TETRAHEDRON_KEAST_RULE
Quadrature Rules for a Tetrahedron

TETRAHEDRON_KEAST_RULE is a C++ 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	0
2	4	1
3	5	2
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 described and 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 FORTRAN90 version and a MATLAB version.

Related Data and Programs:

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

TETRAHEDRON_ARBQ_RULE, a C++ 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 C++ program which investigates the polynomial exactness of a quadrature rule over the interior of a tetrahedron in 3D.

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

TETRAHEDRON_INTEGRALS, a C++ 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 C++ library which uses the Monte Carlo method to estimate integrals over the interior of the unit tetrahedron in 3D.

TETRAHEDRON_NCC_RULE, a C++ library which defines Newton-Cotes Closed (NCC) quadrature rules over the interior of a tetrahedron in 3D.

TETRAHEDRON_NCO_RULE, a C++ library which defines Newton-Cotes Open (NCO) quadrature rules over the interior of a tetrahedron in 3D.

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.

WEDGE_FELIPPA_RULE, a C++ 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.cpp, the source code.
• tetrahedron_keast_rule.hpp, the include file.
• tetrahedron_keast_rule.sh, commands to compile the source code.

Examples and Tests:

• tetrahedron_keast_rule_prb.cpp, a sample calling program.
• tetrahedron_keast_rule_prb.sh, commands to compile and run the sample program.
• tetrahedron_keast_rule_prb_output.txt, the output file.

List of Routines:

• COMP_NEXT computes the compositions of the integer N into K parts.
• I4_MAX returns the maximum of two I4's.
• I4_MIN returns the smaller of two I4's.
• 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 triangle.
• KEAST_ORDER_NUM returns the order of a Keast rule for the triangle.
• KEAST_RULE returns the points and weights of a Keast rule.
• KEAST_RULE_NUM returns the number of Keast rules available.
• 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.
• KEAST_SUBRULE_01 returns a compressed Keast rule 1.
• KEAST_SUBRULE_02 returns a compressed Keast rule 2.
• KEAST_SUBRULE_03 returns a compressed Keast rule 3.
• KEAST_SUBRULE_04 returns a compressed Keast rule 4.
• KEAST_SUBRULE_05 returns a compressed Keast rule 5.
• KEAST_SUBRULE_06 returns a compressed Keast rule 6.
• KEAST_SUBRULE_07 returns a compressed Keast rule 7.
• KEAST_SUBRULE_08 returns a compressed Keast rule 8.
• KEAST_SUBRULE_08 returns a compressed Keast rule 8.
• KEAST_SUBRULE_10 returns a compressed Keast rule 10.
• MONOMIAL_VALUE evaluates a monomial.
• R8_HUGE returns a "huge" R8.
• R8_NINT returns the nearest integer to an R8.
• R8MAT_DET_4D computes the determinant of a 4 by 4 R8MAT.
• R8VEC_DOT computes the dot product of a pair of R8VEC's.
• S_LEN_TRIM returns the length of a string to the last nonblank.
• TETRAHEDRON_REFERENCE_TO_PHYSICAL maps 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.

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