E. A. W. Maunder

A general method for recovering equilibrating element tractions

Abstract A new geometrical presentation of a general method for recovering equilibrating element tractions is developed for 2D finite element models of problems in structural mechanics. The data for the method consists of the prescribed loading and the stress fields resulting from a conventional linear elastic finite element analysis. The geometrical presentation utilises the physical concepts of nodal forces and Maxwell force diagrams based on nodal equilibrium. Only small problems have to be solved for each node. This presentation complements the original algebraic approach of Ladeveze, and extends the method to cover the use of hierarchical p -elements in regular meshes, and general higher-order elements in 1-irregular meshes as encountered in h - p mesh refinement. The same presentation is considered for the recovery of residual tractions as required in the solution of the local residual problem. Comparison is made with related methods for obtaining residual tractions, in particular those proposed by Bank and Weiser and Ainsworth et al. Detailed examples are included to illustrate the presentation. This paper considers only the fundamental aspects of recovering equilibrating tractions. The knowledge gained may be used in various applications such as error analysis, limit analysis, and structural design.

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.

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.