Risk Measures Applied to Robust Aerodynamic Shape Design Optimization

  • Domenico QuagliarellaEmail author
  • Elisa Morales Tirado
  • Andrea Bornaccioni
Conference paper
Part of the Lecture Notes in Applied and Computational Mechanics book series (LNACM, volume 92)


A Robust Design Optimization (RDO) method based on the use of Conditional Value-at-Risk (CVaR) risk measure is briefly described and applied to an aerodynamic shape design problem. The technique leads to optimal design solutions resilient to production tolerances and operating conditions instabilities. The approach is illustrated through the application to an airfoil section design optimization in low transonic conditions with the flow field modeled using an Euler plus boundary layer interactive approach. The results of the robust design are compared to those obtained with a classical deterministic method, and mutual advantages and disadvantages of the two approaches are discussed.



Thanks are due to the prof. M. Drela and MIT for allowing the use of MSES code in this research project.


  1. 1.
    Wiener, N. (1938). The homogeneous chaos. American Journal of Mathematics, 60(4), 897–936.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Xiu, D. (2010). Numerical methods for stochastic computations: A spectral method approach. Princeton University Press.Google Scholar
  3. 3.
    Witteveen, J., & Iaccarino, G. (2010). Simplex elements stochastic collocation for uncertainty propagation in robust design optimization. In 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, American Institute of Aeronautics and Astronautics, January 2010.Google Scholar
  4. 4.
    Heinrich, S. (2001). Multilevel Monte Carlo methods. In S. Margenov, J. Waśniewski, & P. Yalamov (Eds.), Large-Scale Scientific Computing: Third International Conference, LSSC 2001, Sozopol, Bulgaria, 6–10 June 2001. Revised papers (pp. 58–67). Berlin, Heidelberg: Springer.Google Scholar
  5. 5.
    Giles, M. B. (2015). Multilevel Monte Carlo methods. Acta Numerica, 24, 259–328.Google Scholar
  6. 6.
    Quagliarella, D., Petrone, G., & Iaccarino, G. (2015). Reliability-based design optimization with the generalized inverse distribution function. In D. Greiner, B. Galván, J. Périaux, N. Gauger, K. Giannakoglou, & G. Winter (Eds.), Advances in evolutionary and deterministic methods for design, optimization and control in engineering and sciences. Computational methods in applied sciences (Vol. 36, chap. 5, pp. 77–92). Springer. ISBN 978-3-319-11540-5.Google Scholar
  7. 7.
    Quagliarella, D., & Iuliano, E. (2017). Robust design of a supersonic natural laminar flow wing-body. IEEE Computational Intelligence Magazine, 12(4), 14–27.Google Scholar
  8. 8.
    Tyrrell Rockafellar, R., & Uryasev, S. (2002). Conditional value-at-risk for general loss distributions. Journal of Banking and Finance, 26, 1443–1471.Google Scholar
  9. 9.
    Tyrrell Rockafellar, R., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of Risk, 2, 21–41.Google Scholar
  10. 10.
    Quagliarella, D. (2019). Value-at-risk and conditional value-at-risk in optimization under uncertainty. In C. Hirsch, D. Wunsch, J. Szumbarski, Ł. Łaniewski-Wołłk, & J. Pons-Prats (Eds.), Uncertainty management for robust industrial design in aeronautics. Springer.Google Scholar
  11. 11.
    Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203–228.Google Scholar
  12. 12.
    Hansen, N. (2006). The CMA evolution strategy: A comparing review. In J. A. Lozano, P. Larrañaga, I. Inza, & E. Bengoetxea (Eds.), Towards a new evolutionary computation: Advances in the estimation of distribution algorithms (pp. 75–102). Berlin, Heidelberg: Springer.Google Scholar
  13. 13.
    Hansen, N., & Ostermeier, A. (2001). Completely derandomized self-adaptation in evolution strategies. Evolutionary Computation, 9(2), 159–195.CrossRefGoogle Scholar
  14. 14.
    Drela, M., & Giles, M. B. (1987). Viscous-inviscid analysis of transonic and low Reynolds number airfoils. AIAA Journal, 25(10), 1347–1355.Google Scholar
  15. 15.
    Drela, M. (1996). A user’s guide to MSES 2.95. MIT Computational Aerospace Sciences Laboratory.Google Scholar
  16. 16.
    Tollmien, Walter. (1931). Grenzschichttheorie. Handbuch Experimentalphysik, 4, 241–287.zbMATHGoogle Scholar
  17. 17.
    Schlichting, H. (1933). Zur enstehung der turbulenz bei der plattenströmung. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 181–208, 1933.zbMATHGoogle Scholar
  18. 18.
    Van Ingen, J. (2008). The \(e^{N}\) method for transition prediction. Historical review of work at TU Delft. In 38th Fluid Dynamics Conference and Exhibit (p. 3830).Google Scholar
  19. 19.
    Hicks, R., & Henne, P. A. (1978). Wing design by numerical optimization. Journal of Aircraft, 15(7), 407–412.CrossRefGoogle Scholar
  20. 20.
    EADS innovation works.
  21. 21.
    Drela, M., & Youngren, H. (2004). Athena vortex lattice.
  22. 22.
    Efron, B. (1992). Bootstrap methods: Another look at the jackknife. In Breakthroughs in statistics (pp. 569–593). Springer.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Domenico Quagliarella
    • 1
    Email author
  • Elisa Morales Tirado
    • 1
  • Andrea Bornaccioni
    • 2
  1. 1.Centro Italiano Ricerche AerospazialiCapuaItaly
  2. 2.Roma Tre UniversityRomeItaly

Personalised recommendations