Reference documentation for deal.II version 9.6.0
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QTrianglePolar Class Reference

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

Inheritance diagram for QTrianglePolar:

## Public Member Functions

QTrianglePolar (const unsigned int n)

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

## Detailed Description

A quadrature that implements a polar transformation from a square to a triangle to integrate singularities in the origin of the reference simplex. The quadrature is obtained through the following polar transformation:

$\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} \frac{\hat x}{\sin(\theta)+\cos(\theta)} cos(\theta) \\ \frac{\hat x}{\sin(\theta)+\cos(\theta)} sin(\theta) \end{pmatrix} \qquad \theta \dealcoloneq \frac\pi 2 \hat y$

Definition at line 748 of file quadrature_lib.h.

## ◆ QTrianglePolar() [1/2]

Construct a QTrianglePolar quadrature, with different formulas in the radial and angular directions.

Parameters

Definition at line 1534 of file quadrature_lib.cc.

## ◆ QTrianglePolar() [2/2]

 QTrianglePolar::QTrianglePolar ( const unsigned int n )

Call the other constructor, with QGauss<1>(n) for both radial and angular quadrature.

Parameters

Definition at line 1564 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 709 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 726 of file quadrature_lib.cc.

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