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Reference documentation for deal.II version 9.6.0
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#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 > &extents) |
Protected Attributes | |
Table< dim+1, Number > | coefficients |
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 82 of file polynomials_barycentric.h.
BarycentricPolynomial< dim, Number >::BarycentricPolynomial | ( | ) |
Constructor for the zero polynomial.
Definition at line 397 of file polynomials_barycentric.h.
BarycentricPolynomial< dim, Number >::BarycentricPolynomial | ( | const TableIndices< dim+1 > & | powers, |
const Number | coefficient ) |
Constructor for a monomial.
Definition at line 410 of file polynomials_barycentric.h.
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static |
Return the specified monomial.
Definition at line 426 of file polynomials_barycentric.h.
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 438 of file polynomials_barycentric.h.
TableIndices< dim+1 > BarycentricPolynomial< dim, Number >::degrees | ( | ) | const |
Degree of each barycentric polynomial.
Definition at line 472 of file polynomials_barycentric.h.
BarycentricPolynomial< dim, Number > BarycentricPolynomial< dim, Number >::operator- | ( | ) | const |
Unary minus.
Definition at line 484 of file polynomials_barycentric.h.
BarycentricPolynomial< dim, Number > BarycentricPolynomial< dim, Number >::operator+ | ( | const Number2 & | a | ) | const |
Add a scalar.
Definition at line 494 of file polynomials_barycentric.h.
BarycentricPolynomial< dim, Number > BarycentricPolynomial< dim, Number >::operator- | ( | const Number2 & | a | ) | const |
Subtract a scalar.
Definition at line 507 of file polynomials_barycentric.h.
BarycentricPolynomial< dim, Number > BarycentricPolynomial< dim, Number >::operator* | ( | const Number2 & | a | ) | const |
Multiply by a scalar.
Definition at line 517 of file polynomials_barycentric.h.
BarycentricPolynomial< dim, Number > BarycentricPolynomial< dim, Number >::operator/ | ( | const Number2 & | a | ) | const |
Divide by a scalar.
Definition at line 539 of file polynomials_barycentric.h.
BarycentricPolynomial< dim, Number > BarycentricPolynomial< dim, Number >::operator+ | ( | const BarycentricPolynomial< dim, Number > & | augend | ) | const |
Add another barycentric polynomial.
Definition at line 549 of file polynomials_barycentric.h.
BarycentricPolynomial< dim, Number > BarycentricPolynomial< dim, Number >::operator- | ( | const BarycentricPolynomial< dim, Number > & | augend | ) | const |
Subtract another barycentric polynomial.
Definition at line 577 of file polynomials_barycentric.h.
BarycentricPolynomial< dim, Number > BarycentricPolynomial< dim, Number >::operator* | ( | const BarycentricPolynomial< dim, Number > & | multiplicand | ) | const |
Multiply by another barycentric polynomial.
Definition at line 587 of file polynomials_barycentric.h.
BarycentricPolynomial< dim, Number > BarycentricPolynomial< dim, Number >::barycentric_derivative | ( | const unsigned int | coordinate | ) | const |
Differentiate in barycentric coordinates.
Definition at line 623 of file polynomials_barycentric.h.
BarycentricPolynomial< dim, Number > BarycentricPolynomial< dim, Number >::derivative | ( | const unsigned int | coordinate | ) | const |
Differentiate in Cartesian coordinates.
Definition at line 653 of file polynomials_barycentric.h.
Number BarycentricPolynomial< dim, Number >::value | ( | const Point< dim > & | point | ) | const |
Evaluate the polynomial.
Definition at line 664 of file polynomials_barycentric.h.
std::size_t BarycentricPolynomial< dim, Number >::memory_consumption | ( | ) | const |
Return an estimate, in bytes, of the memory usage of the object.
Definition at line 701 of file polynomials_barycentric.h.
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staticprotected |
Utility function for barycentric polynomials: it is 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 710 of file polynomials_barycentric.h.
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protected |
Coefficients of the polynomial. The exponents are the integer indexes.
Definition at line 197 of file polynomials_barycentric.h.