Gauss Legendre Quadrature Formulas for Polygons Using an All Quadrilateral Finite Element Mesh of Triangular Surfaces (International Journal of Engineering and Computer Science, 2017)
This paper presents a numerical integration formula for the evaluation of , where and is any polygonal domain in That is a domain with boundary composed of piecewise straight lines.We then expressin which  is a polygonal domain of N oriented edges with end points    and . We have also assumed that can be discretised into a set of M triangles,  and each triangle  is further discretised into three special quadrilaterals  which are obtained by joining the centroid to the midpoint of its sides. We choose  an arbitrary triangle with vertices  in Cartesian space We have shown that an efficient formula for this purpose is given by                                                                    where,    u= u(ξ,η)=(1-ξ)(5+η)/24,v=v(ξ,η)=(1-η)(5+ξ)/24   and is the standard 2 square in space. This 2 square S in  is discritised into four 1squares in (ξ,η) space.We then use four linear transformations sing the Gauss Legendre Quadrature Rules of order 5(5)40, we obtain the weight coefficients and sampling points which can be used for any polygonal domain,  or  or  (m=3n-2,3n-1,3n;b=1,2,3,4)The present composite integration scheme integrates gives the same accuracy for half the number of triangles for each discretisation  of a polygon used in our earlier work[16].