## On the Application of two Gauss Legendre Quadrature Rules for composite Numerical Integration over a Tetrahedral Region

Publication Type : Journal Article

Publisher : Applied Mathematics and Computation

Source : Applied Mathematics and Computation, Vol.190, 21 - 39

Campus : Bengaluru

School : School of Engineering

Department : Mathematics

Year : 2007

Abstract : 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. Keywords: Finite element method; Composite numerical integration; Tetrahedral regions; Gauss–Legendre quadrature rules; Triangular prisms; Standard 2-cube; Standard tetrahedron

Cite this Research Publication : H.T. Rathod, K.V. Nagaraja, B. Venkatesudu,"On the application of two symmetric Gauss Legendre quadrature rules for composite numerical integration over a triangular surface,", Applied Mathematics and Computation, Volume 190, Issue 1, 2007, Pages 21-39, ISSN 0096-3003,