Numerical integrations over an arbitrary quadrilateral region

In this paper, double integrals over an arbitrary quadrilateral are evaluated exploiting finite element method. The physical region is transformed into a standard quadrilateral finite element using the basis functions in local space. Then the standard quadrilateral is subdivided into two triangles, and each triangle is further discretized into 4 × n2 right isosceles triangles, with area 12n2, and thus composite numerical integration is employed. In addition, the affine transformation over each discretized triangle and the use of linearity property of integrals are applied. Finally, each isosceles triangle is transformed into a 2-square finite element to compute new n2 extended symmetric Gauss points and corresponding weight coefficients, where n is the lower order conventional Gauss Legendre quadratures. These new Gauss points and weights are used to compute the double integral. Examples are considered over an arbitrary domain, and rational and irrational integrals which can not be evaluated analytically.