PolynomialsBernardiRaugel< dim > Class Template Reference

#include <deal.II/base/polynomials_bernardi_raugel.h>

Inheritance diagram for PolynomialsBernardiRaugel< dim >:

Public Member Functions
	PolynomialsBernardiRaugel (const unsigned int k)
std::string	name () const override
void	evaluate (const Point< dim > &unit_point, std::vector< Tensor< 1, dim > > &values, std::vector< Tensor< 2, dim > > &grads, std::vector< Tensor< 3, dim > > &grad_grads, std::vector< Tensor< 4, dim > > &third_derivatives, std::vector< Tensor< 5, dim > > &fourth_derivatives) const override
virtual std::unique_ptr< TensorPolynomialsBase< dim > >	clone () const override
unsigned int	n () const
unsigned int	degree () const

Static Public Member Functions
static unsigned int	n_polynomials (const unsigned int k)

Static Private Member Functions
static std::vector< std::vector< Polynomials::Polynomial< double > > >	create_polynomials_Q ()
static std::vector< std::vector< Polynomials::Polynomial< double > > >	create_polynomials_bubble ()

Private Attributes
const AnisotropicPolynomials< dim >	polynomial_space_Q
const AnisotropicPolynomials< dim >	polynomial_space_bubble
const unsigned int	polynomial_degree
const unsigned int	n_pols

Detailed Description

template<int dim>
class PolynomialsBernardiRaugel< dim >

This class implements the Bernardi-Raugel polynomials similarly to the description in the Mathematics of Computation paper from 1985 by Christine Bernardi and Geneviève Raugel.

The Bernardi-Raugel polynomials are originally defined as an enrichment of the \((P_1)^d\) elements on simplicial meshes for Stokes problems by the addition of bubble functions, yielding a locking-free finite element which is a subset of \((P_2)^d\) elements. This implementation is an enrichment of \((Q_1)^d\) elements which is a subset of \((Q_2)^d\) elements for quadrilateral and hexahedral meshes.

The \(BR_1\) bubble functions are defined to have magnitude 1 at the center of face \(e_i\) and direction \(\mathbf{n}_i\) normal to face \(e_i\), and magnitude 0 on all other vertices and faces. Ordering is consistent with the face numbering in GeometryInfo. The vector \(\mathbf{n}_i\) points in the positive axis direction and not necessarily normal to the element for consistent orientation across edges.

2d bubble functions (in order)

\(x=0\) edge: \(\mathbf{p}_1 = \mathbf{n}_1 (1-x)(y)(1-y)\)

 @f$x=1@f$ edge: @f$\mathbf{p}_2 = \mathbf{n}_2 (x)(y)(1-y)@f$

 @f$y=0@f$ edge: @f$\mathbf{p}_3 = \mathbf{n}_3 (x)(1-x)(1-y)@f$

 @f$y=1@f$ edge: @f$\mathbf{p}_4 = \mathbf{n}_4 (x)(1-x)(y)@f$

3d bubble functions (in order)

\(x=0\) edge: \(\mathbf{p}_1 = \mathbf{n}_1 (1-x)(y)(1-y)(z)(1-z)\)

 @f$x=1@f$ edge: @f$\mathbf{p}_2 = \mathbf{n}_2 (x)(y)(1-y)(z)(1-z)@f$

 @f$y=0@f$ edge: @f$\mathbf{p}_3 = \mathbf{n}_3 (x)(1-x)(1-y)(z)(1-z)@f$

 @f$y=1@f$ edge: @f$\mathbf{p}_4 = \mathbf{n}_4 (x)(1-x)(y)(z)(1-z)@f$

 @f$z=0@f$ edge: @f$\mathbf{p}_5 = \mathbf{n}_5 (x)(1-x)(y)(1-y)(1-z)@f$

 @f$z=1@f$ edge: @f$\mathbf{p}_6 = \mathbf{n}_6 (x)(1-x)(y)(1-y)(z)@f$

Then the \(BR_1(E)\) polynomials are defined on quadrilaterals and hexahedra by \(BR_1(E) = Q_1(E) \oplus \mbox{span}\{\mathbf{p}_i, i=1,...,2d\}\).

Definition at line 84 of file polynomials_bernardi_raugel.h.

Constructor & Destructor Documentation

PolynomialsBernardiRaugel()

template<int dim>

PolynomialsBernardiRaugel< dim >::PolynomialsBernardiRaugel

(

const unsigned int

k

)

Constructor. Creates all basis functions for Bernardi-Raugel polynomials of given degree.

• k The degree of the Bernardi-Raugel-space, which is currently limited to the case k=1.

Definition at line 24 of file polynomials_bernardi_raugel.cc.

Member Function Documentation

name()

template<int dim>

std::string PolynomialsBernardiRaugel< dim >::name ( ) const

inlineoverridevirtual

Return the name of the space, which is BernardiRaugel.

Implements TensorPolynomialsBase< dim >.

Definition at line 165 of file polynomials_bernardi_raugel.h.

evaluate()

template<int dim>

void PolynomialsBernardiRaugel< dim >::evaluate ( const Point< dim > & unit_point, std::vector< Tensor< 1, dim > > & values, std::vector< Tensor< 2, dim > > & grads, std::vector< Tensor< 3, dim > > & grad_grads, std::vector< Tensor< 4, dim > > & third_derivatives, std::vector< Tensor< 5, dim > > & fourth_derivatives ) const

overridevirtual

Compute the value and derivatives of each Bernardi-Raugel polynomial at unit_point.

The size of the vectors must either be zero or equal n(). In the first case, the function will not compute these values.

Implements TensorPolynomialsBase< dim >.

Definition at line 70 of file polynomials_bernardi_raugel.cc.

n_polynomials()

template<int dim>

unsigned int PolynomialsBernardiRaugel< dim >::n_polynomials ( const unsigned int k )

static

Return the number of polynomials in the space BR(degree) without requiring to build an object of PolynomialsBernardiRaugel. This is required by the FiniteElement classes.

Definition at line 237 of file polynomials_bernardi_raugel.cc.

clone()

template<int dim>

std::unique_ptr< TensorPolynomialsBase< dim > > PolynomialsBernardiRaugel< dim >::clone ( ) const

overridevirtual

A sort of virtual copy constructor, this function returns a copy of the polynomial space object. Derived classes need to override the function here in this base class and return an object of the same type as the derived class.

Some places in the library, for example the constructors of FE_PolyTensor, need to make copies of polynomial spaces without knowing their exact type. They do so through this function.

Implements TensorPolynomialsBase< dim >.

Definition at line 253 of file polynomials_bernardi_raugel.cc.

create_polynomials_Q()

template<int dim>

std::vector< std::vector< Polynomials::Polynomial< double > > > PolynomialsBernardiRaugel< dim >::create_polynomials_Q ( )

staticprivate

A static member function that creates the polynomial space we use to initialize the polynomial_space_Q member variable.

Definition at line 55 of file polynomials_bernardi_raugel.cc.

create_polynomials_bubble()

template<int dim>

std::vector< std::vector< Polynomials::Polynomial< double > > > PolynomialsBernardiRaugel< dim >::create_polynomials_bubble ( )

staticprivate

A static member function that creates the polynomial space we use to initialize the polynomial_space_bubble member variable.

Definition at line 33 of file polynomials_bernardi_raugel.cc.

n()

template<int dim>

unsigned int TensorPolynomialsBase< dim >::n ( ) const

inlineinherited

Return the number of polynomials.

Definition at line 151 of file tensor_polynomials_base.h.

degree()

template<int dim>

unsigned int TensorPolynomialsBase< dim >::degree ( ) const

inlineinherited

Return the highest polynomial degree of polynomials represented by this class. A derived class may override this if its value is different from my_degree.

Definition at line 160 of file tensor_polynomials_base.h.

Member Data Documentation

polynomial_space_Q

template<int dim>

const AnisotropicPolynomials<dim> PolynomialsBernardiRaugel< dim >::polynomial_space_Q

private

An object representing the polynomial space of Q functions which forms the BR polynomials through outer products of these with the corresponding unit ijk vectors.

Definition at line 138 of file polynomials_bernardi_raugel.h.

polynomial_space_bubble

template<int dim>

const AnisotropicPolynomials<dim> PolynomialsBernardiRaugel< dim >::polynomial_space_bubble

private

An object representing the polynomial space of bubble functions which forms the BR polynomials through outer products of these with the corresponding normals.

Definition at line 145 of file polynomials_bernardi_raugel.h.

polynomial_degree

template<int dim>

const unsigned int TensorPolynomialsBase< dim >::polynomial_degree

privateinherited

The highest polynomial degree of this functions represented by this object.

Definition at line 139 of file tensor_polynomials_base.h.

n_pols

template<int dim>

const unsigned int TensorPolynomialsBase< dim >::n_pols

privateinherited

The number of polynomials represented by this object.

Definition at line 144 of file tensor_polynomials_base.h.

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

• include/deal.II/base/polynomials_bernardi_raugel.h
• source/base/polynomials_bernardi_raugel.cc