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)dxdydz, where f(x, y, z) is an analytic function in x, y, z and T is the standard tetrahedral region: {(x, y, z)∣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 I=∫-11∫-11∫-11f(x(ξ,η,ζ),y(ξ,η,ζ),z(ξ,η,ζ))∂(x,y,z)∂(ξ,η,ζ)dξdηdζ over the standard 2-cube in (ξ, η, ζ) space: {(ξ, η, ζ)∣ −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 Tic (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 Tic and the linearity property of integrals leads to the result:I=∑i=14∫∫Tic∫f(x,y,z)dxdydz=14∫∫T∫G(X,Y,Z)dXdYdZ,whereG(X,Y,Z)=1p3∑k=14f(x(k)(X,Y,Z),y(k)(X,Y,Z),z(k)(X,Y,Z)),x(k)=x(k)(X,Y,Z),y(k)=y(k)(X,Y,Z)andz(k)=z(k)(X,Y,Z)refer to an affine transformations which map each Tic into the standard tetrahedral region T.We then writeI=∫∫T∫G(X,Y,Z)dXdYdZ=∫01∫01-ξ∫01-ξ-ηG(X(ξ,η,ζ),Y(ξ,η,ζ),Z(ξ,η,ζ))∂(X,Y,Z)∂(ξ,η,ζ)dξdηdζand a composite rule of integration is thus obtained. We next propose the discretisation of the standard tetrahedral region T into p3 tetrahedra Ti (i = 1(1)p3) each of which has volume equal to 1/(6p3) units. We have again shown that the use of affine transformations over each Ti and the use of linearity property of integrals leads to the result:∫∫T∫f(x,y,z)dxdydz=∑i=1p3∫∫Tic∫f(x,y,z)dxdydz=∑α=1p3∫∫Tα(p)∫f(x(α,p),y(α,p),z(α,p))dx(α,p)dy(α,p)dz(α,p)=1p3∫∫T∫H(X,Y,Z)dXdYdZ,whereH(X,Y,Z)=∑α=1P3f(x(α,P)(X,Y,Z),y(α,P)(X,Y,Z),z(α,P)(X,Y,Z)),x(α,p)=x(α,p)(X,Y,Z),y(α,p)=y(α,p)(X,Y,Z)andz(α,p)=z(α,p)(X,Y,Z)refer to the affine transformations which map each Ti 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)dXdYdZ, this amounts to the application of composite numerical integration of T into p3 and 4p3 tetrahedra of equal sizes. We have demonstrated this aspect by applying the above composite integration method to some typical triple integrals.