Shape optimization of underwater wings with a new multi-fidelity bi-level strategy

  • Siqing Sun
  • Baowei Song
  • Peng WangEmail author
  • Huachao Dong
  • Xiao Chen
Research Paper


This paper proposes a new multi-fidelity bi-level optimization (MFBLO) strategy for shape designs of underwater wings. Firstly, hydrodynamic analyses of the wing planform and sections are decoupled for constructing a bi-level shape optimization frame, which includes an upper-level task merely concerning the wing planform design and several lower-level tasks only related to the section designs. By doing this, the shape design optimization gets remarkable benefits from the reduction of dimension and computational costs. Secondly, the bridge function method combined with three multi-fidelity data fusion approaches CC1, CC2, and CC3 are proposed to conduct the bi-level optimization, respectively. After comparison analyses, CC2 shows higher computational efficiency and accuracy, which is more appropriate for the bi-level shape optimization frame. Finally, compared with the single-level optimization with the fixed planform or sections and the conventional high-dimensional optimization, the proposed MFBLO needs less computation budget and gets higher lift-drag ratio, showing its outstanding advantages in handling the shape optimization of underwater wings.


High-dimensional expensive problem Bi-level optimization Multi-fidelity surrogate models Underwater wing design 

Nomenclature of important variables


u, l

Upper and lower

i, j






Optimal values



Wing span,

cr, ct

Root and tip chords


Leading edge angle


Planform area


Number of sections


Dimension of problems




Angle of attack (AOA)

F, f, G, g, H, h, x

Objective functions, constraints and design variables of bi-level optimizations

ind, eff

Induced and effective,


Profile and far field


High and low fidelities


Predicted values

c, t/c

Chord, relative thickness of sections


Spanwise position of sections

wli_ j

jth Parameter of ith section shape curve function


Induced velocities in VLM


Area weight coefficients


Looping index of optimizations

L, D, CL, CD

Lift, drag, and their coefficients of UWs

l, d, Cl, Cd

Lift, drag, and their coefficients of sections



The authors are also grateful to members of the research group for the implementation of some existing multi-fidelity optimization algorithms.

Funding information

Supports are from the National Natural Science Foundation of China (Grant No. 51875466 and Grant No. 51805436) and China Postdoctoral Science Foundation (Grant No. 2018M643726).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Supplementary material

158_2019_2362_MOESM1_ESM.docx (41 kb)
ESM 1 (DOCX 41 kb)


  1. Bohn B, Garcke J, Griebel M (2016) A sparse grid based method for generative dimensionality reduction of high-dimensional data. J Comput Phys 309:1–17MathSciNetCrossRefzbMATHGoogle Scholar
  2. Chernukhin O, Zingg DW (2013) Multimodality and global optimization in aerodynamic design. AIAA J 51(6):1342–1354CrossRefGoogle Scholar
  3. Choi S, Alonso JJ, Kroo IM, Wintzer M (2004) Multi-fidelity design optimization of low-boom supersonic business jets, 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Palm Springs, CA, AIAA Paper 2004-1530Google Scholar
  4. Dong H, Song B, Dong Z, Wang P (2016) Multi-start space reduction (mssr) surrogate-based global optimization method. Struct Multidiscip Optim 54(4):1–20CrossRefGoogle Scholar
  5. Drela M (1989) XFoil: an analysis and design system for low Reynolds number airfoils. In: Conference on low Reynolds number airfoil aerodynamics. University of Notre Dame, Notre Dame, IndianaGoogle Scholar
  6. Drela M, Giles MB (1987) Viscous-inviscid analysis of transonic and low Reynolds number airfoils. Am Inst Aeronaut Astronaut J 25(10):1347–1355CrossRefzbMATHGoogle Scholar
  7. Elham A, Tooren MJLV, Sobieszczanski-Sobieski J (2014) Bilevel optimization strategy for aircraft wing design using parallel computing. AIAA J 52(8):1770–1783CrossRefGoogle Scholar
  8. Graf K, Hoeve AV, Watin S (2014) Comparison of full 3d-rans simulations with 2d-rans/lifting line method calculations for the flow analysis of rigid wings for high performance multihulls. Ocean Eng 90:49–61CrossRefGoogle Scholar
  9. Harada T et al (2006) Screening parameters of pulmonary and cardiovascular integrated model with sensitivity analysis. In: Proceedings of the 28th IEEE EMBS Annual International Conference, New York City, USA, 30 Aug–3 Sept 2006Google Scholar
  10. Harrington HA, Gorder RAV (2015) Reduction of dimension for nonlinear dynamical systems. Nonlinear Dyn 88:1), 1–1),20MathSciNetGoogle Scholar
  11. Hartwig L, Bestle D (2017) Compressor blade design for stationary gas turbines using dimension reduced surrogate modeling. Evolutionary Computation. IEEE, pp 1595–1602Google Scholar
  12. Huang E, Xu J, Zhang S, Chen C-H (2015) Multi-fidelity model integration for engineering design. Proc Comput Sci 44:336–344CrossRefGoogle Scholar
  13. Islam MM, Singh HK, Ray T (2017) A surrogate assisted approach for single-objective bi-level optimization. IEEE Trans Evol Comput (99):1–1Google Scholar
  14. Javaid MY, Ovinis M, Nagarajan T, Hashim FBM (2014) Underwater gliders: a review, vol 13. EDP Sciences, p 02020Google Scholar
  15. Jeroslow RG (1985) The polynomial hierarchy and a simple model for competitive analysis. Math Program 32(2):146–164MathSciNetCrossRefzbMATHGoogle Scholar
  16. Jones DR, Schonlau M, Welch WJ (1998) Efficient global optimization of expensive black-box functions. J Glob Optim 13(4):455–492MathSciNetCrossRefzbMATHGoogle Scholar
  17. Koch PN, Simpson TW, Allen JK, Mistree F (1999) Statistical approximations for multidisciplinary design optimization: the problem of the size. J Aircr 36(1):275–286CrossRefGoogle Scholar
  18. Koo D, Zingg DW (2017) Investigation into aerodynamic shape optimization of planar and nonplanar wings. AIAA J 56:1), 1–1),14Google Scholar
  19. Kulfan BM (2008) Universal parametric geometry representation method. J Aircr 45(1):142–158. CrossRefGoogle Scholar
  20. Leifsson L, Koziel S (2015) Aerodynamic shape optimization by variable-fidelity computational fluid dynamics models: a review of recent progress. J Comput Sci 10:45–54MathSciNetCrossRefGoogle Scholar
  21. Li J, Chen JB (2006) The dimension-reduction strategy via mapping for probability density evolution analysis of nonlinear stochastic systems. Probab Eng Mech 21(4):442–453CrossRefGoogle Scholar
  22. Li C, Wang P, Dong H (2018) Kriging-based multi-fidelity optimization via information fusion with uncertainty. J Mech Sci Technol 32(1):245–259CrossRefGoogle Scholar
  23. 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 Multidiscip Optim 55(3):925–943CrossRefGoogle Scholar
  24. Luo W, Lyu W (2015) An application of multidisciplinary design optimization to the hydrodynamic performances of underwater robots. Ocean Eng 104(23):686–697CrossRefGoogle Scholar
  25. Lyu Z, Kenway GKW, Martins JRRA (2014) Aerodynamic shape optimization investigations of the common research model wing benchmark. AIAA J 53(4):968–985CrossRefGoogle Scholar
  26. Mariens J, Elham A, Tooren MJLV (2014) Quasi-three-dimensional aerodynamic solver for multidisciplinary design optimization of lifting surfaces. J Aircr 51(2):547–558CrossRefGoogle Scholar
  27. Masters DA, Taylor NJ, Rendall TCS, Allen CB, Poole DJ (2017) Geometric comparison of aerofoil shape parameterization methods. AIAA J 55(5):1575–1589. CrossRefGoogle Scholar
  28. Mei Y, Omidvar MN, Li X, Yao X (2016) A competitive divide-and-conquer algorithm for unconstrained large-scale black-box optimization. ACM Trans Math Softw 42(2):13MathSciNetCrossRefGoogle Scholar
  29. Mirjalili S, Mirjalili SM, Lewis A (2014) Grey wolf optimizer. Adv Eng Softw 69(3):46–61CrossRefGoogle Scholar
  30. Mitsos A, Chachuat B, Barton PI (2009) Towards global bilevel dynamic optimization. J Glob Optim 45(1):63MathSciNetCrossRefzbMATHGoogle Scholar
  31. Molland AF, Bahaj AS, Chaplin JR, Batten WMJ (2004) Measurements and predictions of forces, pressures and cavitation on 2-d sections suitable for marine current turbines. Proc Inst Mech Eng M 218(2):127–138CrossRefGoogle Scholar
  32. Moritz S, Oliver PC, Kilian O (2016) Spectral proper orthogonal decomposition. J Fluid Mech 792(7):798–828MathSciNetzbMATHGoogle Scholar
  33. Nguyen NV, Choi SM, Kim WS, Lee JW, Kim S, Neufeld D et al (2013) Multidisciplinary unmanned combat air vehicle system design using multi-fidelity model. Aerosp Sci Technol 26(1):200–210CrossRefGoogle Scholar
  34. Oberkampf WL, Trucano TG (2002) Verification and validation in computational fluid dynamics. Adv Mech 38(3):209–272Google Scholar
  35. Quashie M, Marnay C, Bouffard F, Joós G (2018) Optimal planning of microgrid power and operating reserve capacity. Appl Energy 210 pp. 1229–1236.
  36. Ragon SA, Guacute Z, Haftka r RT, Tzong TJ (2015) Bilevel design of a wing structure using response surfaces. J Aircr 40(5):985–992CrossRefGoogle Scholar
  37. Regis RG (2015) Trust regions in surrogate-assisted evolutionary programming for constrained expensive black-box optimization. Infosys Science Foundation, 51–94Google Scholar
  38. Rudnick DL (2016) Ocean research enabled by underwater gliders. Annu Rev Mar Sci 8:519–541. CrossRefGoogle Scholar
  39. Shan S, Wang GG (2010) Survey of modeling and optimization strategies to solve high-dimensional design problems with computationally-expensive black-box functions. Struct Multidiscip Optim 41(2):219–241MathSciNetCrossRefzbMATHGoogle Scholar
  40. Simpson TW, Mauery TM, Korte JJ, Mistree F (2001) Kriging models for global approximation in simulation-based multidisciplinary design optimization. AIAA J 39(12):2233–2241CrossRefGoogle Scholar
  41. Sinha A, Malo P, Deb K (2018) A review on bilevel optimization: from classical to evolutionary approaches and applications. IEEE Trans Evol Comput 22(2):276–295CrossRefGoogle Scholar
  42. van Ingen JL (2008) The en method for transition prediction. Historical review of work at tu delft. Proceedings of the 38th AIAA fluid dynamics conference and exhibit, Seattle, Washington, June 23-26, 2008, p. 1–49; AIAA paper 2008–3830.
  43. Vanderplaats GN (1984) Numerical optimization techniques for engineering design: with applications. McGraw-HillGoogle Scholar
  44. Viswanath, Asha (2010) Dimension reduction for aerodynamic design optimization. (Doctoral dissertation, University of Southampton)Google Scholar
  45. Wang L, Shan S, Wang GG (2004) Mode-pursuing sampling method for global optimization on expensive black-box functions. Eng Optim 36(4):419–438CrossRefGoogle Scholar
  46. Wang X, Chang Y, Zhang P (2018) Traffic signal optimization based on system equilibrium and bi-level multi-objective programming model. In: Wang W, Bengler K, Jiang X (eds) Green intelligent transportation systems. GITSS 2016. Lecture notes in electrical engineering, vol 419. Springer, Singapore.
  47. Yang P, Tang K, Yao X (2018) Turning high-dimensional optimization into computationally expensive optimization. IEEE Trans Evol Comput 22(1):143–156CrossRefGoogle Scholar
  48. Yata K, Aoshima M (2010) Effective PCA for high-dimension, low-sample-size data with singular value decomposition of cross data matrix. Academic Press, IncGoogle Scholar
  49. Zarruk GA, Brandner PA, Pearce BW, Phillips AW (2014) Experimental study of the steady fluid–structure interaction of flexible hydrofoils. J Fluids Struct 51:326–343CrossRefGoogle Scholar
  50. Zhang N (2014) Hilbert-Schmidt independence criterion in sufficient dimension reduction and feature screening. Ph.D. dissertation, Department of Statistics, The University of Georgia, Georgia, Athens, 2014. Accessed 08 Aug 2019

Copyright information

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

Authors and Affiliations

  1. 1.School of Marine Science and TechnologyNorthwestern Polytechnical UniversityXi’anChina

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