Reference documentation for deal.II version Git 08a245e9a2 2021-03-07 21:13:37 +0100
PolynomialsAdini< dim > Class Template Reference

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

Inheritance diagram for PolynomialsAdini< dim >:
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## Public Member Functions

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

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

## Private Attributes

Table< 2, doublecoef

Table< 2, doubledx

Table< 2, doubledy

Table< 2, doubledxx

Table< 2, doubledyy

Table< 2, doubledxy

## Detailed Description

### template<int dim> class PolynomialsAdini< dim >

The cubic polynomial space for the Adini element

This space consists of the cubic space P3 augmented by the functions xy3 and x3y.

The basis of the space is chosen to match the node functionals of the Adini element.

Todo:
This polynomial space is implemented in 2D only and does not compute derivatives of order 3 or higher.

Definition at line 44 of file polynomials_adini.h.

## Constructor & Destructor Documentation

template<int dim>

Constructor for the polynomials of the described space

Definition at line 42 of file polynomials_adini.cc.

## ◆ evaluate()

template<int dim>
 void PolynomialsAdini< 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
override

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 138 of file polynomials_adini.cc.

## ◆ compute_value()

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

Compute the value of the ith polynomial at unit_point.

Definition at line 193 of file polynomials_adini.cc.

## ◆ compute_1st_derivative()

template<int dim>
 Tensor< 1, dim > PolynomialsAdini< 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.

Definition at line 181 of file polynomials_adini.h.

## ◆ compute_2nd_derivative()

template<int dim>
 Tensor< 2, dim > PolynomialsAdini< 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.

Definition at line 192 of file polynomials_adini.h.

## ◆ compute_3rd_derivative()

template<int dim>
 Tensor< 3, dim > PolynomialsAdini< 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.

Definition at line 203 of file polynomials_adini.h.

## ◆ compute_4th_derivative()

template<int dim>
 Tensor< 4, dim > PolynomialsAdini< 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.

Definition at line 214 of file polynomials_adini.h.

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

Compute the gradient of the ith polynomial at unit_point.

Definition at line 209 of file polynomials_adini.cc.

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

Compute the second derivative (grad_grad) of the ith polynomial at unit_point.

Definition at line 233 of file polynomials_adini.cc.

## ◆ name()

template<int dim>
 std::string PolynomialsAdini< dim >::name ( ) const
inlineoverride

Return the name of the space, which is PolynomialsAdini.

Definition at line 225 of file polynomials_adini.h.

## ◆ clone()

template<int dim>
 std::unique_ptr< ScalarPolynomialsBase< dim > > PolynomialsAdini< 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.

Definition at line 262 of file polynomials_adini.cc.

## ◆ coef

template<int dim>
private

Store the coefficients of the polynomials in the order $$1,x,y,x^2,y^2,xy,x^3,y^3,xy^2,x^2y,x^3y,xy^3$$

Definition at line 144 of file polynomials_adini.h.

## ◆ dx

template<int dim>
private

Store the coefficients of the x-derivative of the polynomials in the order $$1,x,y,x^2,y^2,xy,x^3,y^3,xy^2,x^2y,x^3y,xy^3$$

Definition at line 150 of file polynomials_adini.h.

## ◆ dy

template<int dim>
private

Store the coefficients of the y-derivative of the polynomials in the order $$1,x,y,x^2,y^2,xy,x^3,y^3,xy^2,x^2y,x^3y,xy^3$$

Definition at line 156 of file polynomials_adini.h.

## ◆ dxx

template<int dim>
private

Store the coefficients of the second x-derivative of the polynomials in the order $$1,x,y,x^2,y^2,xy,x^3,y^3,xy^2,x^2y,x^3y,xy^3$$

Definition at line 162 of file polynomials_adini.h.

## ◆ dyy

template<int dim>
private

Store the coefficients of the second y-derivative of the polynomials in the order $$1,x,y,x^2,y^2,xy,x^3,y^3,xy^2,x^2y,x^3y,xy^3$$

Definition at line 168 of file polynomials_adini.h.

## ◆ dxy

template<int dim>
Store the coefficients of the second mixed derivative of the polynomials in the order $$1,x,y,x^2,y^2,xy,x^3,y^3,xy^2,x^2y,x^3y,xy^3$$