triangle_witherden_rule

triangle_witherden_rule, a MATLAB 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 MATLAB version and an Octave versionand a Python version.

Related Data and Programs:

triangle_witherden_rule_test

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

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

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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:

• 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.