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    Comparison between three optimization methodsfor the minimization of maximum bending loadand springback in wiping die bending obtainedby an experimental approach

    Riadh Bahloul & Lanouar Ben Ayed & Alain Potiron &Jean-Luis Batoz

    Received: 28 September 2008 /Accepted: 21 September 2009 /Published online: 14 October 2009# The Author(s) 2009. This article is published with open access at

    Abstract The sheet metal bending process is widely usedin the automotive industries, and it is actually one of themost important manufacturing processes. The robustnessand the reliability of the bending operation, like many otherforming operations, depend of several parameters (geome-try, material, and process). In this paper, the die radius andthe clearance between the punch and the sheet areoptimised in order to reduce the maximum bending loadand the springback. Two optimization problems areformulated, and three optimization procedures based onthe response surface method are proposed and used to findthe optimum solutions. Global and local approximations areused to replace the initial optimization problem, which isimplicit by an explicit problem, and the optimum islocalised using two algorithms: a sequential quadraticprogramming and an evolution strategies. The objectivefunctions are evaluated experimentally into a limited pointsnumber, which are defined using a design of experimentstechnique. Good results are obtained from the threeoptimization procedures. The ability of each technique to

    find the optimal solution is evaluated, and the results showa good agreement between those three methods.

    Keywords Sheet metal bending . Optimization . Bendingload . Springback . DoE . Response surface method . SQP.

    Evolution strategies

    AbbreviationsRSM Response surface methodGAp Global approximationLA Local approximationSQP Sequential quadratic programmingES Evolution strategiesDoE Design of experimentsANN Artificial neural networkDACE Design and analysis of computer

    experimentsRBF Radial basis functionMLS Moving least squaresPDEs Partial differential equationsCMA-ES Covariance matrix adaptation-evolution

    strategiesHSLA High-strength low-alloyed steelEP Evolutionary programmingGAl Genetic algorithms

    1 Introduction

    Sheet metal bending process plays a major role in theautomotive industries. The dimensional accuracy of bentcomponents is a critical factor in these industries because itoften determines a customers impression of overall productquality. At the end of the bending operation, when the tools

    R. Bahloul (*)LGM, Ecole Nationale dIngnieurs de Monastir,Avenue Ibn Eljazzar,5019 Monastir, Tunisiae-mail:

    A. PotironLPMI-ERTGI, ENSAM,2 Boulevard du Ronceray, 49035, Angers, France

    L. B. Ayed : J.-L. Batoz(GIP-InSIC) Institut Suprieur dIngnierie de la Conception,27 rue dHellieule,88100 Saint-Di-des-Vosges, France

    Int J Adv Manuf Technol (2010) 48:11851203DOI 10.1007/s00170-009-2332-0

  • are removed, the metal tries to be turned over to its originalshape because of residual forming stress, and that generatesa shift between the desired geometries and that obtained.This phenomenon called springback depends principallyon material properties of geometrical tools parameters andof process parameters. The improvement of the bent partsconformity is crucial and leads to a reduction of assemblyproblems and to an improvement of performances of thefinal product.

    In the last years, the international competition of theindustries is extremely severe; all companies try to reducemanufacturing costs on the one hand and increase produc-tivity, robustness of the forming process, and quality on theother. Experimental and numerical analyses are more andmore used to evaluate the difficulties in sheet metal formingand to achieve these goals. There are numerous studies onsheet metal bending, but a small number of them deal withthe process optimization. Xu et al. [1] analysed thebehaviour of electronic packages under thermal andbending loads by means of quasi-meshless methods. Thisimplicit problem was solved using multi-quadratic responsesurface, in order to approximate the response of finiteelement formulations for each loading condition. In eachcase, a response surface was created based on design ofexperiments (DoE) matrix using Latin hypercube samplingscheme. In 2004, Wu and Altan [2] developed anoptimization procedure in order to improve the quality ofa clutch hub, which was carried out using a deep-drawingprocess. A number of design modifications were evaluatedto determine the optimum parameters for the selectedprocess conditions. The results have been validated withobservations of the actual process.

    In [3], a methodology for the design of plate-formingdies in cylindrical bending using optimization techniqueswas developed in order to reduce the cost of die productionby reducing the trial-and-error procedure in determining thefinal die geometry. The plate thickness is discretised byplane-strain finite elements. The die is assumed to be rigid,and its profile is approximated by Bezier curves, thecontrol-points coordinates of which are the design varia-

    bles. The die profile is varied to minimise the differencebetween the required shape and the shape of the bent plate,considering the springback action. In their recent works,Naceur et al. [4] proposed a new numerical approach tooptimise the shape of the initial blank, which plays animportant role on the quality of the final 3D workpieceobtained by the deep drawing of thin sheets. This newapproach was based on the coupling between the inverseapproach used for the forming simulation and an evolution-ary algorithm. The preliminary results dealing with theoptimization of the blank contour in the case of square cup(the Benchmark test of Numisheet 1993) show the efficiencyand the potential interest of the proposed approach.

    Frequently, the bending process such as wiping diebending (see Fig. 1) induces manufacturing problemsprimarily due to the effect of process parameters onspringback and on the loads applied by the tools. Theinfluence of tools, particularly the bending die design andthe punch-sheet clearance have been detailed in severalworks [5, 6], mentioning that it affects highly the bendingangle value. In 2000, Inamdar et al. [7] described anartificial neural network based on the backpropagation oferror. It is used to improve the sheet metal bending processin which an attempt is made in order to restrict springbackand consequently to obtain the final angle of bend within asmall tolerance. The architecture, established using ananalytical model for training consisted of six inputs, tenhidden, and two outputs nodes (punch displacement andspringback angle). The six inputs were the angle of bend,the punch radius/thickness ratio, the die gap, the die entryradius, the yield strength to Youngs modulus ratio, and thestrain hardening exponent, n. The effect of networkparameters on the mean square error of prediction wasstudied.

    As many manufacturing processes cannot be discrebedexactly by an analytical formulation, the use of design andanalysis of computer experiments has drastically grownduring the last decade, and several methods have beenproposed for the analysis of simulation results or experi-ments, and a survey of the state of the art can be found in

    Fig. 1 Experimental system ofbending

    1186 Int J Adv Manuf Technol (2010) 48:11851203

  • [8] by J.P.C. Kleijnen. Response surfaces based on DoE [911] Kriging methods [12] were developed for the purposeof approximating complex functions arising in differentbranches of applied sciences. In another work, Simpson etal. [13] compared and contrasted the use of two responsesurface models to approach the computer analysis: apolynomial of second order and a Kriging model. Bothmethods were applied to the multidisciplinary design of anaerospike nozzle which consists of a computational fluiddynamics model and a finite-elements model.

    For such implicit optimizatiom problems, the approxima-tion methods have recently been gaining attention. Anoptimization method using Kriging approximation is appliedto an optimization problem by Sakata et al. in 2003 [14]. Themethodology involves two main procedures. The firstconsists to finding the response surface using the Krigingmethod, and the second is an optimization strategy. The useof the Kriging method makes easier to perform theapproximation of the cost function. Good results wereobtained using the Kriging method, and they were thencompared with those obtained by neural network. Severalother works were carried out using the response surfacemethod (RSM) with models and approximation strategiesmore or less different [1517]. Barthelemy and Haftka [18]and Haftka and Scott [19] reported on their survey ofoptimization using RSM. Box and Wilson discussed andshowed the reliability of the polynomial approximation inthe localization of the optimal design using experimentaldata [20]. Hosder et al. [21] reported on the application ofpolynomial response surface approximation to the multi-disciplinary optimization of high-speed civil transport.

    RSM are well established for physical processes asdocumented by Myers and Montgomery [22], while theapplications to simulation models in computational mechanicsform a relatively young research field. Among other recentworks, Roux et al. [23] discuss experimental design techni-ques and the regression equations for structural optimization.An approach to crashworthiness design uses genetic algo-rithms to select optimal set of experiments between rn

    factorial designs is described by Kurtaran et al. [24]. Anapplication to sheet metal forming process simulated bydynamic explicit method is given by Stander [25] with anenphasis on oscillating solutions. RSM combined withstochast