Reference documentation for deal.II version Git 81120c2 2018-01-16 17:31:12 +0100
Public Member Functions | List of all members
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 > &&)=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 ()=default
Quadratureoperator= (const Quadrature< dim > &)
Quadratureoperator= (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
std::conditional< dim==1, std::array< Quadrature< 1 >, dim >, const std::array< Quadrature< 1 >, dim > & >::type get_tensor_basis () const
- Public Member Functions inherited from Subscriptor
 Subscriptor ()
 Subscriptor (const Subscriptor &)
 Subscriptor (Subscriptor &&)
virtual ~Subscriptor ()
Subscriptoroperator= (const Subscriptor &)
Subscriptoroperator= (Subscriptor &&)
void subscribe (const char *identifier=nullptr) const
void unsubscribe (const char *identifier=nullptr) const
unsigned int n_subscriptions () 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 >
typedef Quadrature< dim-1 > SubQuadrature
- Static Public Member Functions inherited from Subscriptor
static::ExceptionBase & ExcInUse (int arg1, char *arg2, std::string &arg3)
static::ExceptionBase & ExcNoSubscriber (char *arg1, char *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 orgin, 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.

Luca Heltai, Nicola Giuliani, 2017.

Definition at line 797 of file quadrature_lib.h.

Constructor & Destructor Documentation

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.

radial_quadratureBase quadrature to use in the radial direction
angular_quadratureBase quadrature to use in the angular direction

Definition at line 1437 of file

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.


Definition at line 1465 of file

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