Quadratic Lagrangian finite element on a quadrilateral reference element. More...

#include <lf/uscalfe/uscalfe.h>

Inheritance diagram for lf::uscalfe::FeLagrangeO2Quad< SCALAR >:

Public Member Functions
	FeLagrangeO2Quad (const FeLagrangeO2Quad &)=default

	FeLagrangeO2Quad (FeLagrangeO2Quad &&) noexcept=default

FeLagrangeO2Quad &	operator= (const FeLagrangeO2Quad &)=default

FeLagrangeO2Quad &	operator= (FeLagrangeO2Quad &&) noexcept=default

	FeLagrangeO2Quad ()=default

	~FeLagrangeO2Quad () override=default

base::RefEl	RefEl () const override
	Tells the type of reference cell underlying the parametric finite element.

unsigned	Degree () const override
	Quadratic Lagrangian finite elements sport polynomials of degree 2.

size_type	NumRefShapeFunctions () const override
	Nine local shape functions are associated with a quadrilateral cell.

size_type	NumRefShapeFunctions (dim_t codim) const override
	One shape function is attached to each node and edge of the quadrilateral. One interior shape function.

size_type	NumRefShapeFunctions (dim_t codim, sub_idx_t) const override
	One shape function is attached to each node and each edge of the quadrilateral. One interior shape function.

Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic >	EvalReferenceShapeFunctions (const Eigen::MatrixXd &refcoords) const override
	Point evaluation of reference local shape functions in unit square.

Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic >	GradientsReferenceShapeFunctions (const Eigen::MatrixXd &refcoords) const override
	Point evaluation of gradient of reference shape functions for quadratic Lagrangian finite elements on quadrilateral cells.

Eigen::MatrixXd	EvaluationNodes () const override
	Evaluation nodes are the vertices, the midpoints of the edges and the center of the quadrilateral.

size_type	NumEvaluationNodes () const override
	Nine evaluation nodes, same as the number of local shape functions.

Public Member Functions inherited from lf::fe::ScalarReferenceFiniteElement< SCALAR >
virtual	~ScalarReferenceFiniteElement ()=default

dim_t	Dimension () const
	Returns the spatial dimension of the reference cell.

virtual Eigen::Matrix< SCALAR, 1, Eigen::Dynamic >	NodalValuesToDofs (const Eigen::Matrix< SCALAR, 1, Eigen::Dynamic > &nodvals) const
	Computes the linear combination of reference shape functions matching function values at evaluation nodes.

Static Private Attributes
static const FeLagrangeO2Segment< SCALAR >	krsf_segment_

static const Eigen::Matrix< int, 9, 2 >	ksegment_to_quad_mapping_

Additional Inherited Members
Public Types inherited from lf::fe::ScalarReferenceFiniteElement< SCALAR >
using	Scalar = SCALAR
	The underlying scalar type.

Protected Member Functions inherited from lf::fe::ScalarReferenceFiniteElement< SCALAR >
	ScalarReferenceFiniteElement ()=default

	ScalarReferenceFiniteElement (const ScalarReferenceFiniteElement &)=default

	ScalarReferenceFiniteElement (ScalarReferenceFiniteElement &&) noexcept=default

ScalarReferenceFiniteElement &	operator= (const ScalarReferenceFiniteElement &)=default

ScalarReferenceFiniteElement &	operator= (ScalarReferenceFiniteElement &&) noexcept=default

Related Symbols inherited from lf::fe::ScalarReferenceFiniteElement< SCALAR >
template<class SCALAR >
void	PrintInfo (std::ostream &o, const ScalarReferenceFiniteElement< SCALAR > &srfe, unsigned int ctrl=0)
	Print information about a ScalarReferenceFiniteElement to the given stream.

template<typename SCALAR >
std::ostream &	operator<< (std::ostream &o, const ScalarReferenceFiniteElement< SCALAR > &fe_desc)
	Stream output operator: just calls the ScalarReferenceFiniteElement::print() method.

Detailed Description

template<typename SCALAR>
class lf::uscalfe::FeLagrangeO2Quad< SCALAR >

Quadratic Lagrangian finite element on a quadrilateral reference element.

This is a specialization of ScalarReferenceFiniteElement. Refer to its documentation.

The shape functions are computed by a tensor product construction: The shape function associated with the interpolation node \( \mathbf{p} =(x_j,y_l)\) is computed as

\[ \hat{b}^{\mathbf{p}}(x_1,x_2)=\hat{b}^j(x_1)*\hat{b}^l(x_2) \]

Where \( \hat{b}^j\) and \( \hat{b}^l \) are the quadratic Lagrangian shape functions on the reference line segment associated to the interpolation nodes \( x_j \) and \( y_l \).

The first four shape functions are associated with the vertices, the next four with the edges of the quadrilateral. The last shape function is an interior shape function.

See also: ScalarReferenceFiniteElement

Definition at line 713 of file lagr_fe.h.

Constructor & Destructor Documentation

◆ FeLagrangeO2Quad() [1/3]

template<typename SCALAR >

lf::uscalfe::FeLagrangeO2Quad< SCALAR >::FeLagrangeO2Quad ( const FeLagrangeO2Quad< SCALAR > & )

default

◆ FeLagrangeO2Quad() [2/3]

template<typename SCALAR >

lf::uscalfe::FeLagrangeO2Quad< SCALAR >::FeLagrangeO2Quad ( FeLagrangeO2Quad< SCALAR > && )

defaultnoexcept

◆ FeLagrangeO2Quad() [3/3]

template<typename SCALAR >

lf::uscalfe::FeLagrangeO2Quad< SCALAR >::FeLagrangeO2Quad ( )

default

◆ ~FeLagrangeO2Quad()

template<typename SCALAR >

lf::uscalfe::FeLagrangeO2Quad< SCALAR >::~FeLagrangeO2Quad ( )

overridedefault

Member Function Documentation

◆ Degree()

template<typename SCALAR >

unsigned lf::uscalfe::FeLagrangeO2Quad< SCALAR >::Degree ( ) const

inlineoverridevirtual

Quadratic Lagrangian finite elements sport polynomials of degree 2.

Implements lf::fe::ScalarReferenceFiniteElement< SCALAR >.

Definition at line 730 of file lagr_fe.h.

◆ EvalReferenceShapeFunctions()

template<typename SCALAR >

Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic > lf::uscalfe::FeLagrangeO2Quad< SCALAR >::EvalReferenceShapeFunctions ( const Eigen::MatrixXd & refcoords ) const

inlineoverridevirtual

Point evaluation of reference local shape functions in unit square.

The nine local shape functions for quadratic Lagrangian finite elements on the unit square can be obtained by forming tensor product of the three local shape functions on the unit interval. This entails taking into account the numbering of the local shape functions: LSF 0-3 belong to the four corners, LSF 4-7 belong to the edges, LSF 8 belongs to the cell. This complies with the numbering convention for local shape functions as explained in Paragraph 2.7.4.11, Example 2.7.4.12.

Here the local shape functions are chosen as cardinal basis of the space of tensor product polynomials of degree 2 with respect to the following nine evaluation nodes: the four corners of the unit square, the four midpoints of its edges, and its center of gravity, see Figure 119.

See also: fe::ScalarReferenceFiniteElement::EvalReferenceShapeFunctions()

Implements lf::fe::ScalarReferenceFiniteElement< SCALAR >.

Definition at line 780 of file lagr_fe.h.

References lf::uscalfe::FeLagrangeO2Quad< SCALAR >::krsf_segment_, and lf::uscalfe::FeLagrangeO2Quad< SCALAR >::ksegment_to_quad_mapping_.

◆ EvaluationNodes()

template<typename SCALAR >

Eigen::MatrixXd lf::uscalfe::FeLagrangeO2Quad< SCALAR >::EvaluationNodes ( ) const

inlineoverridevirtual

Evaluation nodes are the vertices, the midpoints of the edges and the center of the quadrilateral.

Location and numbering of evaluation nodes:

Returns: a 2x9 matrix containing the refrence coordinates of the evaluation nodes in its columns

Implements lf::fe::ScalarReferenceFiniteElement< SCALAR >.

Definition at line 864 of file lagr_fe.h.

◆ GradientsReferenceShapeFunctions()

template<typename SCALAR >

Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic > lf::uscalfe::FeLagrangeO2Quad< SCALAR >::GradientsReferenceShapeFunctions ( const Eigen::MatrixXd & refcoords ) const

inlineoverridevirtual

Point evaluation of gradient of reference shape functions for quadratic Lagrangian finite elements on quadrilateral cells.

Gradients are returned packed in the rows of a 9 x 2P matrix, where P is the number of evaluation points passed as a 2 x P matrix.

See also: FeLagrangeO2Quad::EvalReferenceShapeFunctions(); fe::ScalarReferenceFiniteElement::EvalReferenceShapeFunctions()

Implements lf::fe::ScalarReferenceFiniteElement< SCALAR >.

Definition at line 812 of file lagr_fe.h.

References lf::uscalfe::FeLagrangeO2Quad< SCALAR >::krsf_segment_, and lf::uscalfe::FeLagrangeO2Quad< SCALAR >::ksegment_to_quad_mapping_.

◆ NumEvaluationNodes()

template<typename SCALAR >

size_type lf::uscalfe::FeLagrangeO2Quad< SCALAR >::NumEvaluationNodes ( ) const

inlineoverridevirtual

Nine evaluation nodes, same as the number of local shape functions.

See also: FeLagrangeO2Quad::EvaluationNodes()

Implements lf::fe::ScalarReferenceFiniteElement< SCALAR >.

Definition at line 878 of file lagr_fe.h.

References lf::uscalfe::FeLagrangeO2Quad< SCALAR >::NumRefShapeFunctions().

◆ NumRefShapeFunctions() [1/3]

template<typename SCALAR >

size_type lf::uscalfe::FeLagrangeO2Quad< SCALAR >::NumRefShapeFunctions ( ) const

inlineoverridevirtual

Nine local shape functions are associated with a quadrilateral cell.

Refer to Example 2.6.2.7.

See also: fe::ScalarReferenceFiniteElement::NumRefShapeFunctions()

Reimplemented from lf::fe::ScalarReferenceFiniteElement< SCALAR >.

Definition at line 738 of file lagr_fe.h.

Referenced by lf::uscalfe::FeLagrangeO2Quad< SCALAR >::NumEvaluationNodes().

◆ NumRefShapeFunctions() [2/3]

template<typename SCALAR >

size_type lf::uscalfe::FeLagrangeO2Quad< SCALAR >::NumRefShapeFunctions ( dim_t codim ) const

inlineoverridevirtual

One shape function is attached to each node and edge of the quadrilateral. One interior shape function.

See also: fe::ScalarReferenceFiniteElement::NumRefShapeFunctions(dim_t codim)

Reimplemented from lf::fe::ScalarReferenceFiniteElement< SCALAR >.

Definition at line 745 of file lagr_fe.h.

◆ NumRefShapeFunctions() [3/3]

template<typename SCALAR >

size_type lf::uscalfe::FeLagrangeO2Quad< SCALAR >::NumRefShapeFunctions	(	dim_t	codim,
		sub_idx_t	) const

inlineoverridevirtual

One shape function is attached to each node and each edge of the quadrilateral. One interior shape function.

See also: fe::ScalarReferenceFiniteElement::NumRefShapeFunctions(dim_t, sub_idx_t)

Implements lf::fe::ScalarReferenceFiniteElement< SCALAR >.

Definition at line 756 of file lagr_fe.h.

◆ operator=() [1/2]

template<typename SCALAR >

FeLagrangeO2Quad & lf::uscalfe::FeLagrangeO2Quad< SCALAR >::operator= ( const FeLagrangeO2Quad< SCALAR > & )

default

◆ operator=() [2/2]

template<typename SCALAR >

FeLagrangeO2Quad & lf::uscalfe::FeLagrangeO2Quad< SCALAR >::operator= ( FeLagrangeO2Quad< SCALAR > && )

defaultnoexcept

◆ RefEl()

template<typename SCALAR >

base::RefEl lf::uscalfe::FeLagrangeO2Quad< SCALAR >::RefEl ( ) const

inlineoverridevirtual

Tells the type of reference cell underlying the parametric finite element.

Implements lf::fe::ScalarReferenceFiniteElement< SCALAR >.

Definition at line 724 of file lagr_fe.h.

References lf::base::RefEl::kQuad().

Member Data Documentation

◆ krsf_segment_

template<typename SCALAR >

const FeLagrangeO2Segment< SCALAR > lf::uscalfe::FeLagrangeO2Quad< SCALAR >::krsf_segment_

staticprivate

Initial value:

=

FeLagrangeO2Segment<SCALAR>()

description of the shape functions on the reference line segment \( [0,1] \) that are used in the tensor product construction of the shape functions

Definition at line 886 of file lagr_fe.h.

Referenced by lf::uscalfe::FeLagrangeO2Quad< SCALAR >::EvalReferenceShapeFunctions(), and lf::uscalfe::FeLagrangeO2Quad< SCALAR >::GradientsReferenceShapeFunctions().

◆ ksegment_to_quad_mapping_

template<typename SCALAR >

const Eigen::Matrix< int, 9, 2 > lf::uscalfe::FeLagrangeO2Quad< SCALAR >::ksegment_to_quad_mapping_

staticprivate

Initial value:

=
        
  (Eigen::Matrix<int,9,2>() <<  0, 0,
                                1, 0,
                                1, 1,
                                0, 1,
                                2, 0,
                                1, 2,
                                2, 1,
                                0, 2,
                                2, 2).finished()

The shape functions are computed by a tensor product construction: The shape function associated with the interpolation node \( \mathbf{p} =(x_j,y_l)\) is computed as

\[ \hat{b}^{\mathbf{p}}(x_1,x_2)=\hat{b}^j(x_1)*\hat{b}^l(x_2) \]

Where \( \hat{b}^j\) and \( \hat{b}^l \) are the quadratic Lagrangian shape functions on the reference line segment associated to the interpolation nodes \( x_j \) and \( y_l \).

This matrix of size NumRefShapeFunctions x 2 contains in each row the indices of the "segment" shape functions, that define the reference shape function via the above formula. if \(p\) was the \(k\)-th interpolation node, the \(k\)-th row would contain the indices \(j\) and \(l\).

Definition at line 905 of file lagr_fe.h.

Referenced by lf::uscalfe::FeLagrangeO2Quad< SCALAR >::EvalReferenceShapeFunctions(), and lf::uscalfe::FeLagrangeO2Quad< SCALAR >::GradientsReferenceShapeFunctions().

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

/__w/lehrfempp/lehrfempp/lib/lf/uscalfe/lagr_fe.h

Public Member Functions

Static Private Attributes

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

◆ FeLagrangeO2Quad() [1/3]

◆ FeLagrangeO2Quad() [2/3]

◆ FeLagrangeO2Quad() [3/3]

◆ ~FeLagrangeO2Quad()

Member Function Documentation

◆ Degree()

◆ EvalReferenceShapeFunctions()

◆ EvaluationNodes()

◆ GradientsReferenceShapeFunctions()

◆ NumEvaluationNodes()

◆ NumRefShapeFunctions() [1/3]

◆ NumRefShapeFunctions() [2/3]

◆ NumRefShapeFunctions() [3/3]

◆ operator=() [1/2]

◆ operator=() [2/2]

◆ RefEl()

Member Data Documentation

◆ krsf_segment_

◆ ksegment_to_quad_mapping_