journal of materials processing technology 209 ( 2 0 0 9 ) 289–296 journal homepage: www.elsevier.com/locate/jmatprotec Shape optimization of clinching tools using the response surface methodology with Moving Least-Square approximation M. Oudjene a,* , L. Ben-Ayed b , A. Delam´ ezi` ere b , J.-L. Batoz c a Nancy-Universit´ e/ENSTIB, 27 rue du Merle Blanc-B.P. 1041, 88051 Epinal Cedex 9, France b Institut Sup´ erieur d’Ing´ enierie de la Conception, 27 rue d’Hellieule 88100 Saint-Di´ e-des-Vosges, France c Universit´ e de Technologie de Compi` egne, 60205 Compi` egne Cedex, France article info Article history: Received 7 February 2007 Received in revised form 10 January 2008 Accepted 3 February 2008 Keywords: FEM Design of experiments RSM SQP method Optimization Clinching process abstract A response surface methodology (RSM), based on Moving Least-Square (MLS) approxima- tion and adaptive moving region of interest, is presented for shape optimization problem. To avoid a local optimum and to obtain an accurate solution at low cost, an efficient strategy which allows to improve the RSM accuracy in the vicinity of the global optimum is presented. During the progression of the optimization procedure, the region of interest is moving and the search space is reduced by half around each local optimum. The clinch forming process is considered as an application example using the ABAQUS finite element code. The geome- tries of both the punch and the die are optimized to improve the joints resistance to tensile loading. © 2008 Elsevier B.V. All rights reserved. 1. Introduction Response surface methodology (RSM) is an efficient and widely established method for optimization problems which finds various applications in many of computational mechan- ics fields. The RSM allows replacing a complex model by an approximate one based on results calculated at various points of the design space. When applying RSM to a particular prob- lem, two important issues have to be considered: the choice of the design of experiments (DoE) and the construction of accu- rate function approximations (Breitkopf et al., 2005; Naceur et al., 2007). In this study, the MLS approximation is adopted to build the response surfaces and central composite DoE ∗ Corresponding author. Tel.: +33 3 29 29 61 37; Fax: +33 3 29 29 61 38. E-mail address: marc.oudjene@enstib.uhp-nancy.fr (M. Oudjene). (Schimmerling et al., 1998) with three levels for function eval- uation. The MLS approximation has been largely documented in the literature and used by many authors for optimization prob- lems. Among others, Breitkopf et al. (2005) and Naceur et al. (2003) have presented an extended approach of pattern search algorithms with a fixed pattern panned and zoomed in a con- tinuous manner across the design space. In another work, Naceur et al. (2007) have investigated the use of MLS for regres- sion model, with strategies for progressive selection of points in the design space where designs are evaluated in the way to maximize the accuracy while minimizing the number of function evaluations. The response surface method presented 0924-0136/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2008.02.030