Quadrature Rules for Tetrahedrons

These are some 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
      

Related Data and Programs:

• GM_RULES is a FORTRAN90 library of routines for defining a Grundmann-Moeller rule for quadrature over a triangle, tetrahedron, or general M-dimensional simplex.
• KEAST is a FORTRAN90 library of routines for defining Keast rules for quadrature on a tetrahedron.
• NCC_TETRAHEDRON is a FORTRAN90 library defining Newton-Cotes closed quadrature rules on a tetrahedron.
• NCO_TETRAHEDRON is a FORTRAN90 library defining Newton-Cotes open quadrature rules on a tetrahedron.
• NINT_EXACTBESS_TET is an executable FORTRAN90 program which investigates the polynomial exactness of a quadrature rule for the tetrahedron.
• QUADRATURE_RULES is a dataset directory of quadrature rules for rectangular regions.
• QUADRATURE_RULES_TRI is a dataset directory of quadrature rules for triangular regions.
• QUADRATURE_TEST an executable MATLAB program which reads the definition of a multidimensional quadrature rule from three files, applies the rule to a number of test integrals, and prints the results.
• QUADRULE is a FORTRAN90 library of routines which defines various quadrature rules.
• SIMPACK is a FORTRAN77 library of routines which approximate the integral of a function or vector of functions over a multidimensional simplex, or a region which is the sum of multidimensional simplexes.
• STROUD is a FORTRAN90 library of routines which defines quadrature rules for a variety of geometric figures.
• TABLE is the file format used to store this data;
• TEST_TET_INT is a FORTRAN90 library of functions that can be used to test algorithms for quadrature over a tetrahedron.
• TET_MESH is a FORTRAN90 library of routines for dealing with data on a mesh of tetrahedrons.

References:

1. Hermann Engels,
   Numerical Quadrature and Cubature,
   Academic Press, 1980,
   ISBN: 012238850X,
   LC: QA299.3E5.
2. 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 0.

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

KEAST1, the Keast Rule, order 4, precision 1.

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

KEAST2, the Keast Rule, order 5, degree of precision 2.

• 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.

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