Reference documentation for deal.II version Git 08727cc441 20200702 15:45:42 0400

#include <deal.II/base/tensor_product_polynomials.h>
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 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 
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 
Anisotropic tensor product of given polynomials.
Given onedimensional 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 onedimensional 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 dimdimensional
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 310 of file tensor_product_polynomials.h.
AnisotropicPolynomials< dim >::AnisotropicPolynomials  (  const std::vector< std::vector< Polynomials::Polynomial< double >>> &  base_polynomials  ) 
Constructor. base_polynomials
is a table of onedimensional 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 453 of file tensor_product_polynomials.cc.

override 
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.
Definition at line 579 of file tensor_product_polynomials.cc.

override 
Compute the value of the i
th 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 (onedimensional) 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 499 of file tensor_product_polynomials.cc.
Tensor<order, dim> AnisotropicPolynomials< dim >::compute_derivative  (  const unsigned int  i, 
const Point< dim > &  p  
)  const 
Compute the order
th derivative of the i
th 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 (onedimensional) polynomials is (unnecessarily) computed several times. Instead use the evaluate() 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.
order  The derivative order. 

overridevirtual 
Compute the first derivative of the i
th polynomial at unit point p
.
Consider using evaluate() instead.

overridevirtual 
Compute the second derivative of the i
th polynomial at unit point p
.
Consider using evaluate() instead.

overridevirtual 
Compute the third derivative of the i
th polynomial at unit point p
.
Consider using evaluate() instead.

overridevirtual 
Compute the fourth derivative of the i
th polynomial at unit point p
.
Consider using evaluate() instead.

override 
Compute the grad of the i
th 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 (onedimensional) 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 515 of file tensor_product_polynomials.cc.

override 
Compute the second derivative (grad_grad) of the i
th 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 (onedimensional) 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 543 of file tensor_product_polynomials.cc.

override 
Return the name of the space, which is AnisotropicPolynomials
.

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 738 of file tensor_product_polynomials.cc.

private 
Each tensor product polynomial \(p_i\) is a product of onedimensional polynomials in each space direction. Compute the indices of these one dimensional polynomials for each space direction, given the index i
.
Definition at line 469 of file tensor_product_polynomials.cc.

staticprivate 
Given the input to the constructor, compute n_pols
.
Definition at line 726 of file tensor_product_polynomials.cc.

private 
Copy of the vector pols
of polynomials given to the constructor.
Definition at line 460 of file tensor_product_polynomials.h.