QDuffy Class Reference

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

Inheritance diagram for QDuffy:

Public Member Functions
	QDuffy (const Quadrature< 1 > &radial_quadrature, const Quadrature< 1 > &angular_quadrature, const double beta=1.0)
	QDuffy (const unsigned int n, const double beta)
Public Member Functions inherited from QSimplex< 2 >
	QSimplex (const Quadrature< dim > &quad)
Quadrature< dim >	compute_affine_transformation (const std::array< Point< dim >, dim+1 > &vertices) const
Public Member Functions inherited from Quadrature< dim >
	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
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)

Additional Inherited Members
Public Types inherited from Quadrature< dim >
using	SubQuadrature = Quadrature< dim - 1 >
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)
Protected Attributes inherited from Quadrature< dim >
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

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.

Author

Luca Heltai, Nicola Giuliani, 2017.

Definition at line 726 of file quadrature_lib.h.

Constructor & Destructor Documentation

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

radial_quadrature	Base quadrature to use in the radial direction
angular_quadrature	Base quadrature to use in the angular direction
beta	Exponent used in the transformation

Definition at line 1284 of file quadrature_lib.cc.

QDuffy() [2/2]

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

Call the above constructor with QGauss<1>(n) quadrature formulas for both the radial and angular quadratures.

Parameters

n	Order of QGauss quadrature
beta	Exponent used in the transformation

Definition at line 1312 of file quadrature_lib.cc.

---

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

• deal.II/base/quadrature_lib.h
• source/base/quadrature_lib.cc