TETRAHEDRON_KEAST_RULE
Quadrature Rules for a Tetrahedron

TETRAHEDRON_KEAST_RULE is a FORTRAN90 library which defines ten quadrature rules, of degree of exactness 0 through 8, over the interior of the tetrahedron in 3D.

The ten rules have the following orders and precisions:

Rule	Order	Precision
1	1	1
2	4	2
3	5	3
4	10	3
5	11	4
6	14	4
7	15	5
8	24	6
9	31	7
10	45	8

Licensing:

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

Languages:

TETRAHEDRON_KEAST_RULE is available in a C version and 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 N-dimensions.

TETRAHEDRON_ARBQ_RULE, a FORTRAN90 library which returns quadrature rules, with exactness up to total degree 15, over the interior of a tetrahedron in 3D, by Hong Xiao and Zydrunas Gimbutas.

TETRAHEDRON_EXACTNESS, a FORTRAN90 program which investigates the polynomial exactness of a quadrature rule over the interior of a tetrahedron in 3D.

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

TETRAHEDRON_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit tetrahedron in 3D.

TETRAHEDRON_MONTE_CARLO, a FORTRAN90 program which uses the Monte Carlo method to estimate integrals over the interior of a tetrahedron in 3D.

TETRAHEDRON_NCC_RULE, a FORTRAN90 library which defines Newton-Cotes closed quadrature rules over the interior of a tetrahedron in 3D.

TETRAHEDRON_NCO_RULE, a FORTRAN90 library which defines Newton-Cotes open quadrature rules over the interior of a tetrahedron in 3D.

TRIANGLE_FELIPPA_RULE, a FORTRAN90 library which returns Felippa's quadratures rules for approximating integrals 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. Patrick Keast,
   Moderate Degree Tetrahedral Quadrature Formulas,
   Computer Methods in Applied Mechanics and Engineering,
   Volume 55, Number 3, May 1986, pages 339-348.

Source Code:

• tetrahedron_keast_rule.f90, the source code.

Examples and Tests:

• tetrahedron_keast_rule_test.f90, a sample calling program.
• tetrahedron_keast_rule_test.txt, the output file.

List of Routines:

• COMP_NEXT computes the compositions of the integer N into K parts.
• I4_MODP returns the nonnegative remainder of I4 division.
• I4_WRAP forces an I4 to lie between given limits by wrapping.
• KEAST_DEGREE returns the degree of a Keast rule for the tetrahedron.
• KEAST_ORDER_NUM returns the order of a Keast rule for the tetrahedron.
• KEAST_RULE returns the points and weights of a Keast rule.
• KEAST_RULE_NUM returns the number of Keast rules for the tetrahedron.
• KEAST_SUBORDER returns the suborders for a Keast rule.
• KEAST_SUBORDER_NUM returns the number of suborders for a Keast rule.
• KEAST_SUBRULE returns a compressed Keast rule.
• MONOMIAL_VALUE evaluates a monomial.
• R8MAT_DET_4D computes the determinant of a 4 by 4 R8MAT.
• TETRAHEDRON_REFERENCE_TO_PHYSICAL maps T4 reference points to physical points.
• TETRAHEDRON_VOLUME computes the volume of a tetrahedron in 3D.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.