This paper presents a numerical integration formula for the evaluation of, where and is any polygonal domain inThat is a domain with boundary composed of piecewise straight lines.We then

express

in 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 a=0,1,2 which are obtained by joining the centroid to the midpoint of its sides. We choose  an arbitrary triangle with vertices in Cartesian spaceWe have furthered refined this mesh two times.The first refinement of this mesh was considered in our recent work.In this study we propose a second refinement of the above stated all quadrilateral mesh.

We have shown that when, the triangle is divided into three quadrilaterals  ,e=1,2,3 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