Synthetic division based integration of rational functions of bivariate polynomial numerators with linear denominators over a unit triangle {0 ≤ξ, n ≤ 1, ξ+ n ≤ 1} in the local parametric space ( ξ, n)

摘要

The domain of real problems in mechanics often contains curved boundaries. Curved boundaries are often more accurately modelled by curved finite elements than by straight edged elements, as straight sides are perfectly satisfactory if the domain has a polygonal boundary. Because fewer curved elements are required, the effort needed to obtain a solution is usually reduced. If some parts of the boundary are curved, however, elements with at least one curved side are desirable. Our aim in this paper is to consider the triangular element with two straight sides and one curved side. This paper is concerned with explicit formulae for evaluating integrals of rational functions of bivariate polynomial numerators with linear denominators over a unit triangle {0 ≤ξ, n ≤ 1, ξ+ n ≤ 1} in the local parametric two dimensional space {ξ, n). These integrals arise in finite element formulations of second order linear partial dif ferential equations by use of triangular element with two straight sides and one curved side of quadratic variation which often require relatively large numerical effort to integrate. The curved elements considered here are the four node, six node and ten node triangular elements with one curved side of quadratic variation and the other two sides have straight edges under the isoparametric and sub- parametric transformations, respectively. We have shown that by use of a method similar to synthetic division, the rational integrals of nth order bivariate polynomial numerator with a linear denominator having (n + l)(n + 2)/2 integrals can be reduced to rational i