wedge_exactness

wedge_exactness, a Python code which measures the precision of a quadrature rule over the interior of the unit wedge in 3D.

The interior of the unit wedge in 3D is defined by the constraints:

        0 <= X
        0 <= Y
             X + Y <= 1
       -1 <= Z <= +1
      

Usage:

wedge_exactness filename degree_max

• filename is the common prefix of the filenames containing the abscissas and the weights of the quadrature rule.
• degree_max is the maximum degree of the monomials to be checked.

Licensing:

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

Languages:

wedge_exactness is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

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

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

pyramid_exactness, a Python code which investigates the polynomial exactness of a quadrature rule over the interior of the unit pyramid in 3D.

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

square_exactness, a Python code which investigates the polynomial exactness of quadrature rules for f(x,y) over the interior of a rectangle in 2D.

tetrahedron_exactness, a Python code which investigates the polynomial exactness of a quadrature rule over the interior of a tetrahedron in 3D.

triangle_exactness, a Python code which investigates the polynomial exactness of a quadrature rule over the interior of a triangle in 2D.

wedge_felippa_rule, a Python code which returns a Felippa quadrature rule for approximating integrals over the interior of the unit wedge in 3D.

wedge_grid, a Python code which computes a grid of points over the interior of the unit wedge in 3D.

wedge_integrals, a Python code which returns the exact value of the integral of any monomial over the interior of the unit wedge in 3D.

wedge_monte_carlo, a Python code which uses the Monte Carlo method to estimate integrals 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:

• wedge_exactness.py, the source code.
• wedge_exactness.sh, runs all the tests.
• wedge_exactness.txt, the output file.

WEDGE_FELIPPA_3x2 is a quadrature rule based on a 3 point triangle rule and a 2 point Legendre rule for the line.

• wedge_felippa_3x2_w.txt, the weight file.
• wedge_felippa_3x2_x.txt, the abscissa file.