On the Application of Two Symmetric Gauss Legendre Quadrature Rules for Composite Numerical Integration Over a Tetrahedral Region
H.T. Rathod, B. Venkatesudu, Kallur Venkat Nagaraja · International Journal for Computational Methods in Engineering Science and Mechanics · 2006
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 the transformations x = x(ξ,η,ζ), y = y (ξ,η,ζ) and z = z(ξ,η,ζ) to change the integral I into an equivalent integral I = ∫− 1 1∫− 1 1∫− 1 1f(x(ξ,η,ζ),y(ξ,η,ζ),z(ξ,η,ζ)) dξ dη dζ 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 rule 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 Ti c ( i = 1,2,3,4) of equal size, which are obtained by joining centroid of T, c = (1/4, 1/4, 1/4) to the four vertices of T. Use of the affine transformations defined over each Ti c and the linearity property of integrals leads to the result: where refer to the affine transformations, which map each Ti c into the standard tetrahedral region T. We then write and 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/(6 p3) 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: where 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 4 p3 tetrahedra of equal volume. We have demonstrated this aspect by applying the above composite integration method to some typical triple integrals.