triangle_witherden_rule

triangle_witherden_rule, an Octave code which returns a symmetric Witherden quadrature rule for the triangle, with exactness up to total degree 20.

The data is given for the following triangle:

      (0,1) 
        | \ 
        |  \ 
        |   \ 
        |    \ 
      (0,0)--(1,0)
      

We suppose we are given a triangle T with vertices A, B, C. We call a rule with n points, returning barycentric coordinates a, b, c, and weights w. Then the integral I of f(x,y) over T is approximated by Q as follows:

      (x,y) = a(1:n) * A + b(1:n) * B + c(1:n) * C
      Q = area(T) * sum ( 1 <= i <= n ) w(i) * f(x(i),y(i)) 
     

Licensing:

The computer code and data files made available on this web page are distributed under the MIT license

Languages:

triangle_witherden_rule is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave versionand a Python version.

Related Data and Programs:

triangle_witherden_rule_test

triangle_dunavant_rule, an Octave code which sets up a Dunavant quadrature rule over the interior of a triangle in 2d.

triangle_fekete_rule, an Octave code which returns a Fekete rule for quadrature or interpolation over the interior of a triangle in 2d.

triangle_felippa_rule, an Octave code which returns a Felippa quadrature rule for approximating integrals over the interior of a triangle in 2d.

triangle_integrals, an Octave code which returns the exact value of the integral of any monomial over the interior of the unit triangle in 2d.

triangle_lyness_rule, an Octave code which returns a Lyness-Jespersen quadrature rule over the interior of a triangle in 2d.

triangle_monte_carlo, an Octave code which uses the Monte Carlo method to estimate the integral of a function over the interior of the unit triangle in 2d.

triangle_ncc_rule, an Octave code which defines Newton-Cotes closed quadrature rules on a triangle.

triangle_nco_rule, an Octave code which defines Newton-Cotes open quadrature rules over the interior of a triangle in 2d.

triangle_symq_rule, an Octave code which returns efficient symmetric quadrature rules, with exactness up to total degree 50, over the interior of an arbitrary triangle in 2D, by Hong Xiao and Zydrunas Gimbutas.

triangle_twb_rule, an Octave code which generates the points and weights of quadrature rules over the interior of a triangle in 2D, determined by Taylor, Wingate, and Bos.

triangle_wandzura_rule, an Octave code which returns a Wandzura quadrature rule of exactness 5, 10, 15, 20, 25 and 30 over the interior of the triangle in 2D.

Reference:

1. Freddie Witherden, Peter Vincent,
   On the identification of symmetric quadrature rules for finite element methods,
   Computers and Mathematics with Applications,
   Volume 69, pages 1232-1241, 2015.

Source Code:

• comp_next.m, returns the next composition of an integer.
• monomial_value.m, evaluates a multidimensional monomial.
• rule_order.m, returns the number of points in rules of order 0 through 20.
• rule00.m, returns the rule of degree 0.
• rule01.m, returns the rule of degree 1.
• rule02.m, returns the rule of degree 2.
• rule03.m, returns the rule of degree 3.
• rule04.m, returns the rule of degree 4.
• rule05.m, returns the rule of degree 5.
• rule06.m, returns the rule of degree 6.
• rule07.m, returns the rule of degree 7.
• rule08.m, returns the rule of degree 8.
• rule09.m, returns the rule of degree 9.
• rule10.m, returns the rule of degree 10.
• rule11.m, returns the rule of degree 11.
• rule12.m, returns the rule of degree 12.
• rule13.m, returns the rule of degree 13.
• rule14.m, returns the rule of degree 14.
• rule15.m, returns the rule of degree 15.
• rule16.m, returns the rule of degree 16.
• rule17.m, returns the rule of degree 17.
• rule18.m, returns the rule of degree 18.
• rule19.m, returns the rule of degree 19.
• rule20.m, returns the rule of degree 20.
• triangle_area.m, returns the area of a triangle.
• triangle_unit_area.m, returns the area of a unit triangle.
• triangle_unit_monomial_integral.m, integrates a monomial over the unit triangle.
• triangle_witherden_rule.m, returns a triangle quadrature rule of given precision.

• triangle_witherden_rule_to_matlab.m, a MATLAB code which reads a data file describing one of the rules and writes a function to return points and weights.