×

zbMATH — the first resource for mathematics

Modeling uncertainty in flow simulations via generalized polynomial chaos. (English) Zbl 1047.76111
Summary: We present a new algorithm to model the input uncertainty and its propagation in incompressible flow simulations. The stochastic input is represented spectrally by employing orthogonal polynomial functionals from the Askey scheme as trial basis to represent 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 can be considered as a generalization of the original polynomial chaos expansion, first introduced by N. Wiener [Am. J. Math. 60, 897 (1938; Zbl 0019.35406)]. The original method employs the Hermite polynomials (one of the 13 members of the Askey scheme) as the basis in random space. The algorithm is applied to micro-channel flows with random wall boundary conditions, and to external flows with random freestream. Efficiency and convergence are studied by comparing with exact solutions as well as numerical solutions obtained by Monte Carlo simulations. It is shown that the generalized polynomial chaos method promises a substantial speed-up compared with the Monte Carlo method. The utilization of different type orthogonal polynomials from the Askey scheme also provides a more efficient way to represent general non-Gaussian processes compared with the original Wiener–Hermite expansions.

MSC:
76M35 Stochastic analysis applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Workshop on Validation and Verification of Computational Mechanics Codes, Technical Report, Caltech, December, 1998
[2] Workshop on Predictability of Complex Phenomena, Los Alamos, 6-8 December 1999, Technical Report
[3] Workshop on Decision Making Under Uncertainty, IMA, 16-17 September 1999, Technical Report
[4] Oden, T.J.; Wu, W.; Ainsworth, M., An a posteriori error estimate for finite element approximations of the navier – stokes equations, Comput. meth. appl. mech. eng., 111, 185, (1994) · Zbl 0844.76056
[5] Machiels, L.; Peraire, J.; Patera, A.T., A posteriori finite element output bounds for the incompressible navier – stokes equations; application to a natural convection problem, J. comput. phys., 172, 401-425, (2001) · Zbl 1002.76069
[6] R.G. Hills and T.G. Trucano, Statistical validation of engineering and scientific models: background, Technical Report SAND99-1256, Sandia National Laboratories, 1999
[7] M. Shinozuka and G. Deodatis, Response variability of stochastic finite element systems, Technical Report, Department of Civil Engineering, Columbia University, New York, 1986
[8] Ghanem, R.G.; Spanos, P., Stochastic finite elements: A spectral approach, (1991), Springer Berlin · Zbl 0722.73080
[9] Wiener, N., The homogeneous chaos, Am. J. math., 60, 897-936, (1938) · JFM 64.0887.02
[10] Wiener, N., Nonlinear problems in random theory, (1958), MIT Technology Press and John Wiley and Sons New York · Zbl 0121.12302
[11] Meecham, W.C.; Siegel, A., Wiener – hermite expansion in model turbulence at large Reynolds numbers, Phys. fluids, 7, 1178-1190, (1964) · Zbl 0134.21804
[12] Siegel, A.; Imamura, T.; Meecham, W.C., Wiener – hermite expansion in model turbulence in the late decay stage, J. math. phys., 6, 707-721, (1965)
[13] Meecham, W.C.; Jeng, D.T., Use of the wiener – hermite expansion for nearly normal turbulence, J. fluid mech., 32, 225-249, (1968) · Zbl 0155.55705
[14] Orszag, S.A.; Bissonnette, L.R., Dynamical properties of truncated wiener – hermite expansions, Phys. fluids, 10, 2603, (1967) · Zbl 0166.46204
[15] Crow, S.C.; Canavan, G.H., Relationship between a wiener – hermite expansion and an energy cascade, J. fluid mech., 41, 387-403, (1970) · Zbl 0191.25603
[16] Chorin, A.J., Gaussian fields and random flow, J. fluid mech., 85, 325-347, (1974)
[17] R. Askey, J. Wilson, Some basic hypergeometric polynomials that generalize Jacobi polynomials, Memoirs of the American Mathematical Society, AMS, Providence, RI, 1985, p. 319 · Zbl 0572.33012
[18] Szegö, G., Orthogonal polynomials, (1939), AMS Providence, RI · JFM 65.0278.03
[19] Beckmann, P., Orthogonal polynomials for engineers and physicists, (1973), Golem Press · Zbl 0253.42013
[20] Chihara, T.S., An introduction to orthogonal polynomials, (1978), Gordon and Breach Science Publishers · Zbl 0389.33008
[21] R. Koekoek and R.F. Swarttouw. 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, 1998
[22] Schoutens, W., Stochastic processes and orthogonal polynomials, (2000), Springer New York · Zbl 0960.60076
[23] Cameron, R.H.; Martin, W.T., The orthogonal development of nonlinear functionals in series of fourier- Hermite functionals, Ann. math., 48, 385, (1947) · Zbl 0029.14302
[24] Ghanem, R.G., Stochastic finite elements for heterogeneous media with multiple random non-Gaussian properties, ASCE J. eng. mech., 125, 1, 26-40, (1999)
[25] Ghanem, R.G., Ingredients for a general purpose stochastic finite element formulation, Comput. meth. appl. mech. eng., 168, 19-34, (1999) · Zbl 0943.65008
[26] Ogura, H., Orthogonal functionals of the Poisson process, IEEE trans. info. theory, 18, 473-481, (1972) · Zbl 0244.60044
[27] Loéve, M., Probability theory, (1977), Springer-Verlag Berlin · Zbl 0359.60001
[28] Karniadakis, G.E.; Israeli, M.; Orszag, S.A., High-order splitting methods for incompressible navier – stokes equations, J. comput. phys., 97, 414, (1991) · Zbl 0738.76050
[29] Karniadakis, G.E.; Sherwin, S.J., Spectral/hp element methods for CFD, (1999), Oxford University Press Oxford · Zbl 0954.76001
[30] Williamson, C.H.K., Vortex dynamics in the cylinder wake, Annu. rev. fluid mech., 28, 477-539, (1996) · Zbl 0899.76129
[31] Kaiktsis, L.; Karniadakis, G.E.; Orszag, S.A., Unsteadiness and convective instabilities in two-dimensional flow over a backward-facing step, J. fluid mech., 321, 157-187, (1996) · Zbl 0875.76111
[32] Karniadakis, G.E., Towards an error bar in CFD, J. fluids eng., 117, March, (1995)
[33] D. Gottlieb, S.A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, CBMS-NSF, SIAM, Philadelphia, PA, 1977 · Zbl 0412.65058
[34] Boyd, J.P., The rate of convergence of Hermite function series, Math. comput., 35, 1039-1316, (1980) · Zbl 0459.40005
[35] Tang, T., The Hermite spectral methods for Gaussian-type functions, SIAM J. sci. comput., 14, 594-606, (1993) · Zbl 0782.65110
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.