PYRAMID_RULE
Generate a Quadrature Rule for a Pyramid

PYRAMID_RULE is a FORTRAN90 program which generates a quadrature rule for a pyramid.

The quadrature rules generated by PYRAMID_RULE are all examples of conical product rules, and involve a kind of direct product of the form:

Legendre rule in X * Legendre rule in Y * Jacobi rule in Z

where the Jacobi rule includes a factor of (1-Z)^2.

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_rule legendre_order jacobi_order filename

where

legendre_order is the order of the 1D Legendre quadrature rule to be used. The X and Y dimensions will use a product of this rule.
jacobi_order is the order of the 1D Jacobi quadrature rule to be used. The Z dimension will use this rule which will include a factor of (1-X)^2 which accounts for the narrowing of the pyramid.
filename is the common prefix for the files containing the region, weight and abscissa information of the quadrature rule;

Licensing:

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

Related Data and Programs:

FELIPPA is a FORTRAN90 library which defines quadrature rules for lines, triangles, quadrilaterals, pyramids, wedges, tetrahedrons and hexahedrons.

GEOMETRY is a FORTRAN90 library which performs geometric calculations in 2, 3 and N dimensional space.

JACOBI_RULE, is a FORTRAN90 program which can compute and print a Gauss-Jacobi quadrature rule.

LEGENDRE_RULE, is a FORTRAN90 program which can compute and print a Gauss-Legendre quadrature rule.

LEGENDRE_RULE_FAST, is a FORTRAN90 program which uses a fast (order N) algorithm to compute a Gauss-Legendre quadrature rule of given order.

NINT_EXACTNESS_PYRAMID is a FORTRAN90 program which investigates the polynomial exactness of a quadrature rule for the pyramid.

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

QUADRATURE_RULES_PYRAMID, a dataset directory which contains quadrature rules for a pyramid with a square base.

QUADRULE is a FORTRAN90 library which defines quadrature rules on a variety of intervals with different weight functions.

STROUD is a FORTRAN90 library which defines quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and N-dimensions.

Reference:

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

Source Code:

pyramid_rule.f90, the source code.
pyramid_rule.csh, commands to compile the source code.

List of Routines:

MAIN is the main program for PYRAMID_RULE.
DTABLE_WRITE0 writes a DTABLE file with no headers.
GET_UNIT returns a free FORTRAN unit number.
JACOBI_COMPUTE computes a Gauss-Jacobi quadrature rule.
JACOBI_RECUR finds the value and derivative of a Jacobi polynomial.
JACOBI_ROOT improves an approximate root of a Jacobi polynomial.
LEGENDRE_COMPUTE computes a Gauss-Legendre quadrature rule.
PYRAMID_HANDLE computes the requested pyramid rule and outputs it.
R8_EPSILON returns the R8 roundoff unit.
R8_GAMMA evaluates Gamma(X) for a real argument.
S_TO_I4 reads an I4 from a string.
TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.

Last revised on 23 July 2009.