FELIPPA
FEM Integration Formulas

FELIPPA is a collection of Mathematica routines 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 FORTRAN90 version and a MATHEMATICA version and a MATLAB version.

Related Data and Programs:

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

FEKETE is a FORTRAN90 library which defines Fekete rules for interpolation or quadrature on a triangle.

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

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

NCC_TETRAHEDRON is a FORTRAN90 library which defines Newton-Cotes Closed quadrature rules on a tetrahedron.

NCC_TRIANGLE is a FORTRAN90 library which defines Newton-Cotes Closed quadrature rules on a triangle.

NCO_TETRAHEDRON is a FORTRAN90 library which defines Newton-Cotes Open quadrature rules on a tetrahedron.

NCO_TRIANGLE is a FORTRAN90 library which defines Newton-Cotes Open quadrature rules on a triangle.

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 for 1D domains.

STROUD is a FORTRAN90 library which defines quadrature rules for a variety of multidimensional reqions.

WANDZURA is 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:

• HexaGaussRuleInfo.nb, 5x5x5 quadrature rules for the hexahedron [-1,1]x[-1,1]x[-1,1], products of 1D rules of orders 1, 2, 3, 4, and 5.
• LineGaussRuleInfo.nb, 5 quadrature rules for the line segment [-1,1], of orders 1, 2, 3, 4, and 5.
• PyraGaussRuleInfo.nb, 9 quadrature rules for the pyramid, of orders 1, 5, 6, 8, 8, 9, 13, 18 and 27.
• QuadGaussRuleInfo.nb, 5x5 quadrature rules for the quadrilateral [-1,1]x[-1,1], products of 1D rules of orders 1, 2, 3, 4, and 5.
• TetrGaussRuleInfo.nb, 9 quadrature rules for the tetrahedron, of orders 1, 4, 8, 8, 14, 14, 15, 15, and 24.
• TrigGaussRuleInfo.nb, 7 quadrature rules for the triangle, of orders 1, 3, 3, 6, 6, 7 and 12.
• WedgGaussRuleInfo.nb, 5x7 quadrature rules for the wedge, products of 1D rules for the line and 2D rules for the triangle.

Examples and Tests:

• HexaGaussRuleInfoTest.nb, an example for HexaGaussRuleInfo.
• LineGaussRuleInfoTest.nb, an example for LineGaussRuleInfo.
• PyraGaussRuleInfoTest.nb, an example for PyraGaussRuleInfo.
• QuadGaussRuleInfoTest.nb, an example for QuadGaussRuleInfo.
• TetrGaussRuleInfoTest.nb, an example for TetrGaussRuleInfo.
• TrigGaussRuleInfoTest.nb, an example for TrigGaussRuleInfo.
• WedgGaussRuleInfoTest.nb, an example for WedgGaussRuleInfo.

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