Reference documentation for deal.II version Git 191d06ed00 2021-05-11 21:15:49 -0400
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Public Member Functions | Static Public Member Functions | Static Public Attributes | Protected Member Functions | Private Member Functions | Private Attributes | List of all members
PolynomialsP< dim > Class Template Reference

#include <deal.II/base/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<class 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
 
template<>
unsigned int n_polynomials (const unsigned int)
 
std::string name () const override
 

Static Public Member Functions

static unsigned int n_polynomials (const unsigned int n)
 

Static Public Attributes

static const unsigned int dimension = dim
 

Protected Member Functions

std::array< unsigned int, dim > compute_index (const unsigned int n) const
 
template<>
std::array< unsigned int, 1 > compute_index (const unsigned int n) const
 
template<>
std::array< unsigned int, 2 > compute_index (const unsigned int n) const
 
template<>
std::array< unsigned int, 3 > compute_index (const unsigned int n) const
 
template<>
std::array< unsigned int, 1 > compute_index (const unsigned int i) const
 
template<>
std::array< unsigned int, 2 > compute_index (const unsigned int i) const
 
template<>
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
 
template<>
void create_polynomial_ordering (std::vector< unsigned int > &index_map) const
 
template<>
void create_polynomial_ordering (std::vector< unsigned int > &index_map) const
 
template<>
void create_polynomial_ordering (std::vector< unsigned int > &index_map) const
 

Private Attributes

const unsigned int p
 

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 47 of file polynomials_p.h.

Constructor & Destructor Documentation

◆ PolynomialsP()

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

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

  • p: the degree of the polynomial space

Definition at line 23 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.

Definition at line 104 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 112 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 81 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]

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

Definition at line 36 of file polynomials_p.cc.

◆ create_polynomial_ordering() [3/4]

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

Definition at line 61 of file polynomials_p.cc.

◆ create_polynomial_ordering() [4/4]

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

Definition at line 90 of file polynomials_p.cc.

◆ output_indices()

template<int dim>
template<class StreamType >
void PolynomialSpace< dim >::output_indices ( StreamType &  out) const
inherited

Prints the list of the indices to out.

Definition at line 318 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 114 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
overrideinherited

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.

Definition at line 205 of file polynomial_space.cc.

◆ compute_value()

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

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

Consider using evaluate() instead.

Definition at line 128 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
orderThe order of the derivative.

Definition at line 333 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.

Definition at line 458 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.

Definition at line 468 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.

Definition at line 478 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.

Definition at line 488 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
overrideinherited

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

Consider using evaluate() instead.

Definition at line 145 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
overrideinherited

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

Consider using evaluate() instead.

Definition at line 170 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 27 of file polynomial_space.cc.

◆ n_polynomials() [2/2]

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

Definition at line 41 of file polynomial_space.cc.

◆ name()

template<int dim>
std::string PolynomialSpace< dim >::name ( ) const
inlineoverrideinherited

Return the name of the space, which is PolynomialSpace.

Definition at line 309 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 pn(x,y,z) = pi(x)pj(y)pk(z).

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

◆ compute_index() [2/7]

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

◆ compute_index() [3/7]

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

◆ compute_index() [4/7]

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

◆ compute_index() [5/7]

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

Definition at line 49 of file polynomial_space.cc.

◆ compute_index() [6/7]

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

Definition at line 59 of file polynomial_space.cc.

◆ compute_index() [7/7]

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

Definition at line 85 of file polynomial_space.cc.

Member Data Documentation

◆ dimension

template<int dim>
const unsigned int PolynomialsP< dim >::dimension = dim
static

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

Definition at line 54 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 97 of file polynomials_p.h.


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