Integration of polynomials over an arbitrary tetrahedron in Euclidean three-dimensional space

In this paper, we present explicit integration formulas and algorithms for computing integrals of polynomials over an arbitrary tetrahedron in Euclidean three-dimensional space. Two different approaches are discussed: the first algorithm/formula is obtained by mapping the arbitrary tetrahedron into a unit orthogonal tetrahedron, while the second algorithm/formula computes the required integral as a sum of four integrals over the unit triangle. These algorithms/formulas are followed by an example for which we have explained the detailed computational scheme. The numerical result thus found is in complete agreement with the previous work. Further, it is shown that the present algorithms are much simpler and more economical as well in terms of arithmetic operations

Generalized Gaussian Quadrature Rules over an arbitrary cube in Euclidean Three-Dimensional Space

Abstract This paper presents a Generalize

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

Integration of polynomials over linear polyhedra in Euclidean three-dimensional space

This paper is concerned with explicit formulas and algorithms for computing integrals of polynomials over a linear polyhedron in Euclidean three-dimensional space. Symbolic formulas for surface and volume integration are given. Two different approaches are discussed: The first algorithm is obtained by transforming a volume integral into a surface integral and then into a parametric line integral while the second algorithm is obtained by transforming a volume integral into a surface integral and then into a parametric double integral. These algorithms and formulas are followed by an application-example for which we have explained the detailed computational scheme. The symbolic results presented in this paper may lead to an easy incorporation of global geometric properties of solid objects, for example, the volume, centre of mass, moments of inertia, required in the engineering design process. © 1995

Integration of polynomials over N-dimensional linear polyhedra

This paper is concerned with explicit integration formulae for computing integrals of n-variate polynomials over linear polyhedra in n-dimensional space ℝn. Two different approaches are discussed; the first set of formulae is obtained by mapping the polyhedron in n-dimensional space ℝn into a standard n-simplex in ℝn, while the second set of formulae is obtained by reducing the n-dimensional integral to a sum of n - 1 dimensional integrals which are n + 1 in number. These formulae are followed by an application example for which we have explained the detailed computational scheme. The symbolic integration formulae presented in this paper may lead to an easy and systematic incorporation of global properties of solid objects, such as, for example, volume, centre of mass, moments of inertia etc., required in engineering design problems. © 1997 Elsevier Science Ltd

Boundary integration of polynomials over an arbitrary linear hexahedron in Euclidean three-dimensional space

This paper is concerned with explicit integration formulas and algorithms for computing volume integrals of trivariate polynomials over an arbitrary linear hexahedron in Euclidean three-dimensional space. Three different approaches are discussed. The first algorithm is obtained by transforming a volume integral into a sum of surface integrals and then into convenient and computationally efficient line integrals. The second algorithm is obtained by transforming a volume integral into a sum of surface integrals over the boundary quadrilaterals. The third algorithm is obtained by transforming a volume integral into a sum of surface integrals over the triangulation of boundary. These algorithms and finite integration formulas are then followed by an application example, for which we have explained the detailed computational scheme. The symbolic finite integration formulas presented in this paper may lead to efficient and easy incorporation of integral properties of arbitrary linear polyhedra required in the engineering design process. © 1998 Elsevier Science S.A. All rights reserved