Reference documentation for deal.II version Git 8236ec9b21 2019-11-20 14:24:58 -0500
AnisotropicPolynomials< dim > Class Template Reference

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

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

AnisotropicPolynomials (const std::vector< std::vector< Polynomials::Polynomial< double >>> &base_polynomials)

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

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

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

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

std::string name () const override

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

Public Member Functions inherited from ScalarPolynomialsBase< dim >
ScalarPolynomialsBase (const unsigned int deg, const unsigned int n_polynomials)

ScalarPolynomialsBase (ScalarPolynomialsBase< dim > &&)=default

ScalarPolynomialsBase (const ScalarPolynomialsBase< dim > &)=default

virtual ~ScalarPolynomialsBase ()=default

unsigned int n () const

unsigned int degree () const

## Private Member Functions

void compute_index (const unsigned int i, std::array< unsigned int, dim > &indices) const

## Static Private Member Functions

static unsigned int get_n_tensor_pols (const std::vector< std::vector< Polynomials::Polynomial< double >>> &pols)

## Private Attributes

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

## Detailed Description

### template<int dim> class AnisotropicPolynomials< dim >

Anisotropic tensor product of given polynomials.

Given one-dimensional polynomials $$P^x_1(x), P^x_2(x), \ldots$$ in $$x$$-direction, $$P^y_1(y), P^y_2(y), \ldots$$ in $$y$$-direction, and so on, this class generates polynomials of the form $$Q_{ijk}(x,y,z) = P^x_i(x)P^y_j(y)P^z_k(z)$$. (With obvious generalization if dim is in fact only 2. If dim is in fact only 1, then the result is simply the same set of one-dimensional polynomials passed to the constructor.)

If the elements of each set of base polynomials are mutually orthogonal on the interval $$[-1,1]$$ or $$[0,1]$$, then the tensor product polynomials are orthogonal on $$[-1,1]^d$$ or $$[0,1]^d$$, respectively.

The resulting dim-dimensional tensor product polynomials are ordered as follows: We iterate over the $$x$$ coordinates running fastest, then the $$y$$ coordinate, etc. For example, for dim==2, the first few polynomials are thus $$P^x_1(x)P^y_1(y)$$, $$P^x_2(x)P^y_1(y)$$, $$P^x_3(x)P^y_1(y)$$, ..., $$P^x_1(x)P^y_2(y)$$, $$P^x_2(x)P^y_2(y)$$, $$P^x_3(x)P^y_2(y)$$, etc.

Definition at line 281 of file tensor_product_polynomials.h.

## ◆ AnisotropicPolynomials()

template<int dim>
 AnisotropicPolynomials< dim >::AnisotropicPolynomials ( const std::vector< std::vector< Polynomials::Polynomial< double >>> & base_polynomials )

Constructor. base_polynomials is a table of one-dimensional polynomials. The number of rows in this table (the first index when indexing into base_polynomials) needs to be equal to the space dimension, with the elements of each row (i.e., the second index) giving the polynomials that shall be used in this particular coordinate direction.

Since we want to build anisotropic polynomials, the dim sets of polynomials passed in as arguments may of course be different, and may also vary in number.

The number of tensor product polynomials is Nx*Ny*Nz, or with terms dropped if the number of space dimensions is less than 3.

Definition at line 442 of file tensor_product_polynomials.cc.

## ◆ evaluate()

template<int dim>
 void AnisotropicPolynomials< 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 tensor product polynomial at unit_point.

The size of the vectors must either be equal 0 or equal this->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 ScalarPolynomialsBase< dim >.

Definition at line 568 of file tensor_product_polynomials.cc.

## ◆ compute_value()

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

Compute the value of the ith tensor product polynomial at unit_point. Here i is given in tensor product numbering.

Note, that using this function within a loop over all tensor product polynomials is not efficient, because then each point value of the underlying (one-dimensional) polynomials is (unnecessarily) computed several times. Instead use the compute function, see above, with values.size()==this->n() to get the point values of all tensor polynomials all at once and in a much more efficient way.

Definition at line 488 of file tensor_product_polynomials.cc.

## ◆ compute_derivative()

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

Compute the orderth derivative of the ith tensor product polynomial at unit_point. Here i is given in tensor product numbering.

Note, that using this function within a loop over all tensor product polynomials is not efficient, because then each derivative value of the underlying (one-dimensional) polynomials is (unnecessarily) computed several times. Instead use the compute() function, see above, with the size of the appropriate parameter set to n() to get the point value of all tensor polynomials all at once and in a much more efficient way.

Template Parameters
 order The derivative order.

## ◆ compute_grad()

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

Compute the grad of the ith tensor product polynomial at unit_point. Here i is given in tensor product numbering.

Note, that using this function within a loop over all tensor product polynomials is not efficient, because then each derivative value of the underlying (one-dimensional) polynomials is (unnecessarily) computed several times. Instead use the compute function, see above, with grads.size()==this->n() to get the point value of all tensor polynomials all at once and in a much more efficient way.

Definition at line 504 of file tensor_product_polynomials.cc.

## ◆ compute_grad_grad()

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

Compute the second derivative (grad_grad) of the ith tensor product polynomial at unit_point. Here i is given in tensor product numbering.

Note, that using this function within a loop over all tensor product polynomials is not efficient, because then each derivative value of the underlying (one-dimensional) polynomials is (unnecessarily) computed several times. Instead use the compute function, see above, with grad_grads.size()==this->n() to get the point value of all tensor polynomials all at once and in a much more efficient way.

Definition at line 532 of file tensor_product_polynomials.cc.

## ◆ name()

template<int dim>
 std::string AnisotropicPolynomials< dim >::name ( ) const
overridevirtual

Return the name of the space, which is AnisotropicPolynomials.

Implements ScalarPolynomialsBase< dim >.

## ◆ clone()

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

Definition at line 727 of file tensor_product_polynomials.cc.

## ◆ compute_index()

template<int dim>
 void AnisotropicPolynomials< dim >::compute_index ( const unsigned int i, std::array< unsigned int, dim > & indices ) const
private

Each tensor product polynomial @þ{i} is a product of one-dimensional polynomials in each space direction. Compute the indices of these one- dimensional polynomials for each space direction, given the index i.

Definition at line 458 of file tensor_product_polynomials.cc.

## ◆ get_n_tensor_pols()

template<int dim>
 unsigned int AnisotropicPolynomials< dim >::get_n_tensor_pols ( const std::vector< std::vector< Polynomials::Polynomial< double >>> & pols )
staticprivate

Given the input to the constructor, compute n_pols.

Definition at line 715 of file tensor_product_polynomials.cc.

## ◆ polynomials

template<int dim>
 const std::vector > > AnisotropicPolynomials< dim >::polynomials
private

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

Definition at line 403 of file tensor_product_polynomials.h.

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