AN EXPLICIT INTEGRATION SCHEME BASED ON RECURSION AND MATRIX MULTIPLICATION FOR THE LINEAR CONVEX QUADRILATERAL ELEMENTS

In this paper we present some analytical integration formulae for computing integrals of rational functions of bivariate polynomial numerator with linear and bilinear denominator over a 2-square |xi| = 1, |eta| = 1 in the local parametric space (xi,eta). These integrals arise in finite element formulations of second order partial differential equations for plane and axisymmetric problems in continuum mechanics to compute the components of element stiffness matrices. In case of a rational integrals of n-th degree bivariate polynomial numerator with a linear or bilinear denominator there are exactly 1/2(n+1)(n+2) rational integrals of monomial numerators with the same linear denominator. By an expansion it is shown that these 1/2(n+1)(n+2) integrals can be computed in two ways, accordingly we have presented an explicit and a recursive scheme. By use of the recursive scheme 1/2n(n+1) such integrals can be computed efficiently with less computational effort whenever (n+1) integrals of order zero to n in one of the variates are known by explicit integration formulae. Integration formulae from zeroth to octic order are, for clarity and reference summarized in tabular forms. Finally to show the application of the derived formulae three application examples to compute the Prandtl stress function values and the torsional constant k are considered. A computer code based on the present integration scheme to obtain the element stiffness matrices for plane problems is also developed.