Gauss Legendre-Gauss Jacobi quadrature rules over a tetrahedral region

This paper presents a Gaussian Quadrature method for the evaluation of the triple integral View the MathML source, where f(x,y,z) is an analytic function in x, y, z and T refers to the standard tetrahedral region: {(x,y,z)|0⩽x,y,z⩽1,x+y+z⩽1} in three space (x,y,z). Mathematical transformation from (x,y,z) space to (U,V,W) space map the standard tetrahedron T in (x,y,z) space to a standard 1-cube: {(U,V,W)/0⩽U,V,W⩽1} in (U,V,W) space. Then we use the product of Gauss Legendre and Gauss Jacobi weight coefficients and abscissas to arrive at an efficient quadrature rule over the standard tetrahedral region T. We have then demonstrated the application of the derived quadrature rules by considering the evaluation of some typical triple integrals over the region T

Gauss Legendre - Gauss Jacobi quadrature rules over a Tetrahedral region

This paper presents a Gaussian quadrature method for the evaluation of the triple integral I = â«â«â«/T f (x, y, z) dxdydz, where f (x, y, z) is an analytic function in x, y, z and T refers to the standard tetrahedral region:(x, y, z) | 0 ⤠x, y, z â¤1, x + y + z â¤1 in three space (x, y, z). Mathematical transformation from (x, y, z) space to (u, v, w) space maps the standard tetrahedron T in (x, y, z) space to a standard 1-cube: (u,v,w) / 0 ⤠u, v, w â¤1 in (u, v, w) space. Then we use the product of Gauss-Legendre and Gauss-Jacobi weight coefficients and abscissas to arrive at an efficient quadrature rule over the standard tetrahedral region T. We have then demonstrated the application of the derived quadrature rules by considering the evaluation of some typical triple integrals over the region T

Gauss Legendre Quadrature Formulae for Tetrahedra

In this paper 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 ≤ ξ, η, ζ ≤ 1} in the (ξ, η, ζ) space. This overcomes the difficulties associated with the derivation of new weight co-efficients and sampling points. The effectiveness of the formulae is demonstrated by applying them to the integration of three nonpolynomial and three polynomial functions

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

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

Numerical integration of some functions over an arbitrary linear tetrahedra in Euclidean three-dimensional space

In this paper it is proposed to compute the volume integral of certain functions whose antiderivates with respect to one of the variates (say either x or y or z ) is available. Then by use of the well known Gauss Divergence theorem, it can be shown that the volume integral of such a function is expressible as sum of four integrals over the unit triangle. The present method can also evaluate the triple integrals of trivariate polynomials over an arbitrary tetrahedron as a special case. It is also demonstrated that certain integrals which are nonpolynomial functions of trivariates x,y,z can be computed by the proposed method. We have applied Gauss Legendre Quadrature rules which were recently derived by Rathod et al. [H.T. Rathod, K.V. Nagaraja, B. Venkatesudu, N.L. Ramesh, Gauss Legendre Quadrature over a Triangle, J. Indian Inst. Sci. 84 (2004) 183–188] to evaluate the typical integrals governed by the proposed method

Efficient Tensor-Product Spectral-Element Operators with the Summation-by-Parts Property on Curved Triangles and Tetrahedra

We present an extension of the summation-by-parts (SBP) framework to tensor-product spectral-element operators in collapsed coordinates. The proposed approach enables the construction of provably stable discretizations of arbitrary order which combine the geometric flexibility of unstructured triangular and tetrahedral meshes with the efficiency of sum-factorization algorithms. Specifically, a methodology is developed for constructing triangular and tetrahedral spectral-element operators of any order which possess the SBP property (i.e. satisfying a discrete analogue of integration by parts) as well as a tensor-product decomposition. Such operators are then employed within the context of discontinuous spectral-element methods based on nodal expansions collocated at the tensor-product quadrature nodes as well as modal expansions employing Proriol-Koornwinder-Dubiner polynomials, the latter approach resolving the time step limitation associated with the singularity of the collapsed coordinate transformation. Energy-stable formulations for curvilinear meshes are obtained using a skew-symmetric splitting of the metric terms, and a weight-adjusted approximation is used to efficiently invert the curvilinear modal mass matrix. The proposed schemes are compared to those using non-tensorial multidimensional SBP operators, and are found to offer comparable accuracy to such schemes in the context of smooth linear advection problems on curved meshes, but at a reduced computational cost for higher polynomial degrees.Comment: 26 pages, 5 figure