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 fðn; gÞj0 6 n; g 6 1; n þ g 6 1g in the local parametric space ðn; gÞ. 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 {ðn þ 1Þðn þ 2Þ=2} rational integrals of monomial numerators with the same linear denominator. By an expansion it is shown that these {ðn þ 1Þðn þ 2Þ=2} integrals can be computed in two ways. We have presented a recursive scheme to compute {nðn þ 1Þ=2} such integrals efficiently whenever ðn þ 1Þ 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. Ó

The use of parabolic arcs in matching curved boundaries by point transformations for some higher order triangular elements

“…(4) reduces to a pair of parametric equations for x and y along the curved boundary and they are quadratic polynomials, either in or (parametric variates, 0 , 1). Let us assume that the given curved boundary can be approximated by a general conic [14], that is, the equation, (say)…”

“…(3a)-(3d) as derived in the previous section. The quadratic and cubic point transformations have been the subject of intensive research in the earlier works by several authors [11][12][13][14].…”

“…Now choose t 13 , t 14 and t 15 such that a (t)=0, so that the above Eq. (28) simplifies to the quadratic form:…”

“…Further, they also present a simple and systematic procedure to choose the parameters of the cubic curves so that the implicit equations of the curves always represent the parabola passing through four points of the original curves and so is a reasonable approximation to it. The development is put to practical use in the recent works of Rathod and Karim [13,14]. It is the purpose of this paper to find equations for point transformation of quartic and quintic arcs using isoparametric coordinate transformations and also to choose the parameters (coordinates of the points on the curved side) in a systematic way so that the implied curves are always a parabola passing through four points (quartic and quintic arcs) of the original curves.…”

Um programa de elementos finitos em GPU e orientado a objetos para análise dinâmica não linear de estruturas.

“…São produtos de uma extensa linha de estudo que, na segunda metade da década de 90, avaliou-se ser composta por mais de 56000 artigos, 380 livros e 400 anais de congressos [148]. É possível afirmar que o Método dos Elementos Finitos aplicado à análise de estruturas sólidas está em constante evolução, com o desenvolvimento de formulações de elementos mais aperfeiçoadas [79,86,89,105,135,182,197,203,232,246], novos modelos de materiais [14,67,168], propostas de algoritmos numéricos [171,200,251], otimizações [44,53,181], notas de aplicação [3,206], entre outros tópicos.…”