We present a generalized polynomial chaos algorithm to model the input uncertainty and its propagation in flow-structure interactions. The stochastic input is represented spectrally by employing orthogonal polynomial functionals from the Askey scheme as the trial basis in the random space. A standard Galerkin projection is applied in the random dimension to obtain the equations in the weak form. The resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode. This approach is a generalization of the original polynomial chaos expansion, which was first introduced by N. Wiener (1938) and employs the Hermite polynomials (a subset of the Askey scheme) as the basis in random space. The algorithm is first applied to second-order oscillators to demonstrate convergence, and subsequently is coupled to incompressible Navier-Stokes equations. Error bars are obtained, similar to laboratory experiments, for the pressure distribution on the surface of a cylinder subject to vortex-induced vibrations.
Issue Section:
Technical Papers
1.
Ziada, S. and Staubli, T., (editors), 2000, “Flow-Induced Vibration,” Proc. 7th Int. Conf., FIV2000, Lucerne, Switzerland, 19–22 June, A. A. Balkema.
2.
Leweke, T. (editor), 2001, “Special Issue on Proc. of IUTAM Symposium on Bluff Body Wakes and Vortex-Induced Vibrations, Marseille, France, 13–16 June 2000,” J. Fluids Struct. .
3.
Shlesinger, M. F. and Swean, T., 1998, Stochastically Excited Nonlinear Ocean Structures, World Scientific.
4.
Hills
, R. G.
, and Trucano
, T. G.
, 1999
, “Statistical Validation of Engineering and Scientific Models: Background,” Technical Report
SAND99-1256, Sandia National Laboratories.
5.
Shinozuka
, M.
, and Deodatis
, G.
, 1986
, “Response Variability of Stochastic Finite Element Systems,” Technical Report, Dept. of Civil Engineering, Columbia University, New York.
6.
Ghanem, R. G., and Spanos, P., 1991, Stochastic Finite Elements: A Spectral Approach, Springer-Verlag.
7.
Wiener
, N.
, 1938
, “The Homogeneous Chaos
,” Am. J. Math.
, 60
, pp. 897
–936
.8.
Wiener, N., 1958, Nonlinear Problems in Random Theory, MIT Technology Press Wiley, New York.
9.
Meecham
, W. C.
, and Siegel
, A.
, 1964
, “Wiener-Hermite Expansion in Model Turbulence at Large Reynolds Numbers
,” Phys. Fluids
, 7
, pp. 1178
–1190
.10.
Siegel
, A.
, Imamura
, T.
, and Meecham
, W. C.
, 1965
, “Wiener-Hermite Expansion in Model Turbulence in the Late Decay Stage
,” J. Math. Phys.
, 6
, pp. 707
–721
.11.
Meecham
, W. C.
, and Jeng
, D. T.
, 1968
, “Use of the Wiener-Hermite Expansion for Nearly Normal Turbulence
,” J. Fluid Mech.
, 32
, pp. 225
–249
.12.
Orszag
, S. A.
, and Bissonnette
, L. R.
, 1967
, “Dynamical Properties of Truncated Wiener-Hermite Expansions
,” Phys. Fluids
, 10
, p. 2603
2603
.13.
Crow
, S. C.
, and Canavan
, G. H.
, 1970
, “Relationship Between a Wiener-Hermite Expansion and an Energy Cascade
,” J. Fluid Mech.
, 41
, pp. 387
–403
.14.
Chorin
, A. J.
, 1974
, “Gaussian Fields and Random Flow
,” J. Fluid Mech.
, 85
, pp. 325
–347
.15.
Xiu, D., and Karniadakis, G. E., 2001, “The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations,” SIAM J. Sci. Comput. (USA), submitted.
16.
Xiu, D., and Karniadakis, G. E., 2001, “Modeling Uncertainty in Flow Simulations Via Generalized Polynomial Chaos,” J. Comput. Phys., to appear.
17.
Askey
, R.
, and Wilson
, J.
, 1985
, “Some Basic Hypergeometric Polynomials that Generalize Jacobi Polynomials
,” Memoirs Amer. Math. Soc., AMS, Providence RI
, 54
, No. 319
, pp. 1
–55
.18.
Hou
, Z.
, Zhou
, Y.
, Dimentberg
, M. F.
, and Noori
, M.
, 1996
, “A Stationary Model for Periodic Excitation with Uncorrelated Random Disturbances
,” Probab. Eng. Mech.
, 11
, pp. 191
–203
.19.
Yim, S. C. S., and Lin, H., 1998, “A Methodology for Analysis and Design of Sensitive Nonlinear Ocean Systems, Stochastically Excited Nonlinear Ocean Structures, World Scientific, p. 105.
20.
Soong, T. T., and Grigoriu, M., 1993, Random Vibration of Mechanical and Structural Systems, Prentice Hall.
21.
Fei, C.-Y., and Vandiver, J. K., 1995, “A Gaussian Model for Predicting the Effect of Unsteady Windspeed on the Vortex-Induced Vibration Response of Structural Members,” J. Struct. Mech., ASME, Proc. 14th Int. Conf. on Offshore Mechanics and Arctic Engineering, Book No. H00939:57–65.
22.
Shinozuka
, M.
, Yun
, C.
, and Vaicatis
, R.
, 1977
, “Dynamic Analysis of Fixed Offshore Structures Subject to Wind Generated Waves
,” J. Struct. Mech.
, 5
, No. 2
, pp. 135
–146
.23.
Grigoriu
, M.
, 1984
, “Extremes of Wave Forces
,” J. Eng. Mech.
, 110
, pp. 1731
–1742
.24.
Schoutens, W., 2000, Stochastic Processes and Orthogonal Polynomials, Springer-Verlag New York.
25.
Koekoek
, R.
, and Swarttouw
, R. F.
, 1998
, “The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its Q-Analogue,” Technical Report 98-17, Department of Technical Mathematics and Informatics, Delft University of Technology.26.
Cameron
, R. H.
, and Martin
, W. T.
, 1997
, “The Orthogonal Development of Nonlinear Functionals in Series of Fourier-Hermite Functionals
,” Ann. Math.
, 48
, p. 385
385
.27.
Ogura
, H.
, 1972
, “Orthogonal Functionals of the Poisson Process
,” IEEE Trans. Inf. Theory
, IT-18
, pp. 473
–481
.28.
Loe`ve, M., 1977, Probability Theory, Fourth edition, Springer-Verlag.
29.
Karniadakis, G. E., and Sherwin, S. J., 1999, Spectral/hp Element Methods for CFD, Oxford University Press.
30.
Karniadakis
, G. E.
, Israeli
, M.
, and Orszag
, S. A.
, 1991
, “High-Order Splitting Methods for Incompressible Navier-Stokes Equations
,” J. Comput. Phys.
, 97
, p. 414
414
.31.
Newman
, D. J.
, and Karniadakis
, G. E.
, 1997
, “Simulations of Flow Past a Freely Vibrating Cable
,” J. Fluid Mech.
, 344
, pp. 95
–136
.32.
Karniadakis
, G. E.
, 1995
, “Towards an Error Bar in CFD
,” ASME J. Fluids Eng.
, 117
, pp. 7
–9
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