Abstract class for the description of a general finite element on a reference simplex in dim dimensions. More...

#include <dofhandler.hh>

Inheritance diagram for FiniteElement< dim, spacedim >:

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Collaboration diagram for FiniteElement< dim, spacedim >:

Public Member Functions
	FiniteElement ()
	Constructor. More...

void	init ()
	Clears all internal structures. More...

const unsigned int	n_dofs () const
	Returns the number of degrees of freedom needed by the finite element. More...

const unsigned int	n_object_dofs (unsigned int object_dim, DofMultiplicity multiplicity)
	Returns the number of single dofs/dof pairs/triples/sextuples that lie on a single geometric entity of the dimension `object_dim`. More...

virtual double	basis_value (const unsigned int i, const arma::vec::fixed< dim > &p) const =0
	Calculates the value of the `i-th` raw basis function at the point `p` on the reference element. More...

virtual arma::vec::fixed< dim >	basis_vector (const unsigned int i, const arma::vec::fixed< dim > &p) const =0
	Variant of basis_value() for vectorial finite elements. More...

virtual arma::vec::fixed< dim >	basis_grad (const unsigned int i, const arma::vec::fixed< dim > &p) const =0
	Calculates the gradient of the `i-th` raw basis function at the point `p` on the reference element. More...

virtual arma::mat::fixed< dim, dim >	basis_grad_vector (const unsigned int i, const arma::vec::fixed< dim > &p) const =0
	Variant of basis_grad() for vectorial finite elements. More...

virtual void	compute_node_matrix ()
	Initializes the `node_matrix` for computing the coefficients of the raw basis functions from values at support points. More...

virtual FEInternalData *	initialize (const Quadrature< dim > &q, UpdateFlags flags)
	Calculates the data on the reference cell. More...

virtual UpdateFlags	update_each (UpdateFlags flags)
	Decides which additional quantities have to be computed for each cell. More...

virtual void	fill_fe_values (const Quadrature< dim > &q, FEInternalData &data, FEValuesData< dim, spacedim > &fv_data)
	Computes the shape function values and gradients on the actual cell and fills the FEValues structure. More...

virtual const unsigned int	polynomial_order () const
	Returns the maximum degree of space of polynomials contained in the finite element space. More...

const bool	is_scalar () const
	Indicates whether the finite element function space is scalar or vectorial. More...

const std::vector< arma::vec::fixed< dim > > &	get_generalized_support_points ()
	Returns either the generalized support points (if they are defined) or the unit support points. More...

virtual	~FiniteElement ()
	Destructor. More...

Protected Attributes
unsigned int	number_of_dofs
	Total number of degrees of freedom at one finite element. More...

unsigned int	number_of_single_dofs [dim+1]
	Number of single dofs at one geometrical entity of the given dimension (point, line, triangle, tetrahedron). More...

unsigned int	number_of_pairs [dim+1]
	Number of pairs of dofs at one geometrical entity of the given dimension (applicable to lines and triangles). More...

unsigned int	number_of_triples [dim+1]
	Number of triples of dofs associated to one triangle. More...

unsigned int	number_of_sextuples [dim+1]
	Number of sextuples of dofs associated to one triangle. More...

unsigned int	order
	Polynomial order - to be possibly used in hp methods. More...

bool	is_scalar_fe
	Indicator of scalar versus vectorial finite element. More...

arma::mat	node_matrix
	Matrix that determines the coefficients of the raw basis functions from the values at the support points. More...

std::vector< arma::vec::fixed< dim > >	unit_support_points
	Support points for Lagrangean finite elements. More...

std::vector< arma::vec::fixed< dim > >	generalized_support_points
	Support points for non-Lagrangean finite elements. More...

Detailed Description

template<unsigned int dim, unsigned int spacedim>
class FiniteElement< dim, spacedim >

Abstract class for the description of a general finite element on a reference simplex in dim dimensions.

Description of dofs:

The reference cell consists of lower dimensional entities (points, lines, triangles). Each dof is associated to one of these entities. This means that if the entity is shared by 2 or more neighbouring cells in the mesh then this dof is shared by the finite elements on all of these cells. If a dof is associated to the cell itself then it is not shared with neighbouring cells. The ordering of nodes in the entity may not be appropriate for the finite elements on the neighbouring cells, hence we need to describe how the order of dofs changes with the relative configuration of the entity with respect to the actual cell. For this reason we define the dof multiplicity which allows to group the dofs as described in DofMultiplicity.

Support points:

Sometimes it is convenient to describe the function space using a basis (called the raw basis) that is different from the set of shape functions for the finite element (the actual basis). For this reason we define the support points which play the role of nodal functionals associated to the particular dofs. To convert between the two bases one can use the node_matrix, which is constructed by the method compute_node_matrix(). In the case of non-Lagrangean finite elements the dofs are not associated to nodal functionals but e.g. to derivatives or averages. For that reason we distinguish between the unit support points which are uniquely associated to the dofs and the generalized support points that are auxiliary for the calculation of the dof functionals.

Definition at line 29 of file dofhandler.hh.

Constructor & Destructor Documentation

template<unsigned int dim, unsigned int spacedim>

FiniteElement< dim, spacedim >::FiniteElement ( )

Constructor.

Definition at line 33 of file finite_element.cc.

template<unsigned int dim, unsigned int spacedim>

FiniteElement< dim, spacedim >::~FiniteElement ( )

virtual

Destructor.

Definition at line 175 of file finite_element.cc.

Member Function Documentation

template<unsigned int dim, unsigned int spacedim>

virtual arma::vec::fixed<dim> FiniteElement< dim, spacedim >::basis_grad	(	const unsigned int	i,
		const arma::vec::fixed< dim > &	p
	)		const

pure virtual

Calculates the gradient of the i-th raw basis function at the point p on the reference element.

The gradient components are relative to the reference cell coordinate system.

Parameters

i	Number of the basis function.
p	Point of evaluation.

Implemented in FE_P_disc< degree, dim, spacedim >, FE_P_disc< 1, 2, 3 >, FE_P_disc< 1, 3, 3 >, FE_P_disc< 1, 1, 3 >, FE_P_disc< 0, dim, 3 >, FE_P< degree, dim, spacedim >, FE_RT0< dim, spacedim >, and FE_RT0< dim, 3 >.

template<unsigned int dim, unsigned int spacedim>

virtual arma::mat::fixed<dim,dim> FiniteElement< dim, spacedim >::basis_grad_vector	(	const unsigned int	i,
		const arma::vec::fixed< dim > &	p
	)		const

pure virtual

Variant of basis_grad() for vectorial finite elements.

Calculates the gradient of the i-th vector-valued raw basis function at the point p on the reference element. The gradient components are relative to the reference cell coordinate system.

Parameters

i	Number of the basis function.
p	Point of evaluation.

Implemented in FE_P_disc< degree, dim, spacedim >, FE_P_disc< 1, 2, 3 >, FE_P_disc< 1, 3, 3 >, FE_P_disc< 1, 1, 3 >, FE_P_disc< 0, dim, 3 >, FE_P< degree, dim, spacedim >, FE_RT0< dim, spacedim >, and FE_RT0< dim, 3 >.

template<unsigned int dim, unsigned int spacedim>

virtual double FiniteElement< dim, spacedim >::basis_value	(	const unsigned int	i,
		const arma::vec::fixed< dim > &	p
	)		const

pure virtual

Calculates the value of the i-th raw basis function at the point p on the reference element.

Parameters

i	Number of the basis function.
p	Point of evaluation.

Implemented in FE_P_disc< degree, dim, spacedim >, FE_P_disc< 1, 2, 3 >, FE_P_disc< 1, 3, 3 >, FE_P_disc< 1, 1, 3 >, FE_P_disc< 0, dim, 3 >, FE_P< degree, dim, spacedim >, FE_RT0< dim, spacedim >, and FE_RT0< dim, 3 >.

template<unsigned int dim, unsigned int spacedim>

virtual arma::vec::fixed<dim> FiniteElement< dim, spacedim >::basis_vector	(	const unsigned int	i,
		const arma::vec::fixed< dim > &	p
	)		const

pure virtual

Variant of basis_value() for vectorial finite elements.

Calculates the value i-th vector-valued raw basis function at the point p on the reference element.

Parameters

i	Number of the basis function.
p	Point of evaluation.

Implemented in FE_P_disc< degree, dim, spacedim >, FE_P_disc< 1, 2, 3 >, FE_P_disc< 1, 3, 3 >, FE_P_disc< 1, 1, 3 >, FE_P_disc< 0, dim, 3 >, FE_P< degree, dim, spacedim >, FE_RT0< dim, spacedim >, and FE_RT0< dim, 3 >.

template<unsigned int dim, unsigned int spacedim>

void FiniteElement< dim, spacedim >::compute_node_matrix ( )

inlinevirtual

Initializes the node_matrix for computing the coefficients of the raw basis functions from values at support points.

The method is implemented for the case of Langrangean finite element. In other cases it may be reimplemented.

Reimplemented in FE_RT0< dim, spacedim >, and FE_RT0< dim, 3 >.

Definition at line 80 of file finite_element.cc.

template<unsigned int dim, unsigned int spacedim>

void FiniteElement< dim, spacedim >::fill_fe_values	(	const Quadrature< dim > &	q,
		FEInternalData &	data,
		FEValuesData< dim, spacedim > &	fv_data
	)

inlinevirtual

Computes the shape function values and gradients on the actual cell and fills the FEValues structure.

Parameters

q	Quadrature rule.
data	Precomputed finite element data.
fv_data	Data to be computed.

Reimplemented in FE_RT0< dim, spacedim >, and FE_RT0< dim, 3 >.

Definition at line 138 of file finite_element.cc.

template<unsigned int dim, unsigned int spacedim>

const vector< arma::vec::fixed< dim > > & FiniteElement< dim, spacedim >::get_generalized_support_points ( )

Returns either the generalized support points (if they are defined) or the unit support points.

Definition at line 161 of file finite_element.cc.

template<unsigned int dim, unsigned int spacedim>

void FiniteElement< dim, spacedim >::init ( )

Clears all internal structures.

Definition at line 39 of file finite_element.cc.

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template<unsigned int dim, unsigned int spacedim>

FEInternalData * FiniteElement< dim, spacedim >::initialize	(	const Quadrature< dim > &	q,
		UpdateFlags	flags
	)

virtual

Calculates the data on the reference cell.

Parameters

q	Quadrature rule.
flags	Update flags.

Reimplemented in FE_RT0< dim, spacedim >, and FE_RT0< dim, 3 >.

Definition at line 95 of file finite_element.cc.

template<unsigned int dim, unsigned int spacedim>

const bool FiniteElement< dim, spacedim >::is_scalar ( ) const

inline

Indicates whether the finite element function space is scalar or vectorial.

Definition at line 269 of file finite_element.hh.

template<unsigned int dim, unsigned int spacedim>

const unsigned int FiniteElement< dim, spacedim >::n_dofs ( ) const

inline

Returns the number of degrees of freedom needed by the finite element.

Definition at line 53 of file finite_element.cc.

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template<unsigned int dim, unsigned int spacedim>

const unsigned int FiniteElement< dim, spacedim >::n_object_dofs	(	unsigned int	object_dim,
		DofMultiplicity	multiplicity
	)

inline

Returns the number of single dofs/dof pairs/triples/sextuples that lie on a single geometric entity of the dimension object_dim.

Parameters

object_dim	Dimension of the geometric entity.
multiplicity	Multiplicity of dofs.

Definition at line 59 of file finite_element.cc.

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template<unsigned int dim, unsigned int spacedim>

virtual const unsigned int FiniteElement< dim, spacedim >::polynomial_order ( ) const

inlinevirtual

Returns the maximum degree of space of polynomials contained in the finite element space.

For possible use in hp methods.

Definition at line 261 of file finite_element.hh.

template<unsigned int dim, unsigned int spacedim>

UpdateFlags FiniteElement< dim, spacedim >::update_each ( UpdateFlags flags )

inlinevirtual

Decides which additional quantities have to be computed for each cell.

Parameters

flags Computed update flags.

Reimplemented in FE_RT0< dim, spacedim >, and FE_RT0< dim, 3 >.

Definition at line 127 of file finite_element.cc.

Member Data Documentation

template<unsigned int dim, unsigned int spacedim>

std::vector<arma::vec::fixed<dim> > FiniteElement< dim, spacedim >::generalized_support_points

protected

Support points for non-Lagrangean finite elements.

In case of non-Lagrangean finite elements the meaning of the support points is different, hence we denote the structure as generalized_support_points.

Definition at line 347 of file finite_element.hh.

template<unsigned int dim, unsigned int spacedim>

bool FiniteElement< dim, spacedim >::is_scalar_fe

protected

Indicator of scalar versus vectorial finite element.

Definition at line 321 of file finite_element.hh.

template<unsigned int dim, unsigned int spacedim>

arma::mat FiniteElement< dim, spacedim >::node_matrix

protected

Matrix that determines the coefficients of the raw basis functions from the values at the support points.

Definition at line 327 of file finite_element.hh.

template<unsigned int dim, unsigned int spacedim>

unsigned int FiniteElement< dim, spacedim >::number_of_dofs

protected

Total number of degrees of freedom at one finite element.

Definition at line 289 of file finite_element.hh.

template<unsigned int dim, unsigned int spacedim>

unsigned int FiniteElement< dim, spacedim >::number_of_pairs[dim+1]

protected

Number of pairs of dofs at one geometrical entity of the given dimension (applicable to lines and triangles).

Definition at line 301 of file finite_element.hh.

template<unsigned int dim, unsigned int spacedim>

unsigned int FiniteElement< dim, spacedim >::number_of_sextuples[dim+1]

protected

Number of sextuples of dofs associated to one triangle.

Definition at line 311 of file finite_element.hh.

template<unsigned int dim, unsigned int spacedim>

unsigned int FiniteElement< dim, spacedim >::number_of_single_dofs[dim+1]

protected

Number of single dofs at one geometrical entity of the given dimension (point, line, triangle, tetrahedron).

Definition at line 295 of file finite_element.hh.

template<unsigned int dim, unsigned int spacedim>

unsigned int FiniteElement< dim, spacedim >::number_of_triples[dim+1]

protected

Number of triples of dofs associated to one triangle.

Definition at line 306 of file finite_element.hh.

template<unsigned int dim, unsigned int spacedim>

unsigned int FiniteElement< dim, spacedim >::order

protected

Polynomial order - to be possibly used in hp methods.

Definition at line 316 of file finite_element.hh.

template<unsigned int dim, unsigned int spacedim>

std::vector<arma::vec::fixed<dim> > FiniteElement< dim, spacedim >::unit_support_points

protected

Support points for Lagrangean finite elements.

Support points are points in the reference element where function values determine the dofs. In case of Lagrangean finite elements the dof values are precisely the function values at unit_support_points.

Definition at line 338 of file finite_element.hh.

The documentation for this class was generated from the following files:

/opt/flow123d/flow123d/src/fem/dofhandler.hh
/opt/flow123d/flow123d/src/fem/finite_element.hh
/opt/flow123d/flow123d/src/fem/finite_element.cc

Public Member Functions

Protected Attributes

Detailed Description

template<unsigned int dim, unsigned int spacedim> class FiniteElement< dim, spacedim >

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation

template<unsigned int dim, unsigned int spacedim>
class FiniteElement< dim, spacedim >