QUADRATURE_RULES_TET
Quadrature Rules for Tetrahedrons

QUADRATURE_RULES_TET is a dataset directory which contains examples of quadrature rules for a tetrahedral region. A quadrature rule is a set of n points (x,y) and associated weights w so that the integral of a function f(x,y,z) over the tetrahedron can be approximated by:

Integral f(x,y,z) dx dy dz = Sum ( 1 <= i <= n ) w(i) * f(x(i),y(i),z(i))

Most quadrature rules for the tetrahedron are defined on the unit tetrahedron whose vertices are (0,0,0), (1,0,0), (0,1,0), (0,0,1). There is a standard technique for transforming such a rule if it is needed to be applied to a general tetrahedron.

For this directory, a quadrature rule is stored as three files, containing the weights, the points, and the vertices that define the region.

Example:

Here is an example of a quadrature rule for the unit tetrahedron, of order 10.

Here is the text of a "X" file storing the abscissas of such a rule:


  0.0000000000000000  0.0000000000000000  0.0000000000000000
  1.0000000000000000  0.0000000000000000  0.0000000000000000
  0.0000000000000000  1.0000000000000000  0.0000000000000000
  0.0000000000000000  0.0000000000000000  1.0000000000000000
  0.5000000000000000  0.0000000000000000  0.0000000000000000
  0.0000000000000000  0.5000000000000000  0.0000000000000000
  0.0000000000000000  0.0000000000000000  0.5000000000000000
  0.0000000000000000  0.5000000000000000  0.5000000000000000
  0.5000000000000000  0.0000000000000000  0.5000000000000000
  0.5000000000000000  0.5000000000000000  0.0000000000000000

Here is the text of an "W" file storing the weights of such a rule


 -0.0500000000000000
 -0.0500000000000000
 -0.0500000000000000
 -0.0500000000000000
  0.2000000000000000
  0.2000000000000000
  0.2000000000000000
  0.2000000000000000
  0.2000000000000000
  0.2000000000000000

Here is the text of an "R" file storing the veretices of the triangle:


  0.0  0.0  0.0
  1.0  0.0  0.0
  0.0  1.0  0.0
  0.0  0.0  1.0

Licensing:

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

Related Data and Programs:

GM_RULE, a FORTRAN90 library which defines a Grundmann-Moeller rule for quadrature over a triangle, tetrahedron, or general M-dimensional simplex.

KEAST, a FORTRAN90 library of routines which defines Keast rules for quadrature on a tetrahedron.

NCC_TETRAHEDRON, a FORTRAN90 library which defines Newton-Cotes closed quadrature rules on a tetrahedron.

NCO_TETRAHEDRON, a FORTRAN90 library which defines Newton-Cotes open quadrature rules on a tetrahedron.

QUADRATURE_RULES_PYRAMID, a dataset directory which contains quadrature rules for a pyramid with a square base.

QUADRATURE_RULES_TRI, a dataset directory of quadrature rules for triangular regions.

QUADRATURE_RULES_WEDGE, a dataset directory which contains quadrature rules for a wedge ( triangle x a line ).

SIMPACK, a FORTRAN77 library which approximates the integral of a function or vector of functions over a multidimensional simplex, or a region which is the sum of multidimensional simplexes.

STROUD, a FORTRAN90 library which defines quadrature rules for a variety of geometric figures.

TEST_TET_INT, a FORTRAN90 library which can be used to test algorithms for quadrature over a tetrahedron.

TET_MESH, a FORTRAN90 library which deals with data on a mesh of tetrahedrons.

TETRAHEDRON_EXACTNESS, a C++ program which investigates the polynomial exactness of a quadrature rule for the tetrahedron.

References:

Hermann Engels,
Numerical Quadrature and Cubature,
Academic Press, 1980,
ISBN: 012238850X,
LC: QA299.3E5.
Olgierd Zienkiewicz,
The Finite Element Method,
Sixth Edition,
Butterworth-Heinemann, 2005,
ISBN: 0750663200,
LC: TA640.2.Z54

Sample Files:

KEAST0, the Keast Rule, order 1, degree of precision 1.

keast0_x.txt, the abscissas.
keast0_w.txt, the weights.
keast0_r.txt, the vertices.

KEAST1, the Keast Rule, order 4, precision 2.

keast1_x.txt, the abscissas.
keast1_w.txt, the weights.
keast1_r.txt, the vertices.

KEAST2, the Keast Rule, order 5, degree of precision 3.

keast2_x.txt, the abscissas.
keast2_w.txt, the weights.
keast2_r.txt, the vertices.

KEAST3, the Keast Rule, order 10, degree of precision 3.

keast3_x.txt, the abscissas.
keast3_w.txt, the weights.
keast3_r.txt, the vertices.

KEAST4, the Keast Rule, order 11, degree of precision 4.

keast4_x.txt, the abscissas.
keast4_w.txt, the weights.
keast4_r.txt, the vertices.

KEAST5, the Keast Rule, order 14, degree of precision 4.

keast5_x.txt, the abscissas.
keast5_w.txt, the weights.
keast5_r.txt, the vertices.

KEAST6, the Keast Rule, order 15, degree of precision 5.

keast6_x.txt, the abscissas.
keast6_w.txt, the weights.
keast6_r.txt, the vertices.

KEAST7, the Keast Rule, order 24, degree of precision 6.

keast7_x.txt, the abscissas.
keast7_w.txt, the weights.
keast7_r.txt, the vertices.

KEAST8, the Keast Rule, order 31, degree of precision 7.

keast8_x.txt, the abscissas.
keast8_w.txt, the weights.
keast8_r.txt, the vertices.

KEAST9, the Keast Rule, order 45, degree of precision 8.

keast9_x.txt, the abscissas.
keast9_w.txt, the weights.
keast9_r.txt, the vertices.

NCC0, the Newton Cotes Closed Rule, order 1, degree of precision 0.

ncc0_x.txt, the abscissas.
ncc0_w.txt, the weights.
ncc0_r.txt, the vertices.

NCC1, the Newton Cotes Closed Rule, order 4, precision 1.

ncc1_x.txt, the abscissas.
ncc1_w.txt, the weights.
ncc1_r.txt, the vertices.

NCC2, the Newton Cotes Closed Rule, order 10, degree of precision 2.

ncc2_x.txt, the abscissas.
ncc2_w.txt, the weights.
ncc2_r.txt, the vertices.

NCC3, the Newton Cotes Closed Rule, order 20, degree of precision 3.

ncc3_x.txt, the abscissas.
ncc3_w.txt, the weights.
ncc3_r.txt, the vertices.

NCC4, the Newton Cotes Closed Rule, order 35, degree of precision 4.

ncc4_x.txt, the abscissas.
ncc4_w.txt, the weights.
ncc4_r.txt, the vertices.

NCC5, the Newton Cotes Closed Rule, order 56, degree of precision 5.

ncc5_x.txt, the abscissas.
ncc5_w.txt, the weights.
ncc5_r.txt, the vertices.

NCC6, the Newton Cotes Closed Rule, order 84, degree of precision 6.

ncc6_x.txt, the abscissas.
ncc6_w.txt, the weights.
ncc6_r.txt, the vertices.

NCO0, the Newton Cotes Open Rule, order 1, degree of precision 0.

nco0_x.txt, the abscissas.
nco0_w.txt, the weights.
nco0_r.txt, the vertices.

NCO1, the Newton Cotes Open Rule, order 4, precision 1.

nco1_x.txt, the abscissas.
nco1_w.txt, the weights.
nco1_r.txt, the vertices.

NCO2, the Newton Cotes Open Rule, order 10, degree of precision 2.

nco2_x.txt, the abscissas.
nco2_w.txt, the weights.
nco2_r.txt, the vertices.

NCO3, the Newton Cotes Open Rule, order 20, degree of precision 3.

nco3_x.txt, the abscissas.
nco3_w.txt, the weights.
nco3_r.txt, the vertices.

NCO4, the Newton Cotes Open Rule, order 35, degree of precision 4.

nco4_x.txt, the abscissas.
nco4_w.txt, the weights.
nco4_r.txt, the vertices.

NCO5, the Newton Cotes Open Rule, order 56, degree of precision 5.

nco5_x.txt, the abscissas.
nco5_w.txt, the weights.
nco5_r.txt, the vertices.

NCO6, the Newton Cotes Open Rule, order 84, degree of precision 6.

nco6_x.txt, the abscissas.
nco6_w.txt, the weights.
nco6_r.txt, the vertices.

Z4, Zienkiewicz, order 4, degree of precision 2.

z4_x.txt, the abscissas.
z4_w.txt, the weights.
z4_r.txt, the vertices.

Z5, Zienkiewicz, order 5, degree of precision 3.

z5_x.txt, the abscissas.
z5_w.txt, the weights.
z5_r.txt, the vertices.

You can go up one level to the DATASETS page.