Pre-basis for B-spline basis. More...

#include <dune/functions/functionspacebases/bsplinebasis.hh>

Public Types
using	GridView = GV
	The grid view that the FE space is defined on.

Public Member Functions
	BSplinePreBasis (const GridView &gridView, const std::vector< double > &knotVector, unsigned int order, bool makeOpen=true)
	Construct a B-spline basis for a given grid view and set of knot vectors. More...

	BSplinePreBasis (const GridView &gridView, const FieldVector< double, dim > &lowerLeft, const FieldVector< double, dim > &upperRight, const std::array< unsigned int, dim > &elements, unsigned int order, bool makeOpen=true)
	Construct a B-spline basis for a given grid view with uniform knot vectors. More...

void	initializeIndices ()
	Initialize the global indices.

const GridView &	gridView () const
	Obtain the grid view that the basis is defined on.

void	update (const GridView &gv)
	Update the stored grid view, to be called if the grid has changed.

Node	makeNode () const
	Create tree node.

template<class ST , int i>
size_type	size (const Dune::ReservedVector< ST, i > &prefix) const
	Return number of possible values for next position in multi index.

size_type	dimension () const
	Get the total dimension of the space spanned by this basis.

size_type	maxNodeSize () const
	Get the maximal number of DOFs associated to node for any element.

template<typename It >
It	indices (const Node &node, It it) const
	Maps from subtree index set [0..size-1] to a globally unique multi index in global basis.

unsigned int	size () const
	Total number of B-spline basis functions.

unsigned int	size (size_t d) const
	Number of shape functions in one direction.

void	evaluateFunction (const FieldVector< typename GV::ctype, dim > &in, std::vector< FieldVector< R, 1 > > &out, const std::array< unsigned, dim > &currentKnotSpan) const
	Evaluate all B-spline basis functions at a given point.

void	evaluateJacobian (const FieldVector< typename GV::ctype, dim > &in, std::vector< FieldMatrix< R, 1, dim > > &out, const std::array< unsigned, dim > &currentKnotSpan) const
	Evaluate Jacobian of all B-spline basis functions. More...

template<size_type k>
void	evaluate (const typename std::array< int, k > &directions, const FieldVector< typename GV::ctype, dim > &in, std::vector< FieldVector< R, 1 > > &out, const std::array< unsigned, dim > &currentKnotSpan) const
	Evaluate Derivatives of all B-spline basis functions.

Static Public Member Functions
static std::array< unsigned int, dim >	getIJK (typename GridView::IndexSet::IndexType idx, std::array< unsigned int, dim > elements)
	Compute integer element coordinates from the element index. More...

static void	evaluateFunction (const typename GV::ctype &in, std::vector< R > &out, const std::vector< R > &knotVector, unsigned int order, unsigned int currentKnotSpan)
	Evaluate all one-dimensional B-spline functions for a given coordinate direction. More...

static void	evaluateFunctionFull (const typename GV::ctype &in, DynamicMatrix< R > &out, const std::vector< R > &knotVector, unsigned int order, unsigned int currentKnotSpan)
	Evaluate all one-dimensional B-spline functions for a given coordinate direction. More...

static void	evaluateAll (const typename GV::ctype &in, std::vector< R > &out, bool evaluateJacobian, std::vector< R > &outJac, bool evaluateHessian, std::vector< R > &outHess, const std::vector< R > &knotVector, unsigned int order, unsigned int currentKnotSpan)
	Evaluate the second derivatives of all one-dimensional B-spline functions for a given coordinate direction. More...

Public Attributes
std::array< unsigned int, dim >	order_
	Order of the B-spline for each space dimension.

std::array< std::vector< double >, dim >	knotVectors_
	The knot vectors, one for each space dimension.

std::array< unsigned, dim >	elements_
	Number of grid elements in the different coordinate directions.

Detailed Description

template<typename GV>
class Dune::Functions::BSplinePreBasis< GV >

Pre-basis for B-spline basis.

Template Parameters

GV	The GridView that the space is defined on

The BSplinePreBasis can be used to embed a BSplineBasis in a larger basis for the construction of product spaces.

Constructor & Destructor Documentation

◆ BSplinePreBasis() [1/2]

template<typename GV >

Dune::Functions::BSplinePreBasis< GV >::BSplinePreBasis	(	const GridView &	gridView,
		const std::vector< double > &	knotVector,
		unsigned int	order,
		bool	makeOpen = `true`
	)

inline

Construct a B-spline basis for a given grid view and set of knot vectors.

The grid must match the knot vectors, i.e.:

The grid must be structured and Cartesian, and have cube elements only
The number of elements in each direction must match the number of knot spans in that direction
In fact, the element spacing in any direction must match the knot spacing in that direction (disregarding knot multiplicities)
When ordering the grid elements according to their indices, the resulting order must be lexicographical, with the x-index increasing fastest.

Unfortunately, not all of these conditions can be checked for automatically.

Parameters

knotVector	A single knot vector, which will be used for all coordinate directions
order	B-spline order, will be used for all coordinate directions
makeOpen	If this is true, then knots are prepended and appended to the knot vector to make the knot vector 'open'. i.e., start and end with 'order+1' identical knots. Basis functions from such knot vectors are interpolatory at the end of the parameter interval.

Todo:: maybe test whether the knot vector is already open?

References Dune::Hybrid::accumulate(), Dune::Functions::BSplinePreBasis< GV >::elements_, Dune::Functions::BSplinePreBasis< GV >::knotVectors_, Dune::Functions::BSplinePreBasis< GV >::order_, and Dune::TypeTree::push_back().

◆ BSplinePreBasis() [2/2]

template<typename GV >

Dune::Functions::BSplinePreBasis< GV >::BSplinePreBasis	(	const GridView &	gridView,
		const FieldVector< double, dim > &	lowerLeft,
		const FieldVector< double, dim > &	upperRight,
		const std::array< unsigned int, dim > &	elements,
		unsigned int	order,
		bool	makeOpen = `true`
	)

inline

Construct a B-spline basis for a given grid view with uniform knot vectors.

The grid must match the knot vectors, i.e.:

The grid must be structured and Cartesian, and have cube elements only
Bounding box and number of elements of the grid must match the corresponding arguments given to this constructor.
The element spacing must be uniform
When ordering the grid elements according to their indices, the resulting order must be lexicographical, with the x-index increasing fastest.

Unfortunately, not all of these conditions can be checked for automatically.

Parameters

gridView	The grid we are defining the basis on
lowerLeft	Lower left corner of the structured grid
upperRight	Upper right corner of the structured grid
elements	Number of elements in each coordinate direction
order	B-spline order, will be used for all coordinate directions
makeOpen	If this is true, then knots are prepended and appended to the knot vector to make the knot vector 'open'. i.e., start and end with 'order+1' identical knots. Basis functions from such knot vectors are interpolatory at the end of the parameter interval.

Todo:: maybe test whether the knot vector is already open?

References Dune::Hybrid::accumulate(), Dune::Functions::BSplinePreBasis< GV >::elements_, Dune::Functions::BSplinePreBasis< GV >::knotVectors_, Dune::Functions::BSplinePreBasis< GV >::order_, and Dune::TypeTree::push_back().

Member Function Documentation

◆ evaluateAll()

template<typename GV >

static void Dune::Functions::BSplinePreBasis< GV >::evaluateAll	(	const typename GV::ctype &	in,
		std::vector< R > &	out,
		bool	evaluateJacobian,
		std::vector< R > &	outJac,
		bool	evaluateHessian,
		std::vector< R > &	outHess,
		const std::vector< R > &	knotVector,
		unsigned int	order,
		unsigned int	currentKnotSpan
	)

inlinestatic

Evaluate the second derivatives of all one-dimensional B-spline functions for a given coordinate direction.

Parameters

	in	Scalar(!) coordinate where to evaluate the functions
	enableEvaluations	switches calculation of desired derivatives on
[out]	out	Vector containing the values of all B-spline derivatives at 'in'
[out]	outJac	Vector containing the first derivatives of all B-spline derivatives at 'in' (only if calculation was switched on by enableEvaluations)
[out]	outHess	Vector containing the second derivatives of all B-spline derivatives at 'in' (only if calculation was switched on by enableEvaluations)

References Dune::Functions::BSplinePreBasis< GV >::evaluateFunctionFull(), Dune::Functions::BSplinePreBasis< GV >::evaluateJacobian(), and Dune::DynamicMatrix< K >::resize().

Referenced by Dune::Functions::BSplinePreBasis< GV >::evaluate().

◆ evaluateFunction()

template<typename GV >

static void Dune::Functions::BSplinePreBasis< GV >::evaluateFunction	(	const typename GV::ctype &	in,
		std::vector< R > &	out,
		const std::vector< R > &	knotVector,
		unsigned int	order,
		unsigned int	currentKnotSpan
	)

inlinestatic

Evaluate all one-dimensional B-spline functions for a given coordinate direction.

This implementations was based on the explanations in the book of Cottrell, Hughes, Bazilevs, "Isogeometric Analysis"

Parameters

	in	Scalar(!) coordinate where to evaluate the functions
[out]	out	Vector containing the values of all B-spline functions at 'in'

Todo:: We only hand out function values for those basis functions whose support overlaps the current knot span. However, in the preceding loop we still computed all values_. This won't scale.

◆ evaluateFunctionFull()

template<typename GV >

static void Dune::Functions::BSplinePreBasis< GV >::evaluateFunctionFull	(	const typename GV::ctype &	in,
		DynamicMatrix< R > &	out,
		const std::vector< R > &	knotVector,
		unsigned int	order,
		unsigned int	currentKnotSpan
	)

inlinestatic

Evaluate all one-dimensional B-spline functions for a given coordinate direction.

This implementations was based on the explanations in the book of Cottrell, Hughes, Bazilevs, "Isogeometric Analysis"

Todo:: This method is a hack! I computes the derivatives of ALL B-splines, even the ones that are zero on the current knot span. I need it as an intermediate step to get the derivatives working. It will/must be removed as soon as possible.

Parameters

	in	Scalar(!) coordinate where to evaluate the functions
[out]	out	Vector containing the values of all B-spline functions at 'in'

References Dune::DynamicMatrix< K >::resize().

Referenced by Dune::Functions::BSplinePreBasis< GV >::evaluateAll(), and Dune::Functions::BSplinePreBasis< GV >::evaluateJacobian().

◆ evaluateJacobian()

template<typename GV >

void Dune::Functions::BSplinePreBasis< GV >::evaluateJacobian	(	const FieldVector< typename GV::ctype, dim > &	in,
		std::vector< FieldMatrix< R, 1, dim > > &	out,
		const std::array< unsigned, dim > &	currentKnotSpan
	)		const

inline

Evaluate Jacobian of all B-spline basis functions.

In theory this is easy: just look up the formula in a B-spline text of your choice. The challenge is compute only the values needed for the current knot span.

References Dune::Functions::BSplinePreBasis< GV >::evaluateFunctionFull(), Dune::Functions::BSplinePreBasis< GV >::knotVectors_, Dune::Functions::BSplinePreBasis< GV >::order_, and Dune::Functions::BSplinePreBasis< GV >::size().

Referenced by Dune::Functions::BSplinePreBasis< GV >::evaluateAll().

◆ getIJK()

template<typename GV >

static std::array< unsigned int, dim > Dune::Functions::BSplinePreBasis< GV >::getIJK	(	typename GridView::IndexSet::IndexType	idx,
		std::array< unsigned int, dim >	elements
	)

inlinestatic

Compute integer element coordinates from the element index.

Warning: This method makes strong assumptions about the grid, namely that it is structured, and that indices are given in a x-fastest fashion.

Referenced by Dune::Functions::BSplinePreBasis< GV >::indices().

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

dune/functions/functionspacebases/bsplinebasis.hh

DUNE Distributed and Unified Numerics Environment

Dune Core Modules (2.9.0)

Public Types

Public Member Functions

Static Public Member Functions

Public Attributes

Detailed Description

Constructor & Destructor Documentation

◆ BSplinePreBasis() [1/2]

◆ BSplinePreBasis() [2/2]

Member Function Documentation

◆ evaluateAll()

◆ evaluateFunction()

◆ evaluateFunctionFull()

◆ evaluateJacobian()

◆ getIJK()