A Mixed-Hybrid Finite Element Model Based on Orthogonal Functions

Pereira, E.M.B.R.; Freitas, J. A. Teixeira de

doi:10.1002/(sici)1097-0207(19960430)39:8<1295::aid-nme903>3.0.co;2-h

Cited by 26 publications

(25 citation statements)

References 9 publications

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“…Under certain conditions, these systems are equivalent to the pairs of primal-dual programs presented in Table 8. Their physical interpretation shows that statements (46) and (47) coincide with the fundamental theorems on the minimum co-energy and energy, respectively (Freitas, 1989a;Pereira and Freitas, 1996b;Freitas, 1997). The same conclusions are reached when the solving systems of the alternative formulations for the displacement model are treated in a similar way.…”

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confidence: 64%

“…To attain these objectives, the bases that have been tested are orthogonal polynomial and Fourier series (Pereira and Freitas, 1996b) and digital Walsh and wavelet series (Freitas and Castro, 1992;Castro and Freitas, 1996). They have been applied in the solution of two-and threedimensional elastic and elastoplastic problems and to thin, thick and higher-order plate bending problems (Pereira and Freitas, 1996a), with particular emphasis on modelling of interlaminar stresses in composites and of boundary layer effects (Pereira, 1993;Pereira and Almeida, 1998).…”

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confidence: 99%

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Non-conventional formulations for the finite element method

Freitas¹,

Almeida²,

Pereira³

1999

Computational Mechanics

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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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confidence: 64%

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confidence: 99%

Non-conventional formulations for the finite element method

Freitas¹,

Almeida²,

Pereira³

1999

Computational Mechanics

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“…The generation and manipulation of such functions is fully described in [47,48,58]. The Legendre polynomials can be defined using the Rodriguez formula, Eq.…”

Section: Hybrid Mixed Stress Modelmentioning

confidence: 99%

“…As it is known, hybrid-mixed stress models do not locally verify the compatibility conditions [47][48][49]. Therefore, some displacement discontinuities between elements may occur.…”

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confidence: 99%

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Static and dynamic physically non-linear analysis of concrete structures using a hybrid mixed finite element model

Arruda

Castro

2013

Advances in Engineering Software

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Three-dimensional hybrid-Trefftz stress elements

Freitas

Bussamra

2000

Int. J. Numer. Meth. Engng.

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The stress model of the hybrid-Tre!tz "nite element formulation is applied to the linear elastostatic analysis of solids. The stresses are approximated in the domain of the element and displacements on its boundary. Complete, linearly independent, hierarchical polynomial approximation functions are used in both domain and boundary approximations. The displacement basis is de"ned independently on each inter-element surface. Continuity at the edges and on the corners of the elements is not enforced a priori. The stress basis is constrained to solve locally the Beltrami governing di!erential equation. It is derived from the associated Papkovitch}Neuber elastic displacement solution. Generalized variables are used to ensure that the approximations are independent of the geometric description of the elements. The solving system is derived directly from the fundamental relations of elastostatics. The solving system is symmetric, when the same property applies to the local elasticity condition, sparse, described by boundary integral arrays and well suited to p-re"nement and parallel processing. The numerical implementation of these equations is discussed and numerical tests are presented to illustrate the performance of the "nite element formulation. thus obtained can be recognized in the embracing works of Pian and Tong [3] and Brezzi and Fortin [4].The formulation used here was identi"ed later as the stress model of the hybrid-Tre!tz "nite element formulation and applied to di!erent two-dimensional structural analysis problems [5}8]. It is, therefore, directly related with the displacement frame Tre!tz element developed by Jirousek and Stein and their co-workers [9, 10]. The theoretical basis of the formulation is established in [11], where a comparative analysis with the related T-elements is also presented.In the terminology adopted here, a "nite element derived from the direct approximation of the stress "eld is termed a stress "nite element model. The stress "eld is directly related with two fundamental conditions, namely the equilibrium condition in the domain of the element and the di!usivity (Neumann) condition on the associated boundary tractions (Cauchy stresses). Three classes of formulations can be established depending on the equilibrium constraints enforced a priori on the stress basis, namely the hybrid-mixed, the hybrid and the hybrid-Tre!tz formulations of the "nite element stress model [12,13].The hybrid-mixed formulation is the most general. No constraints (besides linear independence and completeness) are placed on the stress approximation basis. Besides the stresses, the displacements are also approximated, and independently, in the domain and on the boundary of the element. These bases are used to enforce on average the equilibrium and the di!usivity conditions, respectively. The hybrid formulation is obtained by constraining a priori the stress approximation to satisfy locally the equilibrium condition. Consequently, the displacement approximation in the domain of the element becomes redundant an...

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A Mixed-Hybrid Finite Element Model Based on Orthogonal Functions

Cited by 26 publications

References 9 publications

Non-conventional formulations for the finite element method

Non-conventional formulations for the finite element method

Static and dynamic physically non-linear analysis of concrete structures using a hybrid mixed finite element model

Three-dimensional hybrid-Trefftz stress elements

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