Quadrature Rules for Triangles

These are some examples of quadrature rules for a triangular 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) over the triangle can be approximated by:

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

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

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

Example:

Here is an example of a quadrature rule for the unit triangle, of order 6.

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

        0.659027622374092  0.231933368553031
        0.659027622374092  0.109039009072877
        0.231933368553031  0.659027622374092
        0.231933368553031  0.109039009072877
        0.109039009072877  0.659027622374092
        0.109039009072877  0.231933368553031
      

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

        0.16666666666666666667
        0.16666666666666666667
        0.16666666666666666667
        0.16666666666666666667
        0.16666666666666666667
        0.16666666666666666667  
      

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

        0.0  0.0 
        1.0  0.0
        0.0  1.0  
      

Related Data and Programs:

• DUNAVANT is a FORTRAN90 library of routines for defining Dunavant rules for quadrature on a triangle.
• FEKETE is a FORTRAN90 library of routines for defining a Fekete rule for quadrature or interpolation over a triangle.
• 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.
• NCC_TRIANGLE is a FORTRAN90 library defining Newton-Cotes closed quadrature rules on a triangle.
• NCO_TRIANGLE is a FORTRAN90 library defining Newton-Cotes open quadrature rules on a triangle.
• NINT_EXACTBESS_TRI is an executable FORTRAN90 program which investigates the polynomial exactness of a quadrature rule for the triangle.
• QUADRATURE_RULES is a dataset directory of quadrature rules for rectangular 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_TRI_INT is a FORTRAN90 library of functions that can be used to test algorithms for quadrature over a triangle.
• TRIANGULATION is a FORTRAN90 library of routines for dealing with triangulated data.

References:

1. Jarle Berntsen, Terje Espelid,
   Algorithm 706,
   DCUTRI: an algorithm for adaptive cubature over a collection of triangles,
   ACM Transactions on Mathematical Software,
   Volume 18, Number 3, September 1992, pages 329-342.
2. Elise deDoncker, Ian Robinson,
   Algorithm 612: Integration over a Triangle Using Nonlinear Extrapolation,
   ACM Transactions on Mathematical Software,
   Volume 10, Number 1, March 1984, pages 17-22.
3. Dirk Laurie,
   Algorithm 584, CUBTRI, Automatic Cubature Over a Triangle,
   ACM Transactions on Mathematical Software,
   Volume 8, Number 2, 1982, pages 210-218.
4. James Lyness, Dennis Jespersen,
   Moderate Degree Symmetric Quadrature Rules for the Triangle,
   Journal of the Institute of Mathematics and its Applications,
   Volume 15, Number 1, February 1975, pages 19-32.
5. Hans Rudolf Schwarz,
   Finite Element Methods,
   Academic Press, 1988,
   ISBN: 0126330107,
   LC: TA347.F5.S3313.
6. Gilbert Strang, George Fix,
   An Analysis of the Finite Element Method,
   Cambridge, 1973,
   ISBN: 096140888X,
   LC: TA335.S77.
7. Arthur Stroud,
   Approximate Calculation of Multiple Integrals,
   Prentice Hall, 1971,
   ISBN: 0130438936,
   LC: QA311.S85.
8. Olgierd Zienkiewicz,
   The Finite Element Method,
   Sixth Edition,
   Butterworth-Heinemann, 2005,
   ISBN: 0750663200,
   LC: TA640.2.Z54

Sample Files:

CENTROID, the centroid rule, order 1, degree of precision 1.

• centroid_x.txt, the abscissas for the rule.
• centroid_w.txt, the weights for the rule.
• centroid_r.txt, the vertices of the triangle.

GAUSS4X4, order 16, degree of precision 7, (essentially a product of two 4 point 1D Gauss-Legendre rules).

• gauss4x4_x.txt, the abscissas for the rule.
• gauss4x4_w.txt, the weights for the rule.
• gauss4x4_r.txt, the vertices of the triangle.

GAUSS8X8, order 64, degree of precision 15, (essentially a product of two 8 point 1D Gauss-Legendre rules).

• gauss8x8_x.txt, the abscissas for the rule.
• gauss8x8_w.txt, the weights for the rule.
• gauss8x8_r.txt, the vertices of the triangle.

SEVEN_POINT, order 7, degree of precision 3.

• seven_point_x.txt, the abscissas for the rule.
• seven_point_w.txt, the weights for the rule.
• seven_point_r.txt, the vertices of the triangle.

STRANG1, order 3, degree of precision 2.

• strang1_x.txt, the abscissas for the rule.
• strang1_w.txt, the weights for the rule.
• strang1_r.txt, the vertices of the triangle.

STRANG2, order 3, degree of precision 2.

• strang2_x.txt, the abscissas for the rule.
• strang2_w.txt, the weights for the rule.
• strang2_r.txt, the vertices of the triangle.

STRANG3, order 4, degree of precision 3.

• strang3_x.txt, the abscissas for the rule.
• strang3_w.txt, the weights for the rule.
• strang3_r.txt, the vertices of the triangle.

STRANG4, order 6, degree of precision 3.

• strang4_x.txt, the abscissas for the rule.
• strang4_w.txt, the weights for the rule.
• strang4_r.txt, the vertices of the triangle.

STRANG5, order 6, degree of precision 4.

• strang5_x.txt, the abscissas for the rule.
• strang5_w.txt, the weights for the rule.
• strang5_r.txt, the vertices of the triangle.

STRANG6, order 7, degree of precision 4.

• strang6_x.txt, the abscissas for the rule.
• strang6_w.txt, the weights for the rule.
• strang6_r.txt, the vertices of the triangle.

STRANG7, order 7, degree of precision 5.

• strang7_x.txt, the abscissas for the rule.
• strang7_w.txt, the weights for the rule.
• strang7_r.txt, the vertices of the triangle.

STRANG8, order 9, degree of precision 6.

• strang8_x.txt, the abscissas for the rule.
• strang8_w.txt, the weights for the rule.
• strang8_r.txt, the vertices of the triangle.

STRANG9, order 12, degree of precision 6.

• strang9_x.txt, the abscissas for the rule.
• strang9_w.txt, the weights for the rule.
• strang9_r.txt, the vertices of the triangle.

STRANG10, order 13, degree of precision 7.

• strang10_x.txt, the abscissas for the rule.
• strang10_w.txt, the weights for the rule.
• strang10_r.txt, the vertices of the triangle.

TOMS584_19, order 19, degree of precision 8, a rule from ACM TOMS algorithm #584.

• toms584_19_x.txt, the abscissas for the rule.
• toms584_19_w.txt, the weights for the rule.
• toms584_19_r.txt, the vertices of the triangle.

TOMS612_19, order 19, degree of precision 9, a rule from ACM TOMS algorithm #612.

• toms612_19_x.txt, the abscissas for the rule.
• toms612_19_w.txt, the weights for the rule.
• toms612_19_r.txt, the vertices of the triangle.

TOMS612_28, order 28, degree of precision 11, a rule from ACM TOMS algorithm #612.

• toms612_28_x.txt, the abscissas for the rule.
• toms612_28_w.txt, the weights for the rule.
• toms612_28_r.txt, the vertices of the triangle.

TOMS706_37, order 37, degree of precision 13, a rule from ACM TOMS algorithm #706.

• toms706_37_x.txt, the abscissas for the rule.
• toms706_37_w.txt, the weights for the rule.
• toms706_37_r.txt, the vertices of the triangle.

VERTEX, the vertex rule, order 3, degree of precision 1.

• vertex_x.txt, the abscissas for the rule.
• vertex_w.txt, the weights for the rule.
• vertex_r.txt, the vertices of the triangle.

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