A class for Gauss quadrature with weighting function. This formula can be used to integrate on the reference element , where is a smooth function without singularities, and is the distance from the point to the vertex , 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 . 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 , 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 , or simply , with the factor already included in the quadrature weights.
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.
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:
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:
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.
This function returns true if the quadrature object is a tensor product of one-dimensional formulas and the quadrature points are sorted lexicographically.
std::conditional< dim==1, std::array< Quadrature< 1 >, dim >, conststd::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,
std::array<Quadrature<1>, dim>,
const std::array<Quadrature<1>, dim> &>::type
The type is abbreviated in the online documentation to improve readability of this page.
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.