Non-conventional formulations for the finite element method

Abstract

This paper reports on hybrid formulations being developed by the Structural Analysis Research Group of Instituto Superior Te Âcnico. Three alternative sets of hybrid ®nite element formulations are presented. They are termed hybrid-mixed, hybrid and hybrid-Trefftz and differ essentially on the ®eld conditions that the approximation functions are constrained to satisfy locally. Two models, namely the displacement and the stress models, are obtained for each formulation depending on whether the tractions or the boundary displacements are the ®eld chosen to implement inter-element continuity. Because they are derived from a strict hybrid approach released from the conventional node conformity concepts, these formulations allow different ®elds to be independently approximated, within certain consistency criteria, and enhance the use of a wide range of approximation functions. For simplicity and objectivity, the description of the approach followed in the derivation of the alternative formulations and models is based on the elementary linear elastostatic problem of structural analysis. Their fundamental properties are identi®ed and their patterns of convergence are analysed and compared. IntroductionIt is generally accepted that the hybrid formulations are one of the most accurate variants of the ®nite element method. However, three decades after the pioneering work of Pian (1967) they are yet to succeed in challenging the dominant position of single-®eld based elements as the standard tool in ®nite element computations. The situation is rather similar in the competition between equilibrium and compatibility elements, launched by de Veubeke (1965) also in the mid-sixties. The dominance of the conforming displacement elements is still absolute despite their known limitations, particularly in what concerns accuracy and safety in stress estimates.Different reasons are raised to explain these facts, namely added conceptual complexities, dif®culties in establishing appropriate approximations bases and higher computational costs. There is, however, a renewed research interest on stress elements and on hybrid and mixed ®nite element formulations, consequent upon the recent developments in computer hardware, in particular in what concerns parallel processing, and the motivation to model increasingly more complex structural problems. If this revival in the interest on the hybrid and mixed formulations is to succeed, it is essential to accept them as alternative concepts that should be developed per se, instead of insisting in establishing the basis for their development by mimicking the conventional displacement method.This is the approach adopted by the authors in the development of the hybrid and mixed ®nite element formulations reported in the present paper. The four basic aspects in this approach (Freitas, 1989a) are the following: ± to depart from the relevant ®rst principles of mechanics; ± to use nodeless, hierarchical approximation functions; ± to enforce energetically consistent de®nitions for the discrete generalised...

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