Numerical Integration Over A Linear Convex Polyhedron Using An All Hexahedral Discretisation And Gauss Legendre Formulas

Abstract

Numerical integration is an important ingradient within many techniques of applied mathematics,engineering and scinietific applications, this is due to the need for accurate and efficient integration schemes over complex integration domains and the arbitrary functions as their corresponding integrands. In this paper,we propose a method to discretise the physical domain in the shape of a linear polyhedron into an assemblage of all hexahedral finite elements. The idea is to generate a coarse mesh of all tetrahedrons for the given domain,Then divide each of these tetrahedron further into a refined mesh of all tetrahedrons, if necessary. Then finally, we divide each of these tetrahedron into four hexahedra.We have further demonstrated that each of these hexahedra can be divided into and hexahedra. This generates an all hexahedral finite element mesh which can be used for various applications In order to achieve this we first establish a relation between the arbitrary linear tetrahedron and the standard tetrahedron.We then decompose the standard tetrahedron into four hexahedra. We transform each of these hexahedra into a 2-cube and discover an interesting fact that the Jacobian of these transformations is same and the transformations are also the same but in different order for all the four hexahedra.This fact can be used with great advantage to generate the numerical integration scheme for the standard tetrahedron and hence for the arbitrary linear tetrahedron. We have proposed three numerical schemes which decompose a arbitrary linear tetrahedron into 4, 4( hexahedra.These numerical schemes are applied to solve typical integrals over a unit cube and irregular heptahedron using Gauss Legendre Quadrature Rules. Matlab codes are developed and appended to this paper.