Reference documentation for deal.II version 9.3.0
Quadrature< dim > Class Template Reference

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

Inheritance diagram for Quadrature< dim >:
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## Public Member Functions

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

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

Quadrature (const Point< dim > &point)

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

template<>

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 ::ExceptionBaseExcInUse (int arg1, std::string arg2, std::string arg3)

static ::ExceptionBaseExcNoSubscriber (std::string arg1, std::string arg2)

## Protected Attributes

std::vector< Point< dim > > quadrature_points

std::vector< doubleweights

bool is_tensor_product_flag

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

## 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 Ndim 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).

Definition at line 83 of file quadrature.h.

## Member Typedef Documentation

template<int dim>

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 90 of file quadrature.h.

## Constructor & Destructor Documentation

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.

template<int dim>

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 130 of file quadrature.cc.

template<int dim>
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 229 of file quadrature.cc.

template<int dim>

Copy constructor.

Definition at line 267 of file quadrature.cc.

template<int dim>
defaultnoexcept

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

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.

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.

template<int dim>

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

Definition at line 86 of file quadrature.cc.

template<int dim>
overridevirtualdefault

Virtual destructor.

template<>

Definition at line 31 of file quadrature.cc.

## ◆ operator=() [1/2]

template<int dim>

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

Definition at line 282 of file quadrature.cc.

## ◆ operator=() [2/2]

template<int 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 302 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

## ◆ point()

template<int dim>
 const Point& Quadrature< dim >::point ( const unsigned int i ) const

Return the ith quadrature point.

## ◆ get_points()

template<int dim>
 const std::vector >& 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& 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 ( ) const

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

Definition at line 311 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 >, const std::array< Quadrature< 1 >, dim > & >::type Quadrature< dim >::get_tensor_basis ( ) const

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,

Definition at line 323 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 301 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 318 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.

## Member Data Documentation

template<int dim>
protected

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

Definition at line 283 of file quadrature.h.

## ◆ weights

template<int dim>
protected

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

Definition at line 289 of file quadrature.h.

## ◆ is_tensor_product_flag

template<int dim>
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 298 of file quadrature.h.

## ◆ tensor_basis

template<int dim>
 std::unique_ptr, 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 304 of file quadrature.h.

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