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 \( (1-x)^\alpha (1+x)^\beta \).

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.

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 that the QuadRule object should have.

Returns: A QuadRule object for the given reference element and with a degree >= 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

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.

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().

Classes

Typedefs

Functions

Detailed Description

Typedef Documentation

◆ quadDegree_t

Function Documentation

◆ GaussJacobi()

◆ GaussLegendre() [1/2]

◆ GaussLegendre() [2/2]

◆ make_QuadRule()

◆ make_QuadRuleNodal()

◆ operator<<()