tetrahedron_witherden_rule


tetrahedron_witherden_rule, a C++ 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 computer code and data files made available on this web page are 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

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tetrahedron_monte_carlo, a C++ code which uses the Monte Carlo method to estimate the integral of a function over the interior of the unit tetrahedron in 3d.

tetrahedron_ncc_rule, a C++ code which defines Newton-Cotes Closed (NCC) quadrature rules over the interior of a tetrahedron in 3D.

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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:


Last revised on 25 April 2023.