DUNAVANT
Quadrature Rules for the Triangle

DUNAVANT is a MATLAB library, using double precision arithmetic, which defines the weights and abscisass for a sequence of 20 quadrature rules on a triangle, which are exact for polynomials up to degree 20.

These rules are almost optimal, in the sense that, for each polynomial degree, the number of points used in the rule is close to, or equal to, the theoretical minimum possible value.

A few of the rules include one or two points which are "slightly" outside the triangle; a few of the rules include weights which are negative. Both of these occurrences are generally undesirable.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Related Data and Programs:

CLENSHAW_CURTIS is a MATLAB library which sets up a Clenshaw Curtis quadrature grid in multiple dimensions.

DUNAVANT is also available in a C++ version and a FORTRAN90 version

FEKETE is a MATLAB library which defines Fekete rules for quadrature or interpolation on a triangle.

FELIPPA is a MATHEMATICA library which defines quadrature rules for lines, triangles, quadrilaterals, pyramids, wedges, tetrahedrons and hexahedrons.

FEM_SAMPLE is a MATLAB library which evaluates a finite element function defined on an order 3 or order 6 triangulation.

FEMPACK is a MATLAB library which contains routines useful for finite element calculations.

GM_RULES is a MATLAB library which defines Grundmann-Moeller rules for quadrature over a triangle, tetrahedron, or general M-dimensional simplex.

NCC_TRIANGLE is a MATLAB library defining Newton-Cotes closed quadrature rules on a triangle.

NCO_TRIANGLE is a MATLAB library defining Newton-Cotes open quadrature rules on a triangle.

NINT_EXACTNESS_TRI is a MATLAB program which investigates the polynomial exactness of a quadrature rule for the triangle.

NINTLIB is a MATLAB library containing a variety of routines for numerical estimation of integrals in multiple dimensions.

QUADRULE is a MATLAB library for defining quadrature rules on a variety of intervals with different weight functions.

QUADRULE_FAST is a MATLAB library of routines defining efficient versions of a few 1D quadrature rules.

STROUD is a MATLAB library containing quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and N-dimensions.

TEST_TRI_INT is a MATLAB library of functions which can be used to test algorithms for quadrature over a triangle.

TOMS612 is a FORTRAN77 library of routines which can estimate the integral of a function over a triangle.

TOMS706 is a FORTRAN77 library which estimates the integral of a function over a triangulated region.

WANDZURA is a MATLAB library of routines for defining Dunavant rules for quadrature on a triangle.

Reference:

1. David Dunavant,
   High Degree Efficient Symmetrical Gaussian Quadrature Rules for the Triangle,
   International Journal for Numerical Methods in Engineering,
   Volume 21, 1985, pages 1129-1148.
2. James Lyness, Dennis Jespersen,
   Moderate Degree Symmetric Quadrature Rules for the Triangle,
   Journal of the Institute of Mathematics and its Applications,
   Volume 15, Number 1, February 1975, pages 19-32.

Tar File:

A GZIP'ed TAR file of the contents of this directory is available. This is only done as a convenience for users who want ALL the files, and don't want to download them individually. This is not a convenience for me, so don't be surprised if the tar file is somewhat out of date.

• dunavant_m_src.tar.csh the commands used to create the GZIP'ed TAR file.
• dunavant_m_src.tar.gz the GZIP'ed TAR file.

Source Code:

• dunavant_degree.m returns the degree of a Dunavant rule for the triangle.
• dunavant_order_num.m returns the order of a given Dunavant rule for the triangle.
• dunavant_rule.m returns the points and weights of a Dunavant rule.
• dunavant_rule_num.m returns the number of available Dunavant rules.
• dunavant_suborder.m returns the suborders for a Dunavant rule.
• dunavant_suborder_num.m returns the number of suborders for a Dunavant rule.
• dunavant_subrule.m returns a compressed Dunavant rule.
• file_name_inc.m increments a partially numeric filename.
• i4_modp.m returns the nonnegative remainder of integer division.
• i4_wrap.m forces an integer to lie between given limits by wrapping.
• reference_to_physical_t3.m maps T3 reference points to physical points.
• timestamp.m prints the current YMDHMS date as a time stamp.
• timestring.m writes the current YMDHMS date into a string.
• triangle_area.m computes the area of a triangle.
• triangle_points_plot.m plots a triangle and some points.

Examples and Tests:

• dunavant_test.m, runs all the tests.
• dunavant_test01.m, tests DUNAVANT_ORDER_NUM.
• dunavant_test02.m, tests DUNAVANT_RULE by summing the weights.
• dunavant_test03.m, tests DUNAVANT_RULE by summing the coordinates.
• dunavant_test04.m, tests DUNAVANT_ORDER by integrating monomials in the unit triangle.
• dunavant_test05.m, tests DUNAVANT_RULE by plotting the points.
• dunavant_test06.m, tests REFERENCE_TO_PHYSICAL_T3 by transforming a rule from the unit triangle to another triangle.
• dunavant_test_output.txt, the output file.

One of the tests in the sample calling program creates EPS files of the Dunavant points in the unit triangle. These have been converted to PNG files for display here.

• dunavant_rule_01.png, a plot of rule 1.
• dunavant_rule_02.png, a plot of rule 2.
• dunavant_rule_03.png, a plot of rule 3.
• dunavant_rule_04.png, a plot of rule 4.
• dunavant_rule_05.png, a plot of rule 5.
• dunavant_rule_06.png, a plot of rule 6.
• dunavant_rule_07.png, a plot of rule 7.
• dunavant_rule_08.png, a plot of rule 8.
• dunavant_rule_09.png, a plot of rule 9.
• dunavant_rule_10.png, a plot of rule 10.

You can go up one level to the MATLAB source codes.