J. P. Moitinho de Almeida

Non-conventional formulations for the finite element method

Summary This paper reports on hybrid formulations being developed by the Structural Analysis Research Group of Instituto Superior Técnico. Three alternative sets of hybrid finite element formulations are presented. They are termed hybrid-mixed, hybrid and hybrid-Trefftz and differ essentially on the field 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 field 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 fields 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 identified and their patterns of convergence are analysed and compared.

Alternative approach to the formulation of hybrid equilibrium finite elements

Abstract A formulation appropriate to the development of equilibrated hybrid finite elements is presented. Techniques to overcome the problems traditionally associated with these elements are discussed along with the characteristics of the elements obtained. Numerical tests on the elements obtained using this formulation are presented.

A SET OF HYBRID EQUILIBRIUM FINITE ELEMENT MODELS FOR THE ANALYSIS OF THREE-DIMENSIONAL SOLIDS

In this paper an approach to the formulation of equilibrium elements for the analysis of three-dimensional elasticity problems is presented. This formulation is an extension of the approach previously proposed for the analysis of two-dimensional elasticity problems. The general aspects of the formulation remain unchanged when applied to the new problem, but new points are considered, namely the way to perform volume integrations for general elements and the techniques used to obtain the self-equilibrated three-dimensional stress approximation functions. The numerical behaviour of such elements is presented and discussed.

Adaptive methods for hybrid equilibrium finite element models

This paper presents initial results of work directed at the development of practical adaptive methods for equilibrium finite element models of elastostatic problems in solid mechanics. The formulation of hybrid elements is reviewed, with particular emphasis on recent developments for equilibrium elements. Compatibility defaults for 3D and 2D models are proposed, together with an explicit expression for an error indicator based on such defaults. Two further error indicators are defined which are based on the availability of dual conforming displacement solutions. These solutions are derived from two forms of dual analysis. The one may be considered as being in parallel, the other as being sequential with conforming displacements recovered locally from an equilibrium solution. The two latter error indicators provide upper bounds to the global error. A self-adaptive strategy is proposed to exploit any of these error indicators and estimators, and numerical results are presented and compared for a simple plane stress problem.

A GENERAL FORMULATION OF EQUILIBRIUM MACRO‐ELEMENTS WITH CONTROL OF SPURIOUS KINEMATIC MODES: THE EXORCISM OF AN OLD CURSE

This paper illustrates a method whereby a family of robust equilibrium elements can be formulated in a general manner. The effects of spurious kinematic modes, present to some extent in all primitive equilibrium elements, are eliminated by judicious assembly into macro-equilibrium elements. These macroelements are formulated with sufficient generality so as to retain the polynomial degree of the stress field as a variable. Such a family of macro-elements is a new development, and results for polynomials of degree greater than two have not been seen before. The quality of results for macro-equilibrium elements with varying degrees of polynomial is demonstrated by numerical examples.

An accurate, robust, and easy-to-implement method for integration over arbitrary polyhedra: Application to embedded interface methods

We present an accurate method for the numerical integration of polynomials over arbitrary polyhedra. Using the divergence theorem, the method transforms the domain integral into integrals evaluated over the facets of the polyhedra. The necessity of performing symbolic computation during such transformation is eliminated by using one dimensional Gauss quadrature rule. The facet integrals are computed with the help of quadratures available for triangles and quadrilaterals. Numerical examples, in which the proposed method is used to integrate the weak form of the Navier-Stokes equations in an embedded interface method (EIM), are presented. The results show that our method is as accurate and generalized as the most widely used volume decomposition based methods. Moreover, since the method involves neither volume decomposition nor symbolic computations, it is much easier for computer implementation. Also, the present method is more efficient than other available integration methods based on the divergence theorem. Efficiency of the method is also compared with the volume decomposition based methods and moment fitting methods. To our knowledge, this is the first article that compares both accuracy and computational efficiency of methods relying on volume decomposition and those based on the divergence theorem.

A nonlinear projection method for constrained optimization

Abstract An algorithm is presented for the solution of nonlinear optimization problems involving locally differentiable functions with known analytical expressions. The algorithm is based on perturbation methods of system analysis and develops from a set of easy to implement procedures designed to detect and solve the activation and deactivation of constraints while selecting the steepest feasible trajectory and the largest step length. Numerical applications are presented to illustrate the performance of the algorithm.

Automatic drawing of stress trajectories in plane systems

Abstract An approximate method for plotting the principal stress trajectories in plane systems, using a computer and knowing the stress tensor at every point of a finite element mesh, is described. This method can be used to plot any second-order tensor field in a plane system.

Data structures for the distributed iterative solution of non-conventional finite element models

A class of specialised data structures designed for the distributed solution of non-conventional finite element formulations, which are equally effective when used in conjunction with conventional formulations, is presented. We begin by briefly discussing how the non-conventional finite element formulations being developed within the structural analysis group at IST [Freitas JAT, Almeida JPM, Pereira EMBR. Non-conventional formulations for the finite element method. Comput Mech 1999;23(5-6):488-501] lead to systems of equations that appear to be naturally suited for parallel processing, but we also recognise that to take full advantage of the characteristics of these systems - large dimension, non-overlapping block structure and sparsity - it is necessary to use appropriate data structures. The approach presented, which references the logical subdivisions of the system matrices, was designed to fulfil these objectives. Examples of parallel performance and efficiency on an homogeneous distributed platform are presented.

Curious convergence with hypostatic hybrid equilibrium models

The paper describes an unexpected type of convergence behaviour occurring for a single, variable degree, primitive-type equilibrium element. For this element the number of independent stress fields is less than the number of independent boundary displacement variables that do not correspond to rigid element modes of displacement. This leads to the conclusion that the element is hypostatic and that, in the absence of self-stressing modes, no convergence can occur. Such ‘conventional’ counting procedures do not, however, reveal the whole picture, and numerical determination of the rank of the coefficient matrix of the equilibrium equations shows that, in practice, self-stressing modes can and do exist in a model which would conventionally be described as hypostatic. The rank deficiency in the coefficient matrix is shown to be due to the fact that, upon transformation, independent stress fields do not necessarily lead to independent boundary tractions. Generalization to conventionally iso- and hyperstatic models demonstrates that, whenever the coefficient matrix is rank-deficient, spurious kinematic modes coexist with self-stressing modes. The problem which reveals the curious convergence characteristics for the primitive-type element is resolved using a macro-type element, and it is seen that, with the larger degree of hyperstaticity available to this element, strictly monotonic convergence characteristics are observed.