FELIPPA
Finite Element Quadrature Rules

FELIPPA is a C++ library which can generate quadrature rules (points and weights) 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 C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

PYRAMID_RULE, a C++ program which computes a quadrature rule over the interior of the unit pyramid in 3D.

SIMPLEX_GM_RULE, a C++ library which defines Grundmann-Moeller quadrature rules over the interior of a triangle in 2D, a tetrahedron in 3D, or over the interior of the simplex in M dimensions.

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

TETRAHEDRON_EXACTNESS, a C++ program which investigates the monomial exactness of a quadrature rule over the interior of a tetrahedron in 3D.

TETRAHEDRON_INTEGRALS, a C++ library which returns the exact value of the integral of any monomial over the interior of the unit tetrahedron in 3D.

TETRAHEDRON_KEAST_RULE, a C++ library which defines ten quadrature rules, with exactness degrees 0 through 8, over the interior of a tetrahedron in 3D.

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

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

TETRAHEDRON_NCO_RULE, a C++ library which defines Newton-Cotes Open (NCO) quadrature rules over the interior of a tetrahedron in 3D.

TRIANGLE_DUNAVANT_RULE, a C++ library which defines Dunavant rules for quadrature on a triangle.

TRIANGLE_FEKETE, a C++ library which defines Fekete rules for interpolation or quadrature on a triangle.

TRIANGLE_LYNESS_RULE, a C++ library which returns Lyness-Jespersen quadrature rules for the triangle.

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

TRIANGLE_NCC_RULE, a C++ library which defines Newton-Cotes Closed (NCC) quadrature rules over the interior of a triangle in 2D.

TRIANGLE_NCO_RULE, a C++ library which defines Newton-Cotes Open (NCO) quadrature rules over the interior of a triangle in 2D.

TRIANGLE_WANDZURA_RULE, a C++ 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.cpp, the source code.
• felippa.hpp, the include file.
• felippa.sh, commands to compile the source code.

Examples and Tests:

• felippa_test.cpp, a sample calling program.
• felippa_test.sh, commands to compile and run the sample program.
• felippa_test.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.
• I4_MAX returns the maximum of two I4's.
• I4_MIN returns the minimum of two I4's.
• I4VEC_ODD_ANY is TRUE if any entry of an I4VEC is odd.
• I4VEC_PRODUCT multiplies the entries of an I4VEC.
• 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_ABS returns the absolute value of an R8.
• 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 R8MAT.
• R8MAT_WRITE writes an R8MAT file with no header.
• R8VEC_COPY copies an R8VEC.
• R8VEC_DIRECT_PRODUCT creates a direct product of R8VEC's.
• R8VEC_DIRECT_PRODUCT2 creates a direct product of R8VEC's.
• R8VEC_DOT_PRODUCT computes the dot product of a pair 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 returns the "volume" of the unit triangle in 2D.
• 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.
• WEDG_UNIT_WRITE_TEST writes out some rules for the unit wedge.

You can go up one level to the C++ source codes.