INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2009; 78:254–274 Published online 6 November 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2483 Limit analysis and convex programming: A decomposition approach of the kinematic mixed method Franck Pastor 1, ∗, † , Etienne Loute 2 and Joseph Pastor 3 1 Laboratoire de M´ ecanique de Lille, UMR CNRS 8107, Boulevard Paul Langevin, 59655 Villeneuve d’Ascq C´ edex, France 2 Facult´ es Universitaires Saint-Louis and Louvain School of Management, 43 boulevard du Jardin Botanique, 1000 Brussels, Belgium 3 Universit´ e de Savoie, POLYTECH’Savoie, Laboratoire LOCIE, Le Bourget du Lac 73376, France SUMMARY This paper proposes an original decomposition approach to the upper bound method of limit analysis. It is based on a mixed finite element approach and on a convex interior point solver using linear or quadratic discontinuous velocity fields. Presented in plane strain, this method appears to be rapidly convergent, as verified in the Tresca compressed bar problem in the linear velocity case. Then, using discontinuous quadratic velocity fields, the method is applied to the celebrated problem of the stability factor of a Tresca vertical slope: the upper bound is lowered to 3.7776—value to be compared with the best published lower bound 3.772—by succeeding in solving non-linear optimization problems with millions of variables and constraints. Copyright 2008 John Wiley & Sons, Ltd. Received 16 April 2008; Revised 5 September 2008; Accepted 9 September 2008 KEY WORDS: limit analysis; kinematic method; decomposition approach; convex optimization 1. INTRODUCTION The Lysmer paper [1] was the first to consider the static method of limit analysis (LA) in geotechnics using a finite element method (FEM) for the Coulomb plasticity criterion. Based on this paper an improved static approach was proposed in [2], also restricted to a bounded mechanical system; this drawback was eliminated in [3] with the definition of infinite extension zones allowing the stress fields to remain admissible everywhere beyond the finite element mesh. Pastor and Turgeman [2] also proposed a kinematic method based on linearization of the criterion. More recently, in [4, 5], this work was extended to reinforced soils treated by homogenization— with Coulomb’s soil as a special case—where all inner edges of the mesh were potential arbitrary ∗ Correspondence to: Franck Pastor, Laboratoire de M´ ecanique de Lille, UMR CNRS 8107, Boulevard Paul Langevin, 59655 Villeneuve d’Ascq C´ edex, France. † E-mail: franck.pastor@skynet.be, franck.pastor@uclouvain.be Copyright 2008 John Wiley & Sons, Ltd.