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A two-®eld dual-mixed variational formulation of three-dimensional elasticity in terms of the non-symmetric stress tensor and the skew-symmetric rotation tensor is considered in this paper. The translational equilibrium equations are satis®ed a priori by introducing the tensor of ®rst-order stress functions. It is pointed out that the use of six properly chosen ®rst-order stress function components leads to a (three-dimensional) weak formulation which is analogous to the displacementpressure formulation of elasticity and the velocity-pressure formulation of Stokes¯ow. Selection of stable mixed hp ®nite element spaces is based on this analogy. Basic issues of constructing curvilinear dual-mixed p ®nite elements with higher-order stress approximation and continuous surface tractions are discussed in the two-dimensional case where the number of independent variables reduces to three, namely two components of a ®rst-order stress function vector and a scalar rotation. Numerical performance of three quadrilateral dual-mixed hp ®nite elements is presented and compared to displacement-based hp ®nite elements when the Poisson's ratio converges to the incompressible limit of 0.5. It is shown that the dual-mixed elements developed in this paper are free from locking in the energy norm as well as in the stress computations, both for h-and p-extensions.

This paper is dedicated to Professor Barna A. Szabà o on the occasion of his 65th birthday SUMMARY A complementary energy-based, dimensionally reduced plate model using a two-ÿeld dual-mixed variational principle of non-symmetric stresses and rotations is derived. Both the membrane and bending equilibrium equations, expressed in terms of non-symmetric mid-surface stress components, are satisÿed a priori introducing ÿrst-order stress functions. It is pointed out that (i) the membrane-, shear-and bending energies of the plate written in terms of ÿrst-order stress functions are decoupled, (ii) although unmodiÿed 3-D constitutive equations are applied, the energy parts do not contain the 1=(1 − 2 ) term for isotropic, linearly elastic materials. These facts mean that the ÿnite element formulation based on the present plate model should be free from shear locking when the thickness tends to zero and free from incompressibility locking when the Poisson ratio converges to 0.5, irrespective of low-order h-, or higher-order p elements are used.Curvilinear dual-mixed hp ÿnite elements with higher-order stress approximation and continuous surface tractions are developed and presented for the membrane (2-D elasticity) problem. In this case the formulation requires the approximation of three independent variables: two components of a ÿrst-order stress function vector and a scalar rotation. Numerical performance of three quadrilateral dualmixed elements is presented and compared to displacement-based hp ÿnite elements when the Poisson ratio converges to the incompressible limit of 0.5. The numerical results show that, as expected, the dual-mixed elements developed in this paper are free from locking in the energy norm as well as in the stress computations, for both h-and p-extensions.

The problem of ÿnding a nearly optimal distribution of polynomial degrees on a ÿxed ÿnite element mesh is discussed. An a posteriori error estimator based on the minimum complementary energy principle is proposed which utilizes the displacement vector ÿeld computed from the ÿnite element solution. This estimator, designed for p-and hp-extensions, is conceptually di erent from estimators based on residuals or patch recovery which are designed for h-extension procedures. The quality of the error estimator is demonstrated by examples. The results show that the e ectivity index is reasonably close to unity and the sequences of p-distributions obtained with the error indicators closely follow the optimal trajectory. ? 1998 John Wiley & Sons, Ltd.KEY WORDS: ÿnite element method; adaptivity; a posteriori error estimation

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