Reference documentation for deal.II version Git f0919993dd 2020-09-21 18:25:06 -0600
QSimplex< dim > Class Template Reference

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

Inheritance diagram for QSimplex< dim >:
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## Public Member Functions

Quadrature< dim > compute_affine_transformation (const std::array< Point< dim >, dim+1 > &vertices) const

Public Member Functions inherited from Quadrature< dim >

Quadrature (const std::vector< Point< dim >> &points, const std::vector< double > &weights)

Quadrature (const std::vector< Point< dim >> &points)

Quadrature (const Point< dim > &point)

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

template<>

Public Member Functions inherited from Subscriptor
Subscriptor ()

Subscriptor (const Subscriptor &)

Subscriptor (Subscriptor &&) noexcept

virtual ~Subscriptor ()

Subscriptoroperator= (const Subscriptor &)

Subscriptoroperator= (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)

Public Types inherited from Quadrature< dim >

Static Public Member Functions inherited from Subscriptor
static ::ExceptionBaseExcInUse (int arg1, std::string arg2, std::string arg3)

static ::ExceptionBaseExcNoSubscriber (std::string arg1, std::string arg2)

Protected Attributes inherited from Quadrature< dim >
std::vector< Point< dim > > quadrature_points

std::vector< doubleweights

bool is_tensor_product_flag

std::unique_ptr< std::array< Quadrature< 1 >, dim > > tensor_basis

## Detailed Description

### template<int dim> class QSimplex< dim >

Given an arbitrary quadrature formula, return one that chops the quadrature points above the hyper-plane defined by $$\sum_i x_i = 1$$. In other words, it extracts those quadrature points from the base formula that satisfy $$\sum_i (\mathbf x_q)_i \le 1+10^{-12}$$."

In general the resulting quadrature is not very useful, unless the quadrature you started from has been constructed specifically to integrate over triangles or tetrahedra. This class only ensures that the resulting quadrature formula only has quadrature points in the reference simplex or on its boundary.

No transformation is applied to the weights, and the weights referring to points that live outside the reference simplex are simply discarded.

The main use of this quadrature formula is not to chop tensor product quadratures. Ideally you should pass to this class a quadrature formula constructed directly using points and weights in the reference simplex, capable of integrating on triangles or tetrahedra.

For finite elements based on quadrilaterals and hexahedra, a QSimplex quadrature formula is not very useful on its own. This class is typically used in conjunction with other classes, like QSplit, to patch the reference element using several QSimplex quadrature formulas.

Such quadrature formulas are useful to integrate functions with singularities at certain points, or functions that present jumps along a co-dimension one surface inside the reference element, like in the extended finite element method (XFEM).

Definition at line 595 of file quadrature_lib.h.

## ◆ QSimplex()

template<int dim>

Construct a quadrature that only contains the points that are in the lower left reference simplex.

Parameters

Definition at line 1198 of file quadrature_lib.cc.

## ◆ compute_affine_transformation()

template<int dim>
 Quadrature< dim > QSimplex< dim >::compute_affine_transformation ( const std::array< Point< dim >, dim+1 > & vertices ) const

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.

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 1220 of file quadrature_lib.cc.

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