Abstract
Surrogate models can be used to evaluate computationally expensive functions in optimization of nonlinear problems. The models are able to identify areas of interest in a multidimensional design space driven by methods of searching and evaluating new locations in a sparse design space. The use of constraints based on surrogates complicates the optimization problem as the uncertainty within the design space influences both the objective and constraint functions. While some optimization methods consider the mean prediction for the constraints, several methods have been reported on ways to include the uncertainty into the constraints. One way is to use a probability of feasibility to account for the mean prediction and its uncertainty. In this work, this approach is explored to evaluate its feasibility and compared to other alternate methods documented in the literature. Computationally efficient and reliable results were obtained for the mathematical test functions evaluated. The same advantageous trend could not be observed for more complex engineering applications since convergence was not reached within the prescribed computational cost by either this new approach or the established one. Nevertheless, more feasible designs were generated when the probability of feasibility was included in the objective function. For an airfoil drag minimization problem with aerodynamic and geometric constraints, improvements of \(\sim\)70% and \(\sim\)25% were achieved with the proposed methods considering inviscid and viscous flow conditions, respectively. These values are comparable to the ones obtained with an established upper trust bound approach.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
The choice for the number of initial samples depends on the goal of the model. For accuracy over the design space, many initial samples usually are necessary. For an adaptive optimization strategy, typically a low number of initial samples is chosen and the computational budget is spent on infill samples. Therefore, a low number was chosen (\(n_{start} = \text {max}(d+1,5)\)).
For the OFPF this happens less likely but can still occur as the change in objective is small from generation to generation.
References
AGARD (1979) Experimental data base for computer program assessment. Tech. Rep. 9283513231, North Atlantic Treaty Organization
Bartoli N, Lefebvre T, Dubreuil S, Olivanti R, Priem R, Bons N, Martins J, Morlier J (2019) Adaptive modeling strategy for constrained global optimization with application to aerodynamic wing design. Aerosp Sci Technol 90:85–102. https://doi.org/10.1016/j.ast.2019.03.041
Booker AJ, Dennis JE Jr, Torczon V, Trosset V, Trosset MW (1999) A rigorous framework for optimization of expensive functions by surrogates. Struct Multidisc Optim 17:1–13. https://doi.org/10.1007/BF01197708
Bouhlel MA, He S, Martins JRRA (2020) Scalable gradient-enhanced artificial neural networks for airfoil shape design in the subsonic and transonic regimes. Struct Multidisc Optim 61:1363–1376. https://doi.org/10.1007/s00158-020-02488-5
Chen S, Jiang Z, Yang S, Chen W (2017) Multimodel fusion based sequential optimization. AIAA J 55(1):241–254. https://doi.org/10.2514/1.J054729
Chen W, Chiu K, Fuge MD (2020) Airfoil design parameterization and optimization using bézier generative adversarial networks. AIAA J 58(11):4723–4735. https://doi.org/10.2514/1.J059317
Dong H, Song B, Dong Z, Wang P (2018) SCGOSR: Surrogate-based constrained global optimization using space reduction. Appl Soft Comput 65:462–477. https://doi.org/10.1016/j.asoc.2018.01.041
Du X, He P, Martins JR (2021) Rapid airfoil design optimization via neural networks-based parameterization and surrogate modeling. Aerosp Sci Technol 113:106701. https://doi.org/10.1016/j.ast.2021.106701
Forrester AIJ, Keane AJ (2009) Recent advances in surrogate-based optimization. Progress Aerosp Sci 45(1):50–79. https://doi.org/10.1016/j.paerosci.2008.11.001
Forrester AIJ, Sóbester A, Keane AJ (2008) Engineering design via surrogate modelling: a practical guide. Wiley, Chichester
Giselle Fernández-Godino M, Park C, Kim NH, Haftka RT (2019) Issues in deciding whether to use multifidelity surrogates. AIAA J 57(5):2039–2054. https://doi.org/10.2514/1.J057750
Goldberg DE (1989) Genetic algorithms in search, optimization, and machine learning. Addison-Wesley Publishing Company, Reading
Jones DR, Schonlau M (1998) Efficient global optimization of expensive black-box functions. J Global Optim 13:455–492. https://doi.org/10.1023/A:1008306431147
Kenway GK, Mader CA, He P, Martins JR (2019) Effective adjoint approaches for computational fluid dynamics. Progress Aerosp Sci 110:100542. https://doi.org/10.1016/j.paerosci.2019.05.002
Koziel S, Leifsson L (2013) Surrogate-based aerodynamic shape optimization by variable-resolution models. AIAA J 51(1):94–106. https://doi.org/10.2514/1.J051583
Kulfan B, Bussoletti J (2006) “Fundamental” parameteric geometry representations for aircraft component shapes. In: 11th AIAA/ISSMO multidisciplinary analysis and optimization conference, Portsmouth, https://doi.org/10.2514/6.2006-6948
Kulfan BM (2008) Universal parametric geometry representation method. J Aircraft 45(1):142–158. https://doi.org/10.2514/1.29958
Laurenceau J, Meaux M, Montagnac M, Sagaut P (2010) Comparison of gradient-based and gradient-enhanced response-surface-based optimizers. AIAA J 48(5):981–994. https://doi.org/10.2514/1.45331
Lefebvre T, Bartoli N, Dubreuil S, Panzeri M, Lombardi R, Della Vecchia P, Stingo L, Nicolosi F, De Marco A, Ciampa P, Anisimov K, Savelyev A, Mirzoyan A, Isyanov A (2020) Enhancing optimization capabilities using the agile collaborative MDOo framework with application to wing and nacelle design. Progress Aerosp Sci 119:100649. https://doi.org/10.1016/j.paerosci.2020.100649
Li J, Zhang M (2021) On deep-learning-based geometric filtering in aerodynamic shape optimization. Aerosp Sci Technol 112:106603. https://doi.org/10.1016/j.ast.2021.106603
Liem RP, Mader CA, Martins JR (2015) Surrogate models and mixtures of experts in aerodynamic performance prediction for aircraft mission analysis. Aerosp Sci Technol 43:126–151. https://doi.org/10.1016/j.ast.2015.02.019
Liu J, Song WP, Han ZH, Zhang Y (2017) Efficient aerodynamic shape optimization of transonic wings using a parallel infilling strategy and surrogate models. Struct Multidisc Optim 55:925–943. https://doi.org/10.1007/s00158-016-1546-7
Lomax H, Pulliam TH, Zingg DW (2001) Fundamentals of computational fluid dynamics. Springer, Berlin. https://doi.org/10.1007/978-3-662-04654-8
Mackman TJ, Allen CB, Ghoreyshi M, Badcock KJ (2013) Comparison of adaptive sampling methods for generation of surrogate aerodynamic models. AIAA J 51(4):797–808. https://doi.org/10.2514/1.J051607
Martínez IG, Afonso F, Rodrigues S, Lau F (2021) A sequential approach for aerodynamic shape optimization with topology optimization of airfoils. Math Comput Appl 26(2):34. https://doi.org/10.3390/mca26020034
Mckay MD, Beckman RJ, Conover WJ (1979) A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21(2):239–245. https://doi.org/10.1080/00401706.2000.10485979
Parr JM, Keane AJ, Forrester AIJ, Holden CME (2012) Infill sampling criteria for surrogate-based optimization with constraint handling. Eng Optim 44(10):1147–1166. https://doi.org/10.1080/0305215X.2011.637556
Peherstorfer B, Willcox K, Gunzburger M (2018) Survey of multifidelity methods in uncertainty propagation, inference, and optimization. Siam Rev 60(3):550–591. https://doi.org/10.1137/16M1082469
Priem R, Bartoli N, Diouane Y, Sgueglia A (2020) Upper trust bound feasibility criterion for mixed constrained Bayesian optimization with application to aircraft design. Aerosp Sci Technol 105:105980. https://doi.org/10.1016/j.ast.2020.105980
Queipo NV, Haftka RT, Shyy W, Goel T, Vaidyanathan R, Kevin Tucker P (2005) Surrogate-based analysis and optimization. Progress Aerosp Sci 41(1):1–28. https://doi.org/10.1016/j.paerosci.2005.02.001
Raul V, Leifsson L (2021) Surrogate-based aerodynamic shape optimization for delaying airfoil dynamic stall using kriging regression and infill criteria. Aerosp Sci Technol 111:106555. https://doi.org/10.1016/j.ast.2021.106555
Renganathan SA, Maulik R, Ahuja J (2021) Enhanced data efficiency using deep neural networks and gaussian processes for aerodynamic design optimization. Aerosp Sci Technol 111:106522. https://doi.org/10.1016/j.ast.2021.106522
Sacks J, Welch WJ, Mitchell TJ, Wynn HP (1989) Design and analysis of computer experiments. Stat Sci 4(4):409–423
Sasena MJ (2002) Flexibility and efficiency enhancements for constrained global design optimization with kriging approximations. Phd thesis, University of Michigan
Sasena MJ, Papalambros P, Goovaerts P (2002) Exploration of metamodeling sampling criteria for constrained global optimization. Eng Optim 34(3):263–278. https://doi.org/10.1080/03052150211751
Sóbester A, Leary SJ, Keane AJ (2005) On the design of optimization strategies based on global response surface approximation models. J Global Optim 33(1):31–59. https://doi.org/10.1007/s10898-004-6733-1
Sóbester A, Forrester AIJ, Toal DJJ, Tresidder E, Tucker S (2014) Engineering design applications of surrogate-assisted optimization techniques. Optim Eng 15:243–265. https://doi.org/10.1007/s11081-012-9199-x
Tao J, Sun G (2019) Application of deep learning based multi-fidelity surrogate model to robust aerodynamic design optimization. Aerosp Sci Technol 92:722–737. https://doi.org/10.1016/j.ast.2019.07.002
Thanedar PB, Vanderplaats GN (1995) Survey of discrete variable optimization for structural design. J Struct Eng 121(2):301–306. https://doi.org/10.1061/(ASCE)0733-9445(1995)121:2(301)
Watson AG, Barnes RJ (1995) Infill sampling criteria to locate extremes. Math Geol 27(5):589–608. https://doi.org/10.1007/BF02093902
Yondo R, Andrés E, Valero E (2018) A review on design of experiments and surrogate models in aircraft real-time and many-query aerodynamic analyses. Progress Aerosp Sci 96:23–61. https://doi.org/10.1016/j.paerosci.2017.11.003
Zhang X, Xie F, Ji T, Zhu Z, Zheng Y (2021) Multi-fidelity deep neural network surrogate model for aerodynamic shape optimization. Comput Methods Appl Mech Eng 373:113485. https://doi.org/10.1016/j.cma.2020.113485
Acknowledgements
A.S. and F.A. acknowledge Fundação para a Ciência e a Tecnologia (FCT), through IDMEC, under LAETA, project UIDB/50022/2020. A.S. also acknowledges the NSERC Canada Research Chair funding program.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest
The authors declare that they have no conflict of interest.
Replication of results
The data used for the mathematical test functions, the initial samples and the CFD mesh is provided upon request.
Additional information
Responsible Editor: Nestor V. Queipo
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Sohst, M., Afonso, F. & Suleman, A. Surrogate-based optimization based on the probability of feasibility. Struct Multidisc Optim 65, 10 (2022). https://doi.org/10.1007/s00158-021-03134-4
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00158-021-03134-4