Reference documentation for deal.II version Git d902c8c1aa 2019-12-15 00:36:45 -0500
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Public Member Functions | Static Public Member Functions | Static Private Member Functions | Private Attributes | List of all members
PolynomialsBernardiRaugel< dim > Class Template Reference

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

Inheritance diagram for PolynomialsBernardiRaugel< dim >:
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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
 
- Public Member Functions inherited from TensorPolynomialsBase< dim >
 TensorPolynomialsBase (const unsigned int deg, const unsigned int n_polynomials)
 
 TensorPolynomialsBase (TensorPolynomialsBase< dim > &&)=default
 
 TensorPolynomialsBase (const TensorPolynomialsBase< dim > &)=default
 
virtual ~TensorPolynomialsBase ()=default
 
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
 

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

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

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

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

3D bubble functions (in order)

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

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

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

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

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

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

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

Author
Graham Harper
Date
2018

Definition at line 87 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 173 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.

If you need values or derivatives of all tensor product polynomials then use this function, rather than using any of the compute_value, compute_grad or compute_grad_grad functions, see below, in a loop over all tensor product polynomials.

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 239 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 255 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.

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 146 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 153 of file polynomials_bernardi_raugel.h.


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