Abstract
A new modeling method, related to multiple inputs and multiple outputs (MIMO), simultaneously based on Gaussian process (GP), is proposed to optimize the combinations of process parameters and improve the quality control for multi-objective optimization problems in sheet metal forming. In the MIMO surrogate model, for the use of the system information in processing and the accuracy of the model, quantitative and categorical input variables are both taken into account in GP simultaneously. Firstly, a general method is proposed for constructing covariance functions for GP simultaneous MIMO surrogate model based on correlation matrices. These covariance functions must be able to incorporate the valid definitions of both the spatial correlation based on quantitative input variables and the cross-correlation based on categorical input variables. Secondly, the unrestrictive correlation matrices are constructed by the hypersphere decomposition parameterization, thus directly solving optimization problems with positive definite constraints is needless, and the computational complexity is simplified. Compared with independent modeling method, the proposed GP simultaneous MIMO model has higher accuracy and needs less number of estimated parameters. Moreover, the cross-correlation between the outputs (quality indexes) obtained by proposed model provides some reference to further develop quality intelligent control strategies. Finally, a drawing-forming process of auto rear axle housing is taken as an example to validate the proposed method. The results show that the proposed method can effectively decrease the crack and wrinkle in sheet metal forming.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Zheng Y, Wu YG, Li SJ (1996) Development on the numerical simulation of sheet metal forming processes. J Plast Eng 3:34–39
Lei LP, Sang-Moon H, Beom-Soo K (2002) Finite element analysis and design in stainless steel sheet forming and its experimental comparison. J Mater Process Tech 27:22–26
Xie YM, Yu HP, Chen J, Ruan XY (2007) Robust optimization for deep-drawing process of sheet metal based on CAE with Grey relational analysis method. J Shanghai Jiao Tong Univ (Engl Ed) 12(3):301–307
Mkaddem A, Bahloul R (2007) Experimental and numerical optimization of the sheet products geometry using response surface methodology. J Mater Process Tech 189:441–449
Wei L, Yu YY (2008) Multi-objective optimization of sheet metal forming process using Pareto-based genetic algorithm. J Mater Process Tech 208(1–3):499–506
Xia W, Luo B, Liao XP (2011) An enhanced optimization approach based on Gaussian process surrogate model for process control in injection molding. Int J Adv Manuf Technol. doi:10.1007/s00170-011-3227-4, Published online
Naceur H, Ben-Elechi S, Batoz JL, Knopf-Lenoir C (2008) Response surface methodology for the rapid design of aluminum sheet metal forming parameters. Mater Design 29(4):781–790
Breitkopf P, Naceur H, Rassineux A, Villon P (2005) Moving least squares response surface approximation: formulation and metal forming applications. Comput Struct 83(17-18):1411–1428
Liew KM, Tan H, Ray T, Tan MJ (2004) Optimal process design of sheet metal forming for minimum springback via an integrated neural network evolutionary algorithm. Struct Multidiscip O 26(3–4):284–294
Viswanathan V, Kinsey B, Cao J (2003) Experimental implementation of neural network springback control for sheet metal forming. J Eng Mater-T ASME 125(2):141–147
Wang H, Li E, Li GY (2009) The least square support vector regression coupled with parallel sampling scheme metamodeling technique and application in sheet forming optimization. Mater Design 30(5):1468–1479
Currin C, Mitchell T, Morris M, Ylvisaker D (1991) Bayesian prediction of deterministic functions, with applications to the design and analysis of computer experiments. J Am Stat Assoc 416:953–963
Seeger M, Williams CKI, Lawrence ND (2003) Fast forward selection to speed up sparse Gaussian process regression. In: Proceedings of the 9th International Workshop on Artificial Intelligence and Statistics; Key West, FL, USA
Zhou J (2006) Single and multi-objective process optimization of injection molding using numerical simulation with surrogate modeling approaches and genetic algorithms. Dissertation, University of Winconsin-Madison
Zhou J, Turng LS (2007) Adaptive multi-objective optimization of process conditions for injection molding using a Gaussian process approach. Adv Polym Tech 26(2):71–85. doi:10.1002/adv
Liao XP, Xia W, Long FY (2010) Gaussian process modeling of process optimization and parameter correlation for injection molding. J Mater Sci Eng 4(10):35
Kennedy MC, O'Hagan A (2000) Predicting the output from a complex computer code when fast approximations are available. Biometrika 87:1–13
Qian PZG, Wu CFJ (2008) Bayesian hierarchical modeling for integrating low-accuracy and high-accuracy experiments. Technometrics 50:192–204
Yildiz AR (2012) A comparative study of population-based optimization algorithms for turning operations. Inf Sci 210:81–88
Yildiz AR (2013) Comparison of evolutionary-based optimization algorithms for structural design optimization. Eng Appl Artif Intell 26:81–88
Yildiz AR, Solanki KN (2012) Multi-objective optimization of vehicle crashworthiness using a new particle swarm based approach. Int J Adv Manuf Technol 59(1–4):367–376
Yildiz AR (2013) Cuckoo search algorithm for the selection of optimal machining parameters in milling operations. Int J Adv Manuf Technol 64(1–4):55–61
Yildiz AR (2013) Optimization of cutting parameters in multipass turning using artificial bee colony-based approach. Inf Sci 220:399–407
Yildiz AR (2009) A novel particle swarm optimization approach for product design and manufacturing. Int J Adv Manuf Technol 40(5–6):617–628
Yildiz AR (2013) A new hybrid differential evolution algorithm for the selection of optimal machining parameters in milling operations. Appl Soft Comput 13(3):1433–1439
Yildiz AR (2013) A new hybrid differential evolution algorithm for the selection of optimal machining parameters in milling operations. Appl Soft Comput 13(3):1561–1566
Zhou Q (2011) Computer simulation driven statistical modeling and quality control. Dissertation, University of Winconsin-Madison
Han G, Han G, Santner TJ, Notz WI, Bartel DL (2009) Prediction for computer experiments having quantitative and qualitative input variables. Technometrics 51:278–288
Qian PZG, Wu H, Wu CFJ (2008) Gaussian process models for computer experiments with qualitative and quantitative factors. Technometrics 51:283–396
Liu W, Yang YY, Xing ZW (2005) Automatic process optimization of sheet metal forming with multi-objective. NUMISHEET '2005, Detroit, USA: 843–848
Jakumeit J, Herdy M, Nitsche M (2005) Parameter optimization of the sheet metal forming process using an iterative parallel kriging algorithm. Struct Multidiscip O 6:498–507
An ZG (2009) Application of radial basis function metamodel for optimization of sheet metal forming. Dissertation, Chongqing University
Joseph VR, Delaney JD (2007) Functionally induced priors for the analysis of experiments. Technometrics 49:1–11
McMillian NJ, Sacks J, Welch WJ, Gao F (1999) Analysis of protein activity data by Gaussian stochastic process models. J Biopharm Stat 9:145–160
Rebonato R, Jackel P (1999) The most general methodology for creating a valid correlation matrix for risk management and option pricing purposes. J Risk 2:17–27
Xia W, Zhang SH, Liao XP, Yan XL (2011) Parameters' correlation analysis based on Gaussian process for injection molding. Adv Mater Res 154–155:130–136
O'Hagan A (1978) Curve fitting and optimal design for prediction. J Roy Stat Soc B 40:1–42
Sciascio F, Amicarelli AN (2008) Biomass estimation in batch bio-technological processes by Bayesian Gaussian process regression. Comput Chem Eng 32:3264–3273. doi:10.1016/j. compchemeng.2008.05.015
Likar B, Kocijan J (2007) Predictive control of a gas–liquid separation plant based on a Gaussian process model. Compute Chem Eng 31:142–152. doi:10.1016/j.compchemeng.2006. 05.011
Zhou P, Tian F, Chen X, Shang Z (2008) Modeling and prediction of binding affinities between the human amphiphysin SH3 domain and its peptide ligands using genetic algorithm–Gaussian processes. Pept Sci 90:792–802. doi:10.1002/bip.21091
Yuan J, Wang K, Yu T, Fang M (2008) Reliable multi-objective optimization of high-speed WEDM process based on Gaussian process regression. Int J Mach Tool Manuf 48:47–60
Rasmussen CE, Williams CKI (2006) Gaussian processes for machine learning. MIT Press: Cambridge, MA. ISBN026218253X. www.GaussianProcess.org/gpml
Wu CFJ, Hamada M (2009) Experiments: planning, analysis, and optimization. Wiley, New York
Graham A (1981) Kronecker products and matrix calculus with applications. Ellis Horwood, Chichester
Keeler SP, Backofen WA (1965) Plastic instability and fracture in sheets stretched over rigid punches. Trans ASM 56:25–48
Liao XP, Yan XL, Xia W, Luo B (2010) A fast optimal latin hypercube design for gaussian process regression modeling. Third International Workshop on Advanced Computational Intelligence (IWACI 2010), Suzhou: 474–479
Yildiz AR (2012) A new hybrid particle swarm optimization approach for structural design optimization in automotive industry. J Automob Eng 226(D10):1340–1351
Yildiz AR (2009) A novel hybrid immune algorithm for global optimization in design and manufacturing. Robot Comput Integr Manuf 25(2):261–270
Ruan T (2012) Research of Gaussian mutation hybrid genetic algorithms and its application in injection molding. Dissertation, Guangxi University
Yildiz AR (2009) A novel hybrid immune algorithm for global optimization in design and manufacturing. J Mater Process Technol 50(4):224–228
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this license, visit (http://creativecommons.org/licenses/by-nc-nd/2.0/)
About this article
Cite this article
Xia, W., Yang, H., Liao, Xp. et al. A multi-objective optimization method based on Gaussian process simultaneous modeling for quality control in sheet metal forming. Int J Adv Manuf Technol 72, 1333–1346 (2014). https://doi.org/10.1007/s00170-014-5749-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00170-014-5749-z