QUADRATURE_RULES_TRI
Quadrature Rules for Triangles

QUADRATURE_RULES_TRI is a dataset directory which contains 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 a triangle T can be approximated by:

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

The area of a triangle is equal to the integral of the function f(x,y)=1. This implies that every useful quadrature rule should satisfy

        Sum ( 1 <= i <= n ) w(i) = 1
      

(Other tabulations of quadrature rules for triangles might follow a different convention, in which the weights sum to 1/2. Such a convention makes it easier to integrate over the unit triangle, but slightly more complicated to integrate over a general triangle. My impression is that all quadrature rules should sum to 1, so that the dependence on the area of the region is separated and obvious.)

Quadrature rules for the triangle are generally defined over T1, the unit triangle whose vertices are (0,0), (1,0), (0,1). This implies that, for an integral over T1, we have:

        Integral f(x,y) dx dy = Area(T1) * Sum ( 1 <= i <= n ) w(i) * f(x(i),y(i)) 
                              = 1/2      * Sum ( 1 <= i <= n ) w(i) * f(x(i),y(i))
      

For this directory, a quadrature rule is stored as three files, containing the weights w, the points (x,y), and the vertices that define the triangular region (usually (0,0), (1,0), (0,1) ).

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 vertices of the triangle:

        0.0  0.0 
        1.0  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:

DUNAVANT, a FORTRAN90 library which defines Dunavant rules for quadrature on a triangle.

FEKETE, a FORTRAN90 library which defines Fekete rules for quadrature or interpolation over a triangle.

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

NCC_TRIANGLE, a FORTRAN90 library which defines Newton-Cotes closed quadrature rules on a triangle.

NCO_TRIANGLE, a FORTRAN90 library which defines Newton-Cotes open quadrature rules on a triangle.

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

QUADRATURE_RULES_TET, a dataset directory which contains quadrature rules for tetrahedrons, stored as a file of abscissas, a file of weights, and a file of vertices.

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

QUADRULE, a FORTRAN90 library which defines various quadrature rules.

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_TRI_INT, a FORTRAN90 library which can be used to test algorithms for quadrature over a triangle.

TRIANGLE_EXACTNESS, a FORTRAN90 program which investigates the polynomial exactness of a quadrature rule for the triangle.

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.

You can go up one level to the DATASETS page.