Projection-Based Model Order Reduction for Steady Aerodynamics

  • Alexander Vendl
  • Heike Faßbender

Abstract

Nonlinear model order reduction techniques for systems often make use of Proper Orthogonal Decomposition (POD). In this work a method based on POD, called Missing Point Estimation (MPE), is investigated. It is capable of efficiently simulating steady-state flows with the angle of attack as a system parameter. The basic idea of MPE is to project the governing equations onto the POD subspace in such a way that the resulting reduced order model does not have to evaluate the nonlinear right hand side at each and every grid point, but only at a few selected points. While the projection onto the POD subspace yields a reduced order model, the limitation to only few points actually achieves independence from the full order of the governing equations. To demonstrate the effectiveness of this method, numerical results for the simulation of inviscid, steady-state flows past a three-dimensional, complex airplane configuration are given.

Keywords

Model Order Reduction Missing Point Estimation Proper Orthogonal Decomposition CFD Steady Aerodynamics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    European Commission: Flightpath 2050: Europe’s Vision for Aviation. Report of the high level group on aviation research, European Union (2011)Google Scholar
  2. 2.
    Klenner, J., Becker, K., Cross, M., Kroll, N.: Future Simulation Concept. In: CEAS Conference Berlin, Session Numerical Simulation. Paper No. 1 (2007)Google Scholar
  3. 3.
    Rossow, C.C., Kroll, N.: Numerical Simulation - Complementing Theory and Experiment as the Third Pillar in Aerodynamics. In: Radespiel, R., Rossow, C.C., Brinkmann, B.W. (eds.) Hermann Schlichting–100 Years. NNFM, vol. 102, pp. 39–58. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  4. 4.
    Salas, M.D.: Digital Flight: The Last CFD Aeronautical Grand Challenge. J. Sci. Comput. 28(2-3), 479–505 (2006)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Chaturantabut, S., Sorensen, D.C.: Discrete empirical interpolation for nonlinear model reduction. Technical Report TR09-05, CAAM, Rice University (March 2009)Google Scholar
  6. 6.
    Astrid, P., Weiland, S., Willcox, K., Backx, T.: Missing point estimation in models described by proper orthogonal decomposition. IEEE Transactions on Automatic Control 53(10), 2237–2251 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Astrid, P.: Fast reduced order modeling technique for large scale ltv systems. In: Proceedings of the American Control Conference, vol. 1, pp. 762–767 (July 2004)Google Scholar
  8. 8.
    Astrid, P., Verhoeven, A.: Application of Least Squares MPE technique in the reduced order modeling of electrical circuits. In: Proceedings of the 17th Int. Symp. MTNS, pp. 1980–1986 (2006)Google Scholar
  9. 9.
    Cardoso, M.A., Durlofsky, L.J., Sarma, P.: Development and application of reduced-order modeling procedures for subsurface flow simulation. International Journal for Numerical Methods in Engineering 77(9), 1322–1350 (2009)MATHCrossRefGoogle Scholar
  10. 10.
    Vendl, A., Faßbender, H.: Efficient POD-based Model Order Reduction for steady aerodynamic applications. In: Poloni, C., Quagliare, D., Périaux, J., Gauger, N., Giannakoglou, K. (eds.) Evolutionary and Deterministic Methods for Design, Optimization and Control, EUROGEN 2011, CIRA, Capua, Italy, pp. 296–309 (2011)Google Scholar
  11. 11.
    Vendl, A., Faßbender, H.: Missing point estimation for steady aerodynamic applications. PAMM 11(1), 839–840 (2011)CrossRefGoogle Scholar
  12. 12.
    Lucia, D.J., King, P.I., Beran, P.S.: Reduced order modeling of a two-dimensional flow with moving shocks. Computers & Fluids 32(7), 917–938 (2003)MATHCrossRefGoogle Scholar
  13. 13.
    Kalashnikova, I., Barone, M.F.: On the stability and convergence of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far-field boundary treatment. International Journal for Numerical Methods in Engineering 83(10), 1345–1375 (2010)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Rowley, C., Colonius, T., Murray, R.: Model reduction for compressible flows using POD and Galerkin projection. Physica D: Nonlinear Phenomena 189(1-2), 115–129 (2004)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    LeGresley, P., Alonso, J.J.: Investigation of non-linear projection for pod based reduced order models for aerodynamics. In: 39th AIAA Aerospace Sciences Meeting & Exhibit, AIAA Paper 2001-0926, Reno, NV, January 8-11 (2001)Google Scholar
  16. 16.
    LeGresley, P., Alonso, J.J.: Dynamic domain decomposition and error correction for reduced order models. In: 41st AIAA Aerospace Sciences Meeting & Exhibit, AIAA Paper 2003-0250, Reno, NV January 6-9 (2003)Google Scholar
  17. 17.
    Zimmermann, R.: Towards best-practice guidelines for POD-based reduced order modeling of transonic flows. In: Poloni, C., Quagliare, D., Périaux, J., Gauger, N., Giannakoglou, K. (eds.) Evolutionary and Deterministic Methods for Design, Optimization and Control, EUROGEN 2011, CIRA, Capua, Italy, pp. 326–341 (2011)Google Scholar
  18. 18.
    Zimmermann, R., Görtz, S.: Non-linear reduced order models for steady aerodynamics. Procedia Computer Science 1, 165–174 (2010)CrossRefGoogle Scholar
  19. 19.
    Buffoni, M., Telib, H., Iollo, A.: Iterative methods for model reduction by domain decomposition. Computers & Fluids 38(6), 1160–1167 (2009)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Blazek, J.: Computational Fluid Dynamics: Principles and Applications, 1st edn. Elsevier (2001)Google Scholar
  21. 21.
    Holmes, P., Lumley, J.L., Berkooz, G.: Turbulence, Coherent Structures, Dynamical Systems and Symmertry, Cambridge, New York (1996)Google Scholar
  22. 22.
    Pinnau, R.: Model reduction via proper orthogonal decomposition. In: Model Order Reduction: Theory, Research Aspects and Applications, pp. 95–109. Springer (2008)Google Scholar
  23. 23.
    Powell, M.: A hybrid method for nonlinear equations. Numerical Methods for Nonlinear Algebraic Equations 7, 87–114 (1970)Google Scholar
  24. 24.
    Gerhold, T., Friedrich, O., Evans, J., Galle, M.: Calculation of complex three-dimensional configurations employing the DLR-TAU-code. AIAA Paper 167 (1997)Google Scholar
  25. 25.
    Bui-Thanh, T., Damodaran, M., Willcox, K.: Proper orthogonal decomposition extensions for parametric applications in transonic aerodynamics. AIAA Journal 42(8), 1505–1516 (2004)CrossRefGoogle Scholar
  26. 26.
    Forrester, A., Sobester, A., Keane, A.: Engineering Design via Surrogate Modelling: A Practical Guide. Wiley (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Alexander Vendl
    • 1
  • Heike Faßbender
    • 1
  1. 1.Institute Computational Mathematics, AG NumerikTU BraunschweigBraunschweigGermany

Personalised recommendations