Uncertainty quantification and polynomial chaos techniques in computational fluid dynamics

Annual Review of Fluid Mechanics

Volumn 41, Issue , 2009, Pages 35-52

  Najm, Habib N ^a  

^a SANDIA NATIONAL LABORATORIES   (United States)

Author keywords

CFD; PC; Polynomial chaos; UQ

Indexed keywords

APPLICATION AREA; CFD; CFD APPLICATIONS; CFD MODELS; COMPUTATIONAL MODEL; CROSS-CUTTING; FLOW IN POROUS MEDIA; INPUT UNCERTAINTY; MODEL INPUTS; PC; POLYNOMIAL CHAOS; POLYNOMIAL CHAOS TECHNIQUES; PROBABILISTIC UNCERTAINTY; REACTING FLOWS; THERMOFLUIDS; TIME HORIZONS; UNCERTAINTY QUANTIFICATIONS; UQ;

CHAOS THEORY; FLUID DYNAMICS; INCOMPRESSIBLE FLOW; POLYNOMIALS; POROUS MATERIALS;

COMPUTATIONAL FLUID DYNAMICS;

COMPRESSIBLE FLOW; COMPUTATIONAL FLUID DYNAMICS; FLOW MODELING; PREDICTION; PROBABILITY; QUANTITATIVE ANALYSIS; UNCERTAINTY ANALYSIS;

EID: 63449137570     PISSN: 00664189     EISSN: None     Source Type: Book Series    
DOI: 10.1146/annurev.fluid.010908.165248     Document Type: Review

References (88)

1. Askey R, Wilson J. 1985. Some Basic Hypergeometric Polynomials That Generalize Jacobi Polynomials. Mem. Am. Math. Soc. No. 319. Providence, RI: Am Math. Soc.
2. Asokan B, Zabaras N. 2005. Using stochastic analysis to capture unstable equilibrium in natural convection. J. Comput. Phys. 208:134-53
3. Babuska I, Tempone R, Zouraris G. 2004. Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAMJ. Numer. Anal. 42:800-25
4. Babuska I, Nobile F, Tempone R. 2007. Astochastic collocation method for elliptic partial differential equations with random input data. SIAMJ. Numer. Anal. 45:1005-34
5. Bear J. 1972. Dynamics of Fluids in Porous Media. New York: Dover
6. Boyd J. 1980. The rate of convergence of hermite function series. Math. Comput. 35:1309-16
7. Cacuci D, ed. 2003. Sensitivity and Uncertainty Analysis Theory, vol. 1. Boca Raton, FL: Chapman & Hall/CRC
8. Caflisch R. 1998. Monte Carlo and quasi-Monte Carlo methods. Acta Numer. 7:1-49
9. Cameron R, Martin W. 1947. The orthogonal development of nonlinear functionals in series of Fourier-Hermite functionals. Ann. Math. 48:385-92
10. Chen QY, Gottlieb D, Hesthaven J. 2005. Uncertainty analysis for the steady-state flows in a dual throat nozzle. J. Comput. Phys. 204:378-98
11. Chorin A. 1974. Gaussian fields and random flow. J. Fluid Mech. 63:21-32
12. Cox E, ed. 1999. The Fuzzy Systems Handbook: A Practitioner's Guide to Building, Using, and Maintaining Fuzzy Systems. San Diego, CA: AP Prof. Div. Acad. 2nd ed.
13. Crow S, Canavan G. 1970. Relationship between a Wiener-Hermite expansion and an energy cascade. J. Fluid Mech. 41:387-403
14. Daniels M, Kass R. 1999. Nonconjugate Bayesian estimation of covariance matrices and its use in hierarchical models. J. Am. Stat. Assoc. 94:1254-63
15. Das S, Ghanem R, Spall J. 2008. Asymptotic sampling distribution for polynomial chaos representation from data: a maximum entropy and Fisher information approach. SIAM J. Sci. Comput. In press
16. Deb MK, Babuška I, Oden J. 2001. Solution of stochastic partial differential equations using Galerkin finite element techniques. Comput. Methods Appl. Mech. Eng. 190:6359-72
17. Debusschere B, Najm H, Matta A, Knio O, Ghanem R, Le Maître O. 2003. Protein labeling reactions in electrochemical microchannel flow: numerical simulation and uncertainty propagation. Phys. Fluids 15:2238-50
18. Debusschere B, Najm H, Pébay P, Knio O, Ghanem R, Le Maître O. 2004. Numerical challenges in the use of polynomial chaos representations for stochastic processes. SIAM J. Sci. Comput. 26:698-719
19. de Cooman G, Ruan D, Kerre E, eds. 1995. Foundations and Applications of Possibility Theory. Singapore: World Sci.
20. Faragher J. 2004. Probabilistic methods for the quantification of uncertainty and error in computational fluid dynamics simulations. Tech. Rep. DSTO-TR-1633, Aust. Gov., Dept. Def., Def. Sci. Tech. Org.
21. Ganapathysubramanian B, Zabaras N. 2007. Sparse grid collocation schemes for stochastic natural convection problems. J. Comput. Phys. 225:652-85
22. Ghanem R. 1998a. Hybrid stochastic finite elements and generalized Monte Carlo simulation. ASME J. Appl. Mech. 65:1004-9
23. Ghanem R. 1998b. Probabilistic characterization of transport in heterogeneous media. Comput. Methods Appl. Mech. Eng. 158:199-220
24. Ghanem R, Dham S. 1998. Stochastic finite element analysis for multiphase flow in heterogeneous porous media. Transp. Porous Media 32:239-62
25. Ghanem R, Spanos P. 1991. Stochastic Finite Elements: A Spectral Approach. New York: Springer Verlag
26. Ghanem RG, Doostan A. 2006. On the construction and analysis of stochastic models: characterization and propagation of the errors associated with limited data. J. Comput. Phys. 217:63-81
27. Ghiocel D, Ghanem R. 2002. Stochastic finite-element analysis of seismic soil-structure interaction. J. Eng. Mech. 128:66-77
28. Hansen E. 1992. Global Optimization Using Interval Analysis. New York: Marcel Dekker
29. Helton J, Johnson J, Oberkampf W. 2004. An exploration of alternative approaches to the representation of uncertainty in model predictions. Reliab. Eng. Syst. Saf. 85:39-71
30. Hosder S, Walters R, Balch M. 2008. Efficient uncertainty quantification applied to the aeroelastic analysis of a transonic wing. Presented at AIAA Aerosp. Sci. Meet. Exhib, Reno, 46th, Art. No. AIAA-2008-729
31. Hosder S, Walters R, Perez R. 2006. A nonintrusive polynomial chaos method for uncertainty propagation in CFD simulations. Presented at AIAA Aerosp. Sci. Meet. Exhib, 44th, Reno, Art. No. AIAA-2006-0891
32. Hou TY, Luo W, Rozovskii B, Zhou HM. 2006. Wiener chaos expansions and numerical solutions of randomly-forced equations of fluid mechanics. J. Comput. Phys. 216:687-706
33. Isukapalli S, Roy A, Georgopoulos P. 1998. Stochastic response surface methods (SRSMs) for uncertainty propagation: application to environmental and biological systems. Risk Anal. 18:351-63
34. Janson S. 1997. Gaussian Hilbert Spaces. Cambridge, UK: Cambridge Univ. Press
35. Jaynes E. 2003. Probability Theory: The Logic of Science, ed. GL Bretthorst. Cambridge, UK: Cambridge Univ. Press
36. Karhunen K. 1946. Zur spektraltheorie stochastischer prozesse. Ann. Acad. Sci. Fennicae 34:1-7
37. Knio O, Le Maître O. 2006. Uncertainty propagation in CFD using polynomial chaos decomposition. Fluid Dyn. Res. 38:616-40
38. Kozine I. 1999. Imprecise probabilities relating to prior reliability assessments. Presented at Int. Symp. Imprecise Probab. Appl., 1st, Ghent, Belgium
39. Le Maître O. 2006. Polynomial chaos expansion of a Lagrangian model for the flow around an airfoil. C. R. Mec. 334:693-99
40. Le Maître O, Ghanem R, Knio O, Najm H. 2004a. Uncertainty propagation using Wiener-Haar expansions. J. Comput. Phys. 197:28-57
41. Le Maître O, Knio O. 2007. A stochastic particle-mesh scheme for uncertainty propagation in vortical flows. J. Comput. Phys. 226:645-71
42. Le Maître O, Knio O, Najm H, Ghanem R. 2001. A stochastic projection method for fluid flow I. Basic formulation. J. Comput. Phys. 173:481-511
43. Le Maître O, Najm H, Ghanem R, Knio O. 2004b. Multi-resolution analysis of Wiener-type uncertainty propagation schemes. J. Comput. Phys. 197:502-31
44. Le Maître O, Najm H, Pébay P, Ghanem R, Knio O. 2007. Multi-resolution-analysis scheme for uncertainty quantification in chemical systems. SIAMJ. Sci. Comput. 29:864-89
45. Le Maître O, Reagan M, Debusschere B, Najm H, Ghanem R, Knio O. 2004c. Natural convection in a closed cavity under stochastic, non-Boussinesq conditions. SIAMJ. Sci. Comput. 26:375-94
46. Le Maître O, Reagan M, Najm H, Ghanem R, Knio O. 2002. A stochastic projection method for fluid flow II. Random process. J. Comput. Phys. 181:9-44
47. Li R, Ghanem R. 1998. Adaptive polynomial chaos expansions applied to statistics of extremes in nonlinear random vibration. Prob. Eng. Mech. 13:125-36
48. Lin G, Su CH, Karniadakis G. 2006. Predicting shock dynamics in the presence of uncertainties. J. Comput. Phys. 217:260-76
49. Loeve M. 1948. Fonctions aleatoires du seconde ordre. In Processus Stochastiques et Movement Brownien, ed. P Levy. Paris: Hermann
50. Lu Z, Zhang D. 2004. A comparative study on uncertainty quantification for flow in randomly heterogeneous media using Monte Carlo simulations and conventional and KL-based moment-equation approaches. SIAMJ. Sci. Comput. 26:558-77
51. Lucor D, Karniadakis G. 2004. Noisy inflows cause a shedding-mode switching in flow past an oscillating cylinder. Phys. Rev. Lett. 92:154501
52. Mathelin L, Hussaini M. 2003. A stochastic collocation algorithm for uncertainty analysis. NASA Tech. Rep. NASA/CR-2003-212153, Langley Res. Cent., Hampton, Va.
53. Mathelin L, Hussaini M, Zang T. 2005. Stochastic approaches to uncertainty quantification in CFD simulations. Numer. Algorithms 38:209-36
54. Mathelin L, Hussaini M, Zang T, Bataille F. 2003. Uncertainty propagation for turbulent, compressible flow in a quasi-1D nozzle using stochastic methods. Presented at AIAA Comput. Fluid Dyn. Conf., 69th, Orlando, Art. No. AIAA-2003-4240
55. Mathelin L, Hussaini M, Zang T, Bataille F. 2004. Uncertainty propagation for a turbulent, compressible nozzle flow using stochastic methods. AIAA J. 42:1669-76
56. Matthies HG, Keese A. 2005. Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Eng. 194:1295-331
57. Narayanan V, Zabaras N. 2005. Variational multiscale stabilized FEM formulations for transport equations: stochastic advection-diffusion and incompressible stochastic Navier-Stokes equations. J. Comput. Phys. 202:94-133
58. Nobile F, Tempone R, Webster C. 2008a. An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data. SIAMJ. Numer. Anal. In press
59. Nobile F, Tempone R, Webster C. 2008b. A sparse grid stochastic collocation method for partial differential equations with random input data. SIAMJ. Numer. Anal. In press
60. Oberkampf W, Blottner F. 1998. Issues in computational fluid dynamics code verification and validation. AIAA J. 36:687-95
61. Oberkampf W, Helton J. 2005. Evidence theory for engineering applications. In Engineering Design Reliability Handook, ed. E Nikolaidis, D Ghiocel, S Singhal, pp. 10.1-30. Boca Raton, FL: CRC
62. Orszag S, Bissonnette L. 1967. Dynamical properties of truncated Wiener-Hermite expansions. Phys. Fluids 10:2603-13
63. Perez R, Walters R. 2005. An implicit polynomial chaos formulation for the Euler equations. Presented at AIAA Aerosp. Sci. Meet. Exhib, 43rd, Reno, Art. No. AIAA-2005-1406
64. Pettit C, Beran P. 2004. Polynomial chaos expansions applied to airfoil limit cycle oscillations. Presented at AIAA/ASME/ASCE/AHS/ASC Struct., Struct. Dyn., Mater. Conf., 45th, Palm Springs, Art. No. AIAA-2004-1691
65. Pettit CL, Beran PS. 2006. Spectral and multiresolution Wiener expansions of oscillatory stochastic processes. J. Sound Vib. 294:752-79
66. Phenix B, Dinaro J, Tatang M, Tester J, Howard J, McRae G. 1998. Incorporation of parametric uncertainty into complex kinetic mechanisms: application to hydrogen oxidation in supercritical water. Combust. Flame 112:132-46
67. Reagan M, Najm H, Debusschere B, Le Maître O, Knio O, Ghanem R. 2004. Spectral stochastic uncertainty quantification in chemical systems. Combust. Theory Model. 8:607-32
68. Reagan M, Najm H, Ghanem R, Knio O. 2003. Uncertainty quantification in reacting flow simulations through nonintrusive spectral projection. Combust. Flame 132:545-55
69. Reagan M, Najm H, Pebay P, Knio O, Ghanem R. 2005. Quantifying uncertainty in chemical systems modeling. Int. J. Chem. Kinet. 37:368-82
70. Rosenblatt M. 1952. Remarks on a multivariate transformation. Ann. Math. Stat. 23:470-72
71. Saltelli A, Chan K, Scott E, eds. 2000. Sensitivity Analysis. Chichester, NY: Wiley
72. Schoutens W. 2000. Stochastic Processes and Orthogonal Polynomials. New York: Springer
73. Smoljak SA. 1963. Quadrature and interpolation formulas for tensor products of certain classes of functions. Soviet Math. Dokl. 4:240-43
74. Soize C, Ghanem R. 2004. Physical systems with random uncertainties: chaos representations with arbitrary probability measure. SIAMJ. Sci. Comput. 26:395-410
75. Tartakovsky D, Xiu D. 2006. Stochastic analysis of transport in tubes with rough walls. J. Comput. Phys. 217:248-59
76. Tatang M. 1995. Direct incorporation of uncertainty in chemical and environmental engineering systems. PhD thesis. Mass. Inst. Technol.
77. Walters R, Huyse L. 2002. Uncertainty analysis for fluid mechanics with applications. ICASE Rep. No. 2002-1; NASA/CR-2002-211449, Langley Res. Cent., Hampton, Va.
78. Wan X, Karniadakis G. 2006a. Long-term behavior of polynomial chaos in stochastic flow simulations. Comput. Methods Appl. Mech. Eng. 195:5582-96
79. Wan X, Karniadakis G. 2006b. Multi-element generalized polynomial chaos for arbitrary probability measures. SIAMJ. Sci. Comput. 28:901-28
80. Wan X, Karniadakis GE. 2005. An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. J. Comput. Phys. 209:617-42
81. Webster M, Sokolov A. 2000. A methodology for quantifying uncertainty in climate projections. Clim. Change 46:417-46
82. Wiener N. 1938. The homogeneous chaos. Am. J. Math. 60:897-936
83. Wiener N. 1939. The use of statistical theory in the study of turbulence. Proc. Fifth Int. Cong. Appl. Mech., p. 356. New York: Wiley
84. Xiu D, Hesthaven JS. 2005. High-order collocation methods for differential equations with random inputs. SIAMJ. Sci. Comput. 27:1118-39
85. Xiu D, Karniadakis G. 2002. The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24:619-44
86. Xiu D, Karniadakis G. 2003. Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187:137-67
87. Xiu D, Lucor D, Su CH, Karniadakis G. 2002. Stochastic modeling of flow-structure interactions using generalized polynomial chaos. ASMEJ. Fluids Eng. 124:51-59
88. Zang T, Hemsch M, Hilburger M, Kenny S, Luckring J, et al. 2002. Needs and opportunities for uncertainty-based multidisciplinary design methods for aerospace vehicles. Tech. Rep. NASA/TM-2002-211462, Langley Res. Cent., Hampton, Va.