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

Detailed Description

template<int dim>
class ScalarLagrangePolynomialSimplex< dim >

Polynomials defined on simplex entities. This class can be a basis of FE_SimplexP. We first use the Jacobi polynomials given in [131] to construct a modal basis. With the modal basis a Vandermonde matrix is calculated which leads to a nodal basis. For computing the values of the nodal basis the Vandermonde matrix is multiplied with the modal basis vector evaluated at the evaluation point.

Definition at line 37 of file polynomials_simplex.h.

Inheritance diagram for ScalarLagrangePolynomialSimplex< dim >:

[legend]

Public Member Functions
	ScalarLagrangePolynomialSimplex (const unsigned int degree, const std::vector< Point< dim > > &support_points)

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

std::string	name () const override

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

Tensor< 1, dim >	compute_1st_derivative (const unsigned int i, const Point< dim > &p) const override

Tensor< 2, dim >	compute_2nd_derivative (const unsigned int i, const Point< dim > &p) const override

Tensor< 3, dim >	compute_3rd_derivative (const unsigned int i, const Point< dim > &p) const override

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

unsigned int	n () const

virtual unsigned int	degree () const

virtual std::size_t	memory_consumption () const

Static Public Attributes
static constexpr unsigned int	dimension = dim

Protected Member Functions
void	reinit (const std::vector< Point< dim > > &support_points)

Protected Attributes
FullMatrix< double >	vandermonde_matrix_inverse

Private Member Functions
double	evaluate_orthogonal_basis_function_by_degree (const unsigned int i, const unsigned int j, const unsigned int k, const Point< dim > &p) const override

double	evaluate_orthogonal_basis_function (const unsigned int i, const Point< dim > &p) const override

Tensor< 1, dim >	evaluate_orthogonal_basis_derivative_by_degree (const unsigned int i, const unsigned int j, const unsigned int k, const Point< dim > &p) const override

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

virtual Tensor< 2, dim >	evaluate_orthogonal_basis_2nd_derivative_by_degree (const unsigned int i, const unsigned int j, const unsigned int k, const Point< dim > &p) const override

virtual Tensor< 2, dim >	evaluate_orthogonal_basis_2nd_derivative (const unsigned int i, const Point< dim > &p) const override

Private Attributes
const unsigned int	polynomial_degree

const unsigned int	n_pols

Constructor & Destructor Documentation

◆ ScalarLagrangePolynomialSimplex()

template<int dim>

ScalarLagrangePolynomialSimplex< dim >::ScalarLagrangePolynomialSimplex	(	const unsigned int	degree,
		const std::vector< Point< dim > > &	support_points
	)

Definition at line 44 of file polynomials_simplex.cc.

Member Function Documentation

◆ evaluate()

template<int dim>

void ScalarLagrangePolynomialSimplex< 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 derivatives of the polynomials at unit_point.

The size of the vectors must either be zero or equal n(). In the first case, the function will not compute these values.

If you need values or derivatives of all 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.

Note: Currently, only the vectors values, grads and grads_grads are filled.

Implements ScalarPolynomialsBase< dim >.

Definition at line 491 of file polynomials_simplex.cc.

◆ name()

template<int dim>

std::string ScalarLagrangePolynomialSimplex< dim >::name ( ) const

overridevirtual

Return the name of the space.

Implements ScalarPolynomialsBase< dim >.

Definition at line 519 of file polynomials_simplex.cc.

◆ clone()

template<int dim>

std::unique_ptr< ScalarPolynomialsBase< dim > > ScalarLagrangePolynomialSimplex< 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 528 of file polynomials_simplex.cc.

◆ evaluate_orthogonal_basis_function_by_degree()

template<int dim>

double ScalarLagrangePolynomialSimplex< dim >::evaluate_orthogonal_basis_function_by_degree	(	const unsigned int	i,
		const unsigned int	j,
		const unsigned int	k,
		const Point< dim > &	p
	)		const

overrideprivatevirtual

Evaluate the orthogonal basis at point p. The indices i, j and k correspond to the polynomial degrees of the Jacobi polynomials given in [131].

Implements ScalarPolynomialsVandermondeBase< dim >.

Definition at line 73 of file polynomials_simplex.cc.

◆ evaluate_orthogonal_basis_function()

template<int dim>

double ScalarLagrangePolynomialSimplex< dim >::evaluate_orthogonal_basis_function	(	const unsigned int	i,
		const Point< dim > &	p
	)		const

overrideprivatevirtual

Evaluate the orthogonal basis function i at point p. This function determines the corresponding indices for the Jacobi polynomials and calls the function taking all indices as arguments.

Implements ScalarPolynomialsVandermondeBase< dim >.

Definition at line 134 of file polynomials_simplex.cc.

◆ evaluate_orthogonal_basis_derivative_by_degree()

template<int dim>

Tensor< 1, dim > ScalarLagrangePolynomialSimplex< dim >::evaluate_orthogonal_basis_derivative_by_degree	(	const unsigned int	i,
		const unsigned int	j,
		const unsigned int	k,
		const Point< dim > &	p
	)		const

overrideprivatevirtual

Evaluate the derivative of the orthogonal basis at point p. The indices i, j and k correspond to the polynomial degrees of the Jacobi polynomials given in [131].

Implements ScalarPolynomialsVandermondeBase< dim >.

Definition at line 174 of file polynomials_simplex.cc.

◆ evaluate_orthogonal_basis_derivative()

template<int dim>

Tensor< 1, dim > ScalarLagrangePolynomialSimplex< dim >::evaluate_orthogonal_basis_derivative	(	const unsigned int	i,
		const Point< dim > &	p
	)		const

overrideprivatevirtual

Evaluate the derivative of the orthogonal basis function i at point p. This function determines the corresponding indices for the Jacobi polynomials and calls the function taking all indices as arguments.

Implements ScalarPolynomialsVandermondeBase< dim >.

Definition at line 263 of file polynomials_simplex.cc.

◆ evaluate_orthogonal_basis_2nd_derivative_by_degree()

template<int dim>

Tensor< 2, dim > ScalarLagrangePolynomialSimplex< dim >::evaluate_orthogonal_basis_2nd_derivative_by_degree	(	const unsigned int	i,
		const unsigned int	j,
		const unsigned int	k,
		const Point< dim > &	p
	)		const

overrideprivatevirtual

Evaluate the 2nd derivative of the orthogonal basis at point p. The indices i, j and k correspond to the polynomial degrees of the Jacobi polynomials.

Reimplemented from ScalarPolynomialsVandermondeBase< dim >.

Definition at line 303 of file polynomials_simplex.cc.

◆ evaluate_orthogonal_basis_2nd_derivative()

template<int dim>

Tensor< 2, dim > ScalarLagrangePolynomialSimplex< dim >::evaluate_orthogonal_basis_2nd_derivative	(	const unsigned int	i,
		const Point< dim > &	p
	)		const

overrideprivatevirtual

Evaluate the 2nd derivative of the orthogonal basis function i at point p. This function determines the corresponding indices for the Jacobi polynomials and calls the function taking all indices as arguments.

Reimplemented from ScalarPolynomialsVandermondeBase< dim >.

Definition at line 445 of file polynomials_simplex.cc.

◆ compute_value()

template<int dim>

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

overridevirtualinherited

Compute the value of the ith polynomial at unit point p.

Consider using evaluate() instead.

Implements ScalarPolynomialsBase< dim >.

Definition at line 90 of file scalar_polynomials_vandermonde_base.cc.

◆ compute_derivative()

template<int dim>

template<int order>

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

inherited

Compute the orderth derivative of the ith polynomial at unit point p.

Consider using evaluate() instead.

Template Parameters

order The order of the derivative.

Note: Implemented for first derivative, for simplices also the second derivative is implemented.

Definition at line 215 of file scalar_polynomials_vandermonde_base.h.

◆ compute_1st_derivative()

template<int dim>

Tensor< 1, dim > ScalarPolynomialsVandermondeBase< dim >::compute_1st_derivative	(	const unsigned int	i,
		const Point< dim > &	p
	)		const

overridevirtualinherited

Compute the first derivative of the ith polynomial at unit point p.

Consider using evaluate() instead.

Implements ScalarPolynomialsBase< dim >.

Definition at line 185 of file scalar_polynomials_vandermonde_base.cc.

◆ compute_2nd_derivative()

template<int dim>

Tensor< 2, dim > ScalarPolynomialsVandermondeBase< dim >::compute_2nd_derivative	(	const unsigned int	i,
		const Point< dim > &	p
	)		const

overridevirtualinherited

Compute the second derivative of the ith polynomial at unit point p.

Consider using evaluate() instead.

Note: Only implemented for simplices.

Implements ScalarPolynomialsBase< dim >.

Definition at line 196 of file scalar_polynomials_vandermonde_base.cc.

◆ compute_3rd_derivative()

template<int dim>

Tensor< 3, dim > ScalarPolynomialsVandermondeBase< dim >::compute_3rd_derivative	(	const unsigned int	i,
		const Point< dim > &	p
	)		const

overridevirtualinherited

Compute the third derivative of the ith polynomial at unit point p.

Consider using evaluate() instead.

Note: Not implemented yet.

Implements ScalarPolynomialsBase< dim >.

Definition at line 207 of file scalar_polynomials_vandermonde_base.cc.

◆ compute_4th_derivative()

template<int dim>

Tensor< 4, dim > ScalarPolynomialsVandermondeBase< dim >::compute_4th_derivative	(	const unsigned int	i,
		const Point< dim > &	p
	)		const

overridevirtualinherited

Compute the fourth derivative of the ith polynomial at unit point p.

Consider using evaluate() instead.

Note: Not implemented yet.

Implements ScalarPolynomialsBase< dim >.

Definition at line 223 of file scalar_polynomials_vandermonde_base.cc.

◆ compute_grad()

template<int dim>

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

overridevirtualinherited

Compute the gradient of the ith polynomial at unit point p.

Consider using evaluate() instead.

Implements ScalarPolynomialsBase< dim >.

Definition at line 110 of file scalar_polynomials_vandermonde_base.cc.

◆ compute_grad_grad()

template<int dim>

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

overridevirtualinherited

Compute the second derivative (grad_grad) of the ith polynomial at unit point p.

Consider using evaluate() instead.

Note: Only implemented for simplices.

Implements ScalarPolynomialsBase< dim >.

Definition at line 134 of file scalar_polynomials_vandermonde_base.cc.

◆ reinit()

template<int dim>

void ScalarPolynomialsVandermondeBase< dim >::reinit ( const std::vector< Point< dim > > & support_points )

protectedinherited

Definition at line 39 of file scalar_polynomials_vandermonde_base.cc.

◆ n()

template<int dim>

unsigned int ScalarPolynomialsBase< dim >::n ( ) const

inlineinherited

Return the number of polynomials.

Definition at line 254 of file scalar_polynomials_base.h.

◆ degree()

template<int dim>

unsigned int ScalarPolynomialsBase< dim >::degree ( ) const

inlinevirtualinherited

Return the highest polynomial degree of polynomials represented by this class. A derived class may override this if its value is different from my_degree.

Reimplemented in PolynomialsP< dim >.

Definition at line 263 of file scalar_polynomials_base.h.

◆ memory_consumption()

template<int dim>

std::size_t ScalarPolynomialsBase< dim >::memory_consumption ( ) const

virtualinherited

Return an estimate (in bytes) for the memory consumption of this object.

Reimplemented in BarycentricPolynomials< dim >, TensorProductPolynomials< dim, PolynomialType >, and TensorProductPolynomials< dim - 1 >.

Definition at line 36 of file scalar_polynomials_base.cc.

Member Data Documentation

◆ dimension

template<int dim>

constexpr unsigned int ScalarLagrangePolynomialSimplex< dim >::dimension = dim

staticconstexpr

Make the dimension available to the outside.

Definition at line 44 of file polynomials_simplex.h.

◆ vandermonde_matrix_inverse

template<int dim>

FullMatrix<double> ScalarPolynomialsVandermondeBase< dim >::vandermonde_matrix_inverse

protectedinherited

The Vandermonde matrix evaluates each modal basis function at the chosen nodal points. Applying the inverse of the Vandermonde matrix transforms from the modal basis to the nodal basis.

Definition at line 141 of file scalar_polynomials_vandermonde_base.h.

◆ polynomial_degree

template<int dim>

const unsigned int ScalarPolynomialsBase< dim >::polynomial_degree

privateinherited

The highest polynomial degree of this functions represented by this object.

Definition at line 225 of file scalar_polynomials_base.h.

◆ n_pols

template<int dim>

const unsigned int ScalarPolynomialsBase< dim >::n_pols

privateinherited

The number of polynomials represented by this object.

Definition at line 230 of file scalar_polynomials_base.h.

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

include/deal.II/base/polynomials_simplex.h
source/base/polynomials_simplex.cc

Detailed Description

Public Member Functions

Static Public Attributes

Protected Member Functions

Protected Attributes

Private Member Functions

Private Attributes

Constructor & Destructor Documentation

◆ ScalarLagrangePolynomialSimplex()

Member Function Documentation

◆ evaluate()

◆ name()

◆ clone()

◆ evaluate_orthogonal_basis_function_by_degree()

◆ evaluate_orthogonal_basis_function()

◆ evaluate_orthogonal_basis_derivative_by_degree()

◆ evaluate_orthogonal_basis_derivative()

◆ evaluate_orthogonal_basis_2nd_derivative_by_degree()

◆ evaluate_orthogonal_basis_2nd_derivative()

◆ compute_value()

◆ compute_derivative()

◆ compute_1st_derivative()

◆ compute_2nd_derivative()

◆ compute_3rd_derivative()

◆ compute_4th_derivative()

◆ compute_grad()

◆ compute_grad_grad()

◆ reinit()

◆ n()

◆ degree()

◆ memory_consumption()

Member Data Documentation

◆ dimension

◆ vandermonde_matrix_inverse

◆ polynomial_degree

◆ n_pols