Skip to main content
SpringerLink
Log in

Optimal Aerodynamic Design under Uncertainty

  • Chapter
Management and Minimisation of Uncertainties and Errors in Numerical Aerodynamics

Part of the book series: Notes on Numerical Fluid Mechanics and Multidisciplinary Design ((NNFM,volume 122))

Abstract

Recently, optimization has become an integral part of the aerodynamic design process chain. However, because of uncertainties with respect to the flight conditions and geometry uncertainties, a design optimized by a traditional design optimization method seeking only optimality may not achieve its expected performance. Robust optimization deals with optimal designs, which are robust with respect to small (or even large) perturbations of the optimization setpoint conditions. That means, the optimal designs computed should still be good designs, even if the input parameters for the optimization problem formulation are changed by a non-negligible amount. Thus even more experimental or numerical effort can be saved. In this paper, we aim at an improvement of existing simulation and optimization technology, developed in the German collaborative effort MEGADESIGN1, so that numerical uncertainties are identified, quantized and included in the overall optimization procedure, thus making robust design in this sense possible. We introduce two robust formulations of the aerodynamic optimization problem which we numerically compare in a 2d testcase under uncertain flight conditions. Beside the scalar valued uncertainties we consider the shape itself as an uncertainty source and apply a Karhunen-Loève expansion to approximate the infinite-dimensional probability space. To overcome the curse of dimensionality an adaptively refined sparse grid is used in order to compute statistics of the solution.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Becker, R., Braack, M., Meidner, D., Rannacher, R., Vexler, B.: Adaptive finite element methods for pde-constrained optimal control problems. In: Reactive Flows, Diffusion and Transport, pp. 177–205 (2006)

    Google Scholar 

  2. Bock, H.G., Egartner, W., Kappis, W., Schulz, V.: Practical shape optimization for turbine and compressor blades by the use of PRSQP methods. Optimization and Engineering 3(4), 395–414 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bock, H.G., Kostina, E., Schäfer, A., Schlöder, J.P., Schulz, V.H.: Multiple set point partially reduced SQP method for optimal control of PDE. in reactive flows, diffusion and transport. In: Jäger, W., Rannacher, R., Warnatz, J. (eds.) Reactive Flows, Diffusion and Transport. From Experiments via Mathematical Modeling to Numerical Simulation and Optimization Final Report of SFB (Collaborative Research Center) 359 (2007)

    Google Scholar 

  4. Bock, H.G., Schulz, V.: Mathematical aspects of CFD-model based optimization. In: Thevenin, D., Janiga, G. (eds.) Optimization and Computational Fluid Dynamics, pp. 61–78. Springer (2007)

    Google Scholar 

  5. Borzì, A., von Winckel, G.: Multigrid methods and sparse-grid collocation techniques for parabolic optimal control problems with random coefficients. SIAM Journal on Scientific Computing 31(3), 2172–2192 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bungartz, H.-J.: Finite Elements of Higher Order on Sparse Grids. Habilitationsschrift, Fakultät für Informatik, Technische Universität München, Aachen (November 1998)

    Google Scholar 

  7. Bungartz, H.-J., Dirnstorfer, S.: Multivariate quadrature on adaptive sparse grids. Computing 71(1), 89–114 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration, 2nd edn. Dover Publications (November 2007)

    Google Scholar 

  9. Eldred, M.S., Webster, C.G., Constantine, P.: Evaluation of non-intrusive approaches for wiener-askey generalized polynomial chaos (2008)

    Google Scholar 

  10. Christodoulos, A., Floudas, C.A., Stein, O.: The adaptive convexification algorithm: A feasible point method for semi-infinite programming. SIAM J. on Optimization 18(4), 1187–1208 (2007)

    MATH  Google Scholar 

  11. Frauenfelder, P., Schwab, C., Todor, R.A.: Finite elements for elliptic problems with stochastic coefficients. Computer Methods in Applied Mechanics and Engineering 194(2-5), 205–228 (2005); Selected papers from the 11th Conference on the Mathematics of Finite Elements and Applications

    Article  MathSciNet  MATH  Google Scholar 

  12. Garcke, J., Griebel, M., Thess, M.: Data mining with sparse grids. Computing 67(3), 225–253 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Garcke, J.: A dimension adaptive sparse grid combination technique for machine learning. In: Read, W., Larson, J.W., Roberts, A.J. (eds.) Proceedings of the 13th Biennial Computational Techniques and Applications Conference, CTAC 2006. ANZIAM J., vol. 48, pp. C725–C740 (2007)

    Google Scholar 

  14. Gerstner, T., Griebel, M.: Dimension-adaptive tensor-product quadrature. Computing 71(1), 65–87 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Dover Pubn. Inc. (2003)

    Google Scholar 

  16. Gherman, I.: Approximate Partially Reduced SQP Approaches for Aerodynamic Shape Optimization Problems. PhD thesis, University of Trier, Trier, Germany (2008)

    Google Scholar 

  17. Gumbert, C.R., Newman, P.A., Hou, G.J.-W.: High-fidelity computational optimization for 3-d flexible wings: Part ii effect of random geometric uncertainty on design. Optimization and Engineering 6(1), 139–156 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hazra, S.B., Schulz, V.: Simultaneous pseudo-timestepping for aerodynamic shape optimization problems with state constraints. SIAM J. Sci. Comput. 28(3), 1078–1099 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hazra, S.B., Schulz, V., Brezillon, J., Gauger, N.R.: Aerodynamic shape optimization using simultaneous pseudo-timestepping. J. Comput. Phys. 204(1), 46–64 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Henrion, R.: Structural properties of linear probabilistic constraints. Optimization 56(16), 425–440 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Huyse, L., Lewis, R.M., Li, W., Padula, S.L.: Probabilistic approach to free-form airfoil shape optimization under uncertainty. AIAA Journal 40, 1764–1772 (2002)

    Article  Google Scholar 

  22. Kall, P., Wallace, S.W.: Stochastic Programming. Wiley (1994)

    Google Scholar 

  23. Karhunen, K.: Zur Spektraltheorie stochastischer Prozesse. Suomalaisen Tiedeakatemian Toimituksia: Ser. A: 1, Mathematica, Physica 34 (1946)

    Google Scholar 

  24. Keese, A.: A review of recent developments in the numerical solution of stochastic pdes (stochastic finite elements). Technical Report 2003-6, Technische Universitt Braunschweig, Brunswick (2003)

    Google Scholar 

  25. Khoromskij, B.N., Litvinenko, A., Matthies, H.G.: Application of hierarchical matrices for computing the karhunen-loève expansion. Computing 84(1-2), 49–67 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Klimke, A.: Uncertainty modeling using fuzzy arithmetic and sparse grids. PhD thesis, Universität Stuttgart, Shaker Verlag, Aachen (2006)

    Google Scholar 

  27. Klimke, A., Wohlmuth, B.: Algorithm 847: Spinterp: piecewise multilinear hierarchical sparse grid interpolation in matlab. ACM Trans. Math. Softw. 31(4), 561–579 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Knyazev, A.V.: Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method. SIAM J. Sci. Comput. 23, 517–541 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  29. Li, W., Huyse, L., Padula, S.: Robust airfoil optimization to achieve drag reduction over a range of mach numbers. Structural and Multidisciplinary Optimization 24(1), 38–50 (2002)

    Article  Google Scholar 

  30. Li, W., Padula, S.L.: Robust airfoil optimization in high resolution design space. ICASE NASA Langley Research Centre (2003)

    Google Scholar 

  31. Loève, M.: Probability Theory 1, 4th edn. Springer (1994)

    Google Scholar 

  32. Loeven, G.J.A., Bijl, H.: Airfoil analysis with uncertain geometry using the probabilistic collocation method. In: 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, AIAA 2008-2070 (2008)

    Google Scholar 

  33. Ma, X., Zabaras, N.: An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations. J. Comput. Phys. 228(8), 3084–3113 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  34. Mercer, J.: Functions of positive and negative type and their connection with the theory of integral equations. Philos. Trans. Roy. Soc. London (1909)

    Google Scholar 

  35. Najm, H.N.: Uncertainty quantification and polynomial chaos techniques in computational fluid dynamics. Annual Review of Fluid Mechanics 41(1) (2009)

    Google Scholar 

  36. Padula, S., Gumbert, C., Li, W.: Aerospace applications of optimization under uncertainty. In: ISUMA 2003: Proceedings of the 4th International Symposium on Uncertainty Modelling and Analysis, p. 286. IEEE Computer Society, Washington, DC (2003)

    Chapter  Google Scholar 

  37. Prékopa, A.: Stochastic Programming. Kluwer Academic Publishers, Dordrecht (1995)

    Google Scholar 

  38. Putko, M., Newman, P., Taylor III, A., Green, L.: Approach for uncertainty propagation and robust design in cfd using sensitivity derivatives. In: Design in CFD Using Sensitivity Derivatives, AIAA Paper 2001-2528, in AIAA 15th Computational Fluid Dynamics Conference, pp. 2001–2528 (2001)

    Google Scholar 

  39. Schulz, V., Schillings, C.: On the nature and treatment of uncertainties in aerodynamic design. AIAA Journal 47(3), 646–654 (2009)

    Article  Google Scholar 

  40. Smolyak, S.A.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Soviet Mathematics Doklady 4, 240–243 (1963)

    Google Scholar 

  41. Xiu, D., Karniadakis, G.E.: Modeling uncertainty in flow simulations via generalized polynomial chaos. Journal of Computational Physics 187(1), 137–167 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  42. Zang, T.A., Hemsch, M.J., Hilburger, M.W., Kenny, S.P., Luckring, J.M., Maghami, P., Padula, S.L., Jefferson Strout, W.: Needs and opportunities for uncertainty-based multidisciplinary design methods for aerospace vehicles (2002)

    Google Scholar 

  43. Zhang, Y.: A general robust-optimization formulation for nonlinear programming. J. Optim. Theory Appl. (2004)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. FB IV Department of Mathematics, University of Trier, Universitätsring 15, 54296, Trier, Germany

    Volker Schulz & Claudia Schillings

Authors
  1. Volker Schulz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Claudia Schillings
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Volker Schulz .

Editor information

Editors and Affiliations

  1. Flow Technology, German Aerospace Center (DLR), Institute of Aerodynamics and, Lilienthalplatz 7, Braunschweig, 38108, Germany

    Bernhard Eisfeld

  2. Airbus Operations GmbH, Airbus-Allee 1, Bremen, 28199, Germany

    Holger Barnewitz

  3. EADS Deutschland GmbH Cassidian, Rechliner Str., Manching, 85077, Germany

    Willy Fritz

  4. Technische Akustik, Institut für Strömungsmechanik und, Müller-Breslau-Str. 8, Berlin, 10623, Germany

    Frank Thiele

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Schulz, V., Schillings, C. (2013). Optimal Aerodynamic Design under Uncertainty. In: Eisfeld, B., Barnewitz, H., Fritz, W., Thiele, F. (eds) Management and Minimisation of Uncertainties and Errors in Numerical Aerodynamics. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol 122. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36185-2_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-36185-2_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-36184-5

  • Online ISBN: 978-3-642-36185-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation