PolynomialSpace< dim > Class Template Reference

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

Inheritance diagram for PolynomialSpace< dim >:

Public Member Functions
template<class Pol >
	PolynomialSpace (const std::vector< Pol > &pols)
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
std::string	name () const override
virtual std::unique_ptr< ScalarPolynomialsBase< dim > >	clone () const override
unsigned int	n_polynomials (const unsigned int)
unsigned int	n () const
virtual unsigned int	degree () 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 Attributes
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

Detailed Description

template<int dim>
class PolynomialSpace< dim >

Representation of the space of polynomials of degree at most n in higher dimensions.

Given a vector of n one-dimensional polynomials P₀ to P_n, where P_i has degree i, this class generates all dim-dimensional polynomials of the form P_ijk(x,y,z) = P_i(x)P_j(y)P_k(z), where the sum of i, j and k is less than or equal n.

The output_indices() function prints the ordering of the polynomials, i.e. for each dim-dimensional polynomial in the polynomial space it gives the indices i,j,k of the one-dimensional polynomials in x,y and z direction. The ordering of the dim-dimensional polynomials can be changed by using the set_numbering() function.

The standard ordering of polynomials is that indices for the first space dimension vary fastest and the last space dimension is slowest. In particular, if we take for simplicity the vector of monomials x⁰, x¹, x²,..., xⁿ, we get

1d

x⁰, x¹,...,xⁿ

2d:

x⁰y⁰, x¹y⁰,..., xⁿy⁰,
x⁰y¹, x¹y¹,..., x^n-1y¹,
x⁰y²,... x^n-2y²,
...
x⁰y^n-1, x¹y^n-1,
x⁰yⁿ

3d:

x⁰y⁰z⁰,..., xⁿy⁰z⁰,
x⁰y¹z⁰,..., x^n-1y¹z⁰,
...
x⁰yⁿz⁰,
x⁰y⁰z¹,... x^n-1y⁰z¹,
...
x⁰y^n-1z¹,
x⁰y⁰z²,... x^n-2y⁰z²,
...
x⁰y⁰zⁿ

Definition at line 97 of file polynomial_space.h.

Constructor & Destructor Documentation

PolynomialSpace()

template<int dim>

template<class Pol >

PolynomialSpace< dim >::PolynomialSpace

(

const std::vector< Pol > &

pols

)

Constructor. pols is a vector of pointers to one-dimensional polynomials and will be copied into a private member variable. The static type of the template argument pols needs to be convertible to Polynomials::Polynomial<double>, i.e. should usually be a derived class of Polynomials::Polynomial<double>.

Definition at line 287 of file polynomial_space.h.

Member Function Documentation

output_indices()

template<int dim>

template<typename StreamType >

void PolynomialSpace< dim >::output_indices

(

StreamType &

out

)

const

Prints the list of the indices to out.

Definition at line 317 of file polynomial_space.h.

set_numbering()

template<int dim>

void PolynomialSpace< dim >::set_numbering

(

const std::vector< unsigned int > &

renumber

)

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

Definition at line 113 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

overridevirtual

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 204 of file polynomial_space.cc.

compute_value()

template<int dim>

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

overridevirtual

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

Consider using evaluate() instead.

Implements ScalarPolynomialsBase< dim >.

Definition at line 127 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

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 332 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

inlineoverridevirtual

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

Consider using evaluate() instead.

Implements ScalarPolynomialsBase< dim >.

Definition at line 449 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

inlineoverridevirtual

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

Consider using evaluate() instead.

Implements ScalarPolynomialsBase< dim >.

Definition at line 459 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

inlineoverridevirtual

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

Consider using evaluate() instead.

Implements ScalarPolynomialsBase< dim >.

Definition at line 469 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

inlineoverridevirtual

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

Consider using evaluate() instead.

Implements ScalarPolynomialsBase< dim >.

Definition at line 479 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

overridevirtual

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

Consider using evaluate() instead.

Implements ScalarPolynomialsBase< dim >.

Definition at line 144 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

overridevirtual

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

Consider using evaluate() instead.

Implements ScalarPolynomialsBase< dim >.

Definition at line 169 of file polynomial_space.cc.

n_polynomials() [1/2]

template<int dim>

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

static

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 26 of file polynomial_space.cc.

name()

template<int dim>

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

inlineoverridevirtual

Return the name of the space, which is PolynomialSpace.

Implements ScalarPolynomialsBase< dim >.

Definition at line 308 of file polynomial_space.h.

clone()

template<int dim>

std::unique_ptr< ScalarPolynomialsBase< dim > > PolynomialSpace< 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_Poly, need to make copies of polynomial spaces without knowing their exact type. They do so through this function.

Implements ScalarPolynomialsBase< dim >.

Reimplemented in PolynomialsP< dim >.

Definition at line 403 of file polynomial_space.cc.

compute_index() [1/7]

template<int dim>

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

protected

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

protected

compute_index() [3/7]

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

protected

compute_index() [4/7]

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

protected

n_polynomials() [2/2]

unsigned int PolynomialSpace< 0 >::n_polynomials

(

const unsigned int

)

Definition at line 40 of file polynomial_space.cc.

compute_index() [5/7]

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

protected

Definition at line 48 of file polynomial_space.cc.

compute_index() [6/7]

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

protected

Definition at line 58 of file polynomial_space.cc.

compute_index() [7/7]

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

protected

Definition at line 84 of file polynomial_space.cc.

n()

template<int dim>

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

inlineinherited

Return the number of polynomials.

Definition at line 239 of file scalar_polynomials_base.h.

degree()

template<int dim>

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

inlinevirtualinherited

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.

Reimplemented in PolynomialsP< dim >.

Definition at line 248 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 >, BarycentricPolynomials< 1 >, BarycentricPolynomials< 2 >, TensorProductPolynomials< dim, PolynomialType >, and TensorProductPolynomials< dim - 1 >.

Definition at line 38 of file scalar_polynomials_base.cc.

Member Data Documentation

dimension

template<int dim>

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

staticconstexpr

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

Definition at line 104 of file polynomial_space.h.

polynomials

template<int dim>

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

private

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

Definition at line 255 of file polynomial_space.h.

index_map

template<int dim>

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

private

Index map for reordering the polynomials.

Definition at line 260 of file polynomial_space.h.

index_map_inverse

template<int dim>

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

private

Index map for reordering the polynomials.

Definition at line 265 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 227 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 232 of file scalar_polynomials_base.h.

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

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