tetrahedron_keast_rule

tetrahedron_keast_rule, a MATLAB code 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 described and made available on this web page are distributed under the MIT 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:

annulus_rule, a MATLAB code which computes a quadrature rule for estimating integrals of a function over the interior of a circular annulus in 2d.

cube_felippa_rule, a MATLAB code which returns the points and weights of a felippa quadrature rule over the interior of a cube in 3d.

pyramid_felippa_rule, a MATLAB code which returns felippa's quadratures rules for approximating integrals over the interior of a pyramid in 3d.

simplex_gm_rule, a MATLAB code which defines grundmann-moeller quadrature rules over the interior of a simplex in m dimensions.

square_felippa_rule, a MATLAB code which returns the points and weights of a felippa quadrature rule over the interior of a square in 2d.

stroud, a MATLAB code which defines quadrature rules for a variety of unusual areas, surfaces and volumes in 2d, 3d and n-dimensions.

tetrahedron_arbq_rule, a MATLAB code 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 MATLAB code which investigates the monomial exactness of a quadrature rule over the interior of a tetrahedron in 3d.

tetrahedron_felippa_rule, a MATLAB code which returns felippa's quadratures rules for approximating integrals over the interior of a tetrahedron in 3d.

tetrahedron_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit tetrahedron in 3d.

tetrahedron_keast_rule_test

tetrahedron_monte_carlo, a MATLAB code which uses the monte carlo method to estimate integrals over the interior of a tetrahedron in 3d.

tetrahedron_ncc_rule, a MATLAB code which defines newton-cotes closed (ncc) quadrature rules over the interior of a tetrahedron in 3d.

tetrahedron_nco_rule, a MATLAB code which defines newton-cotes open (nco) quadrature rules over the interior of a tetrahedron in 3d.

triangle_fekete_rule, a MATLAB code which defines fekete rules for quadrature or interpolation over the interior of a triangle in 2d.

triangle_felippa_rule, a MATLAB code which returns felippa's quadratures rules for approximating integrals over the interior of a triangle in 2d.

wedge_felippa_rule, a MATLAB code 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:

• comp_next.m returns the next composition of an integer into a given number of parts.
• i4_modp.m returns the nonnegative remainder of integer division.
• i4_wrap.m forces an integer to lie between given limits by wrapping.
• keast_degree.m returns the degree of a Keast rule for the tetrahedron.
• keast_order_num.m returns the order of a Keast rule for the tetrahedron.
• keast_rule.m returns the points and weights of a Keast rule.
• keast_rule_num.m returns the number of Keast rules for the tetrahedron.
• keast_suborder.m returns the suborders for a Keast rule.
• keast_suborder_num.m returns the number of suborders for a Keast rule.
• keast_subrule.m returns a compressed Keast rule.
• monomial_value.m evaluates a monomial function.
• r8mat_det_4d.m computes the determinant of a 4 by 4 R8MAT.
• tetrahedron_reference_to_physical.m maps T4 reference points to physical points.
• tetrahedron_volume.m computes the volume of a tetrahedron in 3D.