TRIANGLE_DUNAVANT_RULE
Quadrature Rules for the Triangle

TRIANGLE_DUNAVANT_RULE is a FORTRAN90 library which defines the weights and abscisass for 20 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:

CUBE_FELIPPA_RULE, a FORTRAN90 library which returns the points and weights of a Felippa quadrature rule over the interior of a cube in 3D.

PYRAMID_FELIPPA_RULE, a FORTRAN90 library which returns Felippa's quadratures rules for approximating integrals over the interior of a pyramid in 3D.

SIMPLEX_GM_RULE, a FORTRAN90 library which defines Grundmann-Moeller quadrature rules over the interior of a simplex in M dimensions.

SQUARE_FELIPPA_RULE, a FORTRAN90 library which returns the points and weights of a Felippa quadrature rule over the interior of a square in 2D.

STROUD, a FORTRAN90 library which contains quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and M-dimensions.

TETRAHEDRON_FELIPPA_RULE, a FORTRAN90 library which returns Felippa's quadratures rules for approximating integrals over the interior of a tetrahedron in 3D.

TRIANGLE_EXACTNESS, a FORTRAN90 program which investigates the polynomial exactness of a quadrature rule over the interior of a triangle in 2D.

TRIANGLE_FEKETE_RULE, a FORTRAN90 library which defines Fekete rules for interpolation or quadrature over the interior of a triangle in 2D.

TRIANGLE_FELIPPA_RULE, a FORTRAN90 library which returns Felippa's quadratures rules for approximating integrals over the interior of a triangle in 2D.

TRIANGLE_LYNESS_RULE, a FORTRAN90 library which returns Lyness-Jespersen quadrature rules over the interior of a triangle in 2D.

TRIANGLE_MONTE_CARLO, a FORTRAN90 program which uses the Monte Carlo method to estimate integrals over a triangle.

TRIANGLE_NCC_RULE, a FORTRAN90 library which defines Newton-Cotes Closed (NCC) quadrature rules over the interior of a triangle in 2D.

TRIANGLE_NCO_RULE, a FORTRAN90 library which defines Newton-Cotes Open (NCO) quadrature rules over the interior of a triangle in 2D.

TRIANGLE_SVG, a FORTRAN90 library which uses Scalable Vector Graphics (SVG) to plot a triangle and any number of points, to illustrate quadrature rules and sampling techniques.

TRIANGLE_SYMQ_RULE, a FORTRAN90 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 FORTRAN90 library which defines Wandzura rules for quadrature over the interior of a triangle in 2D.

WEDGE_FELIPPA_RULE, a FORTRAN90 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.f90, the source code.

Examples and Tests:

• triangle_dunavant_rule_test.f90, a sample calling program.
• triangle_dunavant_rule_test.txt, the output from a run of the sample program.

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.
• I4_MODP returns the nonnegative remainder of I4 division.
• I4_WRAP forces an I4 to lie between given limits by wrapping.
• REFERENCE_TO_PHYSICAL_T3 maps T3 reference points to physical points.
• 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 FORTRAN90 source codes.