Analytic and Numeric Integration for Quadrilaterals

Two-dimensional finite element analysis uses linear edge arbitrary quadrilaterals with hyperbolic paraboloid shape functions as an important element. To obtain the element stiffness matrix, such a procedure leads to evaluation of integrals involving irrational polynomials and analytic evaluation of such expressions presents considerable difficulty. Almost without exception, the evaluation of integrals has either been done or suggested by using a numerical quadrature of the Legendre-Gauss type. However, what is clearly lacking is quantitative information pertaining to the order of the quadrature formula to be used for various degrees of artibrariness in a quadrilateral. In the subsequent development, it is shown how the equations can be manipulated to yield a simple compact form amenable to closed-form solution, yielding simple programmable expressions, and how they compare with the quadrature formula.