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Adaptive Uncertainty Quantification for Computational Fluid Dynamics

  • Richard P. Dwight
  • Jeroen A. S. Witteveen
  • Hester Bijl
Chapter
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 92)

Abstract

Two different approaches to propagating uncertainty are considered, both with application to CFD models, both adaptive in the stochastic space. The first is “Adaptive Stochastic Finite Elements”, which approximates the model response in the stochastic space on a triangular grid with quadratic reconstruction on the elements. Limiting reduces the reconstruction to linear in the presence of discontinuities, and the mesh is refined using the Hessian of the response as an indicator. This construction allows for UQ in the presence of strong shocks in the response, for example in case of a transonic aerofoil with uncertain Mach number, and where variability in surface pressure is of interest. The second approach is “Adaptive Gradient-Enhanced Kriging for UQ”, which uses a Gaussian process model as a surrogate for the CFD model. To deal with the high cost of interpolating in high-dimensional spaces we make use of the adjoint of the CFD code, which provides derivatives in all stochastic directions at a cost independent of dimension. The Gaussian process framework allows this information to be incorporated into the surrogate, as well as providing regression of both CFD output values and derivatives according to error estimates. It also provides an interpolation error estimate on which an adaptivity indicator is based, weighted with the input uncertainty. A transonic aerofoil with four uncertain shape parameters is given as an example case.

Keywords

Response Surface Mach Number Quadrature Point Uncertainty Quantification Transonic Flow 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Bassi, F., Crivellini, A., Rebay, S., Savini, M.: Discontinuous galerkin solution of the Reynolds-averaged Navier-Stokes and k-w turbulence model equations. Computers and Fluids 34(4–5), 507–540 (2005). DOI DOI:10.1016/j.compfluid.2003.08.004Google Scholar
  2. 2.
    Chorin, A., Marsden, J.: A mathematical introduction to fluid mechanics. Springer-Verlag, New York (1979)CrossRefzbMATHGoogle Scholar
  3. 3.
    Demmel, J.W., Eisenstat, S.C., Gilbert, J.R., Li, X.S., Liu, J.W.H.: A supernodal approach to sparse partial pivoting. SIAM J. Matrix Analysis and Applications 20(3), 720–755 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dwight, R.: Efficiency improvements of RANS-based analysis and optimization using implicit and adjoint methods on unstructured grids. Ph.D. thesis, School of Mathematics, University of Manchester (2006)Google Scholar
  5. 5.
    Dwight, R.: Goal-oriented mesh adaptation using a dissipation-based error indicator. International Journal of Numerical Methods in Fluids 56(8), 1193–1200 (2008). DOI: 10.1002/fld.1582MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dwight, R.: Heuristic a posteriori estimation of error due to dissipation in finite volume schemes and application to mesh adaptation. Journal of Computational Physics 227(5), 2845–2863 (2008). DOI: 10.1016/j.jcp.2007.11.020CrossRefzbMATHGoogle Scholar
  7. 7.
    Dwight, R., Brezillon, J.: Effect of approximations of the discrete adjoint on gradient-based optimization. AIAA Journal 44(12), 3022–3071 (2006)CrossRefGoogle Scholar
  8. 8.
    Dwight, R., Brezillon, J.: Effect of various approximations of the discrete adjoint on gradient-based optimization. In: Proceedings of the 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno NV, AIAA-2006-0690 (2006)Google Scholar
  9. 9.
    Giles, M., Duta, M., Muller, J.D., Pierce, N.: Algorithm developments for discrete adjoint methods. AIAA Journal 41(2), 198–205 (2003)CrossRefGoogle Scholar
  10. 10.
    H.-S., C., Alonso, J.: Using gradients to construct cokriging approximation models for high-dimensional design optimization problems. In: AIAA Paper Series, Paper 2002-0317 (2002)Google Scholar
  11. 11.
    Jones, D., Schonlau, M., Welch, W.: Efficient global optimization of expensive black-box functions. Journal of Global Optimization 13, 455–492 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lagarias, J., Reeds, J., Wright, M., Wright, P.: Convergence properties of the Nelder-Mead Simplex method in low dimensions. SIAM Journal on Optimization 9(1), 112–147 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Maitre, O.L., Najm, H., Ghanem, R., Knio, O.: Multi-resolution analysis of Wiener-type uncertainty propagation schemes. J. Comput. Phys. 197, 502–531 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    McKay, M., Conover, W., Beckman, R.: A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21, 239–245 (1979)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Nelder, J., Mead, R.: A simplex method for function minimization. Computer Journal 7(4), 308–313 (1965)CrossRefzbMATHGoogle Scholar
  16. 16.
    O’Hagan, A., Oakley, J.E.: Probability is perfect, but we can’t elicit it perfectly. Reliability Engineering and System Safety 85(1–3), 239–248 (2004). DOI 10.1016/j.ress.2004.03.014CrossRefGoogle Scholar
  17. 17.
    Petras, K.: Fast calculation of coefficients in the smolyak algorithm. Numerical Algorithms 26(2), 93–109 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Rowan, T.: The subplex method for unconstrained optimization. Ph.D. thesis, Department of Computer Sciences, Univ. of Texas (1990)Google Scholar
  19. 19.
    Sacks, J., Welch, W., Mitchell, T., Wynn, H.: Design and analysis of computer experiments (with discussion). Statistical Science 4, 409–435 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Smolyak, S.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Doklady Akademii Nauk SSSR 4, 240–243 (1963)Google Scholar
  21. 21.
    Webster, R., Oliver, M.: Geostatistics for Environmental Scientists, second edn. Wiley (2007). ISBN 0470028580Google Scholar
  22. 22.
    Wikle, C., Berliner, L.: A Bayesian tutorial for data assimilation. Physica D 230, 1–16 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Xiu, D., Hesthaven, J.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27, 1118–1139 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Richard P. Dwight
    • 1
  • Jeroen A. S. Witteveen
    • 2
  • Hester Bijl
    • 1
  1. 1.Faculty of AerospaceTU DelftDelftThe Netherlands
  2. 2.Center for Turbulence ResearchStanford UniversityStanfordUSA

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