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

Inheritance diagram for Quadrature< dim >:

Public Types
using	SubQuadrature = Quadrature< dim==0 ? 0 :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 (std::vector< Point< dim > > &&points, 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

Subscriptor functionality
Classes derived from Subscriptor provide a facility to subscribe to this object. This is mostly used by the SmartPointer class.
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

Static Public Member Functions
static ::ExceptionBase &	ExcInUse (int arg1, std::string arg2, std::string arg3)

static ::ExceptionBase &	ExcNoSubscriber (std::string arg1, std::string arg2)

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

Private Types
using	map_value_type = decltype(counter_map)::value_type

using	map_iterator = decltype(counter_map)::iterator

Private Member Functions
void	check_no_subscribers () const noexcept

Private Attributes
std::atomic< unsigned int >	counter

std::map< std::string, unsigned int >	counter_map

std::vector< std::atomic< bool > * >	validity_pointers

const std::type_info *	object_info

Static Private Attributes
static std::mutex	mutex

Detailed Description

template<int dim>
class Quadrature< dim >

Base class for quadrature formulae in arbitrary dimensions. Quadrature is a means to approximate an integral by evaluating the integrand at specific points \(\mathbf x_q\) and summing the point values with specific weights \(w_q\); that is, quadrature computes

\begin{align*} \int_K f(\mathbf x) \; dx \approx \sum_{q=0,\ldots,Q-1} f(\mathbf x_q) w_q. \end{align*}

This class stores quadrature points \(\mathbf x_q\) and weights \(w_q\) for concrete "quadrature formulas" when \(K\) (the domain we integrate over) is a reference cell. That is, points and weights are expressed in the coordinate system of a reference cell (see the ReferenceCell class) and as such serves to represent quadrature points and weights on the unit line segment \([0,1]\) in 1d, on the unit square or unit triangle in 2d, as well as the unit tetrahedron, cube, pyramid, and wedge reference cells in 3d. Integration over concrete cells is done by coordinate transformation to the reference cell represented by the current class.

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.

At least for quadrilaterals and hexahedra (or, more precisely, since we work on reference cells: for the unit square and the unit cube), quadrature formulas are typically tensor products of one-dimensional formulas (see also 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 on the reference cell the quadrature formula corresponds to. 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+1)st order.

Tensor product quadrature

At least for hypercube reference cells (i.e., squares and cubes), 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.

Other uses of this class

Quadrature objects are used in a number of places within deal.II where integration is performed, most notably via the FEValues and related classes. Some of these classes are also used in contexts where no integrals are involved, but where functions need to be evaluated at specific points, for example to evaluate the solution at individual points or to create graphical output. Examples are the implementation of VectorTools::point_value() and the DataOut and related classes (in particular in connection with the DataPostprocessor class). In such contexts, one often creates specific "Quadrature" objects in which the "quadrature points" are simply the points (in the coordinate system of the reference cell) at which one wants to evaluate the solution. In these kinds of cases, the weights stored by the current class are not used and the name "quadrature object" is interpreted as "list of evaluation points".

Definition at line 122 of file quadrature.h.

Member Typedef Documentation

◆ SubQuadrature

template<int dim>

using Quadrature< dim >::SubQuadrature = Quadrature<dim == 0 ? 0 : 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. A sub quadrature of a 0-dimensional quadrature is defined as still being 0-dimensional.

Definition at line 130 of file quadrature.h.

◆ map_value_type

using Subscriptor::map_value_type = decltype(counter_map)::value_type

privateinherited

The data type used in counter_map.

Definition at line 230 of file subscriptor.h.

◆ map_iterator

using Subscriptor::map_iterator = decltype(counter_map)::iterator

privateinherited

The iterator type used in counter_map.

Definition at line 235 of file subscriptor.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 42 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 146 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.

◆ 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 297 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 317 of file quadrature.cc.

◆ initialize()

template<int dim>

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

std::size_t Quadrature< dim >::memory_consumption

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

Definition at line 326 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 using the BOOST serialization library.

◆ 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 >, conststd::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 338 of file quadrature.cc.

◆ subscribe()

void Subscriptor::subscribe	(	std::atomic< bool > *const	validity,
		const std::string &	identifier = `""`
	)		const

inherited

Subscribes a user of the object by storing the pointer validity. The subscriber may be identified by text supplied as identifier.

Definition at line 136 of file subscriptor.cc.

◆ unsubscribe()

void Subscriptor::unsubscribe	(	std::atomic< bool > *const	validity,
		const std::string &	identifier = `""`
	)		const

inherited

Unsubscribes a user from the object.

Note: The identifier and the validity pointer must be the same as the one supplied to subscribe().

Definition at line 156 of file subscriptor.cc.

◆ n_subscriptions()

unsigned int Subscriptor::n_subscriptions ( ) const

inlineinherited

Return the present number of subscriptions to this object. This allows to use this class for reference counted lifetime determination where the last one to unsubscribe also deletes the object.

Definition at line 300 of file subscriptor.h.

◆ list_subscribers() [1/2]

template<typename StreamType >

void Subscriptor::list_subscribers ( StreamType & stream ) const

inlineinherited

List the subscribers to the input stream.

Definition at line 317 of file subscriptor.h.

◆ list_subscribers() [2/2]

void Subscriptor::list_subscribers ( ) const

inherited

List the subscribers to deallog.

Definition at line 204 of file subscriptor.cc.

◆ check_no_subscribers()

void Subscriptor::check_no_subscribers ( ) const

privatenoexceptinherited

Check that there are no objects subscribing to this object. If this check passes then it is safe to destroy the current object. It this check fails then this function will either abort or print an error message to deallog (by using the AssertNothrow mechanism), but will not throw an exception.

Note: Since this function is just a consistency check it does nothing in release mode.; If this function is called when there is an uncaught exception then, rather than aborting, this function prints an error message to the standard error stream and returns.

The creator (and owner) of an object is counted in the map below if HE manages to supply identification.

We use the mutable keyword in order to allow subscription to constant objects also.

This counter may be read from and written to concurrently in multithreaded code: hence we use the std::atomic class template.

Definition at line 219 of file subscriptor.h.

◆ counter_map

std::map<std::string, unsigned int> Subscriptor::counter_map

mutableprivateinherited

In this map, we count subscriptions for each different identification string supplied to subscribe().

Definition at line 225 of file subscriptor.h.

◆ validity_pointers

std::vector<std::atomic<bool> *> Subscriptor::validity_pointers

mutableprivateinherited

In this vector, we store pointers to the validity bool in the SmartPointer objects that subscribe to this class.

Definition at line 241 of file subscriptor.h.

◆ object_info

const std::type_info* Subscriptor::object_info

mutableprivateinherited

Pointer to the typeinfo object of this object, from which we can later deduce the class name. Since this information on the derived class is neither available in the destructor, nor in the constructor, we obtain it in between and store it here.

Definition at line 249 of file subscriptor.h.

◆ mutex

std::mutex Subscriptor::mutex

staticprivateinherited

A mutex used to ensure data consistency when printing out the list of subscribers.

Definition at line 271 of file subscriptor.h.

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

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

Public Types

Public Member Functions

Static Public Member Functions

Protected Attributes

Private Types

Private Member Functions

Private Attributes

Static Private Attributes

Detailed Description

Mathematical background

Tensor product quadrature

Other uses of this class

Member Typedef Documentation

◆ SubQuadrature

◆ map_value_type

◆ map_iterator

Constructor & Destructor Documentation

◆ Quadrature() [1/9]

◆ Quadrature() [2/9]

◆ Quadrature() [3/9]

◆ Quadrature() [4/9]

◆ Quadrature() [5/9]

◆ Quadrature() [6/9]

◆ Quadrature() [7/9]

◆ Quadrature() [8/9]

◆ Quadrature() [9/9]

◆ ~Quadrature()

Member Function Documentation

◆ operator=() [1/2]

◆ operator=() [2/2]

◆ operator==()

◆ initialize()

◆ size()

◆ point()

◆ get_points()

◆ weight()

◆ get_weights()

◆ memory_consumption()

◆ serialize()

◆ is_tensor_product()

◆ get_tensor_basis()

◆ subscribe()

◆ unsubscribe()

◆ n_subscriptions()

◆ list_subscribers() [1/2]

◆ list_subscribers() [2/2]

◆ check_no_subscribers()

Member Data Documentation

◆ quadrature_points

◆ weights

◆ is_tensor_product_flag

◆ tensor_basis

◆ counter

◆ counter_map

◆ validity_pointers

◆ object_info

◆ mutex