NUMERICAL INTEGRATION OVER A LINEAR CONVEX POLYHEDRON USING AN ALL HEXAHEDRAL DISCRETISATION AND GAUSS LEGENDRE FORMULAS (International Journal of Engineering and Computer Science, 2017)
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.