Multi-fidelity Metamodels Nourished by Reduced Order Models

  • S. Nachar
  • P.-A. BoucardEmail author
  • D. Néron
  • U. Nackenhorst
  • A. Fau
Part of the Lecture Notes in Applied and Computational Mechanics book series (LNACM, volume 93)


Engineering simulation provides better designed products by allowing many options to be quickly explored and tested. In that context, the computational time is a strong issue because using high-fidelity direct resolution solvers is not always suitable. Metamodels are commonly considered to explore design options without computing every possible combination of parameters, but if the behavior is nonlinear, a large amount of data is required to build this metamodel. A possibility is to use further data sources to generate a multi-fidelity surrogate model by using model reduction. Model reduction techniques constitute one of the tools to bypass the limited calculation budget by seeking a solution to a problem on a reduced-order basis (ROB). The purpose of this study is an online method for generating a multi-fidelity metamodel nourished by calculating the quantity of interest from the basis generated on-the-fly with the LATIN-PGD framework for elasto-viscoplastic problems. Low-fidelity (LF) fields are obtained by stopping the solver before convergence, and high-fidelity (HF) information is obtained with converged solutions. In addition, the solver ability to reuse information from previously calculated PGD basis is exploited.


  1. 1.
    Bhattacharyya, M., Fau, A., Nackenhorst, U., Néron, D., & Ladevèze, P. (2017). A LATIN-based model reduction approach for the simulation of cycling damage. Computational Mechanics, 1–19.
  2. 2.
    Chinesta, F., Keunings, R., & Leygue, A. (2014). The proper generalized decomposition for advanced numerical simulations. In SpringerBriefs in applied sciences and technology. Cham: Springer International Publishing.Google Scholar
  3. 3.
    Courrier, N., Boucard, P. A., & Soulier, B. (2016). Variable-fidelity modeling of structural analysis of assemblies. Journal of Global Optimization, 64(3), 577–613. Scholar
  4. 4.
    Forrester, A. I., Bressloff, N. W., & Keane, A. J. (2006). Optimization using surrogate models and partially converged computational fluid dynamics simulations. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 462(2071), 2177–2204. Scholar
  5. 5.
    Forrester, A. I. J., Keane, A. J., & Bressloff, N. W. (2006). Design and analysis of “noisy” computer experiments. AIAA Journal, 44(10), 2331–2339.CrossRefGoogle Scholar
  6. 6.
    Han, Z., Zimmerman, R., & Görtz, S. (2012). Alternative cokriging method for variable fidelity surrogate modeling. AIAA Journal, 50(5), 1205–1210. Scholar
  7. 7.
    Jones, D. R. (2001). A taxonomy of global optimization methods based on response surfaces. Journal of Global Optimization, 21(4), 345–383. Scholar
  8. 8.
    Kramer, B., Marques, A.N., Peherstorfer, B., Villa, U., & Willcox, K. (2019). Multifidelity probability estimation via fusion of estimators. Journal of Computational Physics, 392, 385–402 .
  9. 9.
    Ladevèze, P. (1999). Nonlinear computational structural mechanics: New approaches and non-incremental methods of calculation. In Mechanical engineering series. Springer.Google Scholar
  10. 10.
    Lemaitre, J., & Chaboche, J. L. (1994). Mechanics of solid materials. Cambridge University Press.Google Scholar
  11. 11.
    Maday, Y., & Ronquist, E. (2004). The reduced basis element method: Application to a thermal fin problem. SIAM Journal on Scientific Computing, 26(1), 240–258. Scholar
  12. 12.
    McKay, M. D., Beckman, R. J., & Conover, W. J. (2000). A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics, 42(1), 55–61.CrossRefGoogle Scholar
  13. 13.
    Rasmussen, C. E., & Williams, C. K. (2004). Gaussian processes in machine learning. Lecture Notes in Computer Science, 3176, 63–71.CrossRefGoogle Scholar
  14. 14.
    Relun, N., Néron, D., & Boucard, P. A. (2013). A model reduction technique based on the PGD for elastic-viscoplastic computational analysis. Computational Mechanics, 51(1), 83–92. Scholar
  15. 15.
    Vitse, M. (2016) Model-order reduction for the parametric analysis of damage in reinforced concrete structures (Ph.D. thesis). Université Paris-Saclay.Google Scholar
  16. 16.
    Zimmerman, D. L., & Holland, D. M. (2005). Complementary co-kriging: Spatial prediction using data combined from several environmental monitoring networks. Environmetrics, 16, 219–234.MathSciNetCrossRefGoogle Scholar
  17. 17.
    Zimmermann, R., & Han, Z. H. (2010). Simplified cross-correlation estimation for multi-fidelity surrogate cokriging models. Advances and Applications in Mathematical Sciences, 7(2), 181–201.MathSciNetzbMATHGoogle Scholar

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • S. Nachar
    • 1
  • P.-A. Boucard
    • 1
    Email author
  • D. Néron
    • 1
  • U. Nackenhorst
    • 2
  • A. Fau
    • 2
  1. 1.Université Paris-Saclay, ENS Paris-Saclay, CNRS, LMT - Laboratoire de Mécanique et TechnologieCachan CedexFrance
  2. 2.LUH - Leibniz Universität Hannover [Hannover] 44961Gottfried Wilhelm Leibniz Universität HannoverHannoverGermany

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