Reference documentation for deal.II version GIT b8135fa6eb 2023-03-25 15:55:02+00:00

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

Inheritance diagram for QDuffy:
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## Public Member Functions

QDuffy (const unsigned int n, const double beta)

Quadrature< spacedim > compute_affine_transformation (const std::array< Point< spacedim >, dim+1 > &vertices) const

Quadrature< spacedim > mapped_quadrature (const std::vector< std::array< Point< spacedim >, dim+1 >> &simplices) const

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 ::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

## 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

A quadrature that implements the Duffy transformation from a square to a triangle to integrate singularities in the origin of the reference simplex.

The Duffy transformation is defined as

$\begin{pmatrix} x\\ y \end{pmatrix} = \begin{pmatrix} \hat x^\beta (1-\hat y)\\ \hat x^\beta \hat y \end{pmatrix}$

with determinant of the Jacobian equal to $$J= \beta \hat x^{2\beta-1}$$. Such transformation maps the reference square $$[0,1]\times[0,1]$$ to the reference simplex, by collapsing the left side of the square and squeezing quadrature points towards the origin, and then shearing the resulting triangle to the reference one. This transformation shows good convergence properties when $$\beta = 1$$ with singularities of order $$1/R$$ in the origin, but different $$\beta$$ values can be selected to increase convergence and/or accuracy when higher order Gauss rules are used (see "Generalized Duffy transformation for integrating vertex singularities", S. E. Mousavi, N. Sukumar, Computational Mechanics 2009).

When $$\beta = 1$$, this transformation is also known as the Lachat-Watson transformation.

Definition at line 736 of file quadrature_lib.h.

## Member Typedef Documentation

template<int dim>
inherited

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.

## ◆ QDuffy() [1/2]

 QDuffy::QDuffy ( const Quadrature< 1 > & radial_quadrature, const Quadrature< 1 > & angular_quadrature, const double beta = 1.0 )

Constructor that allows the specification of different quadrature rules along the "radial" and "angular" directions.

Since this quadrature is not based on a Polar change of coordinates, it is not fully proper to talk about radial and angular directions. However, the effect of the Duffy transformation is similar to a polar change of coordinates, since the resulting quadrature points are aligned radially with respect to the singularity.

Parameters

Definition at line 1328 of file quadrature_lib.cc.

## ◆ QDuffy() [2/2]

 QDuffy::QDuffy ( const unsigned int n, const double beta )

Parameters
 n Order of QGauss quadrature beta Exponent used in the transformation

Definition at line 1356 of file quadrature_lib.cc.

## ◆ compute_affine_transformation()

 Quadrature< spacedim > QSimplex< dim >::compute_affine_transformation ( const std::array< Point< spacedim >, dim+1 > & vertices ) const
inherited

Return an affine transformation of this quadrature, that can be used to integrate on the simplex identified by vertices.

Both the quadrature point locations and the weights are transformed, so that you can effectively use the resulting quadrature to integrate on the simplex.

The transformation is defined as

$x = v_0 + B \hat x$

where the matrix $$B$$ is given by $$B_{ij} = v[j][i]-v[0][i]$$.

The weights are scaled with the absolute value of the determinant of $$B$$, that is $$J \dealcoloneq |\text{det}(B)|$$. If $$J$$ is zero, an empty quadrature is returned. This may happen, in two dimensions, if the three vertices are aligned, or in three dimensions if the four vertices are on the same plane. The present function works also in the codimension one and codimension two case. For instance, when dim=2 and spacedim=3, we can map the quadrature points so that they live on the physical triangle embedded in the three dimensional space. In such a case, the matrix $$B$$ is not square anymore.

Parameters
 [in] vertices The vertices of the simplex you wish to integrate on
Returns
A quadrature object that can be used to integrate on the simplex

Definition at line 643 of file quadrature_lib.cc.

 Quadrature< spacedim > QSimplex< dim >::mapped_quadrature ( const std::vector< std::array< Point< spacedim >, dim+1 >> & simplices ) const
inherited

Given a collection of simplices, this function creates a global quadrature rule on the area covered by the simplices, by mapping the current quadrature on each simplex. A simplex is identified by its vertices, which are stored into an array of Points. Hence, this function can provide quadrature rules on polygons (or polyhedra), as they can be split into simplices.

Parameters
 simplices A std::vector where each entry is an array of dim+1 points, which identifies the vertices of a simplex.
Returns
A Quadrature object on the cell.

Definition at line 660 of file quadrature_lib.cc.

## ◆ operator==()

template<int dim>
 bool Quadrature< dim >::operator== ( const Quadrature< dim > & p ) const
inherited

Test for equality of two quadratures.

Definition at line 304 of file quadrature.cc.

## ◆ initialize()

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

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

Definition at line 52 of file quadrature.cc.

## ◆ size()

template<int dim>
 unsigned int Quadrature< dim >::size ( ) const
inherited

## ◆ point()

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

Return the ith quadrature point.

## ◆ get_points()

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

Return a reference to the whole array of quadrature points.

## ◆ weight()

template<int dim>
 double Quadrature< dim >::weight ( const unsigned int i ) const
inherited

Return the weight of the ith quadrature point.

## ◆ get_weights()

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

Return a reference to the whole array of weights.

## ◆ memory_consumption()

template<int dim>
inherited

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

Definition at line 313 of file quadrature.cc.

## ◆ serialize()

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

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
inherited

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
inherited

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 325 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.

Definition at line 53 of file subscriptor.cc.

## Member Data Documentation

template<int dim>
protectedinherited

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

Definition at line 326 of file quadrature.h.

## ◆ weights

template<int dim>
protectedinherited

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

Definition at line 332 of file quadrature.h.

## ◆ is_tensor_product_flag

template<int dim>
protectedinherited

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 341 of file quadrature.h.

## ◆ tensor_basis

template<int dim>
 std::unique_ptr, dim> > Quadrature< dim >::tensor_basis
protectedinherited

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

Definition at line 347 of file quadrature.h.

## ◆ counter

 std::atomic Subscriptor::counter
mutableprivateinherited

Store the number of objects which subscribed to this object. Initially, this number is zero, and upon destruction it shall be zero again (i.e. all objects which subscribed should have unsubscribed again).

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 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 *> 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: