TRIANGLE_DUNAVANT_RULE
Quadrature Rules for the Triangle

TRIANGLE_DUNAVANT_RULE is a C++ library which defines the weights and abscisass for quadrature rules over the interior of a triangle in 2D.

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.

Languages:

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

Related Data and Programs:

ANNULUS_RULE, a C++ library which computes a quadrature rule for estimating integrals of a function over the interior of a circular annulus in 2D.

CUBE_FELIPPA_RULE, a C++ library which returns the points and weights of a Felippa quadrature rule over the interior of a cube in 3D.

PYRAMID_FELIPPA_RULE, a C++ library which returns Felippa's quadratures rules for approximating integrals over the interior of a pyramid in 3D.

SIMPLEX_GM_RULE, a C++ library which defines Grundmann-Moeller quadrature rules over the interior of the simplex in M dimensions.

SQUARE_FELIPPA_RULE, a C++ library which returns the points and weights of a Felippa quadrature rule over the interior of a square in 2D.

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

TETRAHEDRON_FELIPPA_RULE, a C++ library which returns Felippa's quadratures rules for approximating integrals over the interior of a tetrahedron in 3D.

TRIANGLE_ANALYZE, a C++ program which reads a triangle defined in a file, and uses the triangle_properties() library to compute angles, area, centroid, circumcircle, edge lengths, incircle, orientation, orthocenter, and quality.

TRIANGLE_EXACTNESS, a C++ program which investigates the polynomial exactness of a quadrature rule over the interior of a triangle in 2D.

TRIANGLE_FEKETE_RULE, a C++ library which defines Fekete rules for interpolation or quadrature over the interior of a triangle in 2D.

TRIANGLE_FELIPPA_RULE, a C++ library which returns Felippa's quadratures rules for approximating integrals over the interior of a triangle in 2D.

TRIANGLE_LYNESS_RULE, a C++ library which returns Lyness-Jespersen quadrature rules for the triangle.

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

TRIANGLE_NCC_RULE, a C++ library which defines Newton-Cotes Closed (NCC) quadrature rules over the interior of a triangle in 2D.

TRIANGLE_NCO_RULE, a C++ library which defines Newton-Cotes Open (NCO) quadrature rules over the interior of a triangle in 2D.

TRIANGLE_SYMQ_RULE, a C++ library which returns efficient symmetric quadrature rules, with exactness up to total degree 50, over the interior of an arbitrary triangle in 2D, by Hong Xiao and Zydrunas Gimbutas.

TRIANGLE_WANDZURA_RULE, a C++ library which defines Wandzura rules for quadrature over the interior of a triangle in 2D.

WEDGE_FELIPPA_RULE, a C++ library which returns quadratures rules for approximating integrals over the interior of the unit wedge in 3D.

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.

Source Code:

• triangle_dunavant_rule.cpp, the source code.
• triangle_dunavant_rule.hpp, the include file.

Examples and Tests:

• triangle_dunavant_rule_test.cpp, a sample calling program.
• triangle_dunavant_rule_test.txt, the output ffile.

List of Routines:

• DUNAVANT_DEGREE returns the degree of a Dunavant rule for the triangle.
• DUNAVANT_ORDER_NUM returns the order of a Dunavant rule for the triangle.
• DUNAVANT_RULE returns the points and weights of a Dunavant rule.
• DUNAVANT_RULE_NUM returns the number of Dunavant rules available.
• DUNAVANT_SUBORDER returns the suborders for a Dunavant rule.
• DUNAVANT_SUBORDER_NUM returns the number of suborders for a Dunavant rule.
• DUNAVANT_SUBRULE returns a compressed Dunavant rule.
• DUNAVANT_SUBRULE_01 returns a compressed Dunavant rule 1.
• DUNAVANT_SUBRULE_02 returns a compressed Dunavant rule 2.
• DUNAVANT_SUBRULE_03 returns a compressed Dunavant rule 3.
• DUNAVANT_SUBRULE_04 returns a compressed Dunavant rule 4.
• DUNAVANT_SUBRULE_05 returns a compressed Dunavant rule 5.
• DUNAVANT_SUBRULE_06 returns a compressed Dunavant rule 6.
• DUNAVANT_SUBRULE_07 returns a compressed Dunavant rule 7.
• DUNAVANT_SUBRULE_08 returns a compressed Dunavant rule 8.
• DUNAVANT_SUBRULE_09 returns a compressed Dunavant rule 9.
• DUNAVANT_SUBRULE_10 returns a compressed Dunavant rule 10.
• DUNAVANT_SUBRULE_11 returns a compressed Dunavant rule 11.
• DUNAVANT_SUBRULE_12 returns a compressed Dunavant rule 12.
• DUNAVANT_SUBRULE_13 returns a compressed Dunavant rule 13.
• DUNAVANT_SUBRULE_14 returns a compressed Dunavant rule 14.
• DUNAVANT_SUBRULE_15 returns a compressed Dunavant rule 15.
• DUNAVANT_SUBRULE_16 returns a compressed Dunavant rule 16.
• DUNAVANT_SUBRULE_17 returns a compressed Dunavant rule 17.
• DUNAVANT_SUBRULE_18 returns a compressed Dunavant rule 18.
• DUNAVANT_SUBRULE_19 returns a compressed Dunavant rule 19.
• DUNAVANT_SUBRULE_20 returns a compressed Dunavant rule 20.
• I4_MAX returns the maximum of two integers.
• I4_MIN returns the smaller of two integers.
• I4_MODP returns the nonnegative remainder of integer division.
• I4_WRAP forces an integer to lie between given limits by wrapping.
• R8_HUGE returns a "huge" R8.
• R8_NINT returns the nearest integer to an R8.
• REFERENCE_TO_PHYSICAL_T3 maps T3 reference points to physical points.
• S_LEN_TRIM returns the length of a string to the last nonblank.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TRIANGLE_AREA computes the area of a triangle.

You can go up one level to the C++ source codes.