On the application of two Gauss-Legendre quadrature rules for composite numerical integration over a tetrahedral region

摘要

In this paper we first present a Gauss-Legendre quadrature rule for the evaluation of I = ∫ ∫ T ∫ f (x, y, z) d x d y d z, where f(x, y, z) is an analytic function in x, y, z and T is the standard tetrahedral region: {(x, y, z){divides}0 ≤ x, y, z ≤ 1, x + y + z ≤ 1} in three space (x, y, z). We then use a transformation x = x(ξ, η, ζ), y = y(ξ, η, ζ) and z = z(ξ, η, ζ) to change the integral into an equivalent integral {Mathematical expression} over the standard 2-cube in (ξ, η, ζ) space: {(ξ, η, ζ){divides} -1 ≤ ξ, η, ζ ≤ 1}. We then apply the one-dimensional Gauss-Legendre quadrature rules in ξ, η and ζ variables to arrive at an efficient quadrature rule with new weight coefficients and new sampling points. Then a second Gauss-Legendre quadrature rule of composite type is obtained. This rule is derived by discretising the tetrahedral region T into four new tetrahedra T i c (i = 1, 2, 3, 4) of equal size which are obtained by joining the centroid of T, c = (1/4, 1/4, 1/4) to the four vertices of T. By use of the affine transformations defined over each T i c and the linearity property of integrals leads to the result:I = underover(∑, i = 1, 4) ∫ ∫ Tic ∫ f (x, y, z) d x d y d z = frac(1, 4) ∫ ∫ T ∫ G (X, Y, Z) d X d Y d Z,where{Mathematical expression}refer to an affine transformations which map each T i c into the standard tetrahedral region T. We then write{Mathematical expression}and a composite rule of integration is thus obtained. We next propose the discretisation of the standard tetrahedral region T into p 3 tetrahedra T i (i = 1(1)p 3) each of which has volume equal to 1/(6p 3) units. We have again shown that the use of affine transformations over each T i and the use of linearity property of integrals leads to the result:{Mathematical expression}where{Mathematical expression}refer to the affine transformations which map each T i in (x (α,p), y (α,p), z (α,p)) space into a standard tetrahedron T in the (X, Y, Z) space. We can now apply the two rules earlier derived to the integral ∫ ∫ T ∫ H (X, Y, Z) d X d Y d Z, this amounts to the application of composite numerical integration of T into p 3 and 4p 3 tetrahedra of equal sizes. We have demonstrated this aspect by applying the above composite integration method to some typical triple integrals. © 2006 Elsevier Inc. All rights reserved.