell,
const unsigned int face_no,
const unsigned int sub_no,
const Quadrature<dim_1> &quadrature,
const CellSimilarity::Similarity cell_similarity,
const typename Mapping<dim,spacedim>::InternalDataBase &mapping_internal,
const typename FiniteElement<dim,spacedim>::InternalDataBase &fe_data,
const internal::FEValues::MappingRelatedData<dim,spacedim> &mapping_data,
internal::FEValues::FiniteElementRelatedData<dim,spacedim> &output_data) const;
private:
/**
* Value to indicate that a given face or subface number is invalid.
*/
static const unsigned int invalid_face_number = numbers::invalid_unsigned_int;
/**
* Pointers to underlying finite element objects.
*
* This object contains a pointer to each contributing element of a mixed
* discretization and its multiplicity. It is created by the constructor and
* constant afterwards.
*
* The pointers are managed as shared pointers. This ensures that we can use
* the copy constructor of this class without having to manage cloning the
* elements themselves. Since finite element objects do not contain any
* state, this also allows multiple copies of an FESystem object to share
* pointers to the underlying base finite elements. The last one of these
* copies around will then delete the pointer to the base elements.
*/
std::vector<std::pair<std_cxx11::shared_ptr<const FiniteElement<dim,spacedim> >,
unsigned int> >
base_elements;
/**
* Initialize the @p unit_support_points field of the FiniteElement class.
* Called from the constructor.
*/
void initialize_unit_support_points ();
/**
* Initialize the @p unit_face_support_points field of the FiniteElement
* class. Called from the constructor.
*/
void initialize_unit_face_support_points ();
/**
* Initialize the @p adjust_quad_dof_index_for_face_orientation_table field
* of the FiniteElement class. Called from the constructor.
*/
void initialize_quad_dof_index_permutation ();
/**
* This function is simply singled out of the constructors since there are
* several of them. It sets up the index table for the system as well as @p
* restriction and @p prolongation matrices.
*/
void initialize (const std::vector<const FiniteElement<dim,spacedim>*> &fes,
const std::vector<unsigned int> &multiplicities);
/**
* Used by @p initialize.
*/
void build_cell_tables();
/**
* Used by @p initialize.
*/
void build_face_tables();
/**
* Used by @p initialize.
*/
void build_interface_constraints ();
/**
* A function that computes the hp_vertex_dof_identities(),
* hp_line_dof_identities(), or hp_quad_dof_identities(), depending on the
* value of the template parameter.
*/
template <int structdim>
std::vector<std::pair<unsigned int, unsigned int> >
hp_object_dof_identities (const FiniteElement<dim,spacedim> &fe_other) const;
/**
* Usually: Fields of cell-independent data.
*
* However, here, this class does not itself store the data but only
* pointers to @p InternalData objects for each of the base elements.
*/
class InternalData : public FiniteElement<dim,spacedim>::InternalDataBase
{
public:
/**
* Constructor. Is called by the @p get_data function. Sets the size of
* the @p base_fe_datas vector to @p n_base_elements.
*/
InternalData (const unsigned int n_base_elements);
/**
* Destructor. Deletes all @p InternalDatas whose pointers are stored by
* the @p base_fe_datas vector.
*/
~InternalData();
/**
* Gives write-access to the pointer to a @p InternalData of the @p
* base_noth base element.
*/
void set_fe_data(const unsigned int base_no,
typename FiniteElement<dim,spacedim>::InternalDataBase *);
/**
* Gives read-access to the pointer to a @p InternalData of the @p
* base_noth base element.
*/
typename FiniteElement<dim,spacedim>::InternalDataBase &
get_fe_data (const unsigned int base_no) const;
/**
* Gives read-access to the pointer to an object to which into which the
* <code>base_no</code>th base element will write its output when calling
* FiniteElement::fill_fe_values() and similar functions.
*/
internal::FEValues::FiniteElementRelatedData<dim,spacedim> &
get_fe_output_object (const unsigned int base_no) const;
private:
/**
* Pointers to @p InternalData objects for each of the base elements. They
* are accessed to by the @p set_ and @p get_fe_data functions.
*
* The size of this vector is set to @p n_base_elements by the
* InternalData constructor. It is filled by the @p get_data function.
* Note that since the data for each instance of a base class is
* necessarily the same, we only need as many of these objects as there
* are base elements, irrespective of their multiplicity.
*/
typename std::vector<typename FiniteElement<dim,spacedim>::InternalDataBase *> base_fe_datas;
/**
* A collection of objects to which the base elements will write their
* output when we call FiniteElement::fill_fe_values() and related
* functions on them.
*
* The size of this vector is set to @p n_base_elements by the
* InternalData constructor.
*/
mutable std::vector<internal::FEValues::FiniteElementRelatedData<dim,spacedim> > base_fe_output_objects;
};
/*
* Mutex for protecting initialization of restriction and embedding matrix.
*/
mutable Threads::Mutex mutex;
};
DEAL_II_NAMESPACE_CLOSE
/*---------------------------- fe_system.h ---------------------------*/
#endif
/*---------------------------- fe_system.h ---------------------------*/
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