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

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 95 of file 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 > &extents)

Protected Attributes
Table< dim+1, Number >	coefficients