Helper class which stores a ScalarReferenceFiniteElement with precomputed values at the nodes of a quadrature rule. More...

#include <lf/uscalfe/uscalfe.h>

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

Public Member Functions
	PrecomputedScalarReferenceFiniteElement ()=default
	Default constructor which does not initialize this class at all (invalid state). If any method is called upon it, an error is thrown.

	PrecomputedScalarReferenceFiniteElement (const PrecomputedScalarReferenceFiniteElement &)=delete

	PrecomputedScalarReferenceFiniteElement (PrecomputedScalarReferenceFiniteElement &&) noexcept=default

PrecomputedScalarReferenceFiniteElement &	operator= (const PrecomputedScalarReferenceFiniteElement &)=delete

PrecomputedScalarReferenceFiniteElement &	operator= (PrecomputedScalarReferenceFiniteElement &&) noexcept=default

	PrecomputedScalarReferenceFiniteElement (lf::fe::ScalarReferenceFiniteElement< SCALAR > const *fe, quad::QuadRule qr)
	Main constructor performing precomputations.

bool	isInitialized () const
	Tells initialization status of object.

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

size_type	NumRefShapeFunctions (dim_t codim, sub_idx_t subidx) const override
	The number of interior reference shape functions for every sub-entity.

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

Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic >	GradientsReferenceShapeFunctions (const Eigen::MatrixXd &local) 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
	Tell the number of evaluation (interpolation) nodes.

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

const quad::QuadRule &	Qr () const
	Return the Quadrature rule at which the shape functions (and their gradients) have been precomputed.

const Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic > &	PrecompReferenceShapeFunctions () const
	Value of `EvalReferenceShapeFunctions(Qr().Points())`

const Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic > &	PrecompGradientsReferenceShapeFunctions () const
	Value of `EvalGradientsReferenceShapeFunctions(Qr().Points())`

	~PrecomputedScalarReferenceFiniteElement () override=default

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

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

Private Attributes
fe::ScalarReferenceFiniteElement< SCALAR > const *	fe_ = nullptr

quad::QuadRule	qr_

Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic >	shap_fun_

Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic >	grad_shape_fun_

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<class SCALAR>
class lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >

Helper class which stores a ScalarReferenceFiniteElement with precomputed values at the nodes of a quadrature rule.

Template Parameters

SCALAR The scalar type of the shape functions, e.g. double

This class does essentially three things:

It wraps any ScalarReferenceFiniteElement and forwards all calls to the wrapped instance.
It provides access to a quad::QuadratureRule (passed in constructor)
It provides additional member functions to access the precomputed values
the shape functions/gradients at the nodes of the quadrature rule.

Precomputing entity-independent quantities boost efficiency in the context of parametric finite element methods provided that a uniform quadrature rule is used for local computations on all mesh entities of the same topological type.

Detailed explanations can be found in Paragraph 2.8.3.6, Paragraph 2.7.5.8.

Definition at line 44 of file precomputed_scalar_reference_finite_element.h.

Constructor & Destructor Documentation

◆ PrecomputedScalarReferenceFiniteElement() [1/4]

template<class SCALAR >

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

default

Default constructor which does not initialize this class at all (invalid state). If any method is called upon it, an error is thrown.

◆ PrecomputedScalarReferenceFiniteElement() [2/4]

template<class SCALAR >

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

delete

◆ PrecomputedScalarReferenceFiniteElement() [3/4]

template<class SCALAR >

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

defaultnoexcept

◆ PrecomputedScalarReferenceFiniteElement() [4/4]

template<class SCALAR >

lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >::PrecomputedScalarReferenceFiniteElement	(	lf::fe::ScalarReferenceFiniteElement< SCALAR > const *	fe,
		quad::QuadRule	qr )

inline

Main constructor performing precomputations.

Parameters

fe	definition of reference finite element
qr	quadrature rule (nodes and weights on reference element)

Initialization of local data members.

Definition at line 72 of file precomputed_scalar_reference_finite_element.h.

◆ ~PrecomputedScalarReferenceFiniteElement()

template<class SCALAR >

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

overridedefault

virtual destructor

Member Function Documentation

◆ Degree()

template<class SCALAR >

unsigned lf::uscalfe::PrecomputedScalarReferenceFiniteElement< 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 92 of file precomputed_scalar_reference_finite_element.h.

References lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >::fe_.

◆ EvalReferenceShapeFunctions()

template<class SCALAR >

Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic > lf::uscalfe::PrecomputedScalarReferenceFiniteElement< 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 114 of file precomputed_scalar_reference_finite_element.h.

References lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >::fe_.

◆ EvaluationNodes()

template<class SCALAR >

Eigen::MatrixXd lf::uscalfe::PrecomputedScalarReferenceFiniteElement< 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 124 of file precomputed_scalar_reference_finite_element.h.

References lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >::fe_.

◆ GradientsReferenceShapeFunctions()

template<class SCALAR >

Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic > lf::uscalfe::PrecomputedScalarReferenceFiniteElement< 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 119 of file precomputed_scalar_reference_finite_element.h.

References lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >::fe_.

◆ isInitialized()

template<class SCALAR >

bool lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >::isInitialized ( ) const

inline

Tells initialization status of object.

An object is in an undefined state when built by the default constructor

Definition at line 85 of file precomputed_scalar_reference_finite_element.h.

References lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >::fe_.

Referenced by lf::uscalfe::ReactionDiffusionElementMatrixProvider< SCALAR, DIFF_COEFF, REACTION_COEFF >::Eval().

◆ NodalValuesToDofs()

template<class SCALAR >

Eigen::Matrix< SCALAR, 1, Eigen::Dynamic > lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >::NodalValuesToDofs ( const Eigen::Matrix< SCALAR, 1, Eigen::Dynamic > & nodvals ) const

inlineoverridevirtual

Computes the linear combination of reference shape functions matching function values at evaluation nodes.

Parameters

nodvals row vector of function values at evaluation nodes The length of this vector must agree with NumEvaluationNodes().

Returns: The coefficients of the local "nodal interpolant" with respect to the reference shape functions. This is a row vector of length NumRefShapeFunctions().

If the evaluation nodes are interpolation nodes, that is, if the set of reference shape functions forms a cardinal basis with respect to these nodes, then we have NumEvaluationNodes() == NumRefShapeFunctions() and the linear mapping realized by NodalValuesToDofs() is the identity mapping.

Note: default implementation is the identity mapping

Requirement: reproduction of local finite element functions

If the vector of values at the evaluation nodes agrees with a vector of function values of a linear combination of reference shape functions at the evaluation nodes, then this method must return the very coefficients of the linear combination.

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

Definition at line 133 of file precomputed_scalar_reference_finite_element.h.

References lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >::fe_.

◆ NumEvaluationNodes()

template<class SCALAR >

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

inlineoverridevirtual

Tell the number of evaluation (interpolation) nodes.

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

Definition at line 128 of file precomputed_scalar_reference_finite_element.h.

References lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >::fe_.

◆ NumRefShapeFunctions() [1/3]

template<class SCALAR >

size_type lf::uscalfe::PrecomputedScalarReferenceFiniteElement< 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 97 of file precomputed_scalar_reference_finite_element.h.

References lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >::fe_.

Referenced by lf::uscalfe::ReactionDiffusionElementMatrixProvider< SCALAR, DIFF_COEFF, REACTION_COEFF >::Eval().

◆ NumRefShapeFunctions() [2/3]

template<class SCALAR >

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

inlineoverridevirtual

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 102 of file precomputed_scalar_reference_finite_element.h.

References lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >::fe_.

◆ NumRefShapeFunctions() [3/3]

template<class SCALAR >

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

inlineoverridevirtual

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 107 of file precomputed_scalar_reference_finite_element.h.

References lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >::fe_.

◆ operator=() [1/2]

template<class SCALAR >

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

delete

◆ operator=() [2/2]

template<class SCALAR >

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

defaultnoexcept

◆ PrecompGradientsReferenceShapeFunctions()

template<class SCALAR >

const Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic > & lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >::PrecompGradientsReferenceShapeFunctions ( ) const

inline

Value of EvalGradientsReferenceShapeFunctions(Qr().Points())

See fe::ScalarReferenceFiniteElement::GradientsReferenceShapeFunctions() for the packed format in which the gradients are returned.

Definition at line 166 of file precomputed_scalar_reference_finite_element.h.

References lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >::fe_, and lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >::grad_shape_fun_.

Referenced by lf::uscalfe::ReactionDiffusionElementMatrixProvider< SCALAR, DIFF_COEFF, REACTION_COEFF >::Eval().

◆ PrecompReferenceShapeFunctions()

template<class SCALAR >

const Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic > & lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >::PrecompReferenceShapeFunctions ( ) const

inline

Value of EvalReferenceShapeFunctions(Qr().Points())

Definition at line 152 of file precomputed_scalar_reference_finite_element.h.

References lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >::fe_, and lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >::shap_fun_.

Referenced by lf::uscalfe::ReactionDiffusionElementMatrixProvider< SCALAR, DIFF_COEFF, REACTION_COEFF >::Eval().