ORIGINAL ARTICLE Optimization of sheet metal forming processes by the use of space mapping based metamodeling method Wang Hu & Li Enying & G. Y. Li & Z. H. Zhong Received: 9 February 2007 / Accepted: 20 September 2007 / Published online: 14 November 2007 # Springer-Verlag London Limited 2007 Abstract In this study, an adaptive space mapping tech- nique based on response of objective was suggested for solving practical engineering problems. Response surface methodology was engaged in approximation of objective and constraint functions based on coarser model. The fine simulation model is not only used for correction of the coarser simulation model and validation of final solution but also for applied construction of space mapping expression. Finally, genetic algorithm (GA) was used to optimize updated metamodel according to coarse model. The proposed method combines the space mapping tech- nology based on response of coarse model and modification of design of experiment. It guarantees that metamodel based coarser model is stepwise updated in the right searching direction. For demonstrating practicability of developed method, it was applied for optimization of geometric parameters of addendum surface, blank holder force and drawbead restraining force in sheet forming problems. It was confirmed that the corresponding problem can be optimized successfully in remarkably short comput- ing time by proposed optimization method. Keywords Adaptive space mapping . Design of experiment . Genetic algorithm . Metamodel . Response surface methodology . Sheet forming problems 1 Introduction Finite element (FE) simulations are widely used to develop sheet metal forming processes in practice. With increasing of complexion and size of FE model, computational time also correspondingly increases . Thus, optimization proce- dures for sheet metal forming problems [1–5] often lead to numerous expensive functions. This is particularly the case when cost and constraint functions are obtained via complete finite element simulations involving fine meshes, many degrees of freedom (DOF), and strongly nonlinear geometrical, material behaviors. In order to perform successful sheet metal forming operations and avoid shape errors, overcome tearing and wrinkling defects, processes and material variables such as tools geometry, blank holding force, friction, blank shape, sheet thickness and material properties, etc. should be optimized. Methods of process parameter design have been developed since the 1990s in metal forming analysis with FEMs and design optimization theories [6–13]. Today, forming simulations are mainly used in a trial and error scheme to develop a forming process which produces acceptable parts. This may be costly because the trial and error method requires a high user interaction to manually alter the model and to evaluate the result. It must therefore be the objective to eventually develop an efficient automated optimization of the forming process. In order to develop the efficiency of optimization for sheet forming problems, response surface methodology (RSM) is introduced for optimization process. A response surface is essentially a simplified multidimensional surface fit to what is usually a more complex function. Often the function is highly nonlinear, computationally expensive to evaluate, and/or difficult to differentiate explicitly. For further reading about RSM, see other literatures [14–16]. Int J Adv Manuf Technol (2008) 39:642–655 DOI 10.1007/s00170-007-1253-z W. Hu : L. Enying : G. Y. Li(*) : Z. H. Zhong Key Laboratory of Advanced Technology for Vehicle Body Design & Manufacture, M.O.E., Hunan University, Changsha 410082, PR China e-mail: gyli@hnu.cn W. Hu e-mail: wanghuenying@hotmail.com