Class representing the Generalized Duffy quadrature rule in 2d, $x = u^\beta, y = u^\beta v$ , that is for $\beta=1$ the standard duffy integration rule. More...

#include <quadRule.hh>

Inheritance diagram for concepts::QuadratureRule2dQuadDuffy:

Public Member Functions
virtual const Real *	abscissas (uint i) const
	Returns the quadrature abcissas in the i-th direction. More...

virtual const bool	domain () const
	Method delivers if the integration points and weights are computed in [-1,1]^2 or in [0,1]^2. More...

const uint	n (uint i=0) const
	Returns the number of quadrature points. More...

void	printData () const

virtual bool	quadratureData (const uint i, Real3d &q, Real &w) const
	Method delivers the i-th quadrature point. More...

	QuadratureRule2dQuadDuffy (Real beta, uint nx, uint ny, uint noVtx=0)
	Constructor. More...

virtual const bool	tensor () const
	Method delivers information about the quadrature rule structure. More...

virtual const Real *	weights (uint i=0) const
	Returns the i-th weight, belonging to (x_i,y_i) as nontensored. More...

virtual	~QuadratureRule2dQuadDuffy ()

Protected Member Functions
virtual std::ostream &	info (std::ostream &os) const
	Returns information in an output stream. More...

Protected Attributes
const Real *	intX_
	Abscissas. More...

const Real *	intY_

Private Member Functions
void	compute_ (const Real2d S, const Real2d B, const Real2d C, const Real beta, const uint nx, const uint ny, Real &abscXP, Real &abscYP, Real *&wP)
	Routine computes generalized duffy quadrature points on the triangle conv(S,B,C), where S is the coordinate of the singularity vertex. More...

Private Attributes
Real	beta_

uint	n_

const Real *	weights_

Detailed Description

Class representing the Generalized Duffy quadrature rule in 2d, $x = u^\beta, y = u^\beta v$ , that is for $\beta=1$ the standard duffy integration rule.

The quadrature rule is not tensored and is most effective to integrate $1/r^\alpha$ singularities, for $0<\alpha < 2$ .

The weights are given in .

This class is motivated by the paper "Generalized Duffy Transformation for Integrating Vertex Singularites" by Mousavi / Sukumar

The quadrature rule depends on the duffy parameter beta. beta should be chosen such that

$2\beta -1 -\alpha\beta \geq 0$

.

Note that for too big beta, the quadrature points in become very close to the origin. This should be taken into account, regarding mashine precision. Due to this fact the singular local vertex by default is in origin.

The abscissas are computed on .

The internal compute routine computes generalized duffy points on a triangle. With that, the method can be used for arbitrary reference geometries that can be subdivided into triangles with common singularity vertex.

The quadrature points on the transformed triangles are computed with underlying Gauss-Legrendre points, whose number can be chosen individual in each direction in the transformed quad.

Author: Robert Gruhlke, 2014

Definition at line 481 of file quadRule.hh.

Constructor & Destructor Documentation

◆ QuadratureRule2dQuadDuffy()

concepts::QuadratureRule2dQuadDuffy::QuadratureRule2dQuadDuffy	(	Real	beta,
		uint	nx,
		uint	ny,
		uint	noVtx = `0`
	)

Constructor.

The singular vertex is by default set to zero. This is ok, since when building the mesh one can place the geometry s.t. the first elements vertex lies on the singularity.

Parameters

beta	generalized duffy transformation parameter
nx	number of quadrature points of underlying Gauss-Legendre points in x-direction on transformed square.
ny	number of quadrature points of underlying Gauss-Legendre points in y-direction on transformed square.
noVtx	number of local vertex

◆ ~QuadratureRule2dQuadDuffy()

virtual concepts::QuadratureRule2dQuadDuffy::~QuadratureRule2dQuadDuffy ( )

virtual

Member Function Documentation

◆ abscissas()

virtual const Real* concepts::QuadratureRule2d::abscissas ( uint i ) const

inlinevirtualinherited

Returns the quadrature abcissas in the i-th direction.

Parameters

i	i = 0 : x-direction, i = 1 : y-direction

Implements concepts::QuadratureRule.

Definition at line 360 of file quadRule.hh.

◆ compute_()

void concepts::QuadratureRule2dQuadDuffy::compute_	(	const Real2d	S,
		const Real2d	B,
		const Real2d	C,
		const Real	beta,
		const uint	nx,
		const uint	ny,
		Real *&	abscXP,
		Real *&	abscYP,
		Real *&	wP
	)

private

Routine computes generalized duffy quadrature points on the triangle conv(S,B,C), where S is the coordinate of the singularity vertex.

Parameters

S,B,C	Triangle points
beta	duffy parameter
nx	number of points of underlying quadrature in x direction
ny	number of points of underlying quadrature in y direction
abscXP	abscissa points in x direction
abscYP	abscissa points in y direction
wP	weights

◆ domain()

virtual const bool concepts::QuadratureRule2dQuadDuffy::domain ( ) const

inlinevirtual

Method delivers if the integration points and weights are computed in [-1,1]^2 or in [0,1]^2.

Return true if integration points are computed in

$[-1,1]^2$

or false if they are computed in

$[0,1]^2$

.

Implements concepts::QuadratureRule2d.

Definition at line 520 of file quadRule.hh.

◆ info()

virtual std::ostream& concepts::QuadratureRule2dQuadDuffy::info ( std::ostream & os ) const

protectedvirtual

Returns information in an output stream.

Reimplemented from concepts::OutputOperator.

◆ n()

const uint concepts::QuadratureRule2dQuadDuffy::n ( uint i = 0 ) const

inlinevirtual

Returns the number of quadrature points.

Implements concepts::QuadratureRule2d.

Definition at line 511 of file quadRule.hh.

◆ printData()

void concepts::QuadratureRule2dQuadDuffy::printData ( ) const

◆ quadratureData()

virtual bool concepts::QuadratureRule2dQuadDuffy::quadratureData	(	const uint	i,
		Real3d &	q,
		Real &	w
	)		const

virtual

Method delivers the i-th quadrature point.

$q \in [0,1]^n$

and its belonging i-th weight. If i exceeds the numer of quadrature, false is returned.

Implements concepts::QuadratureRule.

◆ tensor()

virtual const bool concepts::QuadratureRule2dQuadDuffy::tensor ( ) const

inlinevirtual

Method delivers information about the quadrature rule structure.

Returns true for tensor structur and false for non-tensor one.

Implements concepts::QuadratureRule2d.

Definition at line 523 of file quadRule.hh.

◆ weights()

virtual const Real* concepts::QuadratureRule2dQuadDuffy::weights ( uint i = 0 ) const

inlinevirtual

Returns the i-th weight, belonging to (x_i,y_i) as nontensored.

Implements concepts::QuadratureRule2d.

Definition at line 503 of file quadRule.hh.

Member Data Documentation

◆ beta_

Real concepts::QuadratureRule2dQuadDuffy::beta_

private

Definition at line 552 of file quadRule.hh.

◆ intX_

const Real* concepts::QuadratureRule2d::intX_

protectedinherited

Abscissas.

Definition at line 382 of file quadRule.hh.

◆ intY_

const Real* concepts::QuadratureRule2d::intY_

protectedinherited

Definition at line 383 of file quadRule.hh.

◆ n_

uint concepts::QuadratureRule2dQuadDuffy::n_

private

Definition at line 554 of file quadRule.hh.

◆ weights_

const Real* concepts::QuadratureRule2dQuadDuffy::weights_

private

Definition at line 550 of file quadRule.hh.

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

integration/quadRule.hh

concepts::QuadratureRule2dQuadDuffy Class Reference

Public Member Functions

Protected Member Functions

Protected Attributes

Private Member Functions

Private Attributes

Detailed Description

Constructor & Destructor Documentation

◆ QuadratureRule2dQuadDuffy()

◆ ~QuadratureRule2dQuadDuffy()

Member Function Documentation

◆ abscissas()

◆ compute_()

◆ domain()

◆ info()

◆ n()

◆ printData()

◆ quadratureData()

◆ tensor()

◆ weights()

Member Data Documentation

◆ beta_

◆ intX_

◆ intY_

◆ n_

◆ weights_