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Adaptive methods for hybrid equilibrium finite element models. (English) Zbl 0991.74072

The authors claim that, since the inception of the finite element analysis of structures, many developments have focused on conforming displacement elements, nodal variables, and stiffness methods, to the detriment of equilibrium models. Since both approaches become simultaneously feasible, the paper is oriented to error estimation and adaptive methods for equilibrium models of linear elastic structures. The hybrid equilibrium finite element approach used here employs two basic ingredients: a) independent approximation of the stress field within each element as a linear combination of nodeless equilibrated functions; b) independent enforcement of interelement traction continuity by means of weighted residual equations. The weight functions approximate the boundary displacements used in the compatibility equations, resulting in a symmetric system where static-kinematic duality is preserved. An adaptive strategy for \(h\)-refinement of the present approach is based on local mesh refinement that allows for irregular meshes without the need for multipoint constraints. Three different procedures are presented for obtaining element error indicators which regulate the adaptive strategy. For a plane stress problem to which this strategy has been applied, the three procedures give similar convergence rates. In spite of the satisfactory results obtained so far, the authors point out that further investigations are required before general conclusions can be drawn with confidence for plane stress problems, and similar remarks hold also for the three-dimensional case.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74B05 Classical linear elasticity
74G15 Numerical approximation of solutions of equilibrium problems in solid mechanics
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