Quadrature< dim > Class Template Reference

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

Inheritance diagram for Quadrature< dim >:

Public Types
using	SubQuadrature = Quadrature< dim - 1 >

Public Member Functions
	Quadrature (const unsigned int n_quadrature_points=0)
	Quadrature (const SubQuadrature &, const Quadrature< 1 > &)
	Quadrature (const Quadrature< dim !=1 ? 1 :0 > &quadrature_1d)
	Quadrature (const Quadrature< dim > &q)
	Quadrature (Quadrature< dim > &&) noexcept=default
	Quadrature (const std::vector< Point< dim >> &points, const std::vector< double > &weights)
	Quadrature (const std::vector< Point< dim >> &points)
	Quadrature (const Point< dim > &point)
virtual	~Quadrature () override=default
Quadrature &	operator= (const Quadrature< dim > &)
Quadrature &	operator= (Quadrature< dim > &&)=default
bool	operator== (const Quadrature< dim > &p) const
void	initialize (const std::vector< Point< dim >> &points, const std::vector< double > &weights)
unsigned int	size () const
const Point< dim > &	point (const unsigned int i) const
const std::vector< Point< dim > > &	get_points () const
double	weight (const unsigned int i) const
const std::vector< double > &	get_weights () const
std::size_t	memory_consumption () const
template<class Archive >
void	serialize (Archive &ar, const unsigned int version)
bool	is_tensor_product () const
const std::array< Quadrature< 1 >, dim > &	get_tensor_basis () const
	Quadrature (const unsigned int n_q)
Public Member Functions inherited from Subscriptor
	Subscriptor ()
	Subscriptor (const Subscriptor &)
	Subscriptor (Subscriptor &&) noexcept
virtual	~Subscriptor ()
Subscriptor &	operator= (const Subscriptor &)
Subscriptor &	operator= (Subscriptor &&) noexcept
void	subscribe (std::atomic< bool > *const validity, const std::string &identifier="") const
void	unsubscribe (std::atomic< bool > *const validity, const std::string &identifier="") const
unsigned int	n_subscriptions () const
template<typename StreamType >
void	list_subscribers (StreamType &stream) const
void	list_subscribers () const
template<class Archive >
void	serialize (Archive &ar, const unsigned int version)

Protected Attributes
std::vector< Point< dim > >	quadrature_points
std::vector< double >	weights
bool	is_tensor_product_flag
std::unique_ptr< std::array< Quadrature< 1 >, dim > >	tensor_basis

Additional Inherited Members
Static Public Member Functions inherited from Subscriptor
static ::ExceptionBase &	ExcInUse (int arg1, std::string arg2, std::string arg3)
static ::ExceptionBase &	ExcNoSubscriber (std::string arg1, std::string arg2)

Detailed Description

template<int dim>
class Quadrature< dim >

Base class for quadrature formulae in arbitrary dimensions. This class stores quadrature points and weights on the unit line [0,1], unit square [0,1]x[0,1], etc.

There are a number of derived classes, denoting concrete integration formulae. Their names are prefixed by Q. Refer to the list of derived classes for more details.

The schemes for higher dimensions are typically tensor products of the one- dimensional formulae, but refer to the section on implementation detail below.

In order to allow for dimension independent programming, a quadrature formula of dimension zero exists. Since an integral over zero dimensions is the evaluation at a single point, any constructor of such a formula initializes to a single quadrature point with weight one. Access to the weight is possible, while access to the quadrature point is not permitted, since a Point of dimension zero contains no information. The main purpose of these formulae is their use in QProjector, which will create a useful formula of dimension one out of them.

Mathematical background

For each quadrature formula we denote by m, the maximal degree of polynomials integrated exactly. This number is given in the documentation of each formula. The order of the integration error is m+1, that is, the error is the size of the cell to the m+1 by the Bramble- Hilbert Lemma. The number m is to be found in the documentation of each concrete formula. For the optimal formulae QGauss we have \(m = 2N-1\), where N is the constructor parameter to QGauss. The tensor product formulae are exact on tensor product polynomials of degree m in each space direction, but they are still only of m+1st order.

Implementation details

Most integration formulae in more than one space dimension are tensor products of quadrature formulae in one space dimension, or more generally the tensor product of a formula in (dim-1) dimensions and one in one dimension. There is a special constructor to generate a quadrature formula from two others. For example, the QGauss<dim> formulae include N^dim quadrature points in dim dimensions, where N is the constructor parameter of QGauss.

Note

Instantiations for this template are provided for dimensions 0, 1, 2, and 3 (see the section on Template instantiations).

Author

Wolfgang Bangerth, Guido Kanschat, 1998, 1999, 2000, 2005, 2009

Definition at line 85 of file quadrature.h.

Member Typedef Documentation

SubQuadrature

template<int dim>

using Quadrature< dim >::SubQuadrature = Quadrature<dim - 1>

Define an alias for a quadrature that acts on an object of one dimension less. For cells, this would then be a face quadrature.

Definition at line 92 of file quadrature.h.

Constructor & Destructor Documentation

Quadrature() [1/9]

template<int dim>

Quadrature< dim >::Quadrature ( const unsigned int  n_quadrature_points = 0 )

explicit

Constructor.

This constructor is marked as explicit to avoid involuntary accidents like in hp::QCollection<dim> q_collection(3) where hp::QCollection<dim> q_collection(QGauss<dim>(3)) was meant.

Definition at line 40 of file quadrature.cc.

Quadrature() [2/9]

template<int dim>

Quadrature< dim >::Quadrature	(	const SubQuadrature< dim > &	q1,
		const Quadrature< 1 > &	q2
	)

Build this quadrature formula as the tensor product of a formula in a dimension one less than the present and a formula in one dimension. This constructor assumes (and tests) that constant functions are integrated exactly, i.e. the sum of the quadrature weights is one.

SubQuadrature<dim>::type expands to Quadrature<dim-1>.

Definition at line 120 of file quadrature.cc.

Quadrature() [3/9]

template<int dim>

Quadrature< dim >::Quadrature ( const Quadrature< dim !=1 ? 1 :0 > &  quadrature_1d )

explicit

Build this quadrature formula as the dim-fold tensor product of a formula in one dimension.

Assuming that the points in the one-dimensional rule are in ascending order, the points of the resulting rule are ordered lexicographically with x running fastest.

In order to avoid a conflict with the copy constructor in 1d, we let the argument be a 0d quadrature formula for dim==1, and a 1d quadrature formula for all other space dimensions.

This constructor does not require that constant functions are integrated exactly. Therefore, it is appropriate if the one-dimensional formula is defined with respect to a weighting function.

Definition at line 224 of file quadrature.cc.

Quadrature() [4/9]

template<int dim>

Quadrature< dim >::Quadrature

(

const Quadrature< dim > &

q

)

Copy constructor.

Definition at line 262 of file quadrature.cc.

Quadrature() [5/9]

template<int dim>

Quadrature< dim >::Quadrature ( Quadrature< dim > &&  )

defaultnoexcept

Move constructor. Construct a new quadrature object by transferring the internal data of another quadrature object.

Quadrature() [6/9]

template<int dim>

Quadrature< dim >::Quadrature	(	const std::vector< Point< dim >> &	points,
		const std::vector< double > &	weights
	)

Construct a quadrature formula from given vectors of quadrature points (which should really be in the unit cell) and the corresponding weights. You will want to have the weights sum up to one, but this is not checked.

Definition at line 61 of file quadrature.cc.

Quadrature() [7/9]

template<int dim>

Quadrature< dim >::Quadrature

(

const std::vector< Point< dim >> &

points

)

Construct a dummy quadrature formula from a list of points, with weights set to infinity. The resulting object is therefore not meant to actually perform integrations, but rather to be used with FEValues objects in order to find the position of some points (the quadrature points in this object) on the transformed cell in real space.

Definition at line 74 of file quadrature.cc.

Quadrature() [8/9]

template<int dim>

Quadrature< dim >::Quadrature

(

const Point< dim > &

point

)

Constructor for a one-point quadrature. Sets the weight of this point to one.

Definition at line 86 of file quadrature.cc.

~Quadrature()

template<int dim>

virtual Quadrature< dim >::~Quadrature ( )

overridevirtualdefault

Virtual destructor.

Quadrature() [9/9]

Quadrature< 0 >::Quadrature

(

const unsigned int

n_q

)

Definition at line 31 of file quadrature.cc.

Member Function Documentation

operator=() [1/2]

template<int dim>

Quadrature< dim > & Quadrature< dim >::operator=

(

const Quadrature< dim > &

q

)

Assignment operator. Copies contents of weights and quadrature_points as well as size.

Definition at line 277 of file quadrature.cc.

operator=() [2/2]

template<int dim>

Quadrature& Quadrature< dim >::operator= ( Quadrature< dim > &&  )

default

Move assignment operator. Moves all data from another quadrature object to this object.

operator==()

template<int dim>

bool Quadrature< dim >::operator==

(

const Quadrature< dim > &

p

)

const

Test for equality of two quadratures.

Definition at line 297 of file quadrature.cc.

initialize()

template<int dim>

void Quadrature< dim >::initialize	(	const std::vector< Point< dim >> &	points,
		const std::vector< double > &	weights
	)

Set the quadrature points and weights to the values provided in the arguments.

Definition at line 50 of file quadrature.cc.

size()

template<int dim>

unsigned int Quadrature< dim >::size

(

)

const

Number of quadrature points.

point()

template<int dim>

const Point<dim>& Quadrature< dim >::point

(

const unsigned int

i

)

const

Return the ith quadrature point.

get_points()

template<int dim>

const std::vector<Point<dim> >& Quadrature< dim >::get_points

(

)

const

Return a reference to the whole array of quadrature points.

weight()

template<int dim>

double Quadrature< dim >::weight

(

const unsigned int

i

)

const

Return the weight of the ith quadrature point.

get_weights()

template<int dim>

const std::vector<double>& Quadrature< dim >::get_weights

(

)

const

Return a reference to the whole array of weights.

memory_consumption()

template<int dim>

std::size_t Quadrature< dim >::memory_consumption

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

Definition at line 306 of file quadrature.cc.

serialize()

template<int dim>

template<class Archive >

void Quadrature< dim >::serialize	(	Archive &	ar,
		const unsigned int	version
	)

Write or read the data of this object to or from a stream for the purpose of serialization.

is_tensor_product()

template<int dim>

bool Quadrature< dim >::is_tensor_product

(

)

const

This function returns true if the quadrature object is a tensor product of one-dimensional formulas and the quadrature points are sorted lexicographically.

get_tensor_basis()

template<int dim>

std::conditional< dim==1, std::array< Quadrature< 1 >, dim >, const std::array< Quadrature< 1 >, dim > & >::type Quadrature< dim >::get_tensor_basis

In case the quadrature formula is a tensor product, this function returns the dim one-dimensional basis objects. Otherwise, calling this function is not allowed.

For dim equal to one, we can not return the std::array as a const reference and have to return it by value. In this case, the array will always contain a single element (this).

Note

The actual return type of this function is

std::conditional<dim == 1,
                 std::array<Quadrature<1>, dim>,
                 const std::array<Quadrature<1>, dim> &>::type

The type is abbreviated in the online documentation to improve readability of this page.

Definition at line 318 of file quadrature.cc.

Member Data Documentation

quadrature_points

template<int dim>

std::vector<Point<dim> > Quadrature< dim >::quadrature_points

protected

List of quadrature points. To be filled by the constructors of derived classes.

Definition at line 284 of file quadrature.h.

weights

template<int dim>

std::vector<double> Quadrature< dim >::weights

protected

List of weights of the quadrature points. To be filled by the constructors of derived classes.

Definition at line 290 of file quadrature.h.

is_tensor_product_flag

template<int dim>

bool Quadrature< dim >::is_tensor_product_flag

protected

Indicates if this object represents quadrature formula that is a tensor product of one-dimensional formulas. This flag is set if dim==1 or the constructors taking a Quadrature<1> (and possibly a Quadrature<dim-1> object) is called. This implies that the quadrature points are sorted lexicographically.

Definition at line 299 of file quadrature.h.

tensor_basis

template<int dim>

std::unique_ptr<std::array<Quadrature<1>, dim> > Quadrature< dim >::tensor_basis

protected

Stores the one-dimensional tensor basis objects in case this object can be represented by a tensor product.

Definition at line 305 of file quadrature.h.

---

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

• include/deal.II/base/quadrature.h
• source/base/quadrature.cc