Sensitivity analysis of the of the square cup stamping process using a polynomial chaos expansion

Sensitivity analysis of the of the square cup stamping process using a polynomial chaos expansion

PEREIRA André F. G., MARQUES Armando E., OLIVEIRA Marta C., PRATES Pedro A.

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Abstract. The stochastic modelling and quantification of the various sources of uncertainty associated with sheet metal forming processes, usually requires a large computational cost to obtain accurate results. In this work, a polynomial chaos expansion metamodel is used in order to reduce the computational cost of the uncertainty quantification (through Sobol’s indices). The metamodel allows to establish mathematical relationships between the square cup forming results and the uncertainty sources associated with the material behaviour and process conditions. Then, sensitivity indices are estimated with the trained metamodel, without resorting to additional numerical simulations. The indices obtained with the metamodel were compared to those obtained with the traditional approach based on a quasi-Monte Carlo method. The metamodel allowed to reduce the computational cost in about 90% when compared to the traditional approach, without compromising the accuracy of the results.

Sobol’s Indices, Polynomial Chaos Expansion, Square Cup, Uncertainty

Published online 4/19/2023, 10 pages
Copyright © 2023 by the author(s)
Published under license by Materials Research Forum LLC., Millersville PA, USA

Citation: PEREIRA André F. G., MARQUES Armando E., OLIVEIRA Marta C., PRATES Pedro A., Sensitivity analysis of the of the square cup stamping process using a polynomial chaos expansion, Materials Research Proceedings, Vol. 28, pp 1183-1192, 2023


The article was published as article 129 of the book Material Forming

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[1] A.E. Marques, P.A. Prates, A.F.G. Pereira, M.C. Oliveira, J.V. Fernandes, B.M. Ribeiro, Performance Comparison of Parametric and Non-Parametric Regression Models for Uncertainty Analysis of Sheet Metal Forming Processes, Metals 10 (2020) 457.
[2] A.F.G. Pereira, P.A. Prates, M.C. Oliveira, J.V. Fernandes, Normal Stress Components during Shear Tests of Metal Sheets, Int. J. Mech. Sci. 164 (2019) 105169.
[3] P.A. Prates, A.S. Adaixo, M.C. Oliveira, J.V. Fernandes, Numerical Study on the Effect of Mechanical Properties Variability in Sheet Metal Forming Processes, Int. J. Adv. Manuf. Technol. 96 (2018) 561-580.
[4] W. Hancock, M. Zayko, M. Autio, D. Ponagajba, Analysis of components of variation in automotive stamping processes, Qual. Eng. 10 (1997) 115-124.
[5] K.D. Majeske, P.C. Hammett, Identifying Sources of Variation in Sheet Metal Stamping, Int. J. Flexib. Manuf. Syst. 15 (2003) 5-18.
[6] A.F.G. Pereira, M.F. Ruivo, M.C. Oliveira, J.V. Fernandes, P.A. Prates, Numerical Study of the Square Cup Stamping Process: A Stochastic Analysis, ESAFORM 2021 (2021).
[7] M. Arnst, J.-P. Ponthot, R. Boman, Comparison of Stochastic and Interval Methods for Uncertainty Quantification of Metal Forming Processes, Comptes Rendus Mécanique 346 (2018) 634-646.
[8] M. Dwivedy, V. Kalluri, The Effect of Process Parameters on Forming Forces in Single Point Incremental Forming, Procedia Manuf. 29 (2019) 120-128.
[9] V.J. Shahi, A. Masoumi, P. Franciosa, D. Ceglarek, Quality-Driven Optimization of Assembly Line Configuration for Multi-Station Assembly Systems with Compliant Non-Ideal Sheet Metal Parts, Procedia CIRP 75 (2018) 45-50.
[10] M.A. Dib, N.J. Oliveira, A.E. Marques, M.C. Oliveira, J.V. Fernandes, B.M. Ribeiro, P.A. Prates, Single and Ensemble Classifiers for Defect Prediction in Sheet Metal Forming under Variability, Neural Comput. Appl. 32 (2019) 12335-12349.
[11] J. Fruth, O. Roustant, Kuhnt, S. Support Indices: Measuring the Effect of Input Variables over Their Supports, Reliab. Eng. Syst. Saf. 187 (2019) 17-27.
[12] P. Zhu, L. Zhang, R. Zhou, L. Chen, B. Yu, Q. Xie, A Novel Sensitivity Analysis Method in Structural Performance of Hydraulic Press, Math. Probl. Eng. 2012 (2012) 1-21.
[13] S.H. Kim, H. Huh, Design Sensitivity Analysis of Sheet Metal Forming Processes with a Direct Differentiation Method, J. Mater. Process. Technol. 130-131 (2002) 504-510.
[14] I.M. Sobol′, Global Sensitivity Indices for Nonlinear Mathematical Models and Their Monte Carlo Estimates, Math. Comput. Simul. 55 (2001) 271-280.
[15] T. Hama, M. Takamura, A. Makinouchi, C. Teodosiu, H. Takuda, Effect of Tool-Modeling Accuracy on Square-Cup Deep-Drawing Simulation, Mater. Trans. 49 (2008) 278-283.
[16] Y.Q. Li, Z.S. Cui, X.Y. Ruan, D.J. Zhang, CAE-Based Six Sigma Robust Optimization for Deep- Drawing Process of Sheet Metal, Int. J. Adv. Manuf. Technol. 30 (2006) 631-637.
[17] J.M. Gutiérrez Regueras, A.M. Camacho López, Investigations on the Influence of Blank Thickness (t) and Length/Wide Punch Ratio (LD) in Rectangular Deep Drawing of Dual-Phase Steels, Comput. Mater. Sci. 91 (2014) 134-145.
[18] E. Bayraktar, S. Altintaş, Square Cup Deep Drawing Experiments, Proceedings of the NUMISHEET ’93: Proceedings of the 2nd International Conference Numerical Simulation of 3-D Sheet Metal Forming Processes, Isehara, Japan, 1993, p. 441.
[19] B. Sudret, Global Sensitivity Analysis Using Polynomial Chaos Expansions, Reliab. Eng. Syst. Saf. 93 (2008) 964-979.
[20] L.F. Menezes, C. Teodosiu, Three-Dimensional Numerical Simulation of the Deep-Drawing Process Using Solid Finite Elements, J. Mater. Process. Technol. 97 (2000) 100-106.
[21] A. Saltelli, M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana, S. Tarantola, Global Sensitivity Analysis, The Primer, 2008.
[22] A. Janon, T. Klein, A. Lagnoux, M. Nodet, C. Prieur, Asymptotic Normality and Efficiency of Two Sobol Index Estimators, ESAIM – Probability and Statistics 18 (2014) 342-364.
[23] I.M. Sobol’, On the Distribution of Points in a Cube and the Approximate Evaluation of Integrals, USSR Computat. Math. Math. Phys. 7 (1967) 86-112.
[24] J. Lebon, G. le Quilliec, R.F. Coelho, P. Breitkopf, P. Villon, Variability and Sensitivity Analysis of U-Shaped Deep Drawn Metal Sheet, Proceedings of the 11e colloque national en calcul des structures, Giens, France, May 2013.
[25] G. Blatman, B. Sudret, Adaptive Sparse Polynomial Chaos Expansion Based on Least Angle Regression, J. Comput. Phys. 230 (2011) 2345-2367.
[26] B. Sudret, Risk and Reliability in Geotechnical Engineering, In Risk and Reliability in Geotechnical Engineering, K.-K. Phoon, J. Ching (Eds.), CRC Press: Boca Raton, 2018, ISBN 9781482227222.