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Modelling of a thermomechanically coupled forming process based on functional outputs from a finite element analysis and from experimental measurements

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Abstract

In this paper, measurements from experiments and results of a finite element analysis (FEA) are combined in order to compute accurate empirical models for the temperature distribution before a thermomechanically coupled forming process. To accomplish this, Design and Analysis of Computer Experiments (DACE) is used to separately compute models for the measurements and the functional output of the FEA. Based on a hierarchical approach, a combined model of the process is computed. In this combined modelling approach, the model for the FEA is corrected by taking into account the systematic deviations from the experimental measurements. The large number of observations based on the functional output hinders the direct computation of the DACE models due to the internal inversion of the correlation matrix. Thus, different techniques for identifying a relevant subset of the observations are proposed. The application of the resulting procedure is presented, and a statistical validation of the empirical models is performed.

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References

  • Bayarri, M.J., Berger, J.O., Cafeo, J.A., Garcia-Donato, G., Liu, F., Palomo, J., Parthasarathy, R.J., Paulo, R., Sacks, J., Walsh, D.: Computer model validation with functional output. Ann. Stat. 5, 1874–1906 (2007)

    Article  MathSciNet  Google Scholar 

  • Broecker, C., Matzenmiller, A., Steinhoff, K.: Process simulation of stub shaft forging with local heating and cooling—an analysis with EFG. Comput. Methods Mater. Sci. 8, 144–153 (2008)

    Google Scholar 

  • Chen, W., Baghdasaryan, L., Buranathiti, T., Cao, J.: Model validation via uncertainty propagation and data transformations. AIAA J. 42(7), 1406–1416 (2006)

    Article  Google Scholar 

  • Cressie, N.A.C.: Statistics for Spatial Data. Wiley, New York (1993). Revised edn

    Google Scholar 

  • Dowding, K.J., Pilch, M., Hills, R.G.: Formulation of the thermal problem. Comput. Methods Appl. Mech. Eng. 197(29–32), 2385–2389 (2007)

    MathSciNet  Google Scholar 

  • Goldstein, M., Rougier, J.C.: Probabilistic formulations for transferring inferences from mathematical models to physical systems. SIAM J. Sci. Comput. 26, 467–487 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  • Haftka, R.T.: Combining global and local approximations. AIAA J. 29, 1523–1525 (1991)

    Article  Google Scholar 

  • Higdon, D., Kennedy, M.C., Cavendish, J.C., Cafeo, J.A., Ryne, R.D.: Combining field data and computer simulations for calibration and prediction. SIAM J. Sci. Comput. 26, 448–466 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  • Higdon, D., Nakhleh, C., Gattiker, J., Williams, B.: A Bayesian calibration approach to the thermal problem. Comput. Methods Appl. Mech. Eng. 197(29–32), 2431–2441 (2007)

    Google Scholar 

  • Higdon, D., Gattiker, J., Williams, B., Rightley, M.: Computer model calibration using high-dimensional output. J. Am. Stat. Assoc. 103(482), 570–583 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  • Huang, D., Allen, T.T., Notz, W.I., Miller, R.A.: Sequential kriging optimization using multiple fidelity evaluations. Struct. Multidiscip. Optim. 32(5), 369–382 (2006a)

    Article  Google Scholar 

  • Huang, D., Allen, T.T., Notz, W.I., Zheng, N.: Global optimization of stochastic black-box systems via sequential kriging meta-models. J. Glob. Optim. 34(4), 441–466 (2006b)

    Article  MATH  MathSciNet  Google Scholar 

  • Jin, R., Chen, W., Sudjianto, A.A.: An efficient algorithm for constructing optimal design of computer experiments. J. Stat. Plan. Inference 134, 268–287 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • Jones, D.R., Schonlau, M., Welch, W.J.: Efficient global optimization of expensive black-box functions. J. Glob. Optim. 13, 455–492 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  • Keane, A.J.: Wing optimization using design of experiments, response surface, and data fusion methods. J. Aircr. 40(4), 741–750 (2003)

    Article  MathSciNet  Google Scholar 

  • Kennedy, M.C., O’Hagan, A.: Predicting the output from a complex computer code when fast approximations are available. Biometrika 87(1), 1–13 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • Kennedy, M.C., O’Hagan, A.: Bayesian calibration of computer models. J. R. Stat. Soc. Ser. B-Stat. Methodol. 63, 425–464 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  • Levy, S., Steinberg, D.M.: Computer experiments: A review. Adv. Stat. Anal. 94(4) (2010), this issue

  • Liu, F., Bayarri, M.J., Berger, J.O., Paulo, R., Sacks, J.: A Bayesian analysis of the thermal challenge problem. Comput. Methods Appl. Mech. Eng. 197(29–32), 2457–2466 (2007)

    Google Scholar 

  • Lophaven, S.N., Nielsen, H.B., Sondergaard, J.: Aspects of the MATLAB toolbox DACE. Report IMM-REP-2002-13, Informatics and Mathematical Modelling, DTU (2002)

  • Maier, H.J., Tschumak, S., Weidig, U., Steinhoff, K.: Functional gradation of low alloy steel by differentially controlled phase transformation. Steel. Res. Int. 79(2), 33–38 (2008)

    Google Scholar 

  • Matzenmiller, A., Broecker, C., Gerlach, S.: FE-analysis of simultaneous hot/cold forging. Steel. Res. Int. 80(2), 130–136 (2009)

    Google Scholar 

  • Petelet, M., Iooss, B., Asserin, O., Loredo, A.: Latin hypercube sampling with inequality constraints. Adv. Stat. Anal. 94(4) (2010) this issue

  • Pistone, G., Vicario, G.: Comparing and generating latin hypercube designs in kriging models. Adv. Stat. Anal. 94(4) (2010) this issue

  • Qian, P.Z.G., Wu, C.F.J.: Bayesian hierarchical modeling for integrating low-accuracy and high-accuracy experiments. Technometrics 50(2), 192–204 (2008)

    Article  MathSciNet  Google Scholar 

  • Qian, P.Z.G., Seepersad, C.C., Joseph, V.R., Allen, J.K., Wu, C.F.J.: Building surrogate models based on detailed and approximate simulations. J. Mech. Des. 128, 668–677 (2006)

    Google Scholar 

  • Reese, C.S., Wilson, A.G., Hamada, M., Martz, H.F., Ryan, K.J.: Integrated analysis of computer and physical experiments. Technometrics 46, 153–164 (2004)

    Article  MathSciNet  Google Scholar 

  • Rougier, J.: Efficient emulators for multivariate deterministic functions. J. Comput. Graph. Stat. 17(4), 827–843 (2008)

    Article  MathSciNet  Google Scholar 

  • Sacks, J., Welch, W.J., Mitchell, T.J., Wynn, H.P.: Design and analysis of computer experiments. Stat. Sci. 4(4), 409–423 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  • Santner, T.J., Williams, B.J., Notz, W.I.: The Design and Analysis of Computer Experiments. Springer, New York (2003)

    MATH  Google Scholar 

  • Watson, P.M., Gupta, K.C.: EM-ANN models for microstrip vias and interconnects in dataset circuits. IEEE Trans. Microw. Theory Tech. 44, 2495–2503 (1996)

    Article  Google Scholar 

  • Weidig, U., Huebner, K., Steinhoff, K.: Bulk steel products with functionally graded properties produced by differential thermo-mechanical processing. Steel Res. Int. 79(1), 59–65 (2008)

    Google Scholar 

  • Wever, F., et al.: Atlas zur Wärmebehandlung der Stähle. Verlag Stahleisen M.B.H., Düsseldorf (1961)

    Google Scholar 

  • Williams, B., Higdon, D., Gattiker, J., Moore, L., McKay, M., Keller-McNulty, S.: Combining experimental data and computer simulations, with an application to flyer plate experiments. Bayesian Anal. 1(4), 765–792 (2006)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Machining Technology (ISF), Technische Universität Dortmund, Baroper Str. 301, Dortmund, 44227, Germany

    Tobias Wagner & Dirk Biermann

  2. Institute of Mechanics, Computational Mechanics, University of Kassel, Mönchebergstr. 7, Kassel, 34125, Germany

    Christoph Bröcker & Anton Matzenmiller

  3. Chair of Metal Forming Technology, University of Kassel, Kurt-Wolters-Str. 3, Kassel, 34125, Germany

    Nicolas Saba & Kurt Steinhoff

Authors
  1. Tobias Wagner
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  2. Christoph Bröcker
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  3. Nicolas Saba
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  4. Dirk Biermann
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  5. Anton Matzenmiller
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  6. Kurt Steinhoff
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Corresponding author

Correspondence to Tobias Wagner.

Additional information

This paper is based on investigations of the collaborative research centre SFB/TR TRR 30, which is kindly supported by the Deutsche Forschungsgemeinschaft (DFG).

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Wagner, T., Bröcker, C., Saba, N. et al. Modelling of a thermomechanically coupled forming process based on functional outputs from a finite element analysis and from experimental measurements. AStA Adv Stat Anal 94, 389–404 (2010). https://doi.org/10.1007/s10182-010-0149-7

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  • DOI: https://doi.org/10.1007/s10182-010-0149-7

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