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

Abstract

This paper is concerned with explicit integration formulas and algorithms for computing integrals of trivariate polynomials over an arbitrary linear tetrahedron in Euclidean three-dimensional space. This basic three-dimensional integral governing the problem is transformed to surface integrals by use of the divergence theorem. The resulting two-dimensional integrals are then transformed into convenient and computationally efficient line integrals. These algorithms and explicit finite integration formulas are followed by an application—example for which we have explained the detailed computational scheme. The numerical result thus found is in complete agreement with previous works. Further, it is shown that the present algorithms are much simpler and more economical as well, in terms of arithmetic operations. The symbolic finite integration formulas presented in this paper may lead to an easy incorporation of geometric properties of solid objects, for example, the centre of mass, moment of inertia, etc. required in the engineering design process as well as several applications of numerical analysis where integration is required, for example in the finite element and boundary integral equation methods.