tetrahedron_witherden_rule

tetrahedron_witherden_rule, a MATLAB code which returns a symmetric Witherden quadrature rule for the tetrahedron, with exactness up to total degree 10.

The data is given for the tetrahedron: with vertices (0,0,0), (1,0,0), (0,1,0), (0,0,1).

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

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

Licensing:

The information on this web page is distributed under the MIT license.

Languages:

tetrahedron_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:

tetrahedron_witherden_rule_test

matlab_rule, a MATLAB 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:

• 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 10.
• 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.
• tetrahedron_unit_monomial_integral.m, integrates a monomial over the unit tetrahedron.
• tetrahedron_unit_volume.m, computes the volume of a unit tetrahedron.
• tetrahedron_volume.m, returns the volume of a tetrahedron.
• tetrahedron_witherden_rule.m, returns a tetrahedron quadrature rule of given precision.

• tetrahedron_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.