An explicit integration scheme based on recursion for the curved triangular finite elements

Abstract

This paper is concerned with explicit formulae for computing integrals of rational functions of bivariate polynomial numerator with linear denominator over a unit triangle {( ξ, η)|0⩽ ξ, η⩽1, ξ+ η⩽1} in the local parametric space ( ξ, η). These integrals arise in a variety of applications governing the second order linear partial differential equations. Explicit evaluation of these integrals produce algebraic finite element relations, provided that the original element geometry is restricted to a triangle with two straight sides and one curved side. For a rational integral of nth degree bivariate polynomial numerator with a linear denominator there are {( n+1)( n+2)/2} rational integrals of monomial numerators with the same linear denominator. By an expansion it is shown that these {( n+1)( n+2)/2} integrals can be computed in two ways. We have presented a recursive scheme to compute { n( n+1)/2} such integrals efficiently whenever ( n+1) integrals of order 0 (zero) to n in one of the variates are known by explicit integration formulae. The analytical quadrature formulae from zeroth to sextic order due to different element geometry are, for clarity and reference, summarized in tabular forms. An integration formula is also presented for the evaluation of integrals that occur in the calculation of stiffness matrices when straight sided triangular finite elements are considered. Finally we have considered two application examples: First example is to compute the Prandtl stress function values and the torsional constant k for an elliptic cross-section with varying ratios of major and minor distances. The other example is concerned with numerical solution for the Poisson equation in 2D. It is observed in the calculation of components of element matrices by using the Gauss quadrature rule (e.g. 7-point and 13-point) a remarkable discrepancy occurs if the element geometry contains a concave curved side. A computer code based on the present integration scheme to obtain the element stiffness matrices for both the curved and straight sided elements is appended.