lf::quad Namespace Reference

Rules for numerical quadrature on reference entity shapes. More...

Classes
class	QuadRule
	Represents a Quadrature Rule over one of the Reference Elements. More...
class	QuadRuleCache
	A cache for make_QuadRule() More...

Typedefs
using	quadDegree_t = unsigned int

Functions
std::tuple< Eigen::VectorXd, Eigen::VectorXd >	GaussLegendre (unsigned num_points)
std::tuple< Eigen::VectorXd, Eigen::VectorXd >	GaussJacobi (quadDegree_t num_points, double alpha, double beta)
	Computes the quadrature points and weights for the interval [-1,1] of a Gauss-Jacobi quadrature rule with weight function \( (1-x)^\alpha (1+x)^\beta \).
std::tuple< Eigen::VectorXd, Eigen::VectorXd >	GaussLegendre (unsigned int num_points)
	Computes the quadrature points and weights for the interval [0,1] of a Gauss-Legendre quadrature rule using a newton algorithm.
QuadRule	make_QuadRule (base::RefEl ref_el, unsigned degree)
	Returns a QuadRule object for the given Reference Element and Degree.
QuadRule	make_TriaQR_MidpointRule ()
	midpoint quadrature rule for triangles
QuadRule	make_TriaQR_EdgeMidpointRule ()
	edge midpoint quadrature rule for reference triangles
QuadRule	make_QuadQR_EdgeMidpointRule ()
	edge midpoint quadrature rule for unit square (= reference quad)
QuadRule	make_TriaQR_P6O4 ()
	Six point triangular quadrature rule of order 4.
QuadRule	make_TriaQR_P7O6 ()
	Seven point triangular quadrature rule of order 6.
QuadRule	make_TriaQR_P1O2 ()
QuadRule	make_TriaQR_P3O3 ()
QuadRule	make_QuadQR_MidpointRule ()
	midpoint quadrature rule for quadrilaterals
QuadRule	make_QuadQR_P1O2 ()
QuadRule	make_QuadQR_P4O2 ()
QuadRule	make_QuadQR_P4O4 ()
	Fourth-order tensor product Gauss rule for quadrilaterals.
QuadRule	make_QuadRuleNodal (base::RefEl ref_el)
	Create a quadrature rule that evaluates the quadrand only at the nodes of the reference element.
std::ostream &	operator<< (std::ostream &stream, const lf::quad::QuadRule &quadrule)
	Output operator for quadrature rules.

Detailed Description

Rules for numerical quadrature on reference entity shapes.

Quadrature rules are specified by

• a sequence of reference coordinates for quadrature points
• a corresponding sequence of real-valued quadrature weights The main interface is provided by the class lf::quad::QuadRule.

Refer to Lecture Document Paragraph 2.7.5.39.

Typedef Documentation

quadDegree_t

using lf::quad::quadDegree_t = unsigned int

Definition at line 19 of file quad_rule.h.

Function Documentation

GaussJacobi()

std::tuple< Eigen::VectorXd, Eigen::VectorXd > lf::quad::GaussJacobi	(	quadDegree_t	num_points,
		double	alpha,
		double	beta )

Computes the quadrature points and weights for the interval [-1,1] of a Gauss-Jacobi quadrature rule with weight function \( (1-x)^\alpha (1+x)^\beta \).

Note

This function is not included by the quad namespace and used by LehrFEM++ internally only.

Parameters

num_points	The number of points/weights that should be used.
alpha	The exponent of the weight function
beta	The exponent of the weight function

Returns

The nodes(first) and weights(second) of the quadrature rule

Definition at line 71 of file gauss_quadrature.cc.

Referenced by make_QuadRule().

GaussLegendre() [1/2]

std::tuple< Eigen::VectorXd, Eigen::VectorXd > lf::quad::GaussLegendre

(

unsigned int

num_points

)

Computes the quadrature points and weights for the interval [0,1] of a Gauss-Legendre quadrature rule using a newton algorithm.

Note

This function is not included by the quad namespace and used by LehrFEM++ internally only.

Parameters

num_points

The number of points/weights that should be used.

Returns

The nodes (first) and weights(second) of the quadrature rule.

GaussLegendre() [2/2]

std::tuple< Eigen::VectorXd, Eigen::VectorXd > lf::quad::GaussLegendre

(

unsigned

num_points

)

Definition at line 16 of file gauss_quadrature.cc.

Referenced by make_QuadRule().

make_QuadRule()

QuadRule lf::quad::make_QuadRule	(	base::RefEl	ref_el,
		unsigned	degree )

Returns a QuadRule object for the given Reference Element and Degree.

Parameters

ref_el	The type of reference element
degree	The minimum degree of polynomials that should be integrated exactly by the created QuadRule object

Returns

A QuadRule object for the given reference element and with a "degree of exactness" >= degree

This method tries to return optimal quadrature rules when possible:

• For Segments it returns Gauss-Legendre quadrature rules
• For Triangles, it returns specific, hard-coded quadrature rules up to order 50, afterwards, the Duffy-Transform to map a tensor Gaussian Quadrature on a Square to the triangle.
• For Quadrilaterals it uses tensor products of Gauss-Legendre rules

Note that the order of a quadrature rule is equal to degree of exactness + 1!

Usage example:

auto ref_tria = lf::base::RefEl::kTria();
 
lf::quad::QuadRule qr = lf::quad::make_QuadRule(ref_tria, 3);

Definition at line 21 of file make_quad_rule.cc.

References GaussJacobi(), GaussLegendre(), lf::base::RefEl::kQuad(), lf::base::RefEl::kSegment(), and lf::base::RefEl::kTria().

Referenced by lf::geometry::test_utils::checkChildGeometryVolume(), lf::quad::QuadRuleCache::Get(), make_QuadQR_MidpointRule(), make_QuadQR_P4O4(), lf::uscalfe::ReactionDiffusionElementMatrixProvider< SCALAR, DIFF_COEFF, REACTION_COEFF >::ReactionDiffusionElementMatrixProvider(), and lf::uscalfe::ScalarLoadElementVectorProvider< SCALAR, MESH_FUNCTION >::ScalarLoadElementVectorProvider().

make_QuadRuleNodal()

QuadRule lf::quad::make_QuadRuleNodal

(

base::RefEl

ref_el

)

Create a quadrature rule that evaluates the quadrand only at the nodes of the reference element.

Parameters

ref_el

The reference element for which the quadrule is.

Returns

A quadrature rule of order 1.

Usage example:

auto ref_tria = lf::base::RefEl::kTria();
 
lf::quad::QuadRule qr = lf::quad::make_QuadRuleNodal(ref_tria);

Definition at line 14 of file make_quad_rule_nodal.cc.

References lf::base::RefEl::kTria(), lf::base::RefEl::NodeCoords(), and lf::base::RefEl::NumNodes().

operator<<()

std::ostream & lf::quad::operator<<	(	std::ostream &	stream,
		const lf::quad::QuadRule &	quadrule )

Output operator for quadrature rules.

Parameters

stream	The stream to which this function should output
quadrule	the quadrature rule to be printed

Returns

The stream itself.

See also

QuadRule::PrintInfo()

Definition at line 19 of file quad_rule.cc.

References lf::quad::QuadRule::PrintInfo().