WANDZURA
Quadrature Rules for Triangles

WANDZURA is a MATLAB library which can return any of six Wandzura rules for quadrature in a triangle.

There are six rules, which have polynomial degree of exactness of 5, 10, 15, 20, 25, and 30.

Licensing:

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

Languages:

WANDZURA is available in a C++ version and a FORTRAN90 version and a MATLAB version

Related Data and Programs:

DUNAVANT, a MATLAB library which defines a Dunavant rule for quadrature in a triangle.

FEKETE, a MATLAB library which defines a Fekete rule for interpolation or quadrature in a triangle.

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

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

LYNESS_RULE, a MATLAB library which returns Lyness-Jespersen quadrature rules for the triangle.

NCC_TRIANGLE, a MATLAB library which defines Newton-Cotes closed quadrature rules on a triangle.

NCO_TRIANGLE, a MATLAB library which defines Newton-Cotes open quadrature rules on a triangle.

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

TEST_TRI_INT, a MATLAB library which tests algorithms for quadrature over a triangle.

TOMS612, a FORTRAN77 library which estimates the integral of a function over a triangle.

TRIANGLE_MONTE_CARLO, a MATLAB program which uses the Monte Carlo method to estimate integrals over a triangle.

Reference:

1. 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.
2. Stephen Wandzura, Hong Xiao,
   Symmetric Quadrature Rules on a Triangle,
   Computers and Mathematics with Applications,
   Volume 45, Number 12, June 2003, pages 1829-1840.

Source Code:

• 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.
• triangle_area.m computes the area of a triangle.
• triangle_points_plot.m plots a triangle and some points.
• wandzura_degree.m returns the degree of a given Wandzura rule for the triangle.
• wandzura_order_num.m returns the order of a given Wandzura rule for the triangle.
• wandzura_rule.m returns the points and weights of a Wandzura rule.
• wandzura_rule_num.m returns the number of Wandzura rules available.
• wandzura_suborder.m returns the suborders for a Wandzura rule.
• wandzura_suborder_num.m returns the number of suborders for a Wandzura rule.
• wandzura_subrule.m returns a compressed Wandzura rule.
• wandzura_subrule2.m returns a compressed Wandzura rule, using the nonstandard reference triangle.

Examples and Tests:

• wandzura_test.m, runs all the tests.
• wandzura_test01.m, tests WANDZURA_RULE_NUM, WANDZURA_DEGREE, and WANDZURA_ORDER_NUM.
• wandzura_test02.m, tests WANDZURA_RULE by summing the weights.
• wandzura_test03.m, tests WANDZURA_RULE by summing the barycentric coordinates.
• wandzura_test04.m, tests WANDZURA_ORDER by integrating monomials in the unit triangle.
• wandzura_test05.m, tests WANDZURA_RULE by plotting the points.
• wandzura_test06.m, tests REFERENCE_TO_PHYSICAL_T3 by transforming a Wandzura rule from the unit triangle to another triangle.
• wandzura_test_output.txt, the output file.

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

• wandzura_rule_1.png, a plot of rule 1.
• wandzura_rule_2.png, a plot of rule 2.
• wandzura_rule_3.png, a plot of rule 3.
• wandzura_rule_4.png, a plot of rule 4.
• wandzura_rule_5.png, a plot of rule 5.
• wandzura_rule_6.png, a plot of rule 6.

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