An adaptive approach to limit analysis Lavinia Borges a, * , Nestor Zouain a , Cyntia Costa a , Raul Feij  oo b a Department of Mechanical Engineering, Universidade Federal do Rio de Janeiro, P.O. Box 68503, 21945-970 Rio de Janeiro, RJ, Brazil b Department of Computational Mechanics, Laborat orio Nacional de Computac ß~ ao Cient  õ®ca, Rua Get  ulio Vargas, 333, 25651-070 Petr opolis, RJ, Brazil Received 27 August 1999; in revised form 3 February 2000 Abstract The main objective of this paper is to propose an adaptive mesh re®nement procedure for ®nite element models in limit analysis. We use an `a posteriori' indicator based on the local directional interpolation error and a recovering scheme to compute second derivatives of the ®nite element solution. The proposed mesh adaptation process gives improved results in localizing regions of rapid or abrupt variations of the variables, whose location is not known a priori. Limit analysis of bodies in plane strain and plane stress is considered in the applications. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Limit analysis; Error estimation; Finite element; Adaptivity 1. Introduction The main objective of this paper is to propose an adaptive mesh re®nement procedure for limit analysis. This procedure uses an a posteriori estimator of the local directional interpolation error and a recovering scheme to compute the ®rst and second derivatives of the ®nite element solution. The strategy presented here is an extension of the one presented by Borges et al. (1998, 1999) and Feij  oo et al. (1997) wherein the estimator and the adaptive process were only de®ned for linear ®nite elements. Here we generalize the indicator and the adaptive procedure to include quadratic triangles. The advantages of adapting meshes are well known. Furthermore, we place particular emphasis on the anisotropic mesh adaptation process, generated by the proposed directional indicator. The goal of that approach is to achieve a mesh-adaptive strategy accounting for mesh size re®nement, as well as rede®nition of the oriented element stretching. This way, along the adaptation process, the mesh turns aligned with the direction of maximum curvature of the function graph. This mesh adaptation procedure gives improved results in localizing regions of rapid or abrupt variations of the variables, whose location is not known a priori (Peir o, 1989; Peraire et al., 1990; Dompierre et al., 1995; Verf  urth, 1996; Almeida et al., 1998; International Journal of Solids and Structures 38 (2001) 1707±1720 www.elsevier.com/locate/ijsolstr * Corresponding author. E-mail address: lavinia@serv.com.ufrj.br (L. Borges). 0020-7683/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII:S0020-7683(00)00131-1