triangle_witherden_rule

triangle_witherden_rule, a Fortran90 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 information on this web page is 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

f90_rule, a Fortran90 code which computes a quadrature rule which estimates the integral of a function f(x), which might be defined over a one dimensional region (a line) or more complex shapes such as a circle, a triangle, a quadrilateral, a polygon, or a higher dimensional region, and which might include an associated weight function w(x).

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.