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

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

Symbolic integration of polynomial functions over a linear polyhedron in euclidean three-dimensional space

The paper concerns analytical integration of polynomial functions over linear polyhedra in three-dimensional space. To the authors' knowledge this is a first presentation of the analytical integration of monomials over a tetrahedral solid in 3D space. A linear polyhedron can be obtained by decomposing it into a set of solid tetrahedrons, but the division of a linear polyhedral solid in 3D space into tetrahedra sometimes presents difficulties of visualization and could easily lead to errors in nodal numbering, etc We have taken this into account and also the linearity property of integration to derive a symbolic integration formula for linear hexahedra in 3D space. We have also used yet another fact that a hexahedron could be built up in two, and only two, distinct ways from five tetrahedral shaped elements These symbolic integration formulas are then followed by an illustrative numerical example for a rectangular prism element, which clearly verifies the formulas derived for the tetrahedron and hexahedron elements

Optical Limiting in Single-walled Carbon Nanotube Suspensions

Optical limiting behaviour of suspensions of single-walled carbon nanotubes in water, ethanol and ethylene glycol is reported. Experiments with 532 nm, 15 nsec duration laser pulses show that optical limiting occurs mainly due to nonlinear scattering. The observed host liquid dependence of optical limiting in different suspensions suggests that the scattering originates from microbubbles formed due to absorption-induced heating.Comment: 10 pages, 5 eps figures, to appear in Chem. Phys. Let