Reference documentation for deal.II version GIT 194dd8bb02 2022-12-03 08:20:02+00:00
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Public Member Functions | Static Public Member Functions | Static Protected Member Functions | Protected Attributes | List of all members
BarycentricPolynomial< dim, Number > Class Template Reference

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

Public Member Functions

 BarycentricPolynomial ()
 
 BarycentricPolynomial (const TableIndices< dim+1 > &powers, const Number coefficient)
 
void print (std::ostream &out) const
 
TableIndices< dim+1 > degrees () const
 
BarycentricPolynomial< dim, Number > operator- () const
 
template<typename Number2 >
BarycentricPolynomial< dim, Number > operator+ (const Number2 &a) const
 
template<typename Number2 >
BarycentricPolynomial< dim, Number > operator- (const Number2 &a) const
 
template<typename Number2 >
BarycentricPolynomial< dim, Number > operator* (const Number2 &a) const
 
template<typename Number2 >
BarycentricPolynomial< dim, Number > operator/ (const Number2 &a) const
 
BarycentricPolynomial< dim, Number > operator+ (const BarycentricPolynomial< dim, Number > &augend) const
 
BarycentricPolynomial< dim, Number > operator- (const BarycentricPolynomial< dim, Number > &augend) const
 
BarycentricPolynomial< dim, Number > operator* (const BarycentricPolynomial< dim, Number > &multiplicand) const
 
BarycentricPolynomial< dim, Number > barycentric_derivative (const unsigned int coordinate) const
 
BarycentricPolynomial< dim, Number > derivative (const unsigned int coordinate) const
 
Number value (const Point< dim > &point) const
 
std::size_t memory_consumption () const
 

Static Public Member Functions

static BarycentricPolynomial< dim, Number > monomial (const unsigned int d)
 

Static Protected Member Functions

static TableIndices< dim+1 > index_to_indices (const std::size_t &index, const TableIndices< dim+1 > &extent)
 

Protected Attributes

Table< dim+1, Number > coefficients
 

Detailed Description

template<int dim, typename Number = double>
class BarycentricPolynomial< dim, Number >

Polynomial implemented in barycentric coordinates.

Barycentric coordinates are a coordinate system defined on simplices that are particularly easy to work with since they express coordinates in the simplex as convex combinations of the vertices. For example, any point in a triangle can be written as

\[ (x, y) = c_0 (x_0, y_0) + c_1 (x_1, y_1) + c_2 (x_2, y_2). \]

where each value \(c_i\) is the relative weight of each vertex (so the centroid is, in 2D, where each \(c_i = 1/3\)). Since we only consider convex combinations we can rewrite this equation as

\[ (x, y) = (1 - c_1 - c_2) (x_0, y_0) + c_1 (x_1, y_1) + c_2 (x_2, y_2). \]

This results in three polynomials that are equivalent to \(P^1\) in 2D. More exactly, this class implements a polynomial space defined with the basis, in 2D, of

\begin{align*} t_0(x, y) &= 1 - x - y \\ t_1(x, y) &= x \\ t_2(x, y) &= y \end{align*}

and, in 3D,

\begin{align*} t_0(x, y) &= 1 - x - y - z \\ t_1(x, y) &= x \\ t_2(x, y) &= y \\ t_2(x, y) &= z \end{align*}

which is, in practice, a very convenient basis for defining simplex polynomials: for example, the fourth basis function of a TRI6 element is

\[ 4 * t_1(x, y) * t_2(x, y). \]

Barycentric polynomials in dim-dimensional space have dim + 1 variables in since t_0 can be written in terms of the other monomials.

Monomials can be conveniently constructed with BarycentricPolynomial::monomial().

Definition at line 83 of file polynomials_barycentric.h.

Constructor & Destructor Documentation

◆ BarycentricPolynomial() [1/2]

template<int dim, typename Number >
BarycentricPolynomial< dim, Number >::BarycentricPolynomial

Constructor for the zero polynomial.

Definition at line 398 of file polynomials_barycentric.h.

◆ BarycentricPolynomial() [2/2]

template<int dim, typename Number >
BarycentricPolynomial< dim, Number >::BarycentricPolynomial ( const TableIndices< dim+1 > &  powers,
const Number  coefficient 
)

Constructor for a monomial.

Definition at line 411 of file polynomials_barycentric.h.

Member Function Documentation

◆ monomial()

template<int dim, typename Number >
BarycentricPolynomial< dim, Number > BarycentricPolynomial< dim, Number >::monomial ( const unsigned int  d)
static

Return the specified monomial.

Definition at line 427 of file polynomials_barycentric.h.

◆ print()

template<int dim, typename Number >
void BarycentricPolynomial< dim, Number >::print ( std::ostream &  out) const

Print the polynomial to the output stream with lowest-order terms first. For example, the first P6 basis function is printed as -1 * t0^1 + 2 * t0^2, where t0 is the first barycentric variable, t1 is the second, etc.

Definition at line 439 of file polynomials_barycentric.h.

◆ degrees()

template<int dim, typename Number >
TableIndices< dim+1 > BarycentricPolynomial< dim, Number >::degrees

Degree of each barycentric polynomial.

Definition at line 473 of file polynomials_barycentric.h.

◆ operator-() [1/3]

template<int dim, typename Number >
BarycentricPolynomial< dim, Number > BarycentricPolynomial< dim, Number >::operator-

Unary minus.

Definition at line 485 of file polynomials_barycentric.h.

◆ operator+() [1/2]

template<int dim, typename Number >
template<typename Number2 >
BarycentricPolynomial< dim, Number > BarycentricPolynomial< dim, Number >::operator+ ( const Number2 &  a) const

Add a scalar.

Definition at line 495 of file polynomials_barycentric.h.

◆ operator-() [2/3]

template<int dim, typename Number >
template<typename Number2 >
BarycentricPolynomial< dim, Number > BarycentricPolynomial< dim, Number >::operator- ( const Number2 &  a) const

Subtract a scalar.

Definition at line 508 of file polynomials_barycentric.h.

◆ operator*() [1/2]

template<int dim, typename Number >
template<typename Number2 >
BarycentricPolynomial< dim, Number > BarycentricPolynomial< dim, Number >::operator* ( const Number2 &  a) const

Multiply by a scalar.

Definition at line 518 of file polynomials_barycentric.h.

◆ operator/()

template<int dim, typename Number >
template<typename Number2 >
BarycentricPolynomial< dim, Number > BarycentricPolynomial< dim, Number >::operator/ ( const Number2 &  a) const

Divide by a scalar.

Definition at line 540 of file polynomials_barycentric.h.

◆ operator+() [2/2]

template<int dim, typename Number >
BarycentricPolynomial< dim, Number > BarycentricPolynomial< dim, Number >::operator+ ( const BarycentricPolynomial< dim, Number > &  augend) const

Add another barycentric polynomial.

Definition at line 550 of file polynomials_barycentric.h.

◆ operator-() [3/3]

template<int dim, typename Number >
BarycentricPolynomial< dim, Number > BarycentricPolynomial< dim, Number >::operator- ( const BarycentricPolynomial< dim, Number > &  augend) const

Subtract another barycentric polynomial.

Definition at line 578 of file polynomials_barycentric.h.

◆ operator*() [2/2]

template<int dim, typename Number >
BarycentricPolynomial< dim, Number > BarycentricPolynomial< dim, Number >::operator* ( const BarycentricPolynomial< dim, Number > &  multiplicand) const

Multiply by another barycentric polynomial.

Definition at line 588 of file polynomials_barycentric.h.

◆ barycentric_derivative()

template<int dim, typename Number >
BarycentricPolynomial< dim, Number > BarycentricPolynomial< dim, Number >::barycentric_derivative ( const unsigned int  coordinate) const

Differentiate in barycentric coordinates.

Definition at line 624 of file polynomials_barycentric.h.

◆ derivative()

template<int dim, typename Number >
BarycentricPolynomial< dim, Number > BarycentricPolynomial< dim, Number >::derivative ( const unsigned int  coordinate) const

Differentiate in Cartesian coordinates.

Definition at line 654 of file polynomials_barycentric.h.

◆ value()

template<int dim, typename Number >
Number BarycentricPolynomial< dim, Number >::value ( const Point< dim > &  point) const

Evaluate the polynomial.

Definition at line 665 of file polynomials_barycentric.h.

◆ memory_consumption()

template<int dim, typename Number >
std::size_t BarycentricPolynomial< dim, Number >::memory_consumption

Return an estimate, in bytes, of the memory usage of the object.

Definition at line 700 of file polynomials_barycentric.h.

◆ index_to_indices()

template<int dim, typename Number >
TableIndices< dim+1 > BarycentricPolynomial< dim, Number >::index_to_indices ( const std::size_t &  index,
const TableIndices< dim+1 > &  extent 
)
staticprotected

Utility function for barycentric polynomials - its convenient to loop over all the indices at once in a dimension-independent way, but we also need to access the actual indices of the underlying Table object. This utility function converts an integral index into the equivalent TableIndices array (which are also the implicitly stored polynomial exponents).

Definition at line 707 of file polynomials_barycentric.h.

Member Data Documentation

◆ coefficients

template<int dim, typename Number = double>
Table<dim + 1, Number> BarycentricPolynomial< dim, Number >::coefficients
protected

Coefficients of the polynomial. The exponents are the integer indexes.

Definition at line 198 of file polynomials_barycentric.h.


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