The Spline module provides basic routines related to spline function and local polynomials. More...

Namespaces
namespace	Bembel::Basis
	The Basis namespace contains classes and functions that are to be used as an interface between the BEM code and the functions in the Spl namespace.

namespace	Bembel::Spl
	The Spl namespace is the backend for basis functions and B-Spline computations, and should not be used by the user in a boundary element code.

Classes
class	Bembel::Basis::PSpecificBasisHandler< P, Scalar >
	The functions above have a fixed compile time polynomial degree. The PSpecificBasis handler is used to convert this to a runtime p through template recursion. More...

struct	Bembel::static_assertKlessseqN< condition >
	Here we hardcode the binomial coefficients through template recursion and perform bounds checking, just in case... More...

class	Bembel::Basis::PSpecificShapeFunctionHandler< P >
	These routines implement a template recursion that allows to choose a compile time instantiation of a basis-evaluation routine with a runtime p. To replace the underlying basis, only these routines should be changed. More...

Functions
template<int P, typename Scalar >
void	Bembel::Basis::phi_ (Eigen::Matrix< Scalar, Eigen::Dynamic, 1 > *c, Scalar w, double x)
	evaluates the 1D basis at x weighted with a quadrature weight w

template<int P, typename Scalar >
void	Bembel::Basis::phi_dx_ (Eigen::Matrix< Scalar, Eigen::Dynamic, 1 > *c, Scalar w, double x)
	evaluates the derivative of phi

template<int P, typename Scalar >
void	Bembel::Basis::phiphi_ (Eigen::Matrix< Scalar, Eigen::Dynamic, 1 > *c, Scalar w, Eigen::Vector2d a)
	evaluates the 2D tensor product basis at a point a in [0,1]^2

template<int P, typename Scalar >
void	Bembel::Basis::phiphi_dx_ (Eigen::Matrix< Scalar, Eigen::Dynamic, 1 > *c, Scalar w, Eigen::Vector2d a)
	evaluates the x-derivative of the 2D tensor product basis at a point a in [0,1]^2

template<int P, typename Scalar >
void	Bembel::Basis::phiphi_dy_ (Eigen::Matrix< Scalar, Eigen::Dynamic, 1 > *c, Scalar w, Eigen::Vector2d a)
	evaluates the y-derivative of the 2D tensor product basis at a point a in [0,1]^2

template<int P, typename Scalar >
void	Bembel::Basis::Phi_times_Phi_ (Eigen::Matrix< Scalar, Eigen::Dynamic, Eigen::Dynamic > *c, Scalar w, Eigen::Vector2d xi, Eigen::Vector2d eta)
	evaluates the interaction of two phiphis, one at xi and one at eta. Used for e.g. gram matrices.

template<int P, typename Scalar >
void	Bembel::Basis::Div_Phi_times_Div_Phi_ (Eigen::Matrix< Scalar, Eigen::Dynamic, Eigen::Dynamic > *c, Scalar weight, Eigen::Vector2d xi, Eigen::Vector2d eta)
	same as above, just using the divergence

template<int N>
constexpr double	Bembel::Basis::BernsteinX (double evaluation_point) noexcept
	Template recursion to produce Bernstein polynomials. This is only limited by the binomial coefficient, see Pascal.hpp.

template<typename T >
Eigen::SparseMatrix< T >	Bembel::Spl::MakeProjection (const std::vector< T > &x_knots, const std::vector< T > &y_knots, const std::vector< T > &x_unique_knots, const std::vector< T > &y_unique_knots, const int polynomial_degree_x, const int polynomial_degree_y) noexcept
	implements Bezier extraction via the solution of an interpolation problem.

template<typename T >
Eigen::Matrix< T, -1, -1 >	Bembel::Spl::DeBoor (Eigen::Matrix< T, -1, -1 > const &control_points, const std::vector< double > &knot_vector, const std::vector< double > &evaluation_points) noexcept
	"By the book" implementations of the Cox-DeBoor formula. Inefficient, do not use at bottlenecks.

template<typename T >
Eigen::Matrix< T, -1, -1 >	Bembel::Spl::DeBoorDer (Eigen::Matrix< T, -1, -1 > const &control_points, std::vector< double > const &knot, std::vector< double > const &evaluation_points) noexcept
	A "by the book" implementation of the derivatives and TP-algos based on the DeBoor Recursion.

std::vector< double >	Bembel::Spl::MakeBezierKnotVector (int polynomial_degree) noexcept
	Here, routines for the creation and processing of knot vectors are defined.

template<typename T >
std::vector< T >	Bembel::Spl::MakeInterpolationPoints (const std::vector< T > &uniq, const std::vector< T > &mask) noexcept
	Creates a T-vector of equidistant itnerpolation points for a given knot vector without any repetitions.

Eigen::Matrix< double, -1, -1 >	Bembel::Spl::GetInterpolationMatrix (int polynomial_degree, const std::vector< double > &mask)
	returns the coefficients to represent a function in the Bernstein basis on [0,1].

template<typename T >
void	Bembel::Spl::GetCoefficients (const int incr, const std::vector< double > &mask, const std::vector< T > &values, T *coefs)
	this solves a generic interpolation problem.

template<typename T >
std::vector< T >	Bembel::Spl::GetCoefficients (const int increments, const std::vector< double > &mask, const std::vector< T > &values)
	this solves a generic interpolation problem.

Detailed Description

The Spline module provides basic routines related to spline function and local polynomials.

Function Documentation

◆ BernsteinX()

template<int N>

constexpr double Bembel::Basis::BernsteinX ( double evaluation_point )

inlineconstexprnoexcept

Template recursion to produce Bernstein polynomials. This is only limited by the binomial coefficient, see Pascal.hpp.

Definition at line 23 of file Bernstein.hpp.

◆ DeBoor()

template<typename T >

Eigen::Matrix< T, -1, -1 > Bembel::Spl::DeBoor	(	Eigen::Matrix< T, -1, -1 > const &	control_points,
		const std::vector< double > &	knot_vector,
		const std::vector< double > &	evaluation_points
	)

noexcept

"By the book" implementations of the Cox-DeBoor formula. Inefficient, do not use at bottlenecks.

ISO C++ forbids variable length array

Definition at line 22 of file DeBoor.hpp.

◆ DeBoorDer()

template<typename T >

Eigen::Matrix< T, -1, -1 > Bembel::Spl::DeBoorDer	(	Eigen::Matrix< T, -1, -1 > const &	control_points,
		std::vector< double > const &	knot,
		std::vector< double > const &	evaluation_points
	)

noexcept

A "by the book" implementation of the derivatives and TP-algos based on the DeBoor Recursion.

Definition at line 22 of file DeBoorTP.hpp.

◆ Div_Phi_times_Div_Phi_()

template<int P, typename Scalar >

void Bembel::Basis::Div_Phi_times_Div_Phi_	(	Eigen::Matrix< Scalar, Eigen::Dynamic, Eigen::Dynamic > *	c,
		Scalar	weight,
		Eigen::Vector2d	xi,
		Eigen::Vector2d	eta
	)

same as above, just using the divergence

Definition at line 171 of file Basis.hpp.

◆ GetCoefficients() [1/2]

template<typename T >

void Bembel::Spl::GetCoefficients	(	const int	incr,
		const std::vector< double > &	mask,
		const std::vector< T > &	values,
		T *	coefs
	)

inline

this solves a generic interpolation problem.

Definition at line 88 of file Localize.hpp.

◆ GetCoefficients() [2/2]

template<typename T >

std::vector< T > Bembel::Spl::GetCoefficients	(	const int	increments,
		const std::vector< double > &	mask,
		const std::vector< T > &	values
	)

inline

this solves a generic interpolation problem.

Definition at line 117 of file Localize.hpp.

◆ GetInterpolationMatrix()

Eigen::Matrix< double, -1, -1 > Bembel::Spl::GetInterpolationMatrix	(	int	polynomial_degree,
		const std::vector< double > &	mask
	)

inline

returns the coefficients to represent a function in the Bernstein basis on [0,1].

Definition at line 67 of file Localize.hpp.

◆ MakeBezierKnotVector()

std::vector< double > Bembel::Spl::MakeBezierKnotVector ( int polynomial_degree )

inlinenoexcept

Here, routines for the creation and processing of knot vectors are defined.

Definition at line 21 of file Knots.hpp.

◆ MakeInterpolationPoints()

template<typename T >

std::vector< T > Bembel::Spl::MakeInterpolationPoints	(	const std::vector< T > &	uniq,
		const std::vector< T > &	mask
	)

inlinenoexcept

Creates a T-vector of equidistant itnerpolation points for a given knot vector without any repetitions.

Definition at line 47 of file Localize.hpp.

◆ MakeProjection()

template<typename T >

Eigen::SparseMatrix< T > Bembel::Spl::MakeProjection	(	const std::vector< T > &	x_knots,
		const std::vector< T > &	y_knots,
		const std::vector< T > &	x_unique_knots,
		const std::vector< T > &	y_unique_knots,
		const int	polynomial_degree_x,
		const int	polynomial_degree_y
	)

noexcept

implements Bezier extraction via the solution of an interpolation problem.

The inverse matrix, for the solition of the linear system to come.

For each basisfunction phi_j solves for the right coeef such that \Sum

Definition at line 23 of file Bezierextraction.hpp.

◆ phi_()

template<int P, typename Scalar >

void Bembel::Basis::phi_	(	Eigen::Matrix< Scalar, Eigen::Dynamic, 1 > *	c,
		Scalar	w,
		double	x
	)

inline

evaluates the 1D basis at x weighted with a quadrature weight w

Definition at line 58 of file Basis.hpp.

◆ phi_dx_()

template<int P, typename Scalar >

void Bembel::Basis::phi_dx_	(	Eigen::Matrix< Scalar, Eigen::Dynamic, 1 > *	c,
		Scalar	w,
		double	x
	)

inline

evaluates the derivative of phi

Definition at line 72 of file Basis.hpp.

◆ Phi_times_Phi_()

template<int P, typename Scalar >

void Bembel::Basis::Phi_times_Phi_	(	Eigen::Matrix< Scalar, Eigen::Dynamic, Eigen::Dynamic > *	c,
		Scalar	w,
		Eigen::Vector2d	xi,
		Eigen::Vector2d	eta
	)

evaluates the interaction of two phiphis, one at xi and one at eta. Used for e.g. gram matrices.

Definition at line 142 of file Basis.hpp.

◆ phiphi_()

template<int P, typename Scalar >

void Bembel::Basis::phiphi_	(	Eigen::Matrix< Scalar, Eigen::Dynamic, 1 > *	c,
		Scalar	w,
		Eigen::Vector2d	a
	)

inline

evaluates the 2D tensor product basis at a point a in [0,1]^2

Definition at line 86 of file Basis.hpp.

◆ phiphi_dx_()

template<int P, typename Scalar >

void Bembel::Basis::phiphi_dx_	(	Eigen::Matrix< Scalar, Eigen::Dynamic, 1 > *	c,
		Scalar	w,
		Eigen::Vector2d	a
	)

inline

evaluates the x-derivative of the 2D tensor product basis at a point a in [0,1]^2

Definition at line 105 of file Basis.hpp.

◆ phiphi_dy_()

template<int P, typename Scalar >

void Bembel::Basis::phiphi_dy_	(	Eigen::Matrix< Scalar, Eigen::Dynamic, 1 > *	c,
		Scalar	w,
		Eigen::Vector2d	a
	)

inline

evaluates the y-derivative of the 2D tensor product basis at a point a in [0,1]^2

Definition at line 124 of file Basis.hpp.

Namespaces

Classes

Functions

Detailed Description

Function Documentation

◆ BernsteinX()

◆ DeBoor()

◆ DeBoorDer()

◆ Div_Phi_times_Div_Phi_()

◆ GetCoefficients() [1/2]

◆ GetCoefficients() [2/2]

◆ GetInterpolationMatrix()

◆ MakeBezierKnotVector()

◆ MakeInterpolationPoints()

◆ MakeProjection()

◆ phi_()

◆ phi_dx_()

◆ Phi_times_Phi_()

◆ phiphi_()

◆ phiphi_dx_()

◆ phiphi_dy_()