FELIPPA
Finite Element Quadrature Rules

FELIPPA is a FORTRAN90 library which generates the points and weights of a quadrature rule for a variety of 1D, 2D and 3D regions of interest for computations involving the finite element method (FEM).

Regions for which rules are available, with a count of Corners, Edges, and Faces, include:

Name	Acronym	C+E+F
Line segment	Line	2+1+0
Triangle	Trig	3+3+1
Quadrilateral	Quad	4+4+1
Tetrahedron	Tetr	4+6+4
Wedge	Wedg	6+9+5
Pyramid	Pyra	5+8+5
Hexahedron	Hexa	8+12+6

Licensing:

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

Languages:

FELIPPA is available in a C++ version and a FORTRAN90 version and a MATHEMATICA version and a MATLAB version.

Related Data and Programs:

DUNAVANT, a FORTRAN90 library which defines Dunavant rules for quadrature on a triangle.

FEKETE, a FORTRAN90 library which defines a Fekete rule for quadrature or interpolation over a triangle.

GM_RULE, a FORTRAN90 library which defines Grundmann-Moeller rules for quadrature over a triangle, tetrahedron, or general M-dimensional simplex.

KEAST, a FORTRAN90 library which defines quadrature rules for a tetrahedron.

LYNESS_RULE, a FORTRAN90 library which returns Lyness-Jespersen quadrature rules for the triangle.

NCC_TETRAHEDRON, a FORTRAN90 library which defines Newton-Cotes closed quadrature rules on a tetrahedron.

NCC_TRIANGLE, a FORTRAN90 library which defines Newton-Cotes closed quadrature rules on a triangle.

NCO_TETRAHEDRON, a FORTRAN90 library which defines Newton-Cotes open quadrature rules on a tetrahedron.

NCO_TRIANGLE, a FORTRAN90 library which defines Newton-Cotes open quadrature rules on a triangle.

PRODUCT_RULE, a FORTRAN90 program which can create a multidimensional quadrature rule as a product of one dimensional rules.

PYRAMID_RULE, a FORTRAN90 program which computes a quadrature rule for a pyramid.

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

QUADRATURE_RULES_TET, a dataset directory which contains quadrature rules for tetrahedrons, stored as a file of abscissas, a file of weights, and a file of vertices.

QUADRATURE_RULES_TRI, a dataset directory which contains quadrature rules for triangles, stored as a file of abscissas, a file of weights, and a file of vertices.

QUADRATURE_RULES_WEDGE, a dataset directory which contains quadrature rules for a wedge (a triangle x a straight line).

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

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

TETRAHEDRON_MONTE_CARLO, a FORTRAN90 program which uses the Monte Carlo method to estimate integrals over a tetrahedron.

TETRAHEDRONS, a dataset directory which contains examplesof tetrahedrons;

TRIANGLE_MONTE_CARLO, a FORTRAN90 program which uses the Monte Carlo method to estimate integrals over a triangle.

WANDZURA, a FORTRAN90 library which defines Wandzura rules for quadrature on a triangle.

Reference:

1. Carlos Felippa,
   A compendium of FEM integration formulas for symbolic work,
   Engineering Computation,
   Volume 21, Number 8, 2004, pages 867-890.

Source Code:

• felippa.f90, the source code.
• felippa.sh, commands to compile the source code.

Examples and Tests:

• felippa_prb.f90, a sample calling program.
• felippa_prb.sh, commands to compile and run the sample program.
• felippa_prb_output.txt, the output file.

List of Routines:

• COMP_NEXT computes the compositions of the integer N into K parts.
• HEXA_UNIT_MONOMIAL integrates a monomial over the unit hexahedron.
• HEXA_UNIT_MONOMIAL_TEST tests HEXA_UNIT_MONOMIAL.
• HEXA_UNIT_QUAD_TEST tests the rules for the unit hexahedron.
• HEXA_UNIT_RULE returns a quadrature rule for the unit hexahedron.
• HEXA_UNIT_VOLUME: volume of a unit hexahedron.
• LINE_UNIT_MONOMIAL: monomial integral in a unit line.
• LINE_UNIT_MONOMIAL_TEST tests LINE_UNIT_MONOMIAL.
• LINE_UNIT_O01 returns a 1 point quadrature rule for the unit line.
• LINE_UNIT_O02 returns a 2 point quadrature rule for the unit line.
• LINE_UNIT_O03 returns a 3 point quadrature rule for the unit line.
• LINE_UNIT_O04 returns a 4 point quadrature rule for the unit line.
• LINE_UNIT_O05 returns a 5 point quadrature rule for the unit line.
• LINE_UNIT_QUAD_TEST tests the rules for the unit line.
• LINE_UNIT_VOLUME: volume of a unit line.
• MONOMIAL_VALUE evaluates a monomial.
• PYRA_UNIT_MONOMIAL: monomial integral in a unit pyramid.
• PYRA_UNIT_MONOMIAL_TEST tests PYRA_UNIT_MONOMIAL.
• PYRA_UNIT_O01 returns a 1 point quadrature rule for the unit pyramid.
• PYRA_UNIT_O05 returns a 5 point quadrature rule for the unit pyramid.
• PYRA_UNIT_O06 returns a 6 point quadrature rule for the unit pyramid.
• PYRA_UNIT_O08 returns an 8 point quadrature rule for the unit pyramid.
• PYRA_UNIT_O08B returns an 8 point quadrature rule for the unit pyramid.
• PYRA_UNIT_O09 returns a 9 point quadrature rule for the unit pyramid.
• PYRA_UNIT_O13 returns a 13 point quadrature rule for the unit pyramid.
• PYRA_UNIT_O18 returns an 18 point quadrature rule for the unit pyramid.
• PYRA_UNIT_O27 returns a 27 point quadrature rule for the unit pyramid.
• PYRA_UNIT_O48 returns a 48 point quadrature rule for the unit pyramid.
• PYRA_UNIT_QUAD_TEST tests the rules for the unit pyramid.
• PYRA_UNIT_VOLUME: volume of a unit pyramid with square base.
• QUAD_UNIT_MONOMIAL integrates a monomial over the unit quadrilateral.
• QUAD_UNIT_MONOMIAL_TEST tests QUAD_UNIT_MONOMIAL.
• QUAD_UNIT_QUAD_TEST tests the rules for the unit quadrilateral.
• QUAD_UNIT_RULE returns a quadrature rule for the unit quadrilateral.
• QUAD_UNIT_VOLUME: volume of a unit quadrilateral.
• R8_CHOOSE computes the binomial coefficient C(N,K) as an R8.
• R8_MOP returns the I-th power of -1 as an R8.
• R8MAT_DET_4D computes the determinant of a 4 by 4 matrix.
• R8MAT_WRITE writes an R8MAT file.
• R8VEC_DIRECT_PRODUCT creates a direct product of R8VEC's.
• R8VEC_DIRECT_PRODUCT2 creates a direct product of R8VEC's.
• SUBCOMP_NEXT computes the next subcomposition of N into K parts.
• TETR_UNIT_MONOMIAL integrates a monomial over the unit tetrahedron.
• TETR_UNIT_MONOMIAL_TEST tests TETR_UNIT_MONOMIAL.
• TETR_UNIT_O01 returns a 1 point quadrature rule for the unit tetrahedron.
• TETR_UNIT_O04 returns a 4 point quadrature rule for the unit tetrahedron.
• TETR_UNIT_O08 returns an 8 point quadrature rule for the unit tetrahedron.
• TETR_UNIT_O08B returns an 8 point quadrature rule for the unit tetrahedron.
• TETR_UNIT_O14 returns a 14 point quadrature rule for the unit tetrahedron.
• TETR_UNIT_O14B returns a 14 point quadrature rule for the unit tetrahedron.
• TETR_UNIT_O15 returns a 15 point quadrature rule for the unit tetrahedron.
• TETR_UNIT_O15B returns a 15 point quadrature rule for the unit tetrahedron.
• TETR_UNIT_O24 returns a 24 point quadrature rule for the unit tetrahedron.
• TETR_UNIT_QUAD_TEST tests the rules for the unit tetrahedron.
• TETR_UNIT_VOLUME returns the volume of the unit tetrahedron.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TRIG_UNIT_MONOMIAL integrates a monomial over the unit triangle.
• TRIG_UNIT_MONOMIAL_TEST tests TRIG_UNIT_MONOMIAL.
• TRIG_UNIT_O01 returns a 1 point quadrature rule for the unit triangle.
• TRIG_UNIT_O03 returns a 3 point quadrature rule for the unit triangle.
• TRIG_UNIT_O03B returns a 3 point quadrature rule for the unit triangle.
• TRIG_UNIT_O06 returns a 6 point quadrature rule for the unit triangle.
• TRIG_UNIT_O06B returns a 6 point quadrature rule for the unit triangle.
• TRIG_UNIT_O07 returns a 7 point quadrature rule for the unit triangle.
• TRIG_UNIT_O12 returns a 12 point quadrature rule for the unit triangle.
• TRIG_UNIT_QUAD_TEST tests the rules for the unit triangle.
• TRIG_UNIT_VOLUME: volume of a unit triangle.
• WEDG_UNIT_MONOMIAL: monomial integral in a unit wedge.
• WEDG_UNIT_MONOMIAL_TEST tests WEDG_UNIT_MONOMIAL.
• WEDG_UNIT_QUAD_TEST tests the rules for the unit wedge.
• WEDG_UNIT_RULE returns a quadrature rule for the unit wedge.
• WEDG_UNIT_VOLUME: volume of a unit wedge.

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