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Surrogate-Based Aerodynamic Shape Optimization of a Wing-Body Transport Aircraft Configuration

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AeroStruct: Enable and Learn How to Integrate Flexibility in Design (AeroStruct 2015)

Part of the book series: Notes on Numerical Fluid Mechanics and Multidisciplinary Design ((NNFM,volume 138))

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Abstract

Aerodynamic shape optimization driven by high-fidelity computational fluid dynamics (CFD) simulations is still challenging, especially for complex aircraft configurations. The main difficulty is not only associated with the extremely large computational cost, but also related to the complicated design space with many local optima and a large number of design variables. Therefore, development of efficient global optimization algorithms is still of great interest. This study focuses on demonstrating surrogate-based optimization (SBO) for a wing-body configuration representative of a modern civil transport aircraft parameterized with as many as 80 design variables, while most previous SBO studies were limited to rather simple configurations with fewer parameters. The freeform deformation (FFD) method is used to control the shape of the wing. A Reynolds-averaged Navier-Stokes (RANS) flow solver is used to compute the aerodynamic coefficients at a set of initial sample points. Kriging is used to build a surrogate model for the drag coefficient, which is to be minimized, based on the initial samples. The surrogate model is iteratively refined based on different sample infill strategies. For 80 design variables, the SBO-type optimizer is shown to converge to an optimal shape with lower drag based on about 300 samples. Several studies are conducted on the influence of the resolution of the computational grid, the number and randomness of the initial samples, and the number of design variables on the final result.

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Abbreviations

\(C_L\), \(C_D\), \(C_M\) :

= lift, drag and pitching moment coefficients

\(\mathbf {F}\) :

= regression matrix for Kriging predictor

L :

= likelihood function

m :

= number of design variables

Ma :

= Mach number

n :

= number of high-fidelity sample points

p :

= parameter for Gaussian correlation function

Re:

= Reynolds number

R :

= spatial correlation function

\(\mathbf {R}\) :

= correlation matrix of Kriging model

\(\mathbf {r}\) :

= correlation vector of Kriging model

\(\hat{s}(\mathbf {x})\) :

= standard deviation of Kriging prediction

\(\mathbf {S}\) :

= sampling sites for high-fidelity functions

\(\mathbf {V}_{krig}\) :

= hierarchical Kriging predictor

\(\mathbf {w}\) :

= Kriging weights

\(\mathbf {x}\),\(\mathbf {x'}\) :

= design variables

\(\mathbf {x}^{(l)}, \mathbf {x}^{(u)}\) :

= lower and upper limit of design variables

\(\mathbf {y}_S\) :

= response values

Y :

= random function

\(Z(\cdot )\) :

= Gaussian random process

\(\alpha \) :

= angle of attack

\(\beta _0\) :

= coefficient of trend model for Kriging predictor

\(\xi \) :

= weighted distance for spatial correlation function

\(\mu \) :

= Lagrange multiplier

\(\theta \) :

= hyper-parameter vector for spatial correlation function

\(\delta \) :

= the trust region radius

\(\sigma ^2\) :

= process variance of Kriging model

init:

= initial value

k :

= index \(\in \left[ 1,m \right] \)

\(\text {S}\) :

= sampled data

(i):

= index \(\in \left[ 1,n \right] \), referring to i-th sample point

\(.'\) :

= new point

\(\hat{.}\) :

= approximated value

\(\widetilde{.}\) :

= redefined value

References

  1. R.M. Hicks, E.M. Munnan, G.N. Vanderplaats, An Assessment of Airfoil Design by Numerical Optimization, NASA TM X-3092 (NASA Ames Research Center, Moffett Field, 1974)

    Google Scholar 

  2. R.M. Hicks, P.A. Henne, Wing design by numerical optimization. J. Aircr. 15(7), 407–412 (1978)

    Article  Google Scholar 

  3. A. Jameson, Aerodynamic design via control theory. J. Sci. Comput. 3(3), 233–260 (1988)

    Article  MATH  Google Scholar 

  4. A. Jameson, Optimum aerodynamic design using CFD and control theory, in AIAA Paper 95-1729 (1995)

    Google Scholar 

  5. J. Reuther, A. Jameson, J. Farmer, L. Martinelli, D. Saunders, Aerodynamic shape optimization of complex aircraft configurations via an adjoint formulation, in AIAA Paper 96-0094 (1996)

    Google Scholar 

  6. W.K. Anderson, V. Venkatakrishnan, Aerodynamic design optimization on unstructured grids with a continuous adjoint formulation, in AIAA paper 97–0643, 35th Aerospace Sciences Meeting and Exhibit (Reno, Nevada, 1997)

    Google Scholar 

  7. R. Dwight, J. Brezillon, Effect of approximations of the discrete adjoint on gradient-based optimization. AIAA J. 44(12), 3022–3071 (2006)

    Article  Google Scholar 

  8. J. Brezillon, N.R. Gauger, 2D and 3D aerodynamic shape optimization using the adjoint approach. Aerosp. Sci. Technol. 8(8), 715–727 (2004)

    Article  MATH  Google Scholar 

  9. D.J Mavripilis, A discrete adjoint-based approach for optimization problems on three-dimensional unstructured grids, in AIAA Paper 2006-50 (2006)

    Google Scholar 

  10. Z. Lyu, G.K.W. Kenway, J.R.R.A. Martins, Aerodynamic shape optimization investigations of the common research model wing benchmark. AIAA J. 53(4), 968–985 (2015). https://doi.org/10.2514/1.J053318

    Article  Google Scholar 

  11. G.K.W. Kenway, J.R.R.A. Martins, Multipoint aerodynamic shape optimization investigations of the common research model wing. AIAA J. 54(1), 112–128 (2016)

    Article  Google Scholar 

  12. M. Abu-Zurayk, J. Brezillon, Aero-Elastic Multi-Point Wing Optimization Using the Coupled Adjoint Approach, New Results in Numerical and Experimental Fluid Mechanics IX, vol. 124, Notes on Numerical Fluid Mechanics and Multidisciplinary Design (Springer, Berlin, 2014), pp. 45–52

    Chapter  Google Scholar 

  13. D. Koo, D.W. Zingg, Progress in aerodynamic shape optimization based on the reynolds-averaged navier-stokes equations, in AIAA Paper 2016–1292, 54th AIAA Aerospace Science Meeting, 4–8 January 2016 (San Diego, California, USA, 2016)

    Google Scholar 

  14. O. Chernukhin, D.W. Zingg, Multimodality and global optimization in aerodynamic design. AIAA J. 51(6), 1342–1354 (2013)

    Article  Google Scholar 

  15. Y.A Tesfahunegn, S. Koziel, J.R. Gramanzini, S. Hosder, Z.-H Han, L. Leifsson, Application of direct and surrogate-based optimization to two-dimensional benchmark aerodynamic problems: a comparative study, in AIAA Paper 2015-0265, 53rd AIAA Aerospace Sciences Meeting, 5–9 January 2015 (Kissimmee, Florida, 2015)

    Google Scholar 

  16. J. Wild, On the Potential of Numerical Optimization of High-Lift Multi-Element Airfoils based on the Solution of the Navier-Stokes-Equations, Computational Fluid Dynamics (Springer, Berlin, 2002), pp. 267–272

    Google Scholar 

  17. W. Song, A.J. Keane, Surrogate-based aerodynamic shape optimization of a civil aircraft engine nacelle. AIAA J. 45(10), 2565–2574 (2007)

    Article  Google Scholar 

  18. N.V. Queipo, R.T. Haftka, W. Shyy, T. Goel, R. Vaidyanathan, P.K. Tucher, Surrogate-based analysis and optimization. Prog. Aerosp. Sci. 41(1), 1–28 (2005)

    Article  Google Scholar 

  19. A.I.J. Forrester, A.J. Keane, Recent advances in surrogate-based optimization. Prog. Aerosp. Sci. 45(1–3), 50–79 (2009). https://doi.org/10.1016/j.paerosci.2008.11.001

    Article  Google Scholar 

  20. Z.-H. Han, K.-S. Zhang, Surrogate Based Optimization, Real-World Application of Genetic Algorithm (Intec Book, London, 2012), p. 343

    Google Scholar 

  21. S. Koziel, D. Echeverria-Ciaurri, L. Leifsson, Surrogate-based methods, in Computational Optimization, Methods and Algorithms, Studies in Computational Intelligence, ed. by S. Koziel, X.S. Yang (Springer, Berlin, 2011), pp. 33–60

    Chapter  Google Scholar 

  22. S. Shan, G.G. Wang, Survey of modeling and optimization strategies for high-dimensional design problems, in AIAA Paper 2008-5842 (2008)

    Google Scholar 

  23. G.G. Wang, S. Shan, Review of metamodeling techniques in support of engineering design optimization. J. Mech. Des. 129(4), 370–380 (2007)

    Article  Google Scholar 

  24. P.N. Koch, T.W. Simpson, J.K. Allen, F. Mistree, Statistical approximations for multidisciplinary design optimization: the problem of the size. J. Aircr. 36(1), 275–286 (1999)

    Article  Google Scholar 

  25. D.R. Jones, M. Schonlau, W.J. Welch, Efficient global optimization of expensive black-box functions. J. Glob. Optim. 13, 455–492 (1998). https://doi.org/10.1023/A:1008306431147

    Article  MathSciNet  MATH  Google Scholar 

  26. D.D. Cox, S. John, SDO: A statistical method for global optimization, in Proceeding of the ICASE/NASA Langley Workshop on Multidisciplinary Analysis and Optimization, vol. 2 (Reston, VA, 1998), pp. 738–748

    Google Scholar 

  27. J. Laurenceau, P. Sagaut, Building efficient response surfaces of aerodynamic functions with kriging and cokriging. AIAA J. 46(2), 498–507 (2008)

    Article  Google Scholar 

  28. L. Leifsson, S. Koziel, Y.A. Tesfahunegn, Multiobjective aerodynamic optimization by variable-fidelity models and response surface surrogates. AIAA J. 54(2), 531–541 (2016)

    Article  Google Scholar 

  29. Y. Lian, M. Liou, Multiobjective optimization using coupled response surface model and evolutionary algorithm. AIAA J. 43(6), 1316–1325 (2005)

    Article  Google Scholar 

  30. S. Jeong, M. Murayama, K. Yamamoto, Efficient optimization design method using kriging model. J. Aircr. 42(2), 413–420 (2005)

    Article  Google Scholar 

  31. A.J. Keane, Statistical improvement criteria for use in multiobjective design optimization. AIAA J. 44(4), 879–891 (2006)

    Article  Google Scholar 

  32. A. Vavalle, N. Qin, Iterative response surface based optimization scheme for transonic airfoil design. J. Aircr. 44(2), 365–376 (2007)

    Article  Google Scholar 

  33. S. Koziel, L. Leifsson, Surrogate-based aerodynamic shape optimization by variable-resolution models. AIAA J. 51(1), 94–106 (2013)

    Article  Google Scholar 

  34. K.-S. Zhang, Z.-H. Han, W.-J. Li, W.-P. Song, Coupled aerodynamic/structural optimization of a subsonic transport wing using a surrogate model. J. Aircr. 45(6), 2167–2171 (2008)

    Article  Google Scholar 

  35. M. Kanazaki, T. Imamura, S. Jeong, K. Yamamoto, High-lift wing design in consideration of sweep angle effect using kriging model, in 46th AIAA Aerospace Sciences Meeting and Exhibit, 7–10 January 2008 (Reno, Nevada, 2008)

    Google Scholar 

  36. J. Laurenceau, M. Meaux, M. Montagnac, P. Sagaut, Comparison of gradient-based and gradient-enhanced response-surface-based optimizers. AIAA J. 48(5), 981–994 (2010)

    Article  Google Scholar 

  37. S.J. Leary, A. Bhaskar, A.J. Keane, Global approximation and optimization using adjoint computational fluid dynamics codes. AIAA J. 42(3), 631–641 (2004)

    Article  Google Scholar 

  38. C. Li, J. Brezillon, S. Görtz, A Hybrid approach for surrogate-based aerodynamic optimization with constraints, in Proceedings International Conference on Evolutionary and Deterministic Methods for Design, Optimization and Control with Applications to Industrial and Societal Problems (Eurogen 2011), 14–16 September 2011 (Capua, Italy, 2011)

    Google Scholar 

  39. C. Li, J. Brezillon, S. Görtz, Efficient Global Optimization of a Natural Laminar Airfoil Based on Surrogate Modeling, in New Results in Numerical and Experimental Fluid Mechanics IX, vol. 124, Notes on Numerical Fluid Mechanics and Multidisciplinary Design, ed. by A. Dillmann, G. Heller, E. Krämer, H.-P. Kreplin, W. Nitsche, U. Rist (Springer, Berlin, 2014), pp. 53–63. (Contributions to the 18th STAB/DGLR Symposium, Stuttgart, Germany 2012)

    Google Scholar 

  40. C. Li, A surrogate-based framework with hybrid refinement strategies for aerodynamic shape optimization, Ph.D. thesis, German Aerospace Center (DLR), Institute Aerodynamics and Flow Technology, DLR-FB-2013-16, 2013

    Google Scholar 

  41. Z.H. Han, R. Zimmermann, S. Görtz, Alternative co-kriging model for variable-fidelity surrogate modeling. AIAA J. 50(5), 1205–1210 (2012)

    Article  Google Scholar 

  42. Z.H. Han, S. Görtz, Hierarchical kriging model for variable-fidelity surrogate modeling. AIAA J. 50(9), 1885–1896 (2012)

    Article  Google Scholar 

  43. A.A. Giunta, S.F. Wojtkiewicz Jr., M.S. Eldred, Overview of modern design of experiments methods for computational simulations, in AIAA paper 2003-649 (2003)

    Google Scholar 

  44. D.G. Krige, A statistical approach to some basic mine valuations problems on the witwatersrand. J. Chem. Metall. Min. Eng. Soc. S. Afr. 52(6), 119–139 (1951)

    Google Scholar 

  45. G.M. Matheron, Principles of geostatistics. Econ. Geol. 58(8), 1246–1266 (1963)

    Article  Google Scholar 

  46. J. Sacks, W.J. Welch, T.J. Mitchell, H.P. Wynn, Design and analysis of computer experiments. Stat. Sci 4, 409–423 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  47. S.N Lophaven, H.B. Nielsen, J. Søndergaard, DACE - A Matlab Kriging Toolbox (Version 2.0), Technical Report IMM-REP-2002-12, Informatics and Mathematical Modelling, DTU (2002), pp. 1–34

    Google Scholar 

  48. Z.-H. Han, S. Görtz, R. Zimmermann, Improving variable-fidelity surrogate modeling via gradient-enhanced kriging and a generalized hybrid bridge function. Aerosp. Sci. Technol. 25(1), 177–189 (2013). https://doi.org/10.1016/j.ast.2012.01.006

    Article  Google Scholar 

  49. T.W. Simpson, T.M. Mauery, J.J. Korte, F. Mistree, Kriging models for global approximation in simulation-based multidisciplinary design optimization. AIAA J. 39(12), 2233–2241 (2001)

    Article  Google Scholar 

  50. J.D. Martin, T.W. Simpson, Use of kriging models to approximate deterministic computer models. AIAA J. 43(4), 853–863 (2005)

    Article  Google Scholar 

  51. D.J.J. Toal, N.W. Bressloff, A.J. Kean, Kriging hyperparameter tuning strategies. AIAA J. 46(5), 1240–1252 (2008)

    Article  Google Scholar 

  52. J. Liu, Z.-H Han, W.-P. Song, Comparison of infill sampling criteria in kriging-based aerodynamic optimization, in 8th Congress of the International Council of the Aeronautical Sciences, 23–28 September 2012 (Brisbane, Australia, 2012). (ICAS2012-11.10ST, 2012)

    Google Scholar 

  53. N.M. Alexandrov, J.E. Dennis Jr, R.M. Lewis, V. Torczon, A trust-region framework for managing the use of approximation models in optimization. Struct. Multidiscip. Optim. 15(1), 16–23 (1998)

    Article  Google Scholar 

  54. N. Kroll, M. Abu-Zurayk, D. Dimitrov, T. Franz, T. Führer, T. Gerhold, S. Görtz, R. Heinrich, C. Ilic, J. Jepsen, J. Jägersküpper, M. Kruse, A. Krumbein, S. Langer, D. Liu, R. Liepelt, L. Reimer, M. Ritter, A. Schwöppe, J. Scherer, F. Spiering, R. Thormann, V. Togiti, D. Vollmer, J.-H. Wendisch, DLR project digital-X: towards virtual aircraft design and flight testing based on high-fidelity methods. CEAS Aeronaut. J. 7(1), 3–27 (2016)

    Article  Google Scholar 

  55. T. Gerhold, V. Hannemann, D. Schwamborn, On the Validation of the DLR-TAU Code, in New Results in Numerical and Experimental Fluid Mechanics, vol. 72, Notes on Numerical Fluid Mechanics, ed. by W. Nitsche, H.-J. Heinemann, R. Hilbig (Springer Vieweg, Berlin, 1999), pp. 426–433. ISBN 3-528-03122-0

    Google Scholar 

  56. N. Kroll, J.K. Fassbender (eds.), MEGAFLOW - Numerical Flow Simulation for Aircraft Design, vol. 89, Notes on Numerical Fluid Mechanics and Multidisciplinary Design (Springer, Berlin, 2005)

    Google Scholar 

  57. D. Schwamborn, T. Gerhold, R. Heinrich, The DLR TAU-code: recent applications in research and industry, invited lecture, in Proceedings of the European Conference on Computational Fluid Dynamics (ECCOMAS CFD 2006), eds by P. Wesseling, E. Oñate, J. Périaux, (TU Delft, The Netherlands, 2006)

    Google Scholar 

  58. Y. Zhang, Z.-H. Han, L.-X. Shi, W.-P. Song, Multi-round surrogate-based optimization for benchmark aerodynamic design problems, in AIAA Paper 2016–1545, 54th AIAA Aerospace Science Meeting, 4–8 January 2016 (San Diego, California, USA, 2016)

    Google Scholar 

  59. U. Siller, C. Voß, E. Nicke, Automated multidisciplinary optimization of a transonic axial compressor, in AIAA Paper 2009-0863 (2009)

    Google Scholar 

  60. T. Wunderlich, S. Dähne, L. Heinrich, L. Reimer, Multidisciplinary Optimization of a NLF Forward Swept Wing in combination with Aeroelastic Tailoring using CFRP, Deutscher Luft- und Raumfahrtkongress 2015 (2015)

    Google Scholar 

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Acknowledgements

This research work was sponsored by the German national aeronautics research project AeroStruct. The authors acknowledge the support of their DLR colleagues Niklas Karcher, Tobias Wunderlich and Dishi Liu.

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Authors and Affiliations

  1. Aerodynamic Surrogates and Optimization Group, DLR Institute of Aerodynamics and Flow Technology, Braunschweig, Germany

    Zhong-Hua Han, Mohammad Abu-Zurayk & Stefan Görtz

  2. Multi-Disciplinary Optimization Group, DLR Institute of Aerodynamics and Flow Technology, Braunschweig, Germany

    Caslav Ilic

Authors
  1. Zhong-Hua Han
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  2. Mohammad Abu-Zurayk
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  3. Stefan Görtz
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  4. Caslav Ilic
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Correspondence to Stefan Görtz .

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  1. DLR, Institute of Aerodynamics and Flow Technology, Braunschweig, Germany

    Ralf Heinrich

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Han, ZH., Abu-Zurayk, M., Görtz, S., Ilic, C. (2018). Surrogate-Based Aerodynamic Shape Optimization of a Wing-Body Transport Aircraft Configuration. In: Heinrich, R. (eds) AeroStruct: Enable and Learn How to Integrate Flexibility in Design. AeroStruct 2015. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol 138. Springer, Cham. https://doi.org/10.1007/978-3-319-72020-3_16

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  • DOI: https://doi.org/10.1007/978-3-319-72020-3_16

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