Reference documentation for deal.II version Git 0b93389 2014-10-27 10:59:35 -0500
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Public Member Functions | Static Private Member Functions | List of all members
QGaussOneOverR< dim > Class Template Reference

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

Inheritance diagram for QGaussOneOverR< dim >:
[legend]

Public Member Functions

 QGaussOneOverR (const unsigned int n, const Point< dim > singularity, const bool factor_out_singular_weight=false)
 
 QGaussOneOverR (const unsigned int n, const unsigned int vertex_index, const bool factor_out_singular_weight=false)
 
template<>
 QGaussOneOverR (const unsigned int n, const unsigned int index, const bool flag)
 
- 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 (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 ()
 
Quadratureoperator= (const Quadrature< dim > &)
 
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)
 
- Public Member Functions inherited from Subscriptor
 Subscriptor ()
 
 Subscriptor (const Subscriptor &)
 
virtual ~Subscriptor ()
 
Subscriptoroperator= (const Subscriptor &)
 
void subscribe (const char *identifier=0) const
 
void unsubscribe (const char *identifier=0) const
 
unsigned int n_subscriptions () const
 
void list_subscribers () const
 
 DeclException3 (ExcInUse, int, char *, std::string &,<< "Object of class "<< arg2<< " is still used by "<< arg1<< " other objects.\n"<< "(Additional information: "<< arg3<< ")\n"<< "Note the entry in the Frequently Asked Questions of "<< "deal.II (linked to from http://www.dealii.org/) for "<< "more information on what this error means.")
 
 DeclException2 (ExcNoSubscriber, char *, char *,<< "No subscriber with identifier \""<< arg2<< "\" did subscribe to this object of class "<< arg1)
 
template<class Archive >
void serialize (Archive &ar, const unsigned int version)
 

Static Private Member Functions

static unsigned int quad_size (const Point< dim > singularity, const unsigned int n)
 

Additional Inherited Members

- Public Types inherited from Quadrature< dim >
typedef Quadrature< dim-1 > SubQuadrature
 
- Protected Attributes inherited from Quadrature< dim >
std::vector< Point< dim > > quadrature_points
 
std::vector< doubleweights
 

Detailed Description

template<int dim>
class QGaussOneOverR< dim >

A class for Gauss quadrature with \(1/R\) weighting function. This formula can be used to to integrate \(1/R \ f(x)\) on the reference element \([0,1]^2\), where \(f\) is a smooth function without singularities, and \(R\) is the distance from the point \(x\) to the vertex \(\xi\), given at construction time by specifying its index. Notice that this distance is evaluated in the reference element.

This quadrature formula is obtained from two QGauss quadrature formulas, upon transforming them into polar coordinate system centered at the singularity, and then again into another reference element. This allows for the singularity to be cancelled by part of the Jacobian of the transformation, which contains \(R\). In practice the reference element is transformed into a triangle by collapsing one of the sides adjacent to the singularity. The Jacobian of this transformation contains \(R\), which is removed before scaling the original quadrature, and this process is repeated for the next half element.

Upon construction it is possible to specify whether we want the singularity removed, or not. In other words, this quadrature can be used to integrate \(g(x) = 1/R\ f(x)\), or simply \(f(x)\), with the \(1/R\) factor already included in the quadrature weights.

Definition at line 369 of file quadrature_lib.h.

Constructor & Destructor Documentation

template<int dim>
QGaussOneOverR< dim >::QGaussOneOverR ( const unsigned int  n,
const Point< dim >  singularity,
const bool  factor_out_singular_weight = false 
)

This constructor takes three arguments: the order of the Gauss formula, the point of the reference element in which the singularity is located, and whether we include the weighting singular function inside the quadrature, or we leave it in the user function to be integrated.

Traditionally, quadrature formulas include their weighting function, and the last argument is set to false by default. There are cases, however, where this is undesirable (for example when you only know that your singularity has the same order of 1/R, but cannot be written exactly in this way).

In other words, you can use this function in either of the following way, obtaining the same result:

QGaussOneOverR singular_quad(order, q_point, false);
// This will produce the integral of f(x)/R
for(unsigned int i=0; i<singular_quad.size(); ++i)
integral += f(singular_quad.point(i))*singular_quad.weight(i);
// And the same here
QGaussOneOverR singular_quad_noR(order, q_point, true);
// This also will produce the integral of f(x)/R, but 1/R has to
// be specified.
for(unsigned int i=0; i<singular_quad.size(); ++i) {
double R = (singular_quad_noR.point(i)-cell->vertex(vertex_id)).norm();
integral += f(singular_quad_noR.point(i))*singular_quad_noR.weight(i)/R;
}
template<int dim>
QGaussOneOverR< dim >::QGaussOneOverR ( const unsigned int  n,
const unsigned int  vertex_index,
const bool  factor_out_singular_weight = false 
)

The constructor takes three arguments: the order of the Gauss formula, the index of the vertex where the singularity is located, and whether we include the weighting singular function inside the quadrature, or we leave it in the user function to be integrated. Notice that this is a specialized version of the previous constructor which works only for the vertices of the quadrilateral.

Traditionally, quadrature formulas include their weighting function, and the last argument is set to false by default. There are cases, however, where this is undesirable (for example when you only know that your singularity has the same order of 1/R, but cannot be written exactly in this way).

In other words, you can use this function in either of the following way, obtaining the same result:

QGaussOneOverR singular_quad(order, vertex_id, false);
// This will produce the integral of f(x)/R
for(unsigned int i=0; i<singular_quad.size(); ++i)
integral += f(singular_quad.point(i))*singular_quad.weight(i);
// And the same here
QGaussOneOverR singular_quad_noR(order, vertex_id, true);
// This also will produce the integral of f(x)/R, but 1/R has to
// be specified.
for(unsigned int i=0; i<singular_quad.size(); ++i) {
double R = (singular_quad_noR.point(i)-cell->vertex(vertex_id)).norm();
integral += f(singular_quad_noR.point(i))*singular_quad_noR.weight(i)/R;
}

Member Function Documentation

template<int dim>
static unsigned int QGaussOneOverR< dim >::quad_size ( const Point< dim >  singularity,
const unsigned int  n 
)
staticprivate

Given a quadrature point and a degree n, this function returns the size of the singular quadrature rule, considering whether the point is inside the cell, on an edge of the cell, or on a corner of the cell.


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