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Structural and Multidisciplinary Optimization

, Volume 60, Issue 6, pp 2373–2389 | Cite as

A multi-objective optimization methodology based on multi-mid-range meta-models for multimodal deterministic/robust problems

  • S. KhalfallahEmail author
  • H. E. Lehtihet
Research Paper
  • 99 Downloads

Abstract

The success of meta-model-based optimization primarily relies on how accurately the black-box functions are being represented. However, sometimes a global meta-model fails to achieve sufficient accuracy. This can be the case in multi-objective deterministic problems involving multimodal functions with competitive local optima or robust problems which require an accurate local description. This paper proposes a new methodology that deals with this type of situations and that provides the required accuracy both locally and globally. We use a set of mid-range meta-models which, in contrast to other works, are not used to construct a global meta-model but are managed both to compete and collaborate to solve the problem. They are defined across overlapping regions of interest generated by a process which resizes and moves adaptively these regions until tracking the Pareto front. The accuracy of these mid-range meta-models is also improved by a new design-of-experiment (DoE) adaptive technique allowing the suppression of some inefficient DoE points. The proposed method is implemented using standard techniques, such as non-dominated sorting genetic algorithm-II (NSGA-II), whereas the optimal shape factor of radial basis functions (RBF) is calculated by combining NSGA-II and particle swarm optimization (PSO). We also use Hager’s method to detect ill-conditioned systems and avoid propagating their outcome, which significantly improves the performance. This method is tested against difficult deterministic and robust multi-objective multimodal benchmarks and is applied to the robust optimization of an aerodynamic design case.

Keywords

Meta-model-based optimization Robust design Multimodal functions CFD 

Notes

Acknowledgments

The authors would like to thank anonymous reviewers for having provided valuable criticisms and recommendations which have greatly helped improve the quality of the paper.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Supplementary material

158_2019_2327_MOESM1_ESM.zip (4.8 mb)
ESM 1 (ZIP 4944 kb)

References

  1. Baudoui V, Klotz P, Hiriart-Urruty JB, Jan S, Morel F (2012) Local uncertainty processing (LOUP) method for multidisciplinary robust design optimization. Struct Multidiscip Optim 46(5):711–726CrossRefGoogle Scholar
  2. Deb K, Gupta H (2006) Introducing robustness in multi-objective optimization. Evol Comput 14(4):463–494CrossRefGoogle Scholar
  3. Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197CrossRefGoogle Scholar
  4. Deb K, Hussein R, Roy P, and Toscano G, (2017) Classifying metamodeling methods for evolutionary multi-objective optimization: first results. EMO 2017 9th International Conference on Evolutionary Multi-Criterion Optimization, 10173: 160–175Google Scholar
  5. Diaz-Manriquez A, Toscano-Pulido G, and Gomez-Flores W. (2011) On the selection of surrogate models in evolutionary optimization algorithms. proceeding of: Evolutionary Computation (CEC), IEEE Congress on, 2155–2162Google Scholar
  6. Dow EA and Wang Q, (2014) The implications of tolerance optimization on compressor blade design. ASME J Turbomach 137(10):101008_1–101008_7Google Scholar
  7. Emmerich MTM, Giannakoglou KC, Naujoks B (2006) Single- and multiobjective evolutionary optimization assisted by Gaussian random field metamodels. IEEE Trans Evol Comput 10(4):421–439CrossRefGoogle Scholar
  8. Fonseca CM, Fleming PJ (1998) Multiobjective optimization and multiple constraint handling with evolutionary algorithms—part II: application example. IEEE Trans Syst Man Cybern Syst Hum 28(1):38–47CrossRefGoogle Scholar
  9. Forrester AIJ, Keane AJ (2009) Recent advances in surrogate-based optimization. Prog Aerosp Sci 45(1–3):50–79CrossRefGoogle Scholar
  10. Giunta AA, and Eldred MS, (2000) Implementation of a trust region model management strategy in the Dakota optimization toolkit. Proceedings of 8th AIAA/USAF/ NASA/ISSMO Symposium on Multi- disciplinary Analysis and Optimization, Long Beach, CA, U.S.A.Google Scholar
  11. Gray WANGG, Shan S (2007) Review of metamodeling techniques in support of engineering design optimization. ASME J Mechan Design 129(4):370–380CrossRefGoogle Scholar
  12. Gu X, Renaud JE, Batill SM, Brach RM, Budhiraja AS (2000) Worst case propagated uncertainty of multidisciplinary systems in robust design optimization. Struct Multidiscip Optim 20(3):190–213CrossRefGoogle Scholar
  13. Gumbert CR, Newman PA, and Hou GJ-W, (2002) Effect of random geometric uncertainty on the computational design of a 3-D Flexible Wing. 20th AIAA Applied Aerodynamics Conference, AIAA 2002–2806Google Scholar
  14. Haftka RT, Nachlas JA, Watson LT, Rizzo T, Desai R (1987) Two-point constraint approximation in structural optimization. Comput Methods Appl Mech Eng 60(3):289–301CrossRefGoogle Scholar
  15. Hager WW (1984) Condition estimates. SIAM J Sci Stat Comput 5(2):311–316MathSciNetCrossRefGoogle Scholar
  16. Havinga J, van den Boogaard AH, Klaseboer G (2017) Sequential improvement for robust optimization using an uncertainty measure for radial basis functions. Struct Multidiscip Optim 55(4):1345–1363MathSciNetCrossRefGoogle Scholar
  17. Janusevskis J, Riche RL (2013) Simultaneous kriging-based estimation and optimization of mean response. J Glob Optim 55(2):313–336MathSciNetCrossRefGoogle Scholar
  18. Jin R, Chen W, Simpson TW (2001) Comparative studies of metamodeling techniques under multiple modeling criteria. Struct Multidiscip Optim 23(1):1–13CrossRefGoogle Scholar
  19. Jin R, Du X, Chen W (2003) The use of metamodeling techniques for optimization under uncertainty. Struct Multidiscip Optim 25(2):99–116CrossRefGoogle Scholar
  20. Jin R, Chen W, and Sudjianto A, (2004) Analytical metamodel-based global sensitivity analysis and uncertainty propagation for robust design. SAE Technical Paper 2004-01-0429Google Scholar
  21. Jin R, Chen W, Sudjianto A (2005) An efficient algorithm for constructing optimal design of computer experiments. J Stat Plann Infer 134(1):268–287MathSciNetCrossRefGoogle Scholar
  22. Jones DR, Schonlau M, Welch WJ (1998) Efficient global optimization of expensive black-box functions. J Glob Optim 13(4):455–492MathSciNetCrossRefGoogle Scholar
  23. Jurecka F, Ganser M, Bletzinger KU (2007) Update scheme for sequential spatial correlation approximations in robust design optimization. Comput Struct 85(10):606–614CrossRefGoogle Scholar
  24. Kennedy J and Eberhart R, (1995) Particle swarm optimization. Proceedings of the IEEE International Conference on Neural Networks, 4: 1942–1948Google Scholar
  25. Khalfallah S, Ghenaiet A, Benini E, Bedon G (2015) Surrogate-based shape optimization of stall margin and efficiency of a centrifugal compressor. AIAA J Propuls Power 31(6):1607–1162CrossRefGoogle Scholar
  26. Koch PN, Simpson TW, Allen JK, Mistree F (1999) Statistical approximations for multidisciplinary optimization: the problem of size. AIAA Journal of Aircraft 36(1):275–286CrossRefGoogle Scholar
  27. Korolev YM and Toropov VV, (2017) Design optimization under uncertainty using the multipoint approximation method, 58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials ConferenceGoogle Scholar
  28. Kumar A, Nair PB, Keane AJ, Shahpar S (2008) Robust design using Bayesian Monte Carlo. Int J Numer Methods Eng 73(11):1497–1517CrossRefGoogle Scholar
  29. Ladson CL, (1988) Effects of independent variation of Mach and Reynolds numbers on the low-speed aerodynamic characteristics of the NACA 0012 airfoil section. Tech. Rep., NASA Langley Research CenterGoogle Scholar
  30. Leary SJ, Bhaskar A, Keane AJ (2003) A knowledge-based approach to response surface modelling in multifidelity optimization. J Glob Optim 26(3):297–319MathSciNetCrossRefGoogle Scholar
  31. Lee KH, Park GJ (2006) A global robust optimization using kriging based approximation model. Jsme Int J Series C-Mechan Syst Mach Elements Manuf 49(3):779–788CrossRefGoogle Scholar
  32. Lehman JS, Santner TJ, Notz WI (2004) Designing computer experiments to determine robust control variables. Stat Sin 14(2):571–590MathSciNetzbMATHGoogle Scholar
  33. Lelièvre N, Beaurepaire P, Mattrand C, Gayton N, Otsmane A (2016) On the consideration of uncertainty in design: optimization - reliability – robustness. Struct Multidiscip Optim 54(6):1423–1437MathSciNetCrossRefGoogle Scholar
  34. Liu H, Chen W, Kokkolaras M, Papalambros PY, Kim HM (2006) Probabilistic analytical target cascading: a moment matching formulation for multilevel optimization under uncertainty. ASME J Mechan Design 128(4):991–1000CrossRefGoogle Scholar
  35. Lock RC, (1970) Test cases for numerical methods in two-dimensional transonic flows. Report of Advisory Group for Aerospace Research and Development (AGARD), No. 575Google Scholar
  36. Ong YS, Nair PB, Keane AJ (2003) Evolutionary optimization of computationally expensive problems via surrogate modeling. AIAA J 41(4):687–696CrossRefGoogle Scholar
  37. Padulo M, Maginot J, Guenov M, and Holden C, (2009) Airfoil design under uncertainty with robust geometric parameterization. 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Structures, Structural Dynamics, and Materials and Co-located Conferences, AIAA 2009–2270Google Scholar
  38. Pedrielli G and Hui Ng S, (2016) G-STAR: a new kriging-based trust region method for global optimization. IEEE Proceedings of the 2016 Winter Simulation Conference, 803–814Google Scholar
  39. Polynkin A and Toropov VV, (2009) Multiple mid-range and global metamodel building based on linear regression. Collection of Technical Papers - AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials ConferenceGoogle Scholar
  40. Polynkin A, Toropov VV (2012) Mid-range metamodel assembly building based on linear regression for large scale optimization problems. Struct Multidiscip Optim 45(4):515–527CrossRefGoogle Scholar
  41. Qin N, Carnie G, Wang Y, Shahpar S (2014) Design optimization of casing grooves using zipper layer meshing. ASME J Turbomach 136(3):031002–1-031002-12CrossRefGoogle Scholar
  42. Rippa S (1999) An algorithm for selecting a good value for the parameter c in radial basis function interpolation. Adv Comput Math 11(2):193–210MathSciNetCrossRefGoogle Scholar
  43. Rumpfkeil MP (2013) Optimizations under uncertainty using gradients, hessians, and surrogate models. AIAA J 51(2):444–451CrossRefGoogle Scholar
  44. Samareh JA, (2000) Multidisciplinary aerodynamic-structural shape optimization using deformation (MASSOUD). 8th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization. AIAA-2000-4911, Long Beach, CAGoogle Scholar
  45. Simpson TW, Mauery TM, Korte JJ (2001) Kriging models for global approximation in simulation-based multidisciplinary design optimization. AIAA J 39(12):2233–2241CrossRefGoogle Scholar
  46. Simpson TW, Toropov V, Balabanov V, and Viana F, (2008) Design and analysis of computer experiments in multidisciplinary design optimization: a review of how far we have come - or not. 12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Multidisciplinary Analysis Optimization Conferences, Victoria, British Columbia, CanadaGoogle Scholar
  47. Toropov VV, Filatov AA, Polynkin AA (1993) Multiparameter structural optimization using FEM and multipoint explicit approximations. Struct Optimization 6(1):7–14CrossRefGoogle Scholar
  48. Venter G, Haftka RT (1999) Using response surface approximations in fuzzy set based design optimization. Struct Multidiscip Optim 18(4):218–227CrossRefGoogle Scholar
  49. Wang XD, Hirsch C, Kang SH, Lacor C (2011) Multi-objective optimization of turbomachinery using improved NSGA-II and approximation model. Comput Methods Appl Mech Eng 200(9–12):883–895MathSciNetCrossRefGoogle Scholar
  50. Wilson B, Cappelleri D, Simpson TW, Frecker M (2001) Efficient Pareto frontier exploration using surrogate approximations. Optim Eng 2(1):31–50MathSciNetCrossRefGoogle Scholar
  51. Yahyaie F, Filizadeh S (2011) A surrogate-model based multi-modal optimization algorithm. Eng Optim 43(7):779–799CrossRefGoogle Scholar
  52. Zhang J, Chowdhury S, Messac A (2012) An adaptive hybrid surrogate model. Struct Multidiscip Optim 46(2):223–238CrossRefGoogle Scholar
  53. Zhao D, Xue D (2011) A multi-surrogate approximation method for metamodeling. Eng Comput 27(2):139–153CrossRefGoogle Scholar
  54. Zhou Z, Ong YS, Nair PB, Keane AJ, Kai-Yew L (2007) Combining global and local surrogate models to accelerate evolutionary optimization. IEEE Trans Syst Man and Cybernet (SMC), Part C 37:66–76CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Laboratory of Turbomachinery, Ecole Militaire PolytechniqueAlgiersAlgeria
  2. 2.Department of Applied Mathematics Ecole Militaire PolytechniqueAlgiersAlgeria

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