Classes | |
class | ExcBoundaryFaceUsed |
class | ExcComponentIndexInvalid |
class | ExcConstraintsVoid |
class | ExcEmbeddingVoid |
class | ExcFEHasNoSupportPoints |
class | ExcFENotPrimitive |
class | ExcInterpolationNotImplemented |
class | ExcJacobiDeterminantHasWrongSign |
class | ExcProjectionVoid |
class | ExcShapeFunctionNotPrimitive |
class | ExcUnitShapeValuesDoNotExist |
class | ExcWrongInterfaceMatrixSize |
class | InternalDataBase |
Public Member Functions | |
FiniteElement (const FiniteElementData< dim > &fe_data, const std::vector< bool > &restriction_is_additive_flags, const std::vector< std::vector< bool > > &nonzero_components) | |
virtual | ~FiniteElement () |
virtual std::string | get_name () const =0 |
const FiniteElement< dim, spacedim > & | operator[] (const unsigned int fe_index) const |
virtual std::vector< std::pair < unsigned int, unsigned int > > | hp_vertex_dof_identities (const FiniteElement< dim, spacedim > &fe_other) const |
virtual std::vector< std::pair < unsigned int, unsigned int > > | hp_line_dof_identities (const FiniteElement< dim, spacedim > &fe_other) const |
virtual std::vector< std::pair < unsigned int, unsigned int > > | hp_quad_dof_identities (const FiniteElement< dim, spacedim > &fe_other) const |
virtual FiniteElementDomination::Domination | compare_for_face_domination (const FiniteElement< dim, spacedim > &fe_other) const |
bool | operator== (const FiniteElement< dim, spacedim > &) const |
virtual unsigned int | memory_consumption () const |
Shape function access | |
virtual double | shape_value (const unsigned int i, const Point< dim > &p) const |
virtual double | shape_value_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const |
virtual Tensor< 1, dim > | shape_grad (const unsigned int i, const Point< dim > &p) const |
virtual Tensor< 1, dim > | shape_grad_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const |
virtual Tensor< 2, dim > | shape_grad_grad (const unsigned int i, const Point< dim > &p) const |
virtual Tensor< 2, dim > | shape_grad_grad_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const |
virtual bool | has_support_on_face (const unsigned int shape_index, const unsigned int face_index) const |
Transfer and constraint matrices | |
const FullMatrix< double > & | get_restriction_matrix (const unsigned int child, const RefinementCase< dim > &refinement_case=RefinementCase< dim >::isotropic_refinement) const |
const FullMatrix< double > & | get_prolongation_matrix (const unsigned int child, const RefinementCase< dim > &refinement_case=RefinementCase< dim >::isotropic_refinement) const |
bool | prolongation_is_implemented () const |
bool | isotropic_prolongation_is_implemented () const |
bool | restriction_is_implemented () const |
bool | isotropic_restriction_is_implemented () const |
bool | restriction_is_additive (const unsigned int index) const |
const FullMatrix< double > & | constraints (const internal::SubfaceCase< dim > &subface_case=internal::SubfaceCase< dim >::case_isotropic) const |
bool | constraints_are_implemented (const internal::SubfaceCase< dim > &subface_case=internal::SubfaceCase< dim >::case_isotropic) const |
virtual bool | hp_constraints_are_implemented () const |
virtual void | get_interpolation_matrix (const FiniteElement< dim, spacedim > &source, FullMatrix< double > &matrix) const |
Functions to support hp | |
virtual void | get_face_interpolation_matrix (const FiniteElement< dim, spacedim > &source, FullMatrix< double > &matrix) const |
virtual void | get_subface_interpolation_matrix (const FiniteElement< dim, spacedim > &source, const unsigned int subface, FullMatrix< double > &matrix) const |
Index computations | |
std::pair< unsigned int, unsigned int > | system_to_component_index (const unsigned int index) const |
unsigned int | component_to_system_index (const unsigned int component, const unsigned int index) const |
std::pair< unsigned int, unsigned int > | face_system_to_component_index (const unsigned int index) const |
unsigned int | adjust_quad_dof_index_for_face_orientation (const unsigned int index, const bool face_orientation, const bool face_flip, const bool face_rotation) const |
unsigned int | adjust_line_dof_index_for_line_orientation (const unsigned int index, const bool line_orientation) const |
const std::vector< bool > & | get_nonzero_components (const unsigned int i) const |
unsigned int | n_nonzero_components (const unsigned int i) const |
bool | is_primitive (const unsigned int i) const |
bool | is_primitive () const |
virtual unsigned int | n_base_elements () const =0 |
virtual const FiniteElement < dim, spacedim > & | base_element (const unsigned int index) const =0 |
virtual unsigned int | element_multiplicity (const unsigned int index) const =0 |
std::pair< std::pair< unsigned int, unsigned int >, unsigned int > | system_to_base_index (const unsigned int index) const |
std::pair< std::pair< unsigned int, unsigned int >, unsigned int > | face_system_to_base_index (const unsigned int index) const |
unsigned int | first_block_of_base (const unsigned int b) const |
std::pair< unsigned int, unsigned int > | component_to_base_index (const unsigned int component) const |
std::pair< unsigned int, unsigned int > | block_to_base_index (const unsigned int block) const |
std::pair< unsigned int, unsigned int > | system_to_block_index (const unsigned int component) const |
unsigned int | component_to_block_index (const unsigned int component) const |
Support points and interpolation | |
const std::vector< Point< dim > > & | get_unit_support_points () const |
bool | has_support_points () const |
virtual Point< dim > | unit_support_point (const unsigned int index) const |
const std::vector< Point< dim-1 > > & | get_unit_face_support_points () const |
bool | has_face_support_points () const |
virtual Point< dim-1 > | unit_face_support_point (const unsigned int index) const |
const std::vector< Point< dim > > & | get_generalized_support_points () const |
bool | has_generalized_support_points () const |
const std::vector< Point< dim-1 > > & | get_generalized_face_support_points () const |
bool | has_generalized_face_support_points () const |
virtual void | interpolate (std::vector< double > &local_dofs, const std::vector< double > &values) const |
virtual void | interpolate (std::vector< double > &local_dofs, const std::vector< Vector< double > > &values, unsigned int offset=0) const |
virtual void | interpolate (std::vector< double > &local_dofs, const VectorSlice< const std::vector< std::vector< double > > > &values) const |
Protected Member Functions | |
void | reinit_restriction_and_prolongation_matrices (const bool isotropic_restriction_only=false, const bool isotropic_prolongation_only=false) |
TableIndices< 2 > | interface_constraints_size () const |
void | compute_2nd (const Mapping< dim, spacedim > &mapping, const typename Triangulation< dim, spacedim >::cell_iterator &cell, const unsigned int offset, typename Mapping< dim, spacedim >::InternalDataBase &mapping_internal, InternalDataBase &fe_internal, FEValuesData< dim, spacedim > &data) const |
virtual UpdateFlags | update_once (const UpdateFlags flags) const =0 |
virtual UpdateFlags | update_each (const UpdateFlags flags) const =0 |
virtual FiniteElement< dim, spacedim > * | clone () const =0 |
Static Protected Member Functions | |
static std::vector< unsigned int > | compute_n_nonzero_components (const std::vector< std::vector< bool > > &nonzero_components) |
Protected Attributes | |
const bool | cached_primitivity |
std::vector< std::vector < FullMatrix< double > > > | restriction |
std::vector< std::vector < FullMatrix< double > > > | prolongation |
FullMatrix< double > | interface_constraints |
std::vector< Point< dim > > | unit_support_points |
std::vector< Point< dim-1 > > | unit_face_support_points |
std::vector< Point< dim > > | generalized_support_points |
std::vector< Point< dim-1 > > | generalized_face_support_points |
Table< 2, int > | adjust_quad_dof_index_for_face_orientation_table |
std::vector< int > | adjust_line_dof_index_for_line_orientation_table |
Private Member Functions | |
virtual Mapping< dim, spacedim > ::InternalDataBase * | get_data (const UpdateFlags flags, const Mapping< dim, spacedim > &mapping, const Quadrature< dim > &quadrature) const =0 |
virtual Mapping< dim, spacedim > ::InternalDataBase * | get_face_data (const UpdateFlags flags, const Mapping< dim, spacedim > &mapping, const Quadrature< dim-1 > &quadrature) const |
virtual Mapping< dim, spacedim > ::InternalDataBase * | get_subface_data (const UpdateFlags flags, const Mapping< dim, spacedim > &mapping, const Quadrature< dim-1 > &quadrature) const |
virtual void | fill_fe_values (const Mapping< dim, spacedim > &mapping, const typename Triangulation< dim, spacedim >::cell_iterator &cell, const Quadrature< dim > &quadrature, typename Mapping< dim, spacedim >::InternalDataBase &mapping_internal, typename Mapping< dim, spacedim >::InternalDataBase &fe_internal, FEValuesData< dim, spacedim > &data, enum CellSimilarity::Similarity &cell_similarity) const =0 |
virtual void | fill_fe_face_values (const Mapping< dim, spacedim > &mapping, const typename Triangulation< dim, spacedim >::cell_iterator &cell, const unsigned int face_no, const Quadrature< dim-1 > &quadrature, typename Mapping< dim, spacedim >::InternalDataBase &mapping_internal, typename Mapping< dim, spacedim >::InternalDataBase &fe_internal, FEValuesData< dim, spacedim > &data) const =0 |
virtual void | fill_fe_subface_values (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, typename Mapping< dim, spacedim >::InternalDataBase &mapping_internal, typename Mapping< dim, spacedim >::InternalDataBase &fe_internal, FEValuesData< dim, spacedim > &data) const =0 |
Private Attributes | |
std::vector< std::pair < unsigned int, unsigned int > > | system_to_component_table |
std::vector< std::pair < unsigned int, unsigned int > > | face_system_to_component_table |
std::vector< std::pair < std::pair< unsigned int, unsigned int >, unsigned int > > | system_to_base_table |
std::vector< std::pair < std::pair< unsigned int, unsigned int >, unsigned int > > | face_system_to_base_table |
std::vector< unsigned int > | first_block_of_base_table |
std::vector< std::pair < std::pair< unsigned int, unsigned int >, unsigned int > > | component_to_base_table |
const std::vector< bool > | restriction_is_additive_flags |
const std::vector< std::vector < bool > > | nonzero_components |
const std::vector< unsigned int > | n_nonzero_components_table |
Static Private Attributes | |
static const double | fd_step_length |
Friends | |
class | InternalDataBase |
class | FEValuesBase< dim, spacedim > |
class | FEValues< dim, spacedim > |
class | FEFaceValues< dim, spacedim > |
class | FESubfaceValues< dim, spacedim > |
class | FESystem |
class | FE_PolyTensor |
class | hp::FECollection< dim, spacedim > |
Base class for finite elements in arbitrary dimensions. This class provides several fields which describe a specific finite element and which are filled by derived classes. It more or less only offers the fields and access functions which makes it possible to copy finite elements without knowledge of the actual type (linear, quadratic, etc). In particular, the functions to fill the data fields of FEValues and its derived classes are declared.
The interface of this class is very restrictive. The reason is that finite element values should be accessed only by use of FEValues objects. These, together with FiniteElement are responsible to provide an optimized implementation.
This class declares the shape functions and their derivatives on the unit cell . The means to transform them onto a given cell in physical space is provided by the FEValues class with a Mapping object.
The different matrices are initialized with the correct size, such that in the derived (concrete) finite element classes, their entries only have to be filled in; no resizing is needed. If the matrices are not defined by a concrete finite element, they should be resized to zero. This way functions using them can find out, that they are missing. On the other hand, it is possible to use finite element classes without implementation of the full functionality, if only part of it is needed. The functionality under consideration here is hanging nodes constraints and grid transfer, respectively.
The spacedim
parameter has to be used if one wants to solve problems in the boundary element method formulation or in an equivalent one, as it is explained in the Triangulation class. If not specified, this parameter takes the default value =dim
so that this class can be used to solve problems in the finite element method formulation.
For vector valued elements shape functions may have nonzero entries in one or several components of the vector valued function. If the element is primitive, there is indeed a single component with a nonzero entry for each shape function. This component can be determined by system_to_component_index(), the number of components is FiniteElementData::n_components().
Furthermore, you may want to split your linear system into blocks for the use in BlockVector, BlockSparseMatrix, BlockMatrixArray and so on. If you use non-primitive elements, you cannot determine the block number by system_to_component_index(). Instead, you can use system_to_block_index(), which will automatically take care of the additional components occupied by vector valued elements. The number of generated blocks can be determined by FiniteElementData::n_blocks().
If you decide to operate by base element and multiplicity, the function first_block_of_base() will be helpful.
Since a FiniteElement does not have information on the actual grid cell, it can only provide support points on the unit cell. Support points on the actual grid cell must be computed by mapping these points. The class used for this kind of operation is FEValues. In most cases, code of the following type will serve to provide the mapped support points.
Quadrature<dim> dummy_quadrature (fe.get_unit_support_points()); FEValues<dim> fe_values (mapping, fe, dummy_quadrature, update_quadrature_points); fe_values.reinit (cell); Point<dim>& mapped_point = fe_values.quadrature_point (i);
Alternatively, the points can be transformed one-by-one:
const vector<Point<dim> >& unit_points = fe.get_unit_support_points(); Point<dim> mapped_point = mapping.transform_unit_to_real_cell (cell, unit_points[i]);
This is a shortcut, and as all shortcuts should be used cautiously. If the mapping of all support points is needed, the first variant should be preferred for efficiency.
The following sections list the information to be provided by derived classes, depending on the dimension. They are followed by a list of functions helping to generate these values.
Finite elements in one dimension need only set the restriction and prolongation matrices. The constructor of this class in one dimension presets the interface_constraints matrix to have dimension zero. Changing this behaviour in derived classes is generally not a reasonable idea and you risk getting into trouble.
In addition to the fields already present in 1D, a constraint matrix is needed, if the finite element has node values located on edges or vertices. These constraints are represented by an -matrix interface_constraints, where m is the number of degrees of freedom on the refined side without the corner vertices (those dofs on the middle vertex plus those on the two lines), and n is that of the unrefined side (those dofs on the two vertices plus those on the line). The matrix is thus a rectangular one. The
size of the interface_constraints matrix can also be accessed through the interface_constraints_size() function.
The mapping of the dofs onto the indices of the matrix on the unrefined side is as follows: let be the number of dofs on a vertex,
that on a line, then
refers to the dofs on vertex zero of the unrefined line,
to those on vertex one,
to those on the line.
Similarly, refers to the dofs on the middle vertex of the refined side (vertex one of child line zero, vertex zero of child line one),
refers to the dofs on child line zero,
refers to the dofs on child line one. Please note that we do not need to reserve space for the dofs on the end vertices of the refined lines, since these must be mapped one-to-one to the appropriate dofs of the vertices of the unrefined line.
It should be noted that it is not possible to distribute a constrained degree of freedom to other degrees of freedom which are themselves constrained. Only one level of indirection is allowed. It is not known at the time of this writing whether this is a constraint itself.
For the interface constraints, almost the same holds as for the 2D case. The numbering for the indices on the mother face is obvious and keeps to the usual numbering of degrees of freedom on quadrilaterals.
The numbering of the degrees of freedom on the interior of the refined faces for the index is as follows: let
and
be as above, and
be the number of degrees of freedom per quadrilateral (and therefore per face), then
denote the dofs on the vertex at the center,
for the dofs on the vertices at the center of the bounding lines of the quadrilateral,
are for the degrees of freedom on the four lines connecting the center vertex to the outer boundary of the mother face,
for the degrees of freedom on the small lines surrounding the quad, and
for the dofs on the four child faces. Note the direction of the lines at the boundary of the quads, as shown below.
The order of the twelve lines and the four child faces can be extracted from the following sketch, where the overall order of the different dof groups is depicted:
* *--15--4--16--* * | | | * 10 19 6 20 12 * | | | * 1--7---0--8---2 * | | | * 9 17 5 18 11 * | | | * *--13--3--14--* *
The numbering of vertices and lines, as well as the numbering of children within a line is consistent with the one described in Triangulation. Therefore, this numbering is seen from the outside and inside, respectively, depending on the face.
The three-dimensional case has a few pitfalls available for derived classes that want to implement constraint matrices. Consider the following case:
* *-------* * / /| * / / | * / / | * *-------* | * | | *-------* * | | / /| * | 1 | / / | * | |/ / | * *-------*-------* | * | | | * * | | | / * | 2 | 3 | / * | | |/ * *-------*-------* *
Now assume that we want to refine cell 2. We will end up with two faces with hanging nodes, namely the faces between cells 1 and 2, as well as between cells 2 and 3. Constraints have to be applied to the degrees of freedom on both these faces. The problem is that there is now an edge (the top right one of cell 2) which is part of both faces. The hanging node(s) on this edge are therefore constrained twice, once from both faces. To be meaningful, these constraints of course have to be consistent: both faces have to constrain the hanging nodes on the edge to the same nodes on the coarse edge (and only on the edge, as there can then be no constraints to nodes on the rest of the face), and they have to do so with the same weights. This is sometimes tricky since the nodes on the edge may have different local numbers.
For the constraint matrix this means the following: if a degree of freedom on one edge of a face is constrained by some other nodes on the same edge with some weights, then the weights have to be exactly the same as those for constrained nodes on the three other edges with respect to the corresponding nodes on these edges. If this isn't the case, you will get into trouble with the ConstraintMatrix class that is the primary consumer of the constraint information: while that class is able to handle constraints that are entered more than once (as is necessary for the case above), it insists that the weights are exactly the same.
Construction of a finite element and computation of the matrices described above may be a tedious task, in particular if it has to be performed for several dimensions. Therefore, some functions in FETools have been provided to help with these tasks.
First, aready the basis of the shape function space may be difficult to implement for arbitrary order and dimension. On the other hand, if the node values are given, then the duality relation between node functionals and basis functions defines the basis. As a result, the shape function space may be defined with arbitrary "raw" basis functions, such that the actual finite element basis is computed from linear combinations of them. The coefficients of these combinations are determined by the duality of node values.
Using this matrix allows the construction of the basis of shape functions in two steps.
Define the space of shape functions using an arbitrary basis wj and compute the matrix M of node functionals Ni applied to these basis functions, such that its entries are mij = Ni(wj).
The function computing the matrix M for you is FETools::compute_node_matrix(). It relies on the existence of generalized_support_points and implementation of interpolate() with VectorSlice argument.
The piece of code in the constructor of a finite element responsible for this looks like
FullMatrix<double> M(this->dofs_per_cell, this->dofs_per_cell); FETools::compute_node_matrix(M, *this); this->inverse_node_matrix.reinit(this->dofs_per_cell, this->dofs_per_cell); this->inverse_node_matrix.invert(M);
Don't forget to make sure that unit_support_points or generalized_support_points are initialized before this!
Once the shape functions are set up, the grid transfer matrices for Multigrid accessed by get_prolongation_matrix() can be computed automatically, using FETools::compute_embedding_matrices().
This can be achieved by
for (unsigned int i=0; i<GeometryInfo<dim>::children_per_cell; ++i) this->prolongation[i].reinit (this->dofs_per_cell, this->dofs_per_cell); FETools::compute_embedding_matrices (*this, this->prolongation);
missing...
Constraint matrices can be computed semi-automatically using FETools::compute_face_embedding_matrices(). This function computes the representation of the coarse mesh functions by fine mesh functions for each child of a face separately. These matrices must be convoluted into a single rectangular constraint matrix, eliminating degrees of freedom on common vertices and edges as well as on the coarse grid vertices. See the discussion above for details.
FiniteElement< dim, spacedim >::FiniteElement | ( | const FiniteElementData< dim > & | fe_data, | |
const std::vector< bool > & | restriction_is_additive_flags, | |||
const std::vector< std::vector< bool > > & | nonzero_components | |||
) |
Constructor
virtual FiniteElement< dim, spacedim >::~FiniteElement | ( | ) | [virtual] |
Virtual destructor. Makes sure that pointers to this class are deleted properly.
virtual std::string FiniteElement< dim, spacedim >::get_name | ( | ) | const [pure virtual] |
Return a string that uniquely identifies a finite element. The general convention is that this is the class name, followed by the dimension in angle brackets, and the polynomial degree and whatever else is necessary in parentheses. For example, FE_Q<2>(3)
is the value returned for a cubic element in 2d.
Systems of elements have their own naming convention, see the FESystem class.
Implemented in FE_ABF< dim >, FE_DGP< dim, spacedim >, FE_DGPMonomial< dim >, FE_DGPNonparametric< dim, spacedim >, FE_DGQ< dim, spacedim >, FE_DGQArbitraryNodes< dim >, FE_Nedelec< dim, spacedim >, FE_Q< dim, spacedim >, FE_Q_Hierarchical< dim >, FE_RaviartThomas< dim >, FE_RaviartThomasNodal< dim >, FESystem< dim, spacedim >, FE_DGQ< dim >, and FE_Q< dim >.
const FiniteElement< dim, spacedim > & FiniteElement< dim, spacedim >::operator[] | ( | const unsigned int | fe_index | ) | const [inline] |
This operator returns a reference to the present object if the argument given equals to zero. While this does not seem particularly useful, it is helpful in writing code that works with both DoFHandler and the hp version hp::DoFHandler, since one can then write code like this:
* dofs_per_cell * = dof_handler->get_fe()[cell->active_fe_index()].dofs_per_cell; *
This code doesn't work in both situations without the present operator because DoFHandler::get_fe() returns a finite element, whereas hp::DoFHandler::get_fe() returns a collection of finite elements that doesn't offer a dofs_per_cell
member variable: one first has to select which finite element to work on, which is done using the operator[]. Fortunately, cell->active_fe_index()
also works for non-hp classes and simply returns zero in that case. The present operator[] accepts this zero argument, by returning the finite element with index zero within its collection (that, of course, consists only of the present finite element anyway).
References Assert.
virtual double FiniteElement< dim, spacedim >::shape_value | ( | const unsigned int | i, | |
const Point< dim > & | p | |||
) | const [virtual] |
Return the value of the ith
shape function at the point p
. p
is a point on the reference element. If the finite element is vector-valued, then 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 derived classes implementing this function should throw an exception of type ExcShapeFunctionNotPrimitive. In that case, use the shape_value_component() function.
An ExcUnitShapeValuesDoNotExist is thrown if the shape values of the FiniteElement under consideration depends on the shape of the cell in real space.
Reimplemented in FE_DGPNonparametric< dim, spacedim >, FE_Poly< POLY, dim, spacedim >, FE_PolyTensor< POLY, dim, spacedim >, FESystem< dim, spacedim >, FE_Poly< PolynomialSpace< dim >, dim, spacedim >, FE_Poly< PolynomialsP< dim >, dim >, FE_Poly< TensorProductPolynomials< dim >, dim >, FE_Poly< TensorProductPolynomials< dim >, dim, dim >, FE_Poly< TensorProductPolynomials< dim >, dim, spacedim >, FE_PolyTensor< PolynomialsRaviartThomas< dim >, dim >, and FE_PolyTensor< PolynomialsABF< dim >, dim >.
virtual double FiniteElement< dim, spacedim >::shape_value_component | ( | const unsigned int | i, | |
const Point< dim > & | p, | |||
const unsigned int | component | |||
) | const [virtual] |
Just like for shape_value(), but this function will be called when the shape function has more than one non-zero vector component. In that case, this function should return the value of the component-th
vector component of the ith
shape function at point p
.
Reimplemented in FE_DGPNonparametric< dim, spacedim >, FE_Nedelec< dim, spacedim >, FE_Poly< POLY, dim, spacedim >, FE_PolyTensor< POLY, dim, spacedim >, FESystem< dim, spacedim >, FE_Poly< PolynomialSpace< dim >, dim, spacedim >, FE_Poly< PolynomialsP< dim >, dim >, FE_Poly< TensorProductPolynomials< dim >, dim >, FE_Poly< TensorProductPolynomials< dim >, dim, dim >, FE_Poly< TensorProductPolynomials< dim >, dim, spacedim >, FE_PolyTensor< PolynomialsRaviartThomas< dim >, dim >, and FE_PolyTensor< PolynomialsABF< dim >, dim >.
virtual Tensor<1,dim> FiniteElement< dim, spacedim >::shape_grad | ( | const unsigned int | i, | |
const Point< dim > & | p | |||
) | const [virtual] |
Return the gradient of the ith
shape function at the point 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. If the finite element is vector-valued, then 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 derived classes implementing this function should throw an exception of type ExcShapeFunctionNotPrimitive. In that case, use the shape_grad_component() function.
An ExcUnitShapeValuesDoNotExist is thrown if the shape values of the FiniteElement under consideration depends on the shape of the cell in real space.
Reimplemented in FE_DGPNonparametric< dim, spacedim >, FE_Poly< POLY, dim, spacedim >, FE_PolyTensor< POLY, dim, spacedim >, FESystem< dim, spacedim >, FE_Poly< PolynomialSpace< dim >, dim, spacedim >, FE_Poly< PolynomialsP< dim >, dim >, FE_Poly< TensorProductPolynomials< dim >, dim >, FE_Poly< TensorProductPolynomials< dim >, dim, dim >, FE_Poly< TensorProductPolynomials< dim >, dim, spacedim >, FE_PolyTensor< PolynomialsRaviartThomas< dim >, dim >, and FE_PolyTensor< PolynomialsABF< dim >, dim >.
virtual Tensor<1,dim> FiniteElement< dim, spacedim >::shape_grad_component | ( | const unsigned int | i, | |
const Point< dim > & | p, | |||
const unsigned int | component | |||
) | const [virtual] |
Just like for shape_grad(), but this function will be called when the shape function has more than one non-zero vector component. In that case, this function should return the gradient of the component-th
vector component of the ith
shape function at point p
.
Reimplemented in FE_DGPNonparametric< dim, spacedim >, FE_Nedelec< dim, spacedim >, FE_Poly< POLY, dim, spacedim >, FE_PolyTensor< POLY, dim, spacedim >, FESystem< dim, spacedim >, FE_Poly< PolynomialSpace< dim >, dim, spacedim >, FE_Poly< PolynomialsP< dim >, dim >, FE_Poly< TensorProductPolynomials< dim >, dim >, FE_Poly< TensorProductPolynomials< dim >, dim, dim >, FE_Poly< TensorProductPolynomials< dim >, dim, spacedim >, FE_PolyTensor< PolynomialsRaviartThomas< dim >, dim >, and FE_PolyTensor< PolynomialsABF< dim >, dim >.
virtual Tensor<2,dim> FiniteElement< dim, spacedim >::shape_grad_grad | ( | const unsigned int | i, | |
const Point< dim > & | p | |||
) | const [virtual] |
Return the tensor of second derivatives of the ith
shape function at point p
on the unit cell. The derivatives are derivatives on the unit cell with respect to unit cell coordinates. If the finite element is vector-valued, then 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 derived classes implementing this function should throw an exception of type ExcShapeFunctionNotPrimitive. In that case, use the shape_grad_grad_component() function.
An ExcUnitShapeValuesDoNotExist is thrown if the shape values of the FiniteElement under consideration depends on the shape of the cell in real space.
Reimplemented in FE_DGPNonparametric< dim, spacedim >, FE_Poly< POLY, dim, spacedim >, FE_PolyTensor< POLY, dim, spacedim >, FESystem< dim, spacedim >, FE_Poly< PolynomialSpace< dim >, dim, spacedim >, FE_Poly< PolynomialsP< dim >, dim >, FE_Poly< TensorProductPolynomials< dim >, dim >, FE_Poly< TensorProductPolynomials< dim >, dim, dim >, FE_Poly< TensorProductPolynomials< dim >, dim, spacedim >, FE_PolyTensor< PolynomialsRaviartThomas< dim >, dim >, and FE_PolyTensor< PolynomialsABF< dim >, dim >.
virtual Tensor<2,dim> FiniteElement< dim, spacedim >::shape_grad_grad_component | ( | const unsigned int | i, | |
const Point< dim > & | p, | |||
const unsigned int | component | |||
) | const [virtual] |
Just like for shape_grad_grad(), but this function will be called when the shape function has more than one non-zero vector component. In that case, this function should return the gradient of the component-th
vector component of the ith
shape function at point p
.
Reimplemented in FE_DGPNonparametric< dim, spacedim >, FE_Nedelec< dim, spacedim >, FE_Poly< POLY, dim, spacedim >, FE_PolyTensor< POLY, dim, spacedim >, FESystem< dim, spacedim >, FE_Poly< PolynomialSpace< dim >, dim, spacedim >, FE_Poly< PolynomialsP< dim >, dim >, FE_Poly< TensorProductPolynomials< dim >, dim >, FE_Poly< TensorProductPolynomials< dim >, dim, dim >, FE_Poly< TensorProductPolynomials< dim >, dim, spacedim >, FE_PolyTensor< PolynomialsRaviartThomas< dim >, dim >, and FE_PolyTensor< PolynomialsABF< dim >, dim >.
virtual bool FiniteElement< dim, spacedim >::has_support_on_face | ( | const unsigned int | shape_index, | |
const unsigned int | face_index | |||
) | const [virtual] |
Check for non-zero values on a face in order to optimize out matrix elements.
This function returns true
, if the shape function shape_index
has non-zero values on the face face_index
.
A default implementation is provided in this basis class which always returns true
. This is the safe way to go.
Reimplemented in FE_ABF< dim >, FE_DGP< dim, spacedim >, FE_DGPMonomial< dim >, FE_DGPNonparametric< dim, spacedim >, FE_DGQ< dim, spacedim >, FE_Nedelec< dim, spacedim >, FE_Q< dim, spacedim >, FE_Q_Hierarchical< dim >, FE_RaviartThomas< dim >, FE_RaviartThomasNodal< dim >, FESystem< dim, spacedim >, FE_DGQ< dim >, FE_Q< dim >, and FE_Q_Hierarchical< dim >.
const FullMatrix<double>& FiniteElement< dim, spacedim >::get_restriction_matrix | ( | const unsigned int | child, | |
const RefinementCase< dim > & | refinement_case = RefinementCase< dim >::isotropic_refinement | |||
) | const |
Projection from a fine grid space onto a coarse grid space. If this projection operator is associated with a matrix P
, then the restriction of this matrix P_i
to a single child cell is returned here.
The matrix P
is the concatenation or the sum of the cell matrices 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 ExcProjectionVoid. You can check whether this is the case by calling the restriction_is_implemented() or the isotropic_restriction_is_implemented() function.
const FullMatrix<double>& FiniteElement< dim, spacedim >::get_prolongation_matrix | ( | const unsigned int | child, | |
const RefinementCase< dim > & | refinement_case = RefinementCase< dim >::isotropic_refinement | |||
) | const |
Embedding matrix between grids.
The identity operator from a coarse grid space into a fine grid space is associated with a matrix P
. The restriction of this matrix P_i
to a single child cell is returned here.
The matrix P
is the concatenation, not the sum of the cell matrices P_i
. That is, if the same non-zero entry j,k
exists in in two different child matrices P_i
, the value should be the same in both matrices and it is copied into the matrix 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 projection matrices are not implemented in the derived finite element class, this function aborts with ExcEmbeddingVoid. You can check whether this is the case by calling the prolongation_is_implemented() or the isotropic_prolongation_is_implemented() function.
bool FiniteElement< dim, spacedim >::prolongation_is_implemented | ( | ) | const |
Return whether this element implements its prolongation matrices. The return value also indicates whether a call to the get_prolongation_matrix() function will generate an error or not.
Note, that this function returns true
only if the prolongation matrices of the isotropic and all anisotropic refinement cases are implemented. If you are interested in the prolongation matrices for isotropic refinement only, use the isotropic_prolongation_is_implemented function instead.
This function is mostly here in order to allow us to write more efficient test programs which we run on all kinds of weird elements, and for which we simply need to exclude certain tests in case something is not implemented. It will in general probably not be a great help in applications, since there is not much one can do if one needs these features and they are not implemented. This function could be used to check whether a call to get_prolongation_matrix()
will succeed; however, one then still needs to cope with the lack of information this just expresses.
bool FiniteElement< dim, spacedim >::isotropic_prolongation_is_implemented | ( | ) | const |
Return whether this element implements its prolongation matrices for isotropic children. The return value also indicates whether a call to the get_prolongation_matrix
function will generate an error or not.
This function is mostly here in order to allow us to write more efficient test programs which we run on all kinds of weird elements, and for which we simply need to exclude certain tests in case something is not implemented. It will in general probably not be a great help in applications, since there is not much one can do if one needs these features and they are not implemented. This function could be used to check whether a call to get_prolongation_matrix()
will succeed; however, one then still needs to cope with the lack of information this just expresses.
bool FiniteElement< dim, spacedim >::restriction_is_implemented | ( | ) | const |
Return whether this element implements its restriction matrices. The return value also indicates whether a call to the get_restriction_matrix() function will generate an error or not.
Note, that this function returns true
only if the restriction matrices of the isotropic and all anisotropic refinement cases are implemented. If you are interested in the restriction matrices for isotropic refinement only, use the isotropic_restriction_is_implemented function instead.
This function is mostly here in order to allow us to write more efficient test programs which we run on all kinds of weird elements, and for which we simply need to exclude certain tests in case something is not implemented. It will in general probably not be a great help in applications, since there is not much one can do if one needs these features and they are not implemented. This function could be used to check whether a call to get_restriction_matrix()
will succeed; however, one then still needs to cope with the lack of information this just expresses.
bool FiniteElement< dim, spacedim >::isotropic_restriction_is_implemented | ( | ) | const |
Return whether this element implements its restriction matrices for isotropic children. The return value also indicates whether a call to the get_restriction_matrix
function will generate an error or not.
This function is mostly here in order to allow us to write more efficient test programs which we run on all kinds of weird elements, and for which we simply need to exclude certain tests in case something is not implemented. It will in general probably not be a great help in applications, since there is not much one can do if one needs these features and they are not implemented. This function could be used to check whether a call to get_restriction_matrix()
will succeed; however, one then still needs to cope with the lack of information this just expresses.
bool FiniteElement< dim, spacedim >::restriction_is_additive | ( | const unsigned int | index | ) | const [inline] |
Access the restriction_is_additive_flags field. See there for more information on its contents.
The index must be between zero and the number of shape functions of this element.
References Assert, FiniteElementData< dim >::dofs_per_cell, and FiniteElement< dim, spacedim >::restriction_is_additive_flags.
const FullMatrix<double>& FiniteElement< dim, spacedim >::constraints | ( | const internal::SubfaceCase< dim > & | subface_case = internal::SubfaceCase< dim >::case_isotropic |
) | const |
Return a readonly reference to the matrix which describes the constraints at the interface between a refined and an unrefined cell.
The matrix is obviously empty in only one dimension, since there are no constraints then.
Note that some finite elements do not (yet) implement hanging node constraints. If this is the case, then this function will generate an exception, since no useful return value can be generated. If you should have a way to live with this, then you might want to use the constraints_are_implemented() function to check up front whethehr this function will succeed or generate the exception.
bool FiniteElement< dim, spacedim >::constraints_are_implemented | ( | const internal::SubfaceCase< dim > & | subface_case = internal::SubfaceCase< dim >::case_isotropic |
) | const |
Return whether this element implements its hanging node constraints. The return value also indicates whether a call to the constraints() function will generate an error or not.
This function is mostly here in order to allow us to write more efficient test programs which we run on all kinds of weird elements, and for which we simply need to exclude certain tests in case hanging node constraints are not implemented. It will in general probably not be a great help in applications, since there is not much one can do if one needs hanging node constraints and they are not implemented. This function could be used to check whether a call to constraints()
will succeed; however, one then still needs to cope with the lack of information this just expresses.
virtual bool FiniteElement< dim, spacedim >::hp_constraints_are_implemented | ( | ) | const [virtual] |
Return whether this element implements its hanging node constraints in the new way, which has to be used to make elements "hp compatible". That means, the element properly implements the get_face_interpolation_matrix and get_subface_interpolation_matrix methods. Therefore the return value also indicates whether a call to the get_face_interpolation_matrix() method and the get_subface_interpolation_matrix() method will generate an error or not.
Currently the main purpose of this function is to allow the make_hanging_node_constraints method to decide whether the new procedures, which are supposed to work in the hp framework can be used, or if the old well verified but not hp capable functions should be used. Once the transition to the new scheme for computing the interface constraints is complete, this function will be superfluous and will probably go away.
Derived classes should implement this function accordingly. The default assumption is that a finite element does not provide hp capable face interpolation, and the default implementation therefore returns false
.
Reimplemented in FE_DGP< dim, spacedim >, FE_DGPMonomial< dim >, FE_DGPNonparametric< dim, spacedim >, FE_DGQ< dim, spacedim >, FE_Q< dim, spacedim >, FE_RaviartThomasNodal< dim >, FESystem< dim, spacedim >, FE_DGQ< dim >, and FE_Q< dim >.
virtual void FiniteElement< dim, spacedim >::get_interpolation_matrix | ( | const FiniteElement< dim, spacedim > & | source, | |
FullMatrix< double > & | matrix | |||
) | const [virtual] |
Return the matrix interpolating from the given finite element to the present one. The size of the matrix is then dofs_per_cell times source.dofs_per_cell
.
Derived elements 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 ExcInterpolationNotImplemented.
virtual void FiniteElement< dim, spacedim >::get_face_interpolation_matrix | ( | const FiniteElement< dim, spacedim > & | source, | |
FullMatrix< double > & | matrix | |||
) | const [virtual] |
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 source.dofs_per_face
times this->dofs_per_face
.
Derived elements 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 ExcInterpolationNotImplemented.
virtual void FiniteElement< dim, spacedim >::get_subface_interpolation_matrix | ( | const FiniteElement< dim, spacedim > & | source, | |
const unsigned int | subface, | |||
FullMatrix< double > & | matrix | |||
) | const [virtual] |
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 source.dofs_per_face
times this->dofs_per_face
.
Derived elements 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 ExcInterpolationNotImplemented.
virtual std::vector<std::pair<unsigned int, unsigned int> > FiniteElement< dim, spacedim >::hp_vertex_dof_identities | ( | const FiniteElement< dim, spacedim > & | fe_other | ) | const [virtual] |
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 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, and returns a list of pairs of global dof indices in identities
. 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> > FiniteElement< dim, spacedim >::hp_line_dof_identities | ( | const FiniteElement< dim, spacedim > & | fe_other | ) | const [virtual] |
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> > FiniteElement< dim, spacedim >::hp_quad_dof_identities | ( | const FiniteElement< dim, spacedim > & | fe_other | ) | const [virtual] |
Same as hp_vertex_dof_indices(), except that the function treats degrees of freedom on quads.
virtual FiniteElementDomination::Domination FiniteElement< dim, spacedim >::compare_for_face_domination | ( | const FiniteElement< dim, spacedim > & | fe_other | ) | const [virtual] |
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 hp paper.
bool FiniteElement< dim, spacedim >::operator== | ( | const FiniteElement< dim, spacedim > & | ) | const |
Comparison operator. We also check for equality of the constraint matrix, which is quite an expensive operation. Do therefore use this function with care, if possible only for debugging purposes.
Since this function is not that important, we avoid an implementational question about comparing arrays and do not compare the matrix arrays restriction and prolongation.
Reimplemented from FiniteElementData< dim >.
std::pair< unsigned int, unsigned int > FiniteElement< dim, spacedim >::system_to_component_index | ( | const unsigned int | index | ) | const [inline] |
Compute vector component and index of this shape function within the shape functions corresponding to this component from the index of a shape function within this finite element.
If the element is scalar, then the component is always zero, and the index within this component is equal to the overall index.
If the shape function referenced has more than one non-zero component, then it cannot be associated with one vector component, and an exception of type ExcShapeFunctionNotPrimitive will be raised.
Note that if the element is composed of other (base) elements, and a base element has more than one component but all its shape functions are primitive (i.e. are non-zero in only one component), then this mapping contains valid information. However, the index of a shape function of this element within one component (i.e. the second number of the respective entry of this array) does not indicate the index of the respective shape function within the base element (since that has more than one vector-component). For this information, refer to the system_to_base_table field and the system_to_base_index() function.
The use of this function is explained extensively in the step-8 and step-20 tutorial programs as well as in the Handling vector valued problems module.
References Assert, FiniteElement< dim, spacedim >::is_primitive(), and FiniteElement< dim, spacedim >::system_to_component_table.
unsigned int FiniteElement< dim, spacedim >::component_to_system_index | ( | const unsigned int | component, | |
const unsigned int | index | |||
) | const [inline] |
Compute the shape function for the given vector component and index.
If the element is scalar, then the component must be zero, and the index within this component is equal to the overall index.
This is the opposite operation from the system_to_component_index() function.
References Assert, and FiniteElement< dim, spacedim >::system_to_component_table.
std::pair< unsigned int, unsigned int > FiniteElement< dim, spacedim >::face_system_to_component_index | ( | const unsigned int | index | ) | const [inline] |
Same as system_to_component_index(), but do it for shape functions and their indices on a face. The range of allowed indices is therefore 0..dofs_per_face.
You will rarely need this function in application programs, since almost all application codes only need to deal with cell indices, not face indices. The function is mainly there for use inside the library.
References Assert, FiniteElement< dim, spacedim >::face_system_to_component_table, FiniteElementData< dim >::face_to_equivalent_cell_index(), and FiniteElement< dim, spacedim >::is_primitive().
unsigned int FiniteElement< dim, spacedim >::adjust_quad_dof_index_for_face_orientation | ( | const unsigned int | index, | |
const bool | face_orientation, | |||
const bool | face_flip, | |||
const bool | face_rotation | |||
) | const |
For faces with non-standard face_orientation in 3D, the dofs on faces (quads) have to be permuted in order to be combined with the correct shape functions. Given a local dof index
on a quad, return the local index, if the face has non-standard face_orientation, face_flip or face_rotation. In 2D and 1D there is no need for permutation and consequently an exception is thrown.
unsigned int FiniteElement< dim, spacedim >::adjust_line_dof_index_for_line_orientation | ( | const unsigned int | index, | |
const bool | line_orientation | |||
) | const |
For lines with non-standard line_orientation in 3D, the dofs on lines have to be permuted in order to be combined with the correct shape functions. Given a local dof index
on a line, return the local index, if the line has non-standard line_orientation. In 2D and 1D there is no need for permutation, so the given index is simply returned.
const std::vector< bool > & FiniteElement< dim, spacedim >::get_nonzero_components | ( | const unsigned int | i | ) | const [inline] |
Return in which of the vector components of this finite element the ith
shape function is non-zero. The length of the returned array is equal to the number of vector components of this element.
For most finite element spaces, the result of this function will be a vector with exactly one element being true
, since for most spaces the individual vector components are independent. In that case, the component with the single zero is also the first element of what system_to_component_index() returns.
Only for those spaces that couple the components, for example to make a shape function divergence free, will there be more than one true
entry.
References Assert, FiniteElementData< dim >::dofs_per_cell, and FiniteElement< dim, spacedim >::nonzero_components.
unsigned int FiniteElement< dim, spacedim >::n_nonzero_components | ( | const unsigned int | i | ) | const [inline] |
Return in how many vector components the ith
shape function is non-zero. This value equals the number of entries equal to true
in the result of the get_nonzero_components() function.
For most finite element spaces, the result will be equal to one. It is not equal to one only for those ansatz spaces for which vector-valued shape functions couple the individual components, for example in order to make them divergence-free.
References Assert, FiniteElementData< dim >::dofs_per_cell, and FiniteElement< dim, spacedim >::n_nonzero_components_table.
bool FiniteElement< dim, spacedim >::is_primitive | ( | const unsigned int | i | ) | const [inline] |
Return whether the ith
shape function is primitive in the sense that the shape function is non-zero in only one vector component. Non-primitive shape functions would then, for example, be those of divergence free ansatz spaces, in which the individual vector components are coupled.
The result of the function is true
if and only if the result of n_nonzero_components(i)
is equal to one.
References Assert, FiniteElement< dim, spacedim >::cached_primitivity, FiniteElementData< dim >::dofs_per_cell, and FiniteElement< dim, spacedim >::n_nonzero_components_table.
bool FiniteElement< dim, spacedim >::is_primitive | ( | ) | const [inline] |
Return whether the entire finite element is primitive, in the sense that all its shape functions are primitive. If the finite element is scalar, then this is always the case.
Since this is an extremely common operation, the result is cached in the cached_primitivity variable which is computed in the constructor.
References FiniteElement< dim, spacedim >::cached_primitivity.
Referenced by FiniteElement< dim, spacedim >::face_system_to_component_index(), and FiniteElement< dim, spacedim >::system_to_component_index().
virtual unsigned int FiniteElement< dim, spacedim >::n_base_elements | ( | ) | const [pure virtual] |
Number of base elements in a mixed discretization.
Note that even for vector valued finite elements, the number of components needs not coincide with the number of base elements, since they may be reused. For example, if you create a FESystem with three identical finite element classes by using the constructor that takes one finite element and a multiplicity, then the number of base elements is still one, although the number of components of the finite element is equal to the multiplicity.
Implemented in FE_ABF< dim >, FE_DGPNonparametric< dim, spacedim >, FE_Nedelec< dim, spacedim >, FE_Poly< POLY, dim, spacedim >, FE_PolyTensor< POLY, dim, spacedim >, FESystem< dim, spacedim >, FE_Poly< PolynomialSpace< dim >, dim, spacedim >, FE_Poly< PolynomialsP< dim >, dim >, FE_Poly< TensorProductPolynomials< dim >, dim >, FE_Poly< TensorProductPolynomials< dim >, dim, dim >, FE_Poly< TensorProductPolynomials< dim >, dim, spacedim >, FE_PolyTensor< PolynomialsRaviartThomas< dim >, dim >, and FE_PolyTensor< PolynomialsABF< dim >, dim >.
virtual const FiniteElement<dim,spacedim>& FiniteElement< dim, spacedim >::base_element | ( | const unsigned int | index | ) | const [pure virtual] |
Access to base element objects. If the element is scalar, then base_element(0)
is this
.
Implemented in FE_ABF< dim >, FE_DGPNonparametric< dim, spacedim >, FE_Nedelec< dim, spacedim >, FE_Poly< POLY, dim, spacedim >, FE_PolyTensor< POLY, dim, spacedim >, FESystem< dim, spacedim >, FE_Poly< PolynomialSpace< dim >, dim, spacedim >, FE_Poly< PolynomialsP< dim >, dim >, FE_Poly< TensorProductPolynomials< dim >, dim >, FE_Poly< TensorProductPolynomials< dim >, dim, dim >, FE_Poly< TensorProductPolynomials< dim >, dim, spacedim >, FE_PolyTensor< PolynomialsRaviartThomas< dim >, dim >, and FE_PolyTensor< PolynomialsABF< dim >, dim >.
virtual unsigned int FiniteElement< dim, spacedim >::element_multiplicity | ( | const unsigned int | index | ) | const [pure virtual] |
This index denotes how often the base element index
is used in a composed element. If the element is scalar, then the result is always equal to one. See the documentation for the n_base_elements() function for more details.
Implemented in FE_ABF< dim >, FE_DGPNonparametric< dim, spacedim >, FE_Nedelec< dim, spacedim >, FE_Poly< POLY, dim, spacedim >, FE_PolyTensor< POLY, dim, spacedim >, FESystem< dim, spacedim >, FE_Poly< PolynomialSpace< dim >, dim, spacedim >, FE_Poly< PolynomialsP< dim >, dim >, FE_Poly< TensorProductPolynomials< dim >, dim >, FE_Poly< TensorProductPolynomials< dim >, dim, dim >, FE_Poly< TensorProductPolynomials< dim >, dim, spacedim >, FE_PolyTensor< PolynomialsRaviartThomas< dim >, dim >, and FE_PolyTensor< PolynomialsABF< dim >, dim >.
std::pair< std::pair< unsigned int, unsigned int >, unsigned int > FiniteElement< dim, spacedim >::system_to_base_index | ( | const unsigned int | index | ) | const [inline] |
Return for shape function index
the base element it belongs to, the number of the copy of this base element (which is between zero and the multiplicity of this element), and the index of this shape function within this base element.
If the element is not composed of others, then base and instance are always zero, and the index is equal to the number of the shape function. If the element is composed of single instances of other elements (i.e. all with multiplicity one) all of which are scalar, then base values and dof indices within this element are equal to the system_to_component_table. It differs only in case the element is composed of other elements and at least one of them is vector-valued itself.
This function returns valid values also in the case of vector-valued (i.e. non-primitive) shape functions, in contrast to the system_to_component_index() function.
References Assert, and FiniteElement< dim, spacedim >::system_to_base_table.
std::pair< std::pair< unsigned int, unsigned int >, unsigned int > FiniteElement< dim, spacedim >::face_system_to_base_index | ( | const unsigned int | index | ) | const [inline] |
Same as system_to_base_index(), but for degrees of freedom located on a face. The range of allowed indices is therefore 0..dofs_per_face.
You will rarely need this function in application programs, since almost all application codes only need to deal with cell indices, not face indices. The function is mainly there for use inside the library.
References Assert, and FiniteElement< dim, spacedim >::face_system_to_base_table.
unsigned int FiniteElement< dim, spacedim >::first_block_of_base | ( | const unsigned int | b | ) | const [inline] |
Given a base element number, return the first block of a BlockVector it would generate.
References Assert, and FiniteElement< dim, spacedim >::first_block_of_base_table.
Referenced by FiniteElement< dim, spacedim >::system_to_block_index().
std::pair< unsigned int, unsigned int > FiniteElement< dim, spacedim >::component_to_base_index | ( | const unsigned int | component | ) | const [inline] |
For each vector component, return which base element implements this component and which vector component in this base element this is. This information is only of interest for vector-valued finite elements which are composed of several sub-elements. In that case, one may want to obtain information about the element implementing a certain vector component, which can be done using this function and the FESystem::base_element() function.
If this is a scalar finite element, then the return value is always equal to a pair of zeros.
References Assert, and FiniteElement< dim, spacedim >::component_to_base_table.
std::pair< unsigned int, unsigned int > FiniteElement< dim, spacedim >::block_to_base_index | ( | const unsigned int | block | ) | const [inline] |
Return the base element for this block and the number of the copy of the base element.
References Assert, FiniteElement< dim, spacedim >::first_block_of_base_table, numbers::invalid_unsigned_int, and FiniteElementData< dim >::n_blocks().
std::pair< unsigned int, unsigned int > FiniteElement< dim, spacedim >::system_to_block_index | ( | const unsigned int | component | ) | const [inline] |
The vector block and the index inside the block for this shape function.
References Assert, FiniteElementData< dim >::dofs_per_cell, FiniteElement< dim, spacedim >::first_block_of_base(), and FiniteElement< dim, spacedim >::system_to_base_table.
unsigned int FiniteElement< dim, spacedim >::component_to_block_index | ( | const unsigned int | component | ) | const |
The vector block for this component.
const std::vector<Point<dim> >& FiniteElement< dim, spacedim >::get_unit_support_points | ( | ) | const |
Return the support points of the trial functions on the unit cell, if the derived finite element defines some. Finite elements that allow some kind of interpolation operation usually have support points. On the other hand, elements that define their degrees of freedom by, for example, moments on faces, or as derivatives, don't have support points. In that case, the returned field is empty.
If the finite element defines support points, then their number equals the number of degrees of freedom of the element. The order of points in the array matches that returned by the cell->get_dof_indices
function.
See the class documentation for details on support points.
bool FiniteElement< dim, spacedim >::has_support_points | ( | ) | const |
Return whether a finite element has defined support points. If the result is true, then a call to the get_unit_support_points() yields a non-empty array.
The result may be false if an element is not defined by interpolating shape functions, for example by P-elements on quadrilaterals. It will usually only be true if the element constructs its shape functions by the requirement that they be one at a certain point and zero at all the points associated with the other shape functions.
In composed elements (i.e. for the FESystem class, the result will be true if all all the base elements have defined support points.
virtual Point<dim> FiniteElement< dim, spacedim >::unit_support_point | ( | const unsigned int | index | ) | const [virtual] |
Return the position of the support point of the indexth
shape function. If it does not exist, raise an exception.
The default implementation simply returns the respective element from the array you get from get_unit_support_points(), but derived elements may overload this function. In particular, note that the FESystem class overloads it so that it can return the support points of individual base elements, of not all the base elements define support points. In this way, you can still ask for certain support points, even if get_unit_support_points() only returns an empty array.
Reimplemented in FESystem< dim, spacedim >.
const std::vector<Point<dim-1> >& FiniteElement< dim, spacedim >::get_unit_face_support_points | ( | ) | const |
Return the support points of the trial functions on the unit face, if the derived finite element defines some. Finite elements that allow some kind of interpolation operation usually have support points. On the other hand, elements that define their degrees of freedom by, for example, moments on faces, or as derivatives, don't have support points. In that case, the returned field is empty
Note that elements that have support points need not necessarily have some on the faces, even if the interpolation points are located physically on a face. For example, the discontinuous elements have interpolation points on the vertices, and for higher degree elements also on the faces, but they are not defined to be on faces since in that case degrees of freedom from both sides of a face (or from all adjacent elements to a vertex) would be identified with each other, which is not what we would like to have). Logically, these degrees of freedom are therefore defined to belong to the cell, rather than the face or vertex. In that case, the returned element would therefore have length zero.
If the finite element defines support points, then their number equals the number of degrees of freedom on the face (dofs_per_face). The order of points in the array matches that returned by the cell->get_dof_indices
function.
See the class documentation for details on support points.
bool FiniteElement< dim, spacedim >::has_face_support_points | ( | ) | const |
Return whether a finite element has defined support points on faces. If the result is true, then a call to the get_unit_face_support_points() yields a non-empty array.
For more information, see the documentation for the has_support_points() function.
virtual Point<dim-1> FiniteElement< dim, spacedim >::unit_face_support_point | ( | const unsigned int | index | ) | const [virtual] |
The function corresponding to the unit_support_point() function, but for faces. See there for more information.
Reimplemented in FESystem< dim, spacedim >.
const std::vector<Point<dim> >& FiniteElement< dim, spacedim >::get_generalized_support_points | ( | ) | const |
Return a support point vector for generalized interpolation.
bool FiniteElement< dim, spacedim >::has_generalized_support_points | ( | ) | const |
Returns true
if the class provides nonempty vectors either from get_unit_support_points() or get_generalized_support_points().
const std::vector<Point<dim-1> >& FiniteElement< dim, spacedim >::get_generalized_face_support_points | ( | ) | const |
bool FiniteElement< dim, spacedim >::has_generalized_face_support_points | ( | ) | const |
Return whether a finite element has defined generalized support points on faces. If the result is true, then a call to the get_generalized_face_support_points yields a non-empty array.
For more information, see the documentation for the has_support_points() function.
virtual void FiniteElement< dim, spacedim >::interpolate | ( | std::vector< double > & | local_dofs, | |
const std::vector< double > & | values | |||
) | const [virtual] |
Interpolate a set of scalar values, computed in the generalized support points.
Reimplemented in FE_ABF< dim >, FE_RaviartThomas< dim >, and FE_RaviartThomasNodal< dim >.
virtual void FiniteElement< dim, spacedim >::interpolate | ( | std::vector< double > & | local_dofs, | |
const std::vector< Vector< double > > & | values, | |||
unsigned int | offset = 0 | |||
) | const [virtual] |
Interpolate a set of vector values, computed in the generalized support points.
Since a finite element often only interpolates part of a vector, offset
is used to determine the first component of the vector to be interpolated. Maybe consider changing your data structures to use the next function.
Reimplemented in FE_ABF< dim >, FE_RaviartThomas< dim >, and FE_RaviartThomasNodal< dim >.
virtual void FiniteElement< dim, spacedim >::interpolate | ( | std::vector< double > & | local_dofs, | |
const VectorSlice< const std::vector< std::vector< double > > > & | values | |||
) | const [virtual] |
Interpolate a set of vector values, computed in the generalized support points.
Reimplemented in FE_ABF< dim >, FE_RaviartThomas< dim >, and FE_RaviartThomasNodal< dim >.
virtual unsigned int FiniteElement< dim, spacedim >::memory_consumption | ( | ) | const [virtual] |
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.
Reimplemented in FE_ABF< dim >, FE_DGP< dim, spacedim >, FE_DGPMonomial< dim >, FE_DGPNonparametric< dim, spacedim >, FE_DGQ< dim, spacedim >, FE_Nedelec< dim, spacedim >, FE_Q< dim, spacedim >, FE_Q_Hierarchical< dim >, FE_RaviartThomas< dim >, FESystem< dim, spacedim >, FE_DGQ< dim >, and FE_Q< dim >.
void FiniteElement< dim, spacedim >::reinit_restriction_and_prolongation_matrices | ( | const bool | isotropic_restriction_only = false , |
|
const bool | isotropic_prolongation_only = false | |||
) | [protected] |
Reinit the vectors of restriction and prolongation matrices to the right sizes: For every refinement case, except for RefinementCase::no_refinement, and for every child of that refinement case the space of one restriction and prolongation matrix is allocated, see the documentation of the restriction and prolongation vectors for more detail on the actual vector sizes.
isotropic_restriction_only,: | only the restriction matrices required for isotropic refinement are reinited to the right size. | |
isotropic_prolongation_only,: | only the prolongation matrices required for isotropic refinement are reinited to the right size. |
TableIndices<2> FiniteElement< dim, spacedim >::interface_constraints_size | ( | ) | const [protected] |
Return the size of interface constraint matrices. Since this is needed in every derived finite element class when initializing their size, it is placed into this function, to avoid having to recompute the dimension-dependent size of these matrices each time.
Note that some elements do not implement the interface constraints for certain polynomial degrees. In this case, this function still returns the size these matrices should have when implemented, but the actual matrices are empty.
void FiniteElement< dim, spacedim >::compute_2nd | ( | const Mapping< dim, spacedim > & | mapping, | |
const typename Triangulation< dim, spacedim >::cell_iterator & | cell, | |||
const unsigned int | offset, | |||
typename Mapping< dim, spacedim >::InternalDataBase & | mapping_internal, | |||
InternalDataBase & | fe_internal, | |||
FEValuesData< dim, spacedim > & | data | |||
) | const [protected] |
Compute second derivatives by finite differences of gradients.
static std::vector<unsigned int> FiniteElement< dim, spacedim >::compute_n_nonzero_components | ( | const std::vector< std::vector< bool > > & | nonzero_components | ) | [static, protected] |
Given the pattern of nonzero components for each shape function, compute for each entry how many components are non-zero for each shape function. This function is used in the constructor of this class.
virtual UpdateFlags FiniteElement< dim, spacedim >::update_once | ( | const UpdateFlags | flags | ) | const [protected, pure virtual] |
Determine the values a finite element should compute on initialization of data for FEValues.
Given a set of flags indicating what quantities are requested from a FEValues object, update_once() and update_each() compute which values must really be computed. Then, the fill_*_values
functions are called with the result of these.
Furthermore, values must be computed either on the unit cell or on the physical cell. For instance, the function values of FE_Q do only depend on the quadrature points on the unit cell. Therefore, this flags will be returned by update_once(). The gradients require computation of the covariant transformation matrix. Therefore, update_covariant_transformation
and update_gradients
will be returned by update_each().
For an example see the same function in the derived class FE_Q.
Implemented in FE_ABF< dim >, FE_DGPNonparametric< dim, spacedim >, FE_Nedelec< dim, spacedim >, FE_Poly< POLY, dim, spacedim >, FE_PolyTensor< POLY, dim, spacedim >, FESystem< dim, spacedim >, FE_Poly< PolynomialSpace< dim >, dim, spacedim >, FE_Poly< PolynomialsP< dim >, dim >, FE_Poly< TensorProductPolynomials< dim >, dim >, FE_Poly< TensorProductPolynomials< dim >, dim, dim >, FE_Poly< TensorProductPolynomials< dim >, dim, spacedim >, FE_PolyTensor< PolynomialsRaviartThomas< dim >, dim >, and FE_PolyTensor< PolynomialsABF< dim >, dim >.
virtual UpdateFlags FiniteElement< dim, spacedim >::update_each | ( | const UpdateFlags | flags | ) | const [protected, pure virtual] |
Complementary function for update_once().
While update_once() returns the values to be computed on the unit cell for yielding the required data, this function determines the values that must be recomputed on each cell.
Refer to update_once() for more details.
Implemented in FE_ABF< dim >, FE_DGPNonparametric< dim, spacedim >, FE_Nedelec< dim, spacedim >, FE_Poly< POLY, dim, spacedim >, FE_PolyTensor< POLY, dim, spacedim >, FESystem< dim, spacedim >, FE_Poly< PolynomialSpace< dim >, dim, spacedim >, FE_Poly< PolynomialsP< dim >, dim >, FE_Poly< TensorProductPolynomials< dim >, dim >, FE_Poly< TensorProductPolynomials< dim >, dim, dim >, FE_Poly< TensorProductPolynomials< dim >, dim, spacedim >, FE_PolyTensor< PolynomialsRaviartThomas< dim >, dim >, and FE_PolyTensor< PolynomialsABF< dim >, dim >.
virtual FiniteElement<dim,spacedim>* FiniteElement< dim, spacedim >::clone | ( | ) | const [protected, pure virtual] |
A sort of virtual copy constructor. Some places in the library, for example the constructors of FESystem as well as the hp::FECollection class, need to make copied of finite elements without knowing their exact type. They do so through this function.
Implemented in FE_ABF< dim >, FE_DGP< dim, spacedim >, FE_DGPMonomial< dim >, FE_DGPNonparametric< dim, spacedim >, FE_DGQ< dim, spacedim >, FE_DGQArbitraryNodes< dim >, FE_Nedelec< dim, spacedim >, FE_Q< dim, spacedim >, FE_Q_Hierarchical< dim >, FE_RaviartThomas< dim >, FE_RaviartThomasNodal< dim >, FESystem< dim, spacedim >, FE_DGQ< dim >, and FE_Q< dim >.
virtual Mapping<dim,spacedim>::InternalDataBase* FiniteElement< dim, spacedim >::get_data | ( | const UpdateFlags | flags, | |
const Mapping< dim, spacedim > & | mapping, | |||
const Quadrature< dim > & | quadrature | |||
) | const [private, pure virtual] |
Prepare internal data structures and fill in values independent of the cell. Returns a pointer to an object of which the caller of this function then has to assume ownership (which includes destruction when it is no more needed).
Implemented in FE_DGPNonparametric< dim, spacedim >, FE_Nedelec< dim, spacedim >, FE_Poly< POLY, dim, spacedim >, FE_PolyTensor< POLY, dim, spacedim >, FESystem< dim, spacedim >, FE_Poly< PolynomialSpace< dim >, dim, spacedim >, FE_Poly< TensorProductPolynomials< dim >, dim, dim >, and FE_Poly< TensorProductPolynomials< dim >, dim, spacedim >.
virtual Mapping<dim,spacedim>::InternalDataBase* FiniteElement< dim, spacedim >::get_face_data | ( | const UpdateFlags | flags, | |
const Mapping< dim, spacedim > & | mapping, | |||
const Quadrature< dim-1 > & | quadrature | |||
) | const [private, virtual] |
Prepare internal data structure for transformation of faces and fill in values independent of the cell. Returns a pointer to an object of which the caller of this function then has to assume ownership (which includes destruction when it is no more needed).
virtual Mapping<dim,spacedim>::InternalDataBase* FiniteElement< dim, spacedim >::get_subface_data | ( | const UpdateFlags | flags, | |
const Mapping< dim, spacedim > & | mapping, | |||
const Quadrature< dim-1 > & | quadrature | |||
) | const [private, virtual] |
Prepare internal data structure for transformation of children of faces and fill in values independent of the cell. Returns a pointer to an object of which the caller of this function then has to assume ownership (which includes destruction when it is no more needed).
virtual void FiniteElement< dim, spacedim >::fill_fe_values | ( | const Mapping< dim, spacedim > & | mapping, | |
const typename Triangulation< dim, spacedim >::cell_iterator & | cell, | |||
const Quadrature< dim > & | quadrature, | |||
typename Mapping< dim, spacedim >::InternalDataBase & | mapping_internal, | |||
typename Mapping< dim, spacedim >::InternalDataBase & | fe_internal, | |||
FEValuesData< dim, spacedim > & | data, | |||
enum CellSimilarity::Similarity & | cell_similarity | |||
) | const [private, pure virtual] |
Fill the fields of FEValues. This function performs all the operations needed to compute the data of an FEValues object.
The same function in mapping
must have been called for the same cell first!
Implemented in FE_DGPNonparametric< dim, spacedim >, FE_Nedelec< dim, spacedim >, FE_Poly< POLY, dim, spacedim >, FE_PolyTensor< POLY, dim, spacedim >, FESystem< dim, spacedim >, FE_Poly< PolynomialSpace< dim >, dim, spacedim >, FE_Poly< TensorProductPolynomials< dim >, dim, dim >, and FE_Poly< TensorProductPolynomials< dim >, dim, spacedim >.
virtual void FiniteElement< dim, spacedim >::fill_fe_face_values | ( | const Mapping< dim, spacedim > & | mapping, | |
const typename Triangulation< dim, spacedim >::cell_iterator & | cell, | |||
const unsigned int | face_no, | |||
const Quadrature< dim-1 > & | quadrature, | |||
typename Mapping< dim, spacedim >::InternalDataBase & | mapping_internal, | |||
typename Mapping< dim, spacedim >::InternalDataBase & | fe_internal, | |||
FEValuesData< dim, spacedim > & | data | |||
) | const [private, pure virtual] |
Fill the fields of FEFaceValues. This function performs all the operations needed to compute the data of an FEFaceValues object.
The same function in mapping
must have been called for the same cell first!
Implemented in FE_DGPNonparametric< dim, spacedim >, FE_Nedelec< dim, spacedim >, FE_Poly< POLY, dim, spacedim >, FE_PolyTensor< POLY, dim, spacedim >, FESystem< dim, spacedim >, FE_Poly< PolynomialSpace< dim >, dim, spacedim >, FE_Poly< TensorProductPolynomials< dim >, dim, dim >, and FE_Poly< TensorProductPolynomials< dim >, dim, spacedim >.
virtual void FiniteElement< dim, spacedim >::fill_fe_subface_values | ( | 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, | |||
typename Mapping< dim, spacedim >::InternalDataBase & | mapping_internal, | |||
typename Mapping< dim, spacedim >::InternalDataBase & | fe_internal, | |||
FEValuesData< dim, spacedim > & | data | |||
) | const [private, pure virtual] |
Fill the fields of FESubfaceValues. This function performs all the operations needed to compute the data of an FESubfaceValues object.
The same function in mapping
must have been called for the same cell first!
Implemented in FE_DGPNonparametric< dim, spacedim >, FE_Nedelec< dim, spacedim >, FE_Poly< POLY, dim, spacedim >, FE_PolyTensor< POLY, dim, spacedim >, FESystem< dim, spacedim >, FE_Poly< PolynomialSpace< dim >, dim, spacedim >, FE_Poly< TensorProductPolynomials< dim >, dim, dim >, and FE_Poly< TensorProductPolynomials< dim >, dim, spacedim >.
friend class InternalDataBase [friend] |
friend class FEValuesBase< dim, spacedim > [friend] |
friend class FEValues< dim, spacedim > [friend] |
friend class FEFaceValues< dim, spacedim > [friend] |
friend class FESubfaceValues< dim, spacedim > [friend] |
friend class FESystem [friend] |
friend class FE_PolyTensor [friend] |
friend class hp::FECollection< dim, spacedim > [friend] |
const bool FiniteElement< dim, spacedim >::cached_primitivity [protected] |
Store whether all shape functions are primitive. Since finding this out is a very common operation, we cache the result, i.e. compute the value in the constructor for simpler access.
Referenced by FiniteElement< dim, spacedim >::is_primitive().
std::vector<std::vector<FullMatrix<double> > > FiniteElement< dim, spacedim >::restriction [protected] |
Vector of projection matrices. See get_restriction_matrix() above. The constructor initializes these matrices to zero dimensions, which can be changed by derived classes implementing them.
Note, that restriction[refinement_case-1][child]
includes the restriction matrix of child child
for the RefinementCase refinement_case
. Here, we use refinement_case-1
instead of refinement_case
as for RefinementCase::no_refinement(=0) there are no restriction matrices available.
std::vector<std::vector<FullMatrix<double> > > FiniteElement< dim, spacedim >::prolongation [protected] |
Vector of embedding matrices. See get_prolongation_matrix()
above. The constructor initializes these matrices to zero dimensions, which can be changed by derived classes implementing them.
Note, that prolongation[refinement_case-1][child]
includes the prolongation matrix of child child
for the RefinementCase refinement_case
. Here, we use refinement_case-1
instead of refinement_case
as for RefinementCase::no_refinement(=0) there are no prolongation matrices available.
FullMatrix<double> FiniteElement< dim, spacedim >::interface_constraints [protected] |
Specify the constraints which the dofs on the two sides of a cell interface underly if the line connects two cells of which one is refined once.
For further details see the general description of the derived class.
This field is obviously useless in one dimension and has there a zero size.
std::vector<Point<dim> > FiniteElement< dim, spacedim >::unit_support_points [protected] |
List of support points on the unit cell, in case the finite element has any. The constructor leaves this field empty, derived classes may write in some contents.
Finite elements that allow some kind of interpolation operation usually have support points. On the other hand, elements that define their degrees of freedom by, for example, moments on faces, or as derivatives, don't have support points. In that case, this field remains empty.
std::vector<Point<dim-1> > FiniteElement< dim, spacedim >::unit_face_support_points [protected] |
Same for the faces. See the description of the get_unit_face_support_points() function for a discussion of what contributes a face support point.
std::vector<Point<dim> > FiniteElement< dim, spacedim >::generalized_support_points [protected] |
Support points used for interpolation functions of non-Lagrangian elements.
std::vector<Point<dim-1> > FiniteElement< dim, spacedim >::generalized_face_support_points [protected] |
Face support points used for interpolation functions of non-Lagrangian elements.
Table<2,int> FiniteElement< dim, spacedim >::adjust_quad_dof_index_for_face_orientation_table [protected] |
For faces with non-standard face_orientation in 3D, the dofs on faces (quads) have to be permuted in order to be combined with the correct shape functions. Given a local dof index
on a quad, return the shift in the local index, if the face has non-standard face_orientation, i.e. old_index + shift = new_index
. In 2D and 1D there is no need for permutation so the vector is empty. In 3D it has the size of dofs_per_quad * 8
, where 8 is the number of orientations, a face can be in (all comibinations of the three bool flags face_orientation, face_flip and face_rotation).
The standard implementation fills this with zeros, i.e. no permuatation at all. Derived finite element classes have to fill this Table with the correct values.
std::vector<int> FiniteElement< dim, spacedim >::adjust_line_dof_index_for_line_orientation_table [protected] |
For lines with non-standard line_orientation in 3D, the dofs on lines have to be permuted in order to be combined with the correct shape functions. Given a local dof index
on a line, return the shift in the local index, if the line has non-standard line_orientation, i.e. old_index + shift = new_index
. In 2D and 1D there is no need for permutation so the vector is empty. In 3D it has the size of dofs_per_line.
The standard implementation fills this with zeros, i.e. no permuatation at all. Derived finite element classes have to fill this vector with the correct values.
std::vector< std::pair<unsigned int, unsigned int> > FiniteElement< dim, spacedim >::system_to_component_table [private] |
Store what system_to_component_index() will return.
Referenced by FiniteElement< dim, spacedim >::component_to_system_index(), and FiniteElement< dim, spacedim >::system_to_component_index().
std::vector< std::pair<unsigned int, unsigned int> > FiniteElement< dim, spacedim >::face_system_to_component_table [private] |
Map between linear dofs and component dofs on face. This is filled with default values in the constructor, but derived classes will have to overwrite the information if necessary.
By component, we mean the vector component, not the base element. The information thus makes only sense if a shape function is non-zero in only one component.
Referenced by FiniteElement< dim, spacedim >::face_system_to_component_index().
std::vector<std::pair<std::pair<unsigned int,unsigned int>,unsigned int> > FiniteElement< dim, spacedim >::system_to_base_table [private] |
For each shape function, store to which base element and which instance of this base element (in case its multiplicity is greater than one) it belongs, and its index within this base element. If the element is not composed of others, then base and instance are always zero, and the index is equal to the number of the shape function. If the element is composed of single instances of other elements (i.e. all with multiplicity one) all of which are scalar, then base values and dof indices within this element are equal to the system_to_component_table. It differs only in case the element is composed of other elements and at least one of them is vector-valued itself.
This array has valid values also in the case of vector-valued (i.e. non-primitive) shape functions, in contrast to the system_to_component_table.
Referenced by FiniteElement< dim, spacedim >::system_to_base_index(), and FiniteElement< dim, spacedim >::system_to_block_index().
std::vector<std::pair<std::pair<unsigned int,unsigned int>,unsigned int> > FiniteElement< dim, spacedim >::face_system_to_base_table [private] |
Likewise for the indices on faces.
Referenced by FiniteElement< dim, spacedim >::face_system_to_base_index().
std::vector<unsigned int> FiniteElement< dim, spacedim >::first_block_of_base_table [private] |
For each base element, store the first block in a block vector it will generate.
Referenced by FiniteElement< dim, spacedim >::block_to_base_index(), and FiniteElement< dim, spacedim >::first_block_of_base().
std::vector<std::pair<std::pair<unsigned int, unsigned int>, unsigned int> > FiniteElement< dim, spacedim >::component_to_base_table [private] |
The base element establishing a component.
For each component number c
, the entries have the following meaning:
table[c].first.first
c
. table[c].first.second
c
. table[c].second
c
. This variable is set to the correct size by the constructor of this class, but needs to be initialized by derived classes, unless its size is one and the only entry is a zero, which is the case for scalar elements. In that case, the initialization by the base class is sufficient.
Referenced by FiniteElement< dim, spacedim >::component_to_base_index().
const std::vector<bool> FiniteElement< dim, spacedim >::restriction_is_additive_flags [private] |
Projection matrices are concatenated or summed up.
This flags decides on how the projection matrices of the children of the same father are put together to one operator. The possible modes are concatenation and summation.
If the projection is defined by an interpolation operator, the child matrices are concatenated, i.e. values belonging to the same node functional are identified and enter the interpolated value only once. In this case, the flag must be false
.
For projections with respect to scalar products, the child matrices must be summed up to build the complete matrix. The flag should be true
.
For examples of use of these flags, see the places in the library where it is queried.
There is one flag per shape function, indicating whether it belongs to the class of shape functions that are additive in the restriction or not.
Note that in previous versions of the library, there was one flag per vector component of the element. This is based on the fact that all the shape functions that belong to the same vector component must necessarily behave in the same way, to make things reasonable. However, the problem is that it is sometimes impossible to query this flag in the vector-valued case: this used to be done with the system_to_component_index function that returns which vector component a shape function is associated with. The point is that since we now support shape functions that are associated with more than one vector component (for example the shape functions of Raviart-Thomas, or Nedelec elements), that function can no more be used, so it can be difficult to find out which for vector component we would like to query the restriction-is-additive flags.
Referenced by FiniteElement< dim, spacedim >::restriction_is_additive().
const std::vector<std::vector<bool> > FiniteElement< dim, spacedim >::nonzero_components [private] |
For each shape function, give a vector of bools (with size equal to the number of vector components which this finite element has) indicating in which component each of these shape functions is non-zero.
For primitive elements, there is only one non-zero component.
Referenced by FiniteElement< dim, spacedim >::get_nonzero_components().
const std::vector<unsigned int> FiniteElement< dim, spacedim >::n_nonzero_components_table [private] |
This array holds how many values in the respective entry of the nonzero_components element are non-zero. The array is thus a short-cut to allow faster access to this information than if we had to count the non-zero entries upon each request for this information. The field is initialized in the constructor of this class.
Referenced by FiniteElement< dim, spacedim >::is_primitive(), and FiniteElement< dim, spacedim >::n_nonzero_components().
const double FiniteElement< dim, spacedim >::fd_step_length [static, private] |
Second derivatives of shapes functions are not computed analytically, but by finite differences of the gradients. This static variable denotes the step length to be used for that. It's value is set to 1e-6.