pyramid_exactness


pyramid_exactness, a MATLAB code which measures the precision of a quadrature rule defined over the interior of a pyramid in 3D.

The integration region is:

       - ( 1 - Z ) <= X <= 1 - Z
       - ( 1 - Z ) <= Y <= 1 - Z
                 0 <= Z <= 1.
      
When Z is zero, the integration region is a square lying in the (X,Y) plane, centered at (0,0,0) with "radius" 1. As Z increases to 1, the radius of the square diminishes, and when Z reaches 1, the square has contracted to the single point (0,0,1).

Usage:

pyramid_exactness ( 'filename', degree_max )
where

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

pyramid_exactness is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

cube_exactness, a MATLAB code which investigates the polynomial exactness of quadrature rules over the interior of a cube in 3d.

hypercube_exactness, a MATLAB code which measures the monomial exactness of an m-dimensional quadrature rule over the interior of the unit hypercube in m dimensions.

pyramid_exactness_test

pyramid_felippa_rule, a MATLAB code which returns felippa's quadratures rules for approximating integrals over the interior of a pyramid in 3d.

pyramid_grid, a MATLAB code which computes a grid of points over the interior of the unit pyramid in 3d;

pyramid_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit pyramid in 3d.

pyramid_monte_carlo, a MATLAB code which applies a monte carlo method to estimate integrals of a function over the interior of the unit pyramid in 3d;

pyramid_rule, a MATLAB code which can compute a quadrature rule over the interior of the unit pyramid in 3d.

quadrature_rules_pyramid, a dataset directory which contains quadrature rules over the interior of the unit pyramid in 3d.

sphere_exactness, a MATLAB code which tests the polynomial exactness of a quadrature rule over the surface of the unit sphere in 3d.

square_exactness, a MATLAB code which investigates the polynomial exactness of quadrature rules for f(x,y) over the interior of a square (rectangle/quadrilateral) in 2d.

tetrahedron_exactness a MATLAB code which investigates the polynomial exactness of a quadrature rule over the interior of a tetrahedron in 3d.

triangle_exactness, a MATLAB code which investigates the monomial exactness quadrature rule over the interior of a triangle in 2d.

wedge_exactness, a MATLAB code which investigates the monomial exactness of a quadrature rule over the interior of the unit wedge in 3d.

Reference:

  1. Carlos Felippa,
    A compendium of FEM integration formulas for symbolic work,
    Engineering Computation,
    Volume 21, Number 8, 2004, pages 867-890.
  2. Arthur Stroud,
    Approximate Calculation of Multiple Integrals,
    Prentice Hall, 1971,
    ISBN: 0130438936,
    LC: QA311.S85.

Source Code:


Last revised on 05 March 2019.