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

Detailed Description

template<int dim>
class PolynomialsP< dim >

This class implements the polynomial space of degree p based on the monomials \({1,x,x^2,...}\). I.e. in d dimensions it constructs all polynomials of the form \(\prod_{i=1}^d x_i^{n_i}\), where \(\sum_i n_i\leq p\). The base polynomials are given a specific ordering, e.g. in 2 dimensions: \({1,x,y,xy,x^2,y^2,x^2y,xy^2,x^3,y^3,...}\). The ordering of the monomials in \(P_k1\) matches the ordering of the monomials in \(P_k2\) for \(k2>k1\).

Definition at line 44 of file polynomials_p.h.

Inheritance diagram for PolynomialsP< dim >:

[legend]

Public Member Functions
	PolynomialsP (const unsigned int p)

virtual unsigned int	degree () const override

std::array< unsigned int, dim >	directional_degrees (unsigned int n) const

std::unique_ptr< ScalarPolynomialsBase< dim > >	clone () const override

template<typename StreamType >
void	output_indices (StreamType &out) const

void	set_numbering (const std::vector< unsigned int > &renumber)

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

double	compute_value (const unsigned int i, const Point< dim > &p) const override

template<int order>
Tensor< order, dim >	compute_derivative (const unsigned int i, const Point< dim > &p) const

virtual Tensor< 1, dim >	compute_1st_derivative (const unsigned int i, const Point< dim > &p) const override

virtual Tensor< 2, dim >	compute_2nd_derivative (const unsigned int i, const Point< dim > &p) const override

virtual Tensor< 3, dim >	compute_3rd_derivative (const unsigned int i, const Point< dim > &p) const override

virtual Tensor< 4, dim >	compute_4th_derivative (const unsigned int i, const Point< dim > &p) const override

Tensor< 1, dim >	compute_grad (const unsigned int i, const Point< dim > &p) const override

Tensor< 2, dim >	compute_grad_grad (const unsigned int i, const Point< dim > &p) const override

unsigned int	n_polynomials (const unsigned int)

std::string	name () const override

unsigned int	n () const

virtual std::size_t	memory_consumption () const

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

Static Public Attributes
static constexpr unsigned int	dimension = dim

Protected Member Functions
std::array< unsigned int, dim >	compute_index (const unsigned int n) const

std::array< unsigned int, 1 >	compute_index (const unsigned int n) const

std::array< unsigned int, 2 >	compute_index (const unsigned int n) const

std::array< unsigned int, 3 >	compute_index (const unsigned int n) const

std::array< unsigned int, 1 >	compute_index (const unsigned int i) const

std::array< unsigned int, 2 >	compute_index (const unsigned int i) const

std::array< unsigned int, 3 >	compute_index (const unsigned int i) const

Private Member Functions
void	create_polynomial_ordering (std::vector< unsigned int > &index_map) const

void	create_polynomial_ordering (std::vector< unsigned int > &index_map) const

void	create_polynomial_ordering (std::vector< unsigned int > &index_map) const

void	create_polynomial_ordering (std::vector< unsigned int > &index_map) const

Private Attributes
const unsigned int	p

const std::vector< Polynomials::Polynomial< double > >	polynomials

std::vector< unsigned int >	index_map

std::vector< unsigned int >	index_map_inverse

const unsigned int	polynomial_degree

const unsigned int	n_pols

Constructor & Destructor Documentation

◆ PolynomialsP()

template<int dim>

PolynomialsP< dim >::PolynomialsP ( const unsigned int p )

Constructor. Creates all basis functions of \(P_p\).

Parameters

p	The degree of the polynomial space

Definition at line 28 of file polynomials_p.cc.

Member Function Documentation

◆ degree()

template<int dim>

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

inlineoverridevirtual

Return the degree p of the polynomial space P_p.

Note, that this number is PolynomialSpace::degree()-1, compare definition in PolynomialSpace.

Reimplemented from ScalarPolynomialsBase< dim >.

Definition at line 102 of file polynomials_p.h.

◆ directional_degrees()

template<int dim>

std::array< unsigned int, dim > PolynomialsP< dim >::directional_degrees ( unsigned int n ) const

inline

For the nth polynomial \(p_n(x,y,z)=x^i y^j z^k\) this function gives the degrees i,j,k in the x,y,z directions.

In 1d and 2d, obviously only i and i,j are returned.

Definition at line 110 of file polynomials_p.h.

◆ clone()

template<int dim>

std::unique_ptr< ScalarPolynomialsBase< dim > > PolynomialsP< dim >::clone ( ) const

inlineoverridevirtual

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_Poly, need to make copies of polynomial spaces without knowing their exact type. They do so through this function.

Reimplemented from PolynomialSpace< dim >.

Definition at line 79 of file polynomials_p.h.

◆ create_polynomial_ordering() [1/4]

template<int dim>

void PolynomialsP< dim >::create_polynomial_ordering ( std::vector< unsigned int > & index_map ) const

private

Fills the index_map.

◆ create_polynomial_ordering() [2/4]

void PolynomialsP< 1 >::create_polynomial_ordering ( std::vector< unsigned int > & index_map ) const

private

Definition at line 41 of file polynomials_p.cc.

◆ create_polynomial_ordering() [3/4]

void PolynomialsP< 2 >::create_polynomial_ordering ( std::vector< unsigned int > & index_map ) const

private

Definition at line 66 of file polynomials_p.cc.

◆ create_polynomial_ordering() [4/4]

void PolynomialsP< 3 >::create_polynomial_ordering ( std::vector< unsigned int > & index_map ) const

private

Definition at line 95 of file polynomials_p.cc.

◆ output_indices()

template<int dim>

template<typename StreamType >

void PolynomialSpace< dim >::output_indices ( StreamType & out ) const

inherited

Prints the list of the indices to out.

Definition at line 315 of file polynomial_space.h.

◆ set_numbering()

template<int dim>

void PolynomialSpace< dim >::set_numbering ( const std::vector< unsigned int > & renumber )

inherited

Set the ordering of the polynomials. Requires renumber.size()==n(). Stores a copy of renumber.

Definition at line 128 of file polynomial_space.cc.

◆ evaluate()

template<int dim>

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

overridevirtualinherited

Compute the value and the first and second derivatives of each polynomial at unit_point.

The size of the vectors must either be equal 0 or equal n(). In the first case, the function will not compute these values, i.e. you indicate what you want to have computed by resizing those vectors which you want filled.

If you need values or derivatives of all 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 polynomials.

Implements ScalarPolynomialsBase< dim >.

Definition at line 219 of file polynomial_space.cc.

◆ compute_value()

template<int dim>

double PolynomialSpace< dim >::compute_value	(	const unsigned int	i,
		const Point< dim > &	p
	)		const

overridevirtualinherited

Compute the value of the ith polynomial at unit point p.

Consider using evaluate() instead.

Implements ScalarPolynomialsBase< dim >.

Definition at line 142 of file polynomial_space.cc.

◆ compute_derivative()

template<int dim>

template<int order>

Tensor< order, dim > PolynomialSpace< dim >::compute_derivative	(	const unsigned int	i,
		const Point< dim > &	p
	)		const

inherited

Compute the orderth derivative of the ith polynomial at unit point p.

Consider using evaluate() instead.

Template Parameters

order The order of the derivative.

Definition at line 330 of file polynomial_space.h.

◆ compute_1st_derivative()

template<int dim>

Tensor< 1, dim > PolynomialSpace< dim >::compute_1st_derivative	(	const unsigned int	i,
		const Point< dim > &	p
	)		const

inlineoverridevirtualinherited

Compute the first derivative of the ith polynomial at unit point p.

Consider using evaluate() instead.

Implements ScalarPolynomialsBase< dim >.

Definition at line 447 of file polynomial_space.h.

◆ compute_2nd_derivative()

template<int dim>

Tensor< 2, dim > PolynomialSpace< dim >::compute_2nd_derivative	(	const unsigned int	i,
		const Point< dim > &	p
	)		const

inlineoverridevirtualinherited

Compute the second derivative of the ith polynomial at unit point p.

Consider using evaluate() instead.

Implements ScalarPolynomialsBase< dim >.

Definition at line 457 of file polynomial_space.h.

◆ compute_3rd_derivative()

template<int dim>

Tensor< 3, dim > PolynomialSpace< dim >::compute_3rd_derivative	(	const unsigned int	i,
		const Point< dim > &	p
	)		const

inlineoverridevirtualinherited

Compute the third derivative of the ith polynomial at unit point p.

Consider using evaluate() instead.

Implements ScalarPolynomialsBase< dim >.

Definition at line 467 of file polynomial_space.h.

◆ compute_4th_derivative()

template<int dim>

Tensor< 4, dim > PolynomialSpace< dim >::compute_4th_derivative	(	const unsigned int	i,
		const Point< dim > &	p
	)		const

inlineoverridevirtualinherited

Compute the fourth derivative of the ith polynomial at unit point p.

Consider using evaluate() instead.

Implements ScalarPolynomialsBase< dim >.

Definition at line 477 of file polynomial_space.h.

◆ compute_grad()

template<int dim>

Tensor< 1, dim > PolynomialSpace< dim >::compute_grad	(	const unsigned int	i,
		const Point< dim > &	p
	)		const

overridevirtualinherited

Compute the gradient of the ith polynomial at unit point p.

Consider using evaluate() instead.

Implements ScalarPolynomialsBase< dim >.

Definition at line 159 of file polynomial_space.cc.

◆ compute_grad_grad()

template<int dim>

Tensor< 2, dim > PolynomialSpace< dim >::compute_grad_grad	(	const unsigned int	i,
		const Point< dim > &	p
	)		const

overridevirtualinherited

Compute the second derivative (grad_grad) of the ith polynomial at unit point p.

Consider using evaluate() instead.

Implements ScalarPolynomialsBase< dim >.

Definition at line 184 of file polynomial_space.cc.

◆ n_polynomials() [1/2]

template<int dim>

unsigned int PolynomialSpace< dim >::n_polynomials ( const unsigned int n )

staticinherited

Return the number of polynomials spanning the space represented by this class. Here, if N is the number of one-dimensional polynomials given, then the result of this function is N in 1d, N(N+1)/2 in 2d, and N(N+1)(N+2)/6 in 3d.

Definition at line 41 of file polynomial_space.cc.

◆ n_polynomials() [2/2]

unsigned int PolynomialSpace< 0 >::n_polynomials ( const unsigned int )

inherited

Definition at line 55 of file polynomial_space.cc.

◆ name()

template<int dim>

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

inlineoverridevirtualinherited

Return the name of the space, which is PolynomialSpace.

Implements ScalarPolynomialsBase< dim >.

Definition at line 306 of file polynomial_space.h.

◆ compute_index() [1/7]

template<int dim>

std::array< unsigned int, dim > PolynomialSpace< dim >::compute_index ( const unsigned int n ) const

protectedinherited

Compute numbers in x, y and z direction. Given an index n in the d-dimensional polynomial space, return the indices i,j,k such that p_n(x,y,z) = p_i(x)p_j(y)p_k(z).

In 1d and 2d, obviously only i and i,j are returned.

◆ compute_index() [2/7]

std::array< unsigned int, 1 > PolynomialSpace< 1 >::compute_index ( const unsigned int n ) const

protectedinherited

◆ compute_index() [3/7]

std::array< unsigned int, 2 > PolynomialSpace< 2 >::compute_index ( const unsigned int n ) const

protectedinherited

◆ compute_index() [4/7]

std::array< unsigned int, 3 > PolynomialSpace< 3 >::compute_index ( const unsigned int n ) const

protectedinherited

◆ compute_index() [5/7]

std::array< unsigned int, 1 > PolynomialSpace< 1 >::compute_index ( const unsigned int i ) const

protectedinherited

Definition at line 63 of file polynomial_space.cc.

◆ compute_index() [6/7]

std::array< unsigned int, 2 > PolynomialSpace< 2 >::compute_index ( const unsigned int i ) const

protectedinherited

Definition at line 73 of file polynomial_space.cc.

◆ compute_index() [7/7]

std::array< unsigned int, 3 > PolynomialSpace< 3 >::compute_index ( const unsigned int i ) const

protectedinherited

Definition at line 99 of file polynomial_space.cc.

◆ n()

template<int dim>

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

inlineinherited

Return the number of polynomials.

Definition at line 254 of file scalar_polynomials_base.h.

◆ memory_consumption()

template<int dim>

std::size_t ScalarPolynomialsBase< dim >::memory_consumption ( ) const

virtualinherited

Return an estimate (in bytes) for the memory consumption of this object.

Reimplemented in BarycentricPolynomials< dim >, TensorProductPolynomials< dim, PolynomialType >, and TensorProductPolynomials< dim - 1 >.

Definition at line 36 of file scalar_polynomials_base.cc.

Member Data Documentation

◆ dimension

template<int dim>

constexpr unsigned int PolynomialsP< dim >::dimension = dim

staticconstexpr

Access to the dimension of this object, for checking and automatic setting of dimension in other classes.

Definition at line 51 of file polynomials_p.h.

◆ p

template<int dim>

const unsigned int PolynomialsP< dim >::p

private

Degree p of the polynomial space \(P_p\), i.e. the number p which was given to the constructor.

Definition at line 95 of file polynomials_p.h.

◆ polynomials

template<int dim>

const std::vector<Polynomials::Polynomial<double> > PolynomialSpace< dim >::polynomials

privateinherited

Copy of the vector pols of polynomials given to the constructor.

Definition at line 253 of file polynomial_space.h.

◆ index_map

template<int dim>

std::vector<unsigned int> PolynomialSpace< dim >::index_map

privateinherited

Index map for reordering the polynomials.

Definition at line 258 of file polynomial_space.h.

◆ index_map_inverse

template<int dim>

std::vector<unsigned int> PolynomialSpace< dim >::index_map_inverse

privateinherited

Index map for reordering the polynomials.

Definition at line 263 of file polynomial_space.h.

◆ polynomial_degree

template<int dim>

const unsigned int ScalarPolynomialsBase< dim >::polynomial_degree

privateinherited

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

Definition at line 225 of file scalar_polynomials_base.h.

◆ n_pols

template<int dim>

const unsigned int ScalarPolynomialsBase< dim >::n_pols

privateinherited

The number of polynomials represented by this object.

Definition at line 230 of file scalar_polynomials_base.h.

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

include/deal.II/base/polynomials_p.h
source/base/polynomials_p.cc

Detailed Description

Public Member Functions

Static Public Member Functions

Static Public Attributes

Protected Member Functions

Private Member Functions

Private Attributes

Constructor & Destructor Documentation

◆ PolynomialsP()

Member Function Documentation

◆ degree()

◆ directional_degrees()

◆ clone()

◆ create_polynomial_ordering() [1/4]

◆ create_polynomial_ordering() [2/4]

◆ create_polynomial_ordering() [3/4]

◆ create_polynomial_ordering() [4/4]

◆ output_indices()

◆ set_numbering()

◆ evaluate()

◆ compute_value()

◆ compute_derivative()

◆ compute_1st_derivative()

◆ compute_2nd_derivative()

◆ compute_3rd_derivative()

◆ compute_4th_derivative()

◆ compute_grad()

◆ compute_grad_grad()

◆ n_polynomials() [1/2]

◆ n_polynomials() [2/2]

◆ name()

◆ compute_index() [1/7]

◆ compute_index() [2/7]

◆ compute_index() [3/7]

◆ compute_index() [4/7]

◆ compute_index() [5/7]

◆ compute_index() [6/7]

◆ compute_index() [7/7]

◆ n()

◆ memory_consumption()

Member Data Documentation

◆ dimension

◆ p

◆ polynomials

◆ index_map

◆ index_map_inverse

◆ polynomial_degree

◆ n_pols