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

#include <lf/uscalfe/uscalfe.h>

Inheritance diagram for lf::uscalfe::FeLagrangeO2Tria< SCALAR >:

Public Member Functions
	FeLagrangeO2Tria (const FeLagrangeO2Tria &)=default

	FeLagrangeO2Tria (FeLagrangeO2Tria &&) noexcept=default

FeLagrangeO2Tria &	operator= (const FeLagrangeO2Tria &)=default

FeLagrangeO2Tria &	operator= (FeLagrangeO2Tria &&) noexcept=default

	FeLagrangeO2Tria ()=default

	~FeLagrangeO2Tria () 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 rely on polynomials of degree 2.

size_type	NumRefShapeFunctions () const override
	Six local shape functions are associated with a triangular cell in the case of quadratic Lagrangian finite elements.

size_type	NumRefShapeFunctions (dim_t codim) const override
	One shape function attached to each node and one to each edge of the triangle. There are no interior shape functions.

size_type	NumRefShapeFunctions (dim_t codim, sub_idx_t) const override
	One shape function attached to each node and one to each edge of the triangle. There are no interior shape functions.

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

Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic >	GradientsReferenceShapeFunctions (const Eigen::MatrixXd &refcoords) const override
	Point evaluations of gradients of reference shape functions.

Eigen::MatrixXd	EvaluationNodes () const override
	Evaluation nodes are the three vertices of the triangle and the three midpoints of its edges.

size_type	NumEvaluationNodes () const override
	Six evaluation nodes, the same number as 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.

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::FeLagrangeO2Tria< SCALAR >

Quadratic Lagrangian finite element on a triangular reference element.

This is a specialization of fe::ScalarReferenceFiniteElement.

The reference shape functions are combinations of the barycentric coordinate functions on the reference triangle, see Example 2.6.1.2.

The first three shape functions are associated to the vertices, the other three shape functions to the edges of the triangle. There are no interior shape functions, see Equation 2.6.1.6:

\begin{eqnarray} \hat{b}^1({\bf x}) &=& 2(1-x_1-x_2)(\frac12 -x_1 -x_2)\;,\\ \hat{b}^2({\bf x}) &=& 2x_1(x_1-\frac12)\;,\\ \hat{b}^3({\bf x}) &=& 2x_2(x_2-\frac12)\;,\\ \hat{b}^4({\bf x}) &=& 4(1-x_1-x_2)x_1\;,\\ \hat{b}^5({\bf x}) &=& 4x_1x_2\;,\\ \hat{b}^6({\bf x}) &=& 4(1-x_1-x_2)x_2\;. \end{eqnarray}

The numbering convention for the local shape functions is explained in Example 2.7.4.12.

See also: fe::ScalarReferenceFiniteElement

Definition at line 400 of file lagr_fe.h.

Constructor & Destructor Documentation

◆ FeLagrangeO2Tria() [1/3]

template<typename SCALAR >

lf::uscalfe::FeLagrangeO2Tria< SCALAR >::FeLagrangeO2Tria ( const FeLagrangeO2Tria< SCALAR > & )

default

◆ FeLagrangeO2Tria() [2/3]

template<typename SCALAR >

lf::uscalfe::FeLagrangeO2Tria< SCALAR >::FeLagrangeO2Tria ( FeLagrangeO2Tria< SCALAR > && )

defaultnoexcept

◆ FeLagrangeO2Tria() [3/3]

template<typename SCALAR >

lf::uscalfe::FeLagrangeO2Tria< SCALAR >::FeLagrangeO2Tria ( )

default

◆ ~FeLagrangeO2Tria()

template<typename SCALAR >

lf::uscalfe::FeLagrangeO2Tria< SCALAR >::~FeLagrangeO2Tria ( )

overridedefault

Member Function Documentation

◆ Degree()

template<typename SCALAR >

unsigned lf::uscalfe::FeLagrangeO2Tria< SCALAR >::Degree ( ) const

inlineoverridevirtual

Quadratic Lagrangian finite elements rely on polynomials of degree 2.

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

Definition at line 419 of file lagr_fe.h.

◆ EvalReferenceShapeFunctions()

template<typename SCALAR >

Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic > lf::uscalfe::FeLagrangeO2Tria< SCALAR >::EvalReferenceShapeFunctions ( const Eigen::MatrixXd & refcoords ) const

inlineoverridevirtual

Point evaluation of reference shape functions.

The formulas of the reference shape functions are given in the class documentation of FeLagrangeO2Tria.

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

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

Definition at line 459 of file lagr_fe.h.

◆ EvaluationNodes()

template<typename SCALAR >

Eigen::MatrixXd lf::uscalfe::FeLagrangeO2Tria< SCALAR >::EvaluationNodes ( ) const

inlineoverridevirtual

Evaluation nodes are the three vertices of the triangle and the three midpoints of its edges.

The reference shape functions as implemented by this class are a cardinal basis of the space of quadratic two-variate polynomials with respect to these evaluation nodes.

The numbering of evluation nodes is fixed by the numbering of the local shape functions.

See also: fe::ScalarReferenceFiniteElement::EvaluationNodes()

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

Definition at line 529 of file lagr_fe.h.

◆ GradientsReferenceShapeFunctions()

template<typename SCALAR >

Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic > lf::uscalfe::FeLagrangeO2Tria< SCALAR >::GradientsReferenceShapeFunctions ( const Eigen::MatrixXd & refcoords ) const