Advertisement

Structural and Multidisciplinary Optimization

, Volume 60, Issue 6, pp 2205–2220 | Cite as

Linear regression-based multifidelity surrogate for disturbance amplification in multiphase explosion

  • M. Giselle Fernández-Godino
  • Sylvain Dubreuil
  • Nathalie Bartoli
  • Christian Gogu
  • S. Balachandar
  • Raphael T. HaftkaEmail author
Research Paper
  • 165 Downloads

Abstract

When simulations are very expensive and many are required, as for optimization or uncertainty quantification, a way to reduce cost is using surrogates. With multiple simulations to predict the quantity of interest, some being very expensive and accurate (high-fidelity simulations) and others cheaper but less accurate (low-fidelity simulations), it may be worthwhile to use multifidelity surrogates (MFSs). Moreover, if we can afford just a few high-fidelity simulations or experiments, MFS becomes necessary. Co-Kriging, which is probably the most popular MFS, replaces both low-fidelity and high-fidelity simulations by a single MFS. A recently proposed linear regression–based MFS (LR-MFS) offers the option to correct the LF simulations instead of correcting the LF surrogate in the MFS. When the low-fidelity simulation is cheap enough for use in an application, such as optimization, this may be an attractive option. In this paper, we explore the performance of LR-MFS using exact and surrogate-replaced low-fidelity simulations. The problem studied is a cylindrical dispersal of 100-μ m-diameter solid particles after detonation and the quantity of interest is a measure of the amplification of the departure from axisymmetry. We find very substantial accuracy improvements for this problem using the LR-MFS with exact low-fidelity simulations. Inspired by these results, we also compare the performance of co-Kriging to the use of Kriging to correct exact low-fidelity simulations and find a similar accuracy improvement when simulations are directly used. For this problem, further improvements in accuracy are achievable by taking advantage of inherent parametric symmetries. These results may alert users of MFSs to the possible advantages of using exact low-fidelity simulations when this is affordable.

Keywords

Multifidelity Surrogates Symmetries Linear regression Kriging Co-Kriging 

Nomenclature

δ(x)

Discrepancy function

\(\hat {\delta }(\mathbf {x})\)

Discrepancy function surrogate, also known as additive correction

ρ

Constant scaling factor

yHF(x)

High-fidelity simulation

\(\hat {y}_{HF}(\mathbf {x})\)

High-fidelity surrogate

yLF(x)

Low-fidelity simulation

\(\hat {y}_{LF}(\mathbf {x})\)

Low-fidelity surrogate

\(\hat {y}_{\hat {add}}(\mathbf {x})\)

Multifidelity surrogate that uses additive correction and where the prediction is performed using a low-fidelity surrogate

\(\hat {y}_{\hat {comp}}(\mathbf {x})\)

Multifidelity surrogate that uses comprehensive correction and where the prediction is performed using a low-fidelity surrogate

\(\hat {y}_{add}(\mathbf {x})\)

Multifidelity surrogate that uses additive correction and where the prediction is performed using low-fidelity simulations

\(\hat {y}_{comp}(\mathbf {x})\)

Multifidelity surrogate that uses comprehensive correction and where the prediction is performed using low-fidelity simulations

Notes

Acknowledgments

We gratefully acknowledge the contribution of a reviewer, Professor Andy Keane, who suggested trying the additive Kriging.

Funding sources

This work was partially supported by the Center for Compressible Multiphase Turbulence, the U.S. Department of Energy, National Nuclear Security Administration, Advanced Simulation and Computing Program, as a Cooperative Agreement under the Predictive Science Academic Alliance Program, under Contract No. DE-NA0002378.

This work was partially supported by the French National Research Agency (ANR) through the ReBReD project under grant ANR-16-CE10-0002 and by a ONERA internal project MUFIN dedicated about multi-fidelity.

This work was partially performed under U.S. Government contract 89233218CNA000001 for Los Alamos National Laboratory (LANL), which is operated by Triad National Security, LLC for the U.S. Department of Energy/National Nuclear Security Administration. Approved for public release LA-UR-19-22491.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Supplementary material

158_2019_2387_MOESM1_ESM.tex (120 kb)
(TEX 119 KB)

References

  1. Annamalai S, Rollin B, Ouellet F, Neal C, Jackson TL, Balachandar S (2016) Effects of initial perturbations in the early moments of an explosive dispersal of particles. J Fluids Eng 138(7):070903.  https://doi.org/10.1115/1.4030954 CrossRefGoogle Scholar
  2. Blatman G, Sudret B (2010) Efficient computation of global sensitivity indices using sparse polynomial chaos expansions. Reliab Eng Syst Saf 95(11):1216–1229.  https://doi.org/10.1016/j.ress.2010.06.015. http://www.sciencedirect.com/science/article/pii/S0951832010001493 CrossRefGoogle Scholar
  3. Blatman G, Sudret B (2011) Adaptive sparse polynomial chaos expansion based on least angle regression. J Comput Phys 230(6):2345–2367.  https://doi.org/10.1016/j.jcp.2010.12.021. http://www.sciencedirect.com/science/article/pii/S0021999110006856 http://www.sciencedirect.com/science/article/pii/S0021999110006856 MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bouhlel MA, Hwang JT, Bartoli N, Lafage R, Morlier J, Martins JRRA (2019) A Python surrogate modeling framework with derivatives. Adv Eng Softw.  https://doi.org/10.1016/j.advengsoft.2019.03.005 CrossRefGoogle Scholar
  5. Cameron RH, Martin WT (1947) The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals. Ann Math 48(2):385–392. http://www.jstor.org/stable/1969178 MathSciNetCrossRefGoogle Scholar
  6. Cressie N (1993) Statistics for spatial data: Wiley series in probability and statistics. Wiley, New York.  https://doi.org/10.1002/9781119115151 CrossRefGoogle Scholar
  7. Dobrat B, Crawford P (1981) Handbook, LLNL explosives. Lawrence Livermore National Laboratory. OSTI Identifier 6530310Google Scholar
  8. Dubreuil S, Berveiller M, Petitjean F, Salaün M (2014) Construction of bootstrap confidence intervals on sensitivity indices computed by polynomial chaos expansion. Reliab Eng Syst Saf 121(Supplement C):263–275.  https://doi.org/10.1016/j.ress.2013.09.011. http://www.sciencedirect.com/science/article/pii/S0951832013002688 CrossRefGoogle Scholar
  9. Fernández-Godino MG, Balachandar S, Haftka RT (2019a) On the use of symmetries in building surrogate models. J Mech Des 141(6):061402.  https://doi.org/10.1115/1.4042047 CrossRefGoogle Scholar
  10. Fernández-Godino MG, Ouellet F, Haftka R, Balachandar S (2019b) Early time evolution of circumferential perturbation of initial particle volume fraction in explosive cylindrical multiphase dispersion. J Fluids Eng 141:0913021–09130220.  https://doi.org/10.1115/1.4043055 CrossRefGoogle Scholar
  11. Fernández-Godino MG, Park C, Kim NH, Haftka RT (2019c) Issues in deciding whether to use multifidelity surrogates. AIAA J 57(5):2039–2054.  https://doi.org/10.2514/1.J057750 CrossRefGoogle Scholar
  12. Kennedy MC, O’Hagan A (2000) Predicting the output from a complex computer code when fast approximations are available. Biometrika 87(1):1–13.  https://doi.org/10.1093/biomet/87.1.1 MathSciNetCrossRefzbMATHGoogle Scholar
  13. Le Gratiet L, Garnier J (2014) Recursive co-kriging model for design of computer experiments with multiple levels of fidelity. Int J Uncertain Quantif 365–386Google Scholar
  14. Myers DE (1982) Matrix formulation of co-kriging. Math Geol 14(3):249–257.  https://doi.org/10.1007/BF01032887 MathSciNetCrossRefGoogle Scholar
  15. Ouellet F, Annamalai S, Rollin B (2017) Effect of a bimodal initial particle volume fraction perturbation in an explosive dispersal of particles. In: AIP Conference proceedings, vol 1793. AIP Publishing, p 150011.  https://doi.org/10.1063/1.4971740
  16. Queipo NV, Haftka RT, Shyy W, Goel T, Vaidyanathan R, Tucker PK (2005) Surrogate-based analysis and optimization. Prog Aerosp Sci 41(1):1–28.  https://doi.org/10.1016/j.paerosci.2005.02.001 CrossRefGoogle Scholar
  17. Seber GA, Lee AJ (2012) Linear regression analysis, vol 329. Wiley, New York. ISBN 978-0-471-41540-4Google Scholar
  18. Sobol IM (2001) Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math Comput Simul 55(1):271–280. http://www.sciencedirect.com/science/article/pii/S0378475400002706 MathSciNetCrossRefGoogle Scholar
  19. Sudret B (2008) Global sensitivity analysis using polynomial chaos expansions. Reliab Eng Syst Saf 93(7):964–979.  https://doi.org/10.1016/j.ress.2007.04.002. http://www.sciencedirect.com/science/article/pii/S0951832007001329. Bayesian Networks in DependabilityCrossRefGoogle Scholar
  20. Vauclin R (2014) Développement de modèles réduits multifidélité en vue de l’optimisation de structures aéronautiques. In: Rapport Institut Supérieur de l’Aéronautique et de l’Espace – École nationale Supérieure des mines de Saint-ÉtienneGoogle Scholar
  21. Zhang Y, Kim NH, Park C, Haftka RT (2018) Multifidelity surrogate based on single linear regression. AIAA J 56(12):4944–4952.  https://doi.org/10.2514/1.J057299 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Los Alamos National LaboratoryLos AlamosUSA
  2. 2.ONERA/DTIS, Université de ToulouseToulouseFrance
  3. 3.Université de Toulouse, CNRS, UPS, INSA, ISAE, Mines Albi, Institut, Clément Ader (ICA)ToulouseFrance
  4. 4.University of FloridaGainesvilleUSA

Personalised recommendations