FELIPPA
Finite Element Quadrature Rules

FELIPPA is a MATLAB 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.

Related Data and Programs:

CLENSHAW_CURTIS is a MATLAB library which sets up a Clenshaw Curtis quadrature grid in multiple dimensions.

DUNAVANT is a MATLAB library which defines Dunavant rules for quadrature on a triangle.

FEKETE is a MATLAB library which defines a Fekete rule for quadrature or interpolation over a triangle.

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

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

KEAST is a MATLAB library which defines quadrature rules for a tetrahedron.

NCC_TETRAHEDRON is a MATLAB library which defines Newton-Cotes closed quadrature rules on a tetrahedron.

NCC_TRIANGLE is a MATLAB library which defines Newton-Cotes closed quadrature rules on a triangle.

NCO_TETRAHEDRON is a MATLAB library which defines Newton-Cotes open quadrature rules on a tetrahedron.

NCO_TRIANGLE is a MATLAB library which defines Newton-Cotes open quadrature rules on a triangle.

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

PYRAMID_RULE, is a MATLAB 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 ( triangle x a line ).

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

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

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

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

WANDZURA is a MATLAB 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.

Tar File:

A GZIP'ed TAR file of the contents of this directory is available.

• felippa_m_src.tar.csh the commands used to create the GZIP'ed TAR file.
• felippa_m_src.tar.gz the GZIP'ed TAR file.

Source Code:

• comp_next.m, computes the compositions of the integer N into K parts.
• hexa_unit_monomial.m, returns the exact integral of a monomial in a unit hexahedron;
• hexa_unit_monomial_test.m, tests HEXA_UNIT_MONOMIAL.
• hexa_unit_quad_test.m, tests the quadrature rules for the unit hexahedron.
• hexa_unit_rule.m, returns a quadrature rule for the unit hexahedron;
• hexa_unit_volume.m, returns the volume of a unit hexahedron;
• line_unit_monomial.m, returns the exact integral of a monomial in a unit line;
• line_unit_monomial_test.m, tests LINE_UNIT_MONOMIAL.
• line_unit_o01.m, returns a 1 point quadrature rule for the unit line.
• line_unit_o02.m, returns a 2 point quadrature rule for the unit line.
• line_unit_o03.m, returns a 3 point quadrature rule for the unit line.
• line_unit_o04.m, returns a 4 point quadrature rule for the unit line.
• line_unit_o05.m, returns a 5 point quadrature rule for the unit line.
• line_unit_quad_test.m, tests the quadrature rules for the unit line.
• line_unit_volume.m, returns the volume of a unit line;
• monomial_value.m, evaluates a monomial.
• pyra_unit_monomial.m, returns the exact integral of a monomial in a unit pyramid;
• pyra_unit_monomial_test.m, tests PYRA_UNIT_MONOMIAL.
• pyra_unit_o01.m, returns a 1 point quadrature rule for the unit pyramid.
• pyra_unit_o05.m, returns a 5 point quadrature rule for the unit pyramid.
• pyra_unit_o06.m, returns a 6 point quadrature rule for the unit pyramid.
• pyra_unit_o08.m, returns an 8 point quadrature rule for the unit pyramid.
• pyra_unit_o08b.m, returns an 8 point quadrature rule for the unit pyramid.
• pyra_unit_o09.m, returns a 9 point quadrature rule for the unit pyramid.
• pyra_unit_o13.m, returns a 13 point quadrature rule for the unit pyramid.
• pyra_unit_o18.m, returns an 18 point quadrature rule for the unit pyramid.
• pyra_unit_o27.m, returns a 27 point quadrature rule for the unit pyramid.
• pyra_unit_o48.m, returns a 48 point quadrature rule for the unit pyramid.
• pyra_unit_quad_test.m, tests the quadrature rules for the unit pyramid.
• pyra_unit_volume.m, returns the volume of a unit pyramid;
• quad_unit_monomial.m, returns the exact integral of a monomial in a unit quadrilateral;
• quad_unit_monomial_test.m, tests QUAD_UNIT_MONOMIAL.
• quad_unit_quad_test.m, tests the quadrature rules for the unit quadrilateral.
• quad_unit_rule.m, returns a quadrature rule for the unit quadrilateral;
• quad_unit_volume.m, returns the volume of a unit quadrilateral;
• r8_choose.m, computes the binomial coefficient C(N,K) as an R8.
• r8_mop.m, returns the I-th power of -1 as an R8.
• r8mat_write.m, writes an R8MAT file.
• r8vec_direct_product.m, creates a direct product of R8VEC's.
• r8vec_direct_product2.m, creates a direct product of R8VEC's.
• subcomp_next.m, computes the next subcomposition of N into K parts.
• tetr_unit_monomial.m, returns the exact integral of a monomial in a unit tetrahedron;
• tetr_unit_monomial_test.m, tests TETR_UNIT_MONOMIAL.
• tetr_unit_o01.m, returns a 1 point quadrature rule for the unit tetrahedron.
• tetr_unit_o04.m, returns a 4 point quadrature rule for the unit tetrahedron.
• tetr_unit_o08.m, returns an 8 point quadrature rule for the unit tetrahedron.
• tetr_unit_o08b.m, returns an 8 point quadrature rule for the unit tetrahedron.
• tetr_unit_o14.m, returns a 14 point quadrature rule for the unit tetrahedron.
• tetr_unit_o14b.m, returns a 14 point quadrature rule for the unit tetrahedron.
• tetr_unit_o15.m, returns a 15 point quadrature rule for the unit tetrahedron.
• tetr_unit_o15.m, returns a 15 point quadrature rule for the unit tetrahedron.
• tetr_unit_o24.m, returns a 24 point quadrature rule for the unit tetrahedron.
• tetr_unit_quad_test.m, tests the quadrature rules for the unit tetrahedron.
• tetr_unit_volume.m, computes the volume of the unit tetrahedron;
• timestamp.m, prints the YMDHMS date as a timestamp;
• trig_unit_monomial.m, returns the exact integral of a monomial in a unit triangle;
• trig_unit_monomial_test.m, tests TRIG_UNIT_MONOMIAL.
• trig_unit_o01.m, returns a 1 point quadrature rule for the unit triangle.
• trig_unit_o03.m, returns a 3 point quadrature rule for the unit triangle.
• trig_unit_o03b.m, returns a 3 point quadrature rule for the unit triangle.
• trig_unit_o06.m, returns a 6 point quadrature rule for the unit triangle.
• trig_unit_o06b.m, returns a 6 point quadrature rule for the unit triangle.
• trig_unit_o07.m, returns a 7 point quadrature rule for the unit triangle.
• trig_unit_o12.m, returns a 12 point quadrature rule for the unit triangle.
• trig_unit_quad_test.m, tests the quadrature rules for the unit triangle.
• trig_unit_volume.m, returns the volume of the unit triangle;
• wedg_unit_monomial.m, returns the exact integral of a monomial in a unit wedge;
• wedg_unit_monomial_test.m, tests WEDG_UNIT_MONOMIAL.
• wedg_unit_quad_test.m, tests the quadrature rules for the unit wedge.
• wedg_unit_rule.m, returns a quadrature rule for the unit wedge;
• wedg_unit_volume.m, returns the volume of a unit wedge;
• wedg_unit_write_test.m, writes some wedge quadrature rules to files.

Examples and Tests:

• felippa_test.m, runs all the tests.
• felippa_test_output.txt, the output file.

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