Integration of rational functions of bivariate polynomial numerators with linear denominators over a (−1,1) square in a local parametric two-dimensional space

Abstract

This paper is concerned with explicit formulas and algorithms for computing integral of rational function of bivariate polynomial numerators with linear denominators over a (−1,1) square in the local parametric space. These integrals arise in finite element formulations of second-order partial differential equations. Explicit evaluation of these integrals produce analytical finite element relations, provided that the original element geometry is restricted to a linear convex quadrilateral. We have also presented two algorithms, either of these can be used to compute n(n + 1) 2 integrals of 0th to nth order bivariate polynomial numerator whenever ( n + 1) such integrals of order 0 (zero) to n in one of the variates are known by explicit integration formulas. The analytical quadrature formulas from cubic to quintic monomial numerator in one of the variâtes due to different element geometry are, for clarity and reference, summarized in tabular forms. All the explicit formulas are constructed with three simple functions of element nodal values. They can be easily coded and save much computation time besides their inherent merit on numerical accuracy associated with analytical integration. We have also derived some integration formulas for the product of global derivatives which has applications to second-order linear partial differential equations in a variety of disciplines. Finally, an application example to compute the torsional constant K for an equilateral triangular cross-section is also considered for which we have explained the detailed computational scheme.