Gauss legendre quadrature formulas over a tetrahedron

Abstract

Abstract In this article we consider the Gauss Legendre Quadrature method for numerical integration over the standard tetrahedron: {( x , y , z )|0 ≤ x , y , z ≤ 1, x + y + z ≤ 1} in the Cartesian three‐dimensional ( x , y , z ) space. The mathematical transformation from the ( x , y , z ) space to (ξ, η, ζ) space is described to map the standard tetrahedron in ( x , y , z ) space to a standard 2‐cube: {(ξ, η, ζ)| − 1 ≤ ζ, η, ζ ≤ 1} in the (ξ, η, ζ) space. This overcomes the difficulties associated with the derivation of new weight coefficients and sampling points. The effectiveness of the formulas is demonstrated by applying them to the integration of three nonpolynomial, three polynomial functions and to the evaluation of integrals for element stiffness matrices in linear three‐dimensional elasticity. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006