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

#include <lf/uscalfe/uscalfe.h>

Inheritance diagram for lf::uscalfe::FeLagrangeO3Quad< SCALAR >:

Public Member Functions
	FeLagrangeO3Quad (const FeLagrangeO3Quad &)=default

	FeLagrangeO3Quad (FeLagrangeO3Quad &&) noexcept=default

FeLagrangeO3Quad &	operator= (const FeLagrangeO3Quad &)=default

FeLagrangeO3Quad &	operator= (FeLagrangeO3Quad &&) noexcept=default

	FeLagrangeO3Quad ()=default

	~FeLagrangeO3Quad () override=default

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

unsigned	Degree () const override
	Request the maximal polynomial degree of the basis functions in this finite element.

size_type	NumRefShapeFunctions () const override
	Total number of reference shape functions associated with the reference cell.

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

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

Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic >	EvalReferenceShapeFunctions (const Eigen::MatrixXd &refcoords) const override
	Evaluation of all reference shape functions in a number of points.

Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic >	GradientsReferenceShapeFunctions (const Eigen::MatrixXd &refcoords) const override
	Computation of the gradients of all reference shape functions in a number of points.

Eigen::MatrixXd	EvaluationNodes () const override
	Returns reference coordinates of "evaluation nodes" for evaluation of parametric degrees of freedom, nodal interpolation in the simplest case.

size_type	NumEvaluationNodes () const override
	Sixteen evaluation nodes.

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 FeLagrangeO3Segment< SCALAR >	krsf_segment_

static const Eigen::Matrix< int, 16, 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::FeLagrangeO3Quad< SCALAR >

Cubic 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 cubic Lagrangian shape functions on the reference line segment associated to the interpolation nodes \( x_j \) and \( y_l \), see Figure 119.

The first four shape functions are associated with the vertices, the next eight = 4*2 with edges of the quadrilateral. The last four shape functions are interior shape functions.

See also: fe::ScalarReferenceFiniteElement

Definition at line 1279 of file lagr_fe.h.

Constructor & Destructor Documentation

◆ FeLagrangeO3Quad() [1/3]

template<typename SCALAR >

lf::uscalfe::FeLagrangeO3Quad< SCALAR >::FeLagrangeO3Quad ( const FeLagrangeO3Quad< SCALAR > & )

default

◆ FeLagrangeO3Quad() [2/3]

template<typename SCALAR >

lf::uscalfe::FeLagrangeO3Quad< SCALAR >::FeLagrangeO3Quad ( FeLagrangeO3Quad< SCALAR > && )

defaultnoexcept

◆ FeLagrangeO3Quad() [3/3]

template<typename SCALAR >

lf::uscalfe::FeLagrangeO3Quad< SCALAR >::FeLagrangeO3Quad ( )

default

◆ ~FeLagrangeO3Quad()

template<typename SCALAR >

lf::uscalfe::FeLagrangeO3Quad< SCALAR >::~FeLagrangeO3Quad ( )

overridedefault

Member Function Documentation

◆ Degree()

template<typename SCALAR >

unsigned lf::uscalfe::FeLagrangeO3Quad< SCALAR >::Degree ( ) const

inlineoverridevirtual

Request the maximal polynomial degree of the basis functions in this finite element.

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

Definition at line 1293 of file lagr_fe.h.

◆ EvalReferenceShapeFunctions()

template<typename SCALAR >

Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic > lf::uscalfe::FeLagrangeO3Quad< SCALAR >::EvalReferenceShapeFunctions ( const Eigen::MatrixXd & refcoords ) const

inlineoverridevirtual

Evaluation of all reference shape functions in a number of points.

Parameters

refcoords coordinates of N points in the reference cell passed as columns of a matrix of size dim x N, where dim is the dimension of the reference element, that is =0 for points, =1 for edges, =2 for triangles and quadrilaterals

Returns: An Eigen Matrix of size NumRefShapeFunctions() x refcoords.cols() which contains the shape functions evaluated at every quadrature point.

Concerning the numbering of local shape functions, please consult the documentation of lf::assemble::DofHandler or the documentation of the class.

Note: shape functions are assumed to be real-valued.

Example: Triangular Linear Lagrangian finite elements

There are three reference shape functions \(\hat{b}^1,\hat{b}^2,\hat{b}^3\) associated with the vertices of the reference triangle. Let us assume that the refcoords argument is a 2x2 matrix \([\mathbf{x}_1\;\mathbf{x}_2]\), which corresponds to passing the coordinates of two points in the reference triangle. Then this method will return a 3x2 matrix:

\[ \begin{pmatrix}\hat{b}^1(\mathbf{x}_1) & \hat{b}^1(\mathbf{x}_2) \\ \hat{b}^2(\mathbf{x}_1) & \hat{b}^2(\mathbf{x}_2) \\ \hat{b}^3(\mathbf{x}_1)\ & \hat{b}^3(\mathbf{x}_2) \end{pmatrix} \]

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

Definition at line 1332 of file lagr_fe.h.

References lf::uscalfe::FeLagrangeO3Quad< SCALAR >::krsf_segment_, and lf::uscalfe::FeLagrangeO3Quad< SCALAR >::ksegment_to_quad_mapping_.

◆ EvaluationNodes()

template<typename SCALAR >

Eigen::MatrixXd lf::uscalfe::FeLagrangeO3Quad< SCALAR >::EvaluationNodes ( ) const

inlineoverridevirtual

Returns reference coordinates of "evaluation nodes" for evaluation of parametric degrees of freedom, nodal interpolation in the simplest case.

Returns: A d x N matrix, where d is the dimension of the reference cell, and N the number of interpolation nodes. The columns of this matrix contain their reference coordinates.

Every parametric scalar finite element implicitly defines a local interpolation operator by duality with the reference shape functions. This interpolation operator can be realized through evaluations at certain evaluation nodes, which are provided by this method.

Unisolvence

The evaluation points must satisfy the following requirement: If the values of a function belonging to the span of the reference shape functions are known in the evaluation nodes, then this function is uniquely determined. This entails that the number of evaluation nodes must be at least as big as the number of reference shape functions.

Note: It is not required that any vector a values at evaluation nodes can be produced by a suitable linear combination of reference shape functions. For instance, this will not be possible, if there are more evaluation points than reference shape functions. If both sets have the same size, however, the interpolation problem has a solution for any vector of values at the evluation points.

Example: Principal lattice

For triangular Lagrangian finite elements of degree p the evaluation nodes, usually called "interpolation nodes" in this context, can be chosen as \(\left(\frac{j}{p},\frac{k}{p}\right),\; 0\leq j,k \leq p, j+k\leq p\).

Moment-based degrees of freedom

For some finite element spaces the interpolation functional may be defined based on integrals over edges. In this case the evaluation nodes will be quadrature nodes for the approximate evaluation of these integrals.

The quadrature rule must be exact for the polynomials contained in the local finite element spaces.

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

Definition at line 1400 of file lagr_fe.h.

◆ GradientsReferenceShapeFunctions()

template<typename SCALAR >

Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic > lf::uscalfe::FeLagrangeO3Quad< SCALAR >::GradientsReferenceShapeFunctions ( const Eigen::MatrixXd & refcoords ) const

inlineoverridevirtual

Computation of the gradients of all reference shape functions in a number of points.

Parameters

refcoords coordinates of N points in the reference cell passed as columns of a matrix of size dim x N.

Returns: An Eigen Matrix of size NumRefShapeFunctions() x (dim * refcoords.cols()) where dim is the dimension of the reference finite element. The gradients are returned in chunks of rows of this matrix.

Concerning the numbering of local shape functions, please consult the documentation of lf::assemble::DofHandler.

Example: Triangular Linear Lagrangian finite elements

There are three reference shape functions \(\hat{b}^1,\hat{b}^2,\hat{b}^3\) associated with the vertices of the reference triangle. Let us assume that the refcoords argument is a 2x2 matrix \([\mathbf{x}_1\;\mathbf{x}_2]\), which corresponds to passing the coordinates of two points in the reference triangle. Then this method will return a 3x4 matrix:

\[ \begin{pmatrix} \grad^T\hat{b}^1(\mathbf{x}_1) & \grad^T\hat{b}^1(\mathbf{x}_2) \\ \grad^T\hat{b}^2(\mathbf{x}_1) & \grad^T\hat{b}^2(\mathbf{x}_2) \\ \grad^T\hat{b}^3(\mathbf{x}_1) & \grad^T\hat{b}^3(\mathbf{x}_2) \end{pmatrix} \]

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

Definition at line 1356 of file lagr_fe.h.

References lf::uscalfe::FeLagrangeO3Quad< SCALAR >::krsf_segment_, and lf::uscalfe::FeLagrangeO3Quad< SCALAR >::ksegment_to_quad_mapping_.

◆ NumEvaluationNodes()

template<typename SCALAR >

size_type lf::uscalfe::FeLagrangeO3Quad< SCALAR >::NumEvaluationNodes ( ) const

inlineoverridevirtual

Sixteen evaluation nodes.

Tell the number of evaluation (interpolation) nodes.

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

Definition at line 1426 of file lagr_fe.h.

References lf::uscalfe::FeLagrangeO3Quad< SCALAR >::NumRefShapeFunctions().

◆ NumRefShapeFunctions() [1/3]

template<typename SCALAR >

size_type lf::uscalfe::FeLagrangeO3Quad< SCALAR >::NumRefShapeFunctions ( ) const

inlineoverridevirtual

Total number of reference shape functions associated with the reference cell.

Note: the total number of shape functions is the sum of the number of interior shape functions for all sub-entities and the entity itself.

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

Definition at line 1295 of file lagr_fe.h.

Referenced by lf::uscalfe::FeLagrangeO3Quad< SCALAR >::NumEvaluationNodes(), and lf::uscalfe::FeLagrangeO3Quad< SCALAR >::NumRefShapeFunctions().

◆ NumRefShapeFunctions() [2/3]

template<typename SCALAR >

size_type lf::uscalfe::FeLagrangeO3Quad< SCALAR >::NumRefShapeFunctions ( dim_t codim ) const

inlineoverridevirtual

One shape function attached to each node and two to each edge of the quadrilateral. Four interior shape functions.

The number of interior reference shape functions for sub-entities of a particular co-dimension.

Parameters

codim co-dimension of the subentity

Returns: number of interior reference shape function belonging to the specified sub-entity.

Note: this method will throw an exception when different numbers of reference shape functions are assigned to different sub-entities of the same co-dimension

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

Definition at line 1303 of file lagr_fe.h.

◆ NumRefShapeFunctions() [3/3]

template<typename SCALAR >

size_type lf::uscalfe::FeLagrangeO3Quad< SCALAR >::NumRefShapeFunctions	(	dim_t	codim,
		sub_idx_t	subidx ) const

inlineoverridevirtual

One shape function attached to each node and two to each edge of the quadrilateral. Four interior shape functions.

The number of interior reference shape functions for every sub-entity.

Parameters

codim	do-dimension of the subentity
subidx	local index of the sub-entity

Returns: number of interior reference shape function belonging to the specified sub-entity.

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

Definition at line 1323 of file lagr_fe.h.

References lf::uscalfe::FeLagrangeO3Quad< SCALAR >::NumRefShapeFunctions().

◆ operator=() [1/2]

template<typename SCALAR >

FeLagrangeO3Quad & lf::uscalfe::FeLagrangeO3Quad< SCALAR >::operator= ( const FeLagrangeO3Quad< SCALAR > & )

default

◆ operator=() [2/2]

template<typename SCALAR >

FeLagrangeO3Quad & lf::uscalfe::FeLagrangeO3Quad< SCALAR >::operator= ( FeLagrangeO3Quad< SCALAR > && )

defaultnoexcept

◆ RefEl()

template<typename SCALAR >

base::RefEl lf::uscalfe::FeLagrangeO3Quad< SCALAR >::RefEl ( ) const

inlineoverridevirtual

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

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

Definition at line 1289 of file lagr_fe.h.

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

Member Data Documentation

◆ krsf_segment_

template<typename SCALAR >

const FeLagrangeO3Segment< SCALAR > lf::uscalfe::FeLagrangeO3Quad< SCALAR >::krsf_segment_

staticprivate

Initial value:

=

FeLagrangeO3Segment<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 1434 of file lagr_fe.h.

Referenced by lf::uscalfe::FeLagrangeO3Quad< SCALAR >::EvalReferenceShapeFunctions(), and lf::uscalfe::FeLagrangeO3Quad< SCALAR >::GradientsReferenceShapeFunctions().

◆ ksegment_to_quad_mapping_

template<typename SCALAR >

const Eigen::Matrix< int, 16, 2 > lf::uscalfe::FeLagrangeO3Quad< SCALAR >::ksegment_to_quad_mapping_

staticprivate

Initial value:

=
        
  (Eigen::Matrix<int,16,2>() << 0, 0,
                                1, 0,
                                1, 1,
                                0, 1,
                                2, 0,
                                3, 0,
                                1, 2,
                                1, 3,
                                3, 1,
                                2, 1,
                                0, 3,
                                0, 2,
                                2, 2,
                                3, 2,
                                3, 3,
                                2, 3).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 cubic 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 1453 of file lagr_fe.h.

Referenced by lf::uscalfe::FeLagrangeO3Quad< SCALAR >::EvalReferenceShapeFunctions(), and lf::uscalfe::FeLagrangeO3Quad< 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

◆ FeLagrangeO3Quad() [1/3]

◆ FeLagrangeO3Quad() [2/3]

◆ FeLagrangeO3Quad() [3/3]

◆ ~FeLagrangeO3Quad()

Member Function Documentation

◆ Degree()

◆ EvalReferenceShapeFunctions()

Example: Triangular Linear Lagrangian finite elements

◆ EvaluationNodes()

Unisolvence

Example: Principal lattice

Moment-based degrees of freedom

◆ GradientsReferenceShapeFunctions()

Example: Triangular Linear Lagrangian finite elements

◆ 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_