CEAS Aeronautical Journal

, Volume 6, Issue 1, pp 121–136

POD approach for unsteady aerodynamic model updating

Original Paper

Abstract

A method for aerodynamic model updating is proposed in this paper. The approach is based upon a correction of the eigenvalues of the reduced-order unsteady aerodynamic matrix through an optimization with objective function defined through the difference in the generalized aerodynamic forces or on the aeroelastic poles. The high-fidelity model in reduced-order form is obtained by the proper orthogonal decomposition (POD) technique applied to the computational fluid dynamics Euler-based formulation. Many of the methods that have been developed in the past years for simpler aeroelastic models that use, for example, doublet-lattice method aerodynamics, can be adopted for this purpose as well. However, this model is not able to capture shocks and flow separation in transonic flow. The proposed approach performs the updating of the aerodynamic model by imposing the minimization of a global error between target aerodynamic performances, namely experimental performances, and an aerodynamic model in reduced-order form via POD approach. After a general presentation of the application of the POD method to the linearized Euler equations, the optimization strategy is presented. First, a simple test on a 2D wing section with theoretical biased data is performed, and then, the performances of different optimization strategies are tested on a 3D model updated by wind tunnel data.

Keywords

Aeroelasticity POD Reduced-order model Unsteady aerodynamic updating WT test data 

List of symbols

Ω

Computational domain volume

\(\bar {W}\)

Mean instantaneous field

F

Flux

Ni

Normal to the cell face

ai

Velocity in the cell face

f, g, h

Flux component vector

\(\psi\)

Proper orthogonal mode

U

Flow field

\(\eta\)

Vector of the components of the disturbance field in the POD base

\(\varphi\)

Structural modes

d

Displacement field

q

Generalized coordinates

\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f}\)

Generalized aerodynamic force

\({\hat{\text{K}}}\)

Stiffness matrix

\({\hat{\text{M}}}\)

Mass matrix

\({\hat{\text{C}}}\)

Damping matrix

M

The instantaneous position of the grid nodes

a0, a1

Fluid system matrix

b0, b1

Coupling vector

\(a\)

Reduced-order aerodynamic matrix

V

Right eigenvector

\(\lambda\)

Aeroelastic poles

\(\mu\)

Complex aerodynamic poles

\(r\)

Design variable for real part

\(s\)

Design variable for complex part

References

  1. 1.
    Mortchelewicz, G.D.: Application of proper orthogonal decomposition to linearized Euler or Reynolds-Averaged Navier-Stokes equation. In: 47th Israel Annual Conference on Aerospace Sciences, Tel Aviv, February 2007. http://publications.onera.fr/
  2. 2.
    Palacios, R., Climent, H., Karlsson, A., Winzell, B.: Assessment of strategies for correcting linear unsteady aerodynamics using CFD or test results. In: International Forum of Aeroelasticity and Structural Dynamics, Madrid, June 2001Google Scholar
  3. 3.
    Giesing, J.P., Kalman, T.P., Rodden, W.P.: Correction factor techniques for improving aerodynamic prediction method. Technical report NASA CR-144967 (1976)Google Scholar
  4. 4.
    Yates, E.C.: Modified-strip-analysis method for predicting wing flutter at subsonic or hypersonic speeds. J. Aircr. 3(1), 25–29 (1966)CrossRefGoogle Scholar
  5. 5.
    Pitt, D.M., Goodman, C.E.: Flutter calculations using Doublet Lattice Aerodynamics modified by the full potential equations. In: 28th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics and Materials Conference, Monterrey, 1987 (Paper AIAA-87-0882-CP)Google Scholar
  6. 6.
    Baker, M.L.: CFD based corrections for linear aerodynamic methods, AGARD R822, October 1997Google Scholar
  7. 7.
    Schulze, S., Tichy, L.: An efficient aero-correction method for transonic flutter calculations, DASA B08/98Google Scholar
  8. 8.
    Pak, C.: Unsteady aerodynamic model tuning for precise flutter prediction. J. Aircr. 48(6), 2178–2184 (2011)CrossRefGoogle Scholar
  9. 9.
    Hu, P., Qu, K., Xue, L., Ni, K., Dowell, E.: Efficient aeroelastic model updating in support of flight testing. In: AIAA Atmospheric Flight Mechanics Conference, Chicago, August 2009 (AIAA-2009-5713)Google Scholar
  10. 10.
    Thomas, J.P., Dowell, E.H., Hall, K.C.: Three-dimensional transonic aeroelasticity using proper-orthogonal decomposition-based reduced-order models. J. Aircr. 40(3), 544–551 (2003)CrossRefGoogle Scholar
  11. 11.
    Bui-Thanh, T., Damodaran, M., Willcox, K.: Proper orthogonal decomposition extensions for parametric applications in transonic aerodynamics. In: 21st AIAA Applied Aerodynamics conference, Orlando, June 2003 (AIAA Paper 2003-4213)Google Scholar
  12. 12.
    Lind, R., Brenner, M.: Incorporating flight data into a robust aeroelastic model. J. Aircr. 35(3), 470–477 (1998)CrossRefGoogle Scholar
  13. 13.
    Maute, K., Nikbay, M., Farhat, C.: Coupled analytical sensitivity analysis and optimization of three-dimensional nonlinear aeroelastic systems. AIAA J. 39(11), 2051–2061 (2011)CrossRefGoogle Scholar
  14. 14.
    Giunta, A.A.: Sensitivity analysis for coupled aero-structural systems. In: NASA Technical memorandum, Hampton, August 1999 (NASA/TM-1999-209367)Google Scholar
  15. 15.
    Thomas, J.P., Hall, K.C., Dowell, E.H.: Reduced-order modelling of unsteady small-disturbance flows using a frequency-domain proper orthogonal decomposition technique. In: 37th AIAA Aerospace Sciences Meeting and Exhibit, Reno, January 1999 (AIAA 99-655)Google Scholar
  16. 16.
    Xiao, M., Breitkopf, P., Filomeno Coelho, R., Knopf-Lenoir, C., Villon, P.: Constrained proper orthogonal decomposition based on QR-factorization for aerodynamical shape optimization. Appl. Math. Comput. (2013). doi:10.1016/j.amc.2013.07.086 MathSciNetGoogle Scholar
  17. 17.
    Fang, F., Pain, C.C., Navon, I.M., Gorman, G.J., Piggott, M.D., Allison, P.A., Goddard, A.J.H: A POD goal-oriented error measure for mesh optimisation. Int. J. Numer. Methods Fluids 63, 185–206 (2010)Google Scholar
  18. 18.
    Mortchelewicz, G.D.: Flutter simulations. Aerospace Sci Technol 4(1), 33–40 (2000). ISSN 1270-9638. http://dx.doi.org/10.1016/S1270-9638(00)00116-4
  19. 19.
    Zona Technology Inc.: ZAERO Theoretical Manual, June 2011Google Scholar
  20. 20.
    Mortchelewicz, G.D.: Aircraft aeroelasticity computed with linearized RANS equations. In: 43rd Israel Annual Conference on Aerospace Sciences, Tel Aviv, February 2003. http://publications.onera.fr/
  21. 21.
    Carlberg, K., Farhat, C.: A low-cost, goal-oriented ‘compact proper orthogonal decomposition’ basis for model reduction of static system. Int. J. Numer. Meth. Eng. 86, 381–402 (2011). doi:10.1002/nme.3074 CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Filomeno Coelho, R., Breitkopf, P., Knopf-Lenoir, C.: Bi-level model reduction for coupled problems. Application to a 3D wing. Struct. Multidiscipl. Optim. 39(4), 401–418 (2009). doi:10.1007/s00158-008-0335-3
  23. 23.
    Lucia, D.J., Beran, P.S., Silva, W.A.: Reduced-order modelling: new approaches for computational physics. Progr. Aerospace Sci. 40(1–2), 51–117 (2004). doi:10.1016/j.paerosci.2003.12.001
  24. 24.
    Vetrano, F., le Garrec, C., Mortchelewicz, G.D., Ohayon, R.: Assessment of strategies for interpolating POD based reduced order model and application to aeroelasticity. J. Aeroelast. Struct. Dyn. 2(2), 85–104 (2011)Google Scholar
  25. 25.
    Gill, P.E., Murray, W., Saunders, M.A.: User’s Guide for SNOPT Version 7: Software for Large-Scale Nonlinear Programming, February 2006Google Scholar
  26. 26.
    Zingel, H. et al.: Measurement of steady and unsteady air loads on a stiffness scaled model of a modern transport aircraft wing. In: International Forum on Aeroelasticity and Structural dynamics, Aachen, June 1991. DGLR 91-06, pp. 120–131 (1991)Google Scholar

Copyright information

© Deutsches Zentrum für Luft- und Raumfahrt e.V. 2014

Authors and Affiliations

  1. 1.Loads and Aeroelasticity DepartmentAirbusToulouseFrance
  2. 2.Mechanical and Aerospace Engineering Department“La Sapienza” University of RomeRomeItaly
  3. 3.Structural Mechanics and Coupled System LaboratoryConservatoire National des Arts et MétiersParisFrance

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