Advertisement

Uncertainty Quantification in Fluid Flow

  • Habib N. Najm
Part of the Fluid Mechanics and Its Applications book series (FMIA, volume 95)

Abstract

This chapter addresses the topic of uncertainty quantification in fluid flow computations. The relevance and utility of this pursuit are discussed, outlining highlights of available methodologies. Particular attention is focused on spectral polynomial chaos methods for uncertainty quantification that have seen significant development over the past two decades. The fundamental structure of these methods is presented, along with associated challenges. We also discuss demonstrations of their use in a number of fluid flow applications covering a range of complexity that is inherent in turbulent combustion.

Keywords

Monte Carlo Uncertain Parameter Uncertainty Propagation Sparse Grid Evidence Theory 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. Dover, New York (1970) Google Scholar
  2. 2.
    Agarwal, N., Aluru, N.: A domain adaptive stochastic collocation approach for analysis of MEMS under uncertainties. J. Comput. Phys. 228, 7662–7688 (2009) zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Askey, R., Wilson, J.: Some basic hypergeometric polynomials that generalize jacobi polynomials. Memoirs Amer. Math. Soc. 319, 1–55 (1985) MathSciNetGoogle Scholar
  4. 4.
    Asokan, B., Zabaras, N.: Using stochastic analysis to capture unstable equilibrium in natural convection. J. Comput. Phys. 208, 134–153 (2005) zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Babuška, I., Tempone, R., Zouraris, G.: Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42, 800–825 (2004) zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Babuška, I., Tempone, R., Zouraris, G.: Solving elliptic boundary value problems with uncertain coefficients by the finite element method: The stochastic formulation. Comput. Methods Appl. Mech. Engrg. 194, 1251–1294 (2005) zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Babuška, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Num. Anal. 45, 1005–1034 (2007) zbMATHCrossRefGoogle Scholar
  8. 8.
    Bae, H.R., Grandhi, R., Canfield, R.: Uncertainty quantification of structural response using evidence theory. AIAA J. 41, 2062–2068 (2003) CrossRefGoogle Scholar
  9. 9.
    Barthelmann, V., Novak, E., Ritter, K.: High-dimensional polynomial interpolation on sparse grids. Adv. Compu. Math. 12, 273–288 (2000) zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Beran, P.S., Pettit, C.L., Millman, D.R.: Uncertainty quantification of limit-cycle oscillations. J. Comput. Phys. 217, 217–247 (2006). zbMATHCrossRefGoogle Scholar
  11. 11.
    Bieri, M., Schwab, C.: Sparse high order FEM for elliptic sPDEs. CMAME 198, 1149–1170 (2009) zbMATHMathSciNetGoogle Scholar
  12. 12.
    Boyd, J.: The rate of convergence of Hermite function series. Math. Comput. 35, 1309–1316 (1980) zbMATHCrossRefGoogle Scholar
  13. 13.
    Cacuci, D. (ed.): Sensitivity and Uncertainty Analysis Theory, vol. 1. Chapman & Hall/CRC, Boca Raton, FL (2003) Google Scholar
  14. 14.
    Cadafalch, J., Pérez-Segarra, C., Cónsul, R., Oliva, A.: Verification of finite volume computations on steady-state fluid flow and heat transfer. ASME J. Fluids Eng. 124, 11–21 (2002) CrossRefGoogle Scholar
  15. 15.
    Cameron, R., Martin, W.: The orthogonal development of nonlinear functionals in series of Fourier-Hermite functionals. Annals Math. 48, 385–392 (1947) MathSciNetCrossRefGoogle Scholar
  16. 16.
    Campolongo, F., Saltelli, A., Sørensen, T., Tarantola, S.: Hitchhiker’s guide to sensitivity analysis. In: A. Saltelli, K. Chan, E. Scott (eds.) Sensitivity Analysis. Wiley, Chicester (2000) Google Scholar
  17. 17.
    Canavan, G.: Some properties of a Lagrangian Wiener-Hermite expansion. J. Fluid Mech. 41, 405–412 (1970) CrossRefGoogle Scholar
  18. 18.
    Celik, I., Li, J.: Assessment of numerical uncertainty for the calculations of turbulent flow over a backward-facing step. Int. J. Numer. Meth. Fluids 49, 1015–1031 (2005) zbMATHCrossRefGoogle Scholar
  19. 19.
    Chen, Q.Y., Gottlieb, D., Hesthaven, J.: Uncertainty analysis for the steady-state flows in a dual throat nozzle. J. Comput. Phys. 204, 378–398 (2005) zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Chorin, A.: A numerical method for solving incompressible viscous flow problems. J. Comput. Phys. 2, 12–26 (1967) zbMATHCrossRefGoogle Scholar
  21. 21.
    Chorin, A.: Gaussian fields and random flow. J. Fluid Mech. 63, 21–32 (1974) zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    de Cooman, G., Ruan, D., Kerre, E. (eds.): Foundations and Applications of Possibility Theory. World Scientific Publishing, Singapore (1995) zbMATHGoogle Scholar
  23. 23.
    Cox, E. (ed.): The Fuzzy Systems Handbook: A Practitioner’s Guide to Building, Using, and Maintaining Fuzzy Systems, 2 edn. AP Professional, Div. of Academic Press, San Diego, CA, USA (1999) Google Scholar
  24. 24.
    Crow, S., Canavan, G.: Relationship between a Wiener-Hermite expansion and an energy cascade. J. Fluid Mech. 41, 387–403 (1970) zbMATHCrossRefGoogle Scholar
  25. 25.
    Deb, M.K., Babuška, I., Oden, J.: Solution of stochastic partial differential equations using Galerkin finite element techniques. Comput. Methods Appl. Mech. Engrg. 190, 6359–6372 (2001) zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Debusschere, B., Najm, H., Matta, A., Knio, O., Ghanem, R., Le Maître, O.: Protein labeling reactions in electrochemical microchannel flow: Numerical simulation and uncertainty propagation. Phys. Fluids 15, 2238–2250 (2003) CrossRefGoogle Scholar
  27. 27.
    Debusschere, B., Najm, H., Pébay, P., Knio, O., Ghanem, R., Le Maître, O.: Numerical challenges in the use of polynomial chaos representations for stochastic processes. SIAM J. Sci. Comput. 26, 698–719 (2004) zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    DeVolder, B., Glimm, J., Grove, J., Kang, Y., Lee, Y., Pao, K., Sharp, D., Ye, K.: Uncertainty quantification for multiscale simulations. ASME J. Fluids Eng. 124, 29–41 (2002) CrossRefGoogle Scholar
  29. 29.
    Faragher, J.: Probabilistic methods for the quantification of uncertainty and error in computational fluid dynamics simulations. Tech. rep., Australian Gov., Dept. of Defense, Defense Sci. and Tech. Org., DSTO-TR-1633 (2004) Google Scholar
  30. 30.
    Frauenfelder, P., Schwab, C., Todor, R.: Finite elements for elliptic problems with stochastic coefficients. Comput. Methods Appl. Mech. Engrg. 194, 205–228 (2005) zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Frenklach, M.: Transforming data into knowledge–Process Informatics for combustion chemistry. Proc. Comb. Inst. 31, 125–140 (2007). http://primekinetics.org CrossRefGoogle Scholar
  32. 32.
    Ganapathysubramanian, B., Zabaras, N.: Sparse grid collocation schemes for stochastic natural convection problems. J. Comput. Phys. 225, 652–685 (2007) zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Gautschi, W.: On generating orthogonal polynomials. SIAM J. Sci. Stat. Comput. 3, 289–317 (1982) zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Gerstner, T., Griebel, M.: Numerical integration using sparse grids. Numerical Algorithms 18, 209–232 (1998). (also as SFB 256 preprint 553, Univ. Bonn, 1998) zbMATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Gerstner, T., Griebel, M.: Dimension adaptive tensor product quadrature. Computing 71, 2003 (2003) MathSciNetCrossRefGoogle Scholar
  36. 36.
    Ghanem, R.: Probabilistic characterization of transport in heterogeneous media. Comput. Methods Appl. Mech. Engrg. 158, 199–220 (1998) zbMATHCrossRefGoogle Scholar
  37. 37.
    Ghanem, R., Dham, S.: Stochastic finite element analysis for multiphase flow in heterogeneous porous media. Trans. Porous Media 32, 239–262 (1998) MathSciNetCrossRefGoogle Scholar
  38. 38.
    Ghanem, R., Spanos, P.: Stochastic Finite Elements: A Spectral Approach. Springer Verlag, New York (1991) zbMATHGoogle Scholar
  39. 39.
    Ghanem, R.G., Doostan, A.: On the construction and analysis of stochastic models: Characterization and propagation of the errors associated with limited data. J. Comput. Phys. 217, 63–81 (2006) zbMATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Guan, J., Bell, D.: Evidence Theory and its Applications, vol. I. Elsevier Science Publishers, Amsterdam, The Netherlands (1991) Google Scholar
  41. 41.
    Helton, J., Johnson, J., Oberkampf, W.: An exploration of alternative approaches to the representation of uncertainty in model predictions. Reliab. Eng. System Safety 85, 39–71 (2004) CrossRefGoogle Scholar
  42. 42.
    Hemsch, M.: Statistical Analysis of Computational Fluid Dynamics Solutions from the Drag Prediction Workshop. J. Aircraft 41, 95–103 (2004) CrossRefGoogle Scholar
  43. 43.
    Hoeting, J., Madigan, D., Raftery, A., Volinsky, C.: Bayesian Model Averaging: A Tutorial. Stat. Sci. 14, 382–417 (1999) zbMATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    Hosder, S., Walters, R., Balch, M.: Efficient uncertainty quantification applied to the aeroelastic analysis of a transonic wing. In: AIAA-2008-729, 46th AIAA Aerospace Sciences Meeting and Exhibit. Reno, NV (2008) Google Scholar
  45. 45.
    Hosder, S., Walters, R., Perez, R.: A non-intrusive polynomial chaos method for uncertainty propagation in cfd simulations. In: Paper AIAA 2006-0891, 44th AIAA Aerospace Sciences Meeting and Exhibit. Reno, NV (2006) Google Scholar
  46. 46.
    Hou, T.Y., Luo, W., Rozovskii, B., Zhou, H.M.: Wiener Chaos expansions and numerical solutions of randomly-forced equations of fluid mechanics. J. Comput. Phys. 216, 687–706 (2006) zbMATHMathSciNetCrossRefGoogle Scholar
  47. 47.
    Jaynes, E.: Probability Theory: The Logic of Science, G.L. Bretthorst, Ed. Cambridge University Press, Cambridge, UK (2003) CrossRefGoogle Scholar
  48. 48.
    Kaipio, J., Somersalo, E.: Statistical and Computational Inverse Problems. Springer (2005) zbMATHGoogle Scholar
  49. 49.
    Kennedy, M., O’Hagan, A.: Predicting the output from a complex computer code when fast approximations are available. Biometrika 87, 1–13 (2000) zbMATHMathSciNetCrossRefGoogle Scholar
  50. 50.
    Kennedy, M.C., O’Hagan, A.: Bayesian calibration of computer models. J. Royal Stat. Soc.: Series B 63, 425–464 (2001) zbMATHMathSciNetCrossRefGoogle Scholar
  51. 51.
    Kim, J., Moin, P.: Application of a fractional-step method to incompressible Navier-Stokes equations. J. Comput. Phys. 59, 308–323 (1985) zbMATHMathSciNetCrossRefGoogle Scholar
  52. 52.
    Knio, O., Le Maître, O.: Uncertainty propagation in cfd using polynomial chaos decomposition. Fluid Dyn. Res. 38, 616–640 (2006) zbMATHCrossRefGoogle Scholar
  53. 53.
    Ko, J., Lucor, D., Sagaut, P.: Sensitivity of two-dimensional spatially developing mixing layers with respect to uncertain inflow conditions. Phys. Fluids 20, 1–20 (2008) CrossRefGoogle Scholar
  54. 54.
    Kozine, I.: Imprecise Probabilities Relating to Prior Reliability Assessments. In: 1st Int. Symp. on Imprecise Probabilities and their Applications. Belgium (1999) Google Scholar
  55. 55.
    Kraichnan, R.: Direct-interaction approximation for a system of several interacting simple shear waves. Phys. Fluids 6, 1603 (1963) zbMATHCrossRefGoogle Scholar
  56. 56.
    Krist, S., Biedron, R., Rumsey, C.: CFL3D User’s Manual (Version 5.0). Tech. rep., NASA TM-1998-208444, NASA Langley Res. Ctr., Hampton, VA (1998) Google Scholar
  57. 57.
    Le Maître, O.: Polynomial chaos expansion of a Lagrangian model for the flow around an airfoil. Comptes Rendus Mec. 334, 693–699 (2006) Google Scholar
  58. 58.
    Le Maître, O., Ghanem, R., Knio, O., Najm, H.: Uncertainty propagation using Wiener-Haar expansions. J. Comput. Phys. 197, 28–57 (2004) zbMATHMathSciNetCrossRefGoogle Scholar
  59. 59.
    Le Maître, O., Knio, O.: A stochastic particle-mesh scheme for uncertainty propagation in vortical flows. J. Comput. Phys. 226, 645–671 (2007) zbMATHMathSciNetCrossRefGoogle Scholar
  60. 60.
    Le Maître, O., Knio, O., Najm, H., Ghanem, R.: A stochastic projection method for fluid flow I. Basic formulation. J. Comput. Phys. 173, 481–511 (2001) zbMATHMathSciNetCrossRefGoogle Scholar
  61. 61.
    Le Maître, O., Najm, H., Ghanem, R., Knio, O.: Multi-resolution analysis of Wiener-type uncertainty propagation schemes. J. Comput. Phys. 197, 502–531 (2004) zbMATHMathSciNetCrossRefGoogle Scholar
  62. 62.
    Le Maître, O., Najm, H., Pébay, P., Ghanem, R., Knio, O.: Multi-resolution-analysis scheme for uncertainty quantification in chemical systems. SIAM J. Sci. Comput. 29, 864–889 (2007) zbMATHMathSciNetCrossRefGoogle Scholar
  63. 63.
    Le Maître, O., Reagan, M., Debusschere, B., Najm, H., Ghanem, R., Knio, O.: Natural convection in a closed cavity under stochastic, non-Boussinesq conditions. SIAM J. Sci. Comput. 26, 375–394 (2004) zbMATHMathSciNetCrossRefGoogle Scholar
  64. 64.
    Le Maître, O., Reagan, M., Najm, H., Ghanem, R., Knio, O.: A stochastic projection method for fluid flow II. Random process. J. Comput. Phys. 181, 9–44 (2002) zbMATHMathSciNetCrossRefGoogle Scholar
  65. 65.
    Li, R., Ghanem, R.: Adaptive polynomial chaos expansions applied to statistics of extremes in nonlinear random vibration. Prob. Engrg. Mech. 13, 125–136 (1998) CrossRefGoogle Scholar
  66. 66.
    Lin, G., Su, C.H., Karniadakis, G.: Predicting shock dynamics in the presence of uncertainties. J. Comput. Phys. 217, 260–276 (2006) zbMATHMathSciNetCrossRefGoogle Scholar
  67. 67.
    Lucor, D., Karniadakis, G.: Noisy inflows cause a shedding-mode switching in flow past an oscillating cylinder. Phys. Rev. Lett. 92, 154,501.1–154,501.4 (2004) CrossRefGoogle Scholar
  68. 68.
    Lucor, D., Meyers, J., Sagaut, P.: Sensitivity analysis of large-eddy simulations to subgrid-scale-model parametric uncertainty using polynomial chaos. J. Fluid Mech. 585, 255–279 (2007) zbMATHMathSciNetCrossRefGoogle Scholar
  69. 69.
    Lucor, D., Triantafyllou, M.: Parametric study of a two degree-of-freedom cylinder subject to vortex-induced vibrations. J. Fluids Struct. 24, 1284–1293 (2008) CrossRefGoogle Scholar
  70. 70.
    Lucor, D., Xiu, D., Su, C., Karniadakis, G.: Predictability and uncertainty in cfd. Int. J. Num. Meth. Fluids 43, 483–505 (2003) zbMATHMathSciNetGoogle Scholar
  71. 71.
    Ma, X., Zabaras, N.: An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations. J. Comput. Phys. 228, 3084–3113 (2009) zbMATHMathSciNetCrossRefGoogle Scholar
  72. 72.
    Marzouk, Y.M., Najm, H.N.: Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems. J. Comput. Phys. 228, 1862–1902 (2009) zbMATHMathSciNetCrossRefGoogle Scholar
  73. 73.
    Marzouk, Y.M., Najm, H.N., Rahn, L.A.: Stochastic spectral methods for efficient Bayesian solution of inverse probelms. J. Comput. Phys. 224, 560–586 (2007) zbMATHMathSciNetCrossRefGoogle Scholar
  74. 74.
    Mathelin, L., Hussaini, M., Zang, T.: Stochastic Approaches to Uncertainty Quantification in CFD Simulations. Num. Algorithms 38, 209–236 (2005) zbMATHMathSciNetGoogle Scholar
  75. 75.
    Mathelin, L., Hussaini, M., Zang, T., Bataille, F.: Uncertainty propagation for turbulent, compressible flow in a quasi-1D nozzle using stochastic methods. In: AIAA-2003-4240, 16th AIAA Comput. Fluid Dyn. Conf. Orlando, FL (2003) Google Scholar
  76. 76.
    Mathelin, L., Hussaini, M., Zang, T., Bataille, F.: Uncertainty propagation for a turbulent, compressible nozzle flow using stochastic methods. AIAA J. 42, 1669–1676 (2004) CrossRefGoogle Scholar
  77. 77.
    Matthies, H.G., Keese, A.: Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput. Meth. Appl. Mech. Eng. 194, 1295–1331 (2005) zbMATHMathSciNetCrossRefGoogle Scholar
  78. 78.
    Millman, D., King, P., Beran, P.: Airfoil pitch and plunge bifurcation behavior with Fourier chaos expansions. J. Aircraft 42, 376–384 (2005) CrossRefGoogle Scholar
  79. 79.
    Millman, D., King, P., Maple, R., Beran, P., Chilton, L.: Uncertainty quantification with a B-spline stochastic projection. AIAA J. 44, 1845–1853 (2006) CrossRefGoogle Scholar
  80. 80.
    Najm, H.: Uncertainty quantification and plynomial chaos techniques in computational fluid dynamics. Ann. Rev. Fluid Mech. 41, 35–52 (2009) MathSciNetCrossRefGoogle Scholar
  81. 81.
    Najm, H., Debusschere, B., Marzouk, Y., Widmer, S., Le Maître, O.: Uncertainty quantification in chemical systems. Int. J. Num. Meth. Eng. 80, 789–814 (2009) zbMATHCrossRefGoogle Scholar
  82. 82.
    Narayanan, V., Zabaras, N.: Variational multiscale stabilized FEM formulations for transport equations: stochastic advection-diffusion and incompressible stochastic Navier-Stokes equations. J. Comput. Phys. 202, 94–133 (2005) zbMATHMathSciNetCrossRefGoogle Scholar
  83. 83.
    Nobile, F., Tempone, R., Webster, C.: A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Num. Anal. 46, 2309–2345 (2008) zbMATHMathSciNetCrossRefGoogle Scholar
  84. 84.
    Nobile, F., Tempone, R., Webster, C.: An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Num. Anal. 46, 2411–2442 (2008) zbMATHMathSciNetCrossRefGoogle Scholar
  85. 85.
    Novak, E., Ritter, K.: High dimensional integration of smooth functions over cubes. Numerische Mathematik 75, 79–97 (1996) zbMATHMathSciNetCrossRefGoogle Scholar
  86. 86.
    Novak, E., Ritter, K.: Simple cubature formulas with high polynomial exactness. Constructive Approximation 15, 499–522 (1999) zbMATHMathSciNetCrossRefGoogle Scholar
  87. 87.
    Novak, E., Ritter, K., Schmitt, R., Steinbauer, A.: On an interpolatory method for high-dimensional integration. J. Comput. Appl. Math. 112, 215–228 (1999) zbMATHMathSciNetCrossRefGoogle Scholar
  88. 88.
    Oakley, J., O’Hagan, A.: Bayesian inference for the uncertainty distribution of computer model outputs. Biometrika 89, 769–784 (2002) CrossRefGoogle Scholar
  89. 89.
    Oakley, J., O’Hagan, A.: Probabilistic sensitivity analysis of complex models: a Bayesian approach. J.R. Statist. Soc. B 66, 751–769 (2004) zbMATHMathSciNetCrossRefGoogle Scholar
  90. 90.
    Oberkampf, W., Barone, M.: Measures of agreement between computation and experiment: Validation metrics. J. Comput. Phys. 217, 5–36 (2006) zbMATHCrossRefGoogle Scholar
  91. 91.
    Oberkampf, W., Blottner, F.: Issues in computational fluid dynamics code verification and validation. AIAA J. 36, 687–695 (1998) CrossRefGoogle Scholar
  92. 92.
    Oberkampf, W., DeLand, S., Rutherford, B., Diegert, K., Alvin, K.: Error and uncertainty in modeling and simulation. Reliab. Eng. Sys. Safety 75, 333–357 (2002) CrossRefGoogle Scholar
  93. 93.
    Oberkampf, W., Helton, J.: Evidence Theory for Engineering Applications. In: E. Nikolaidis, D. Ghiocel, S. Singhal (eds.) Engineering Design Reliability Handbook, pp. 10.1–10.30. CRC Press (2005) Google Scholar
  94. 94.
    Oberkampf, W., Trucano, T.: Validation methodology in computational fluid dynamics. AIAA-2000-2549 (2000) Google Scholar
  95. 95.
    Ogura, H.: Orthogonal Functionals of the Poisson Process. IEEE Trans. Info. Theory 18, 473–481 (1972) zbMATHMathSciNetCrossRefGoogle Scholar
  96. 96.
    Orszag, S., Bissonnette, L.: Dynamical properties of truncated Wiener-Hermite expansions. Phys. Fluids 10, 2603–2613 (1967) zbMATHCrossRefGoogle Scholar
  97. 97.
    Perez, R., Walters, R.: An implicit polynomial chaos formulation for the euler equations. In: Paper AIAA 2005-1406, 43rd AIAA Aerospace Sciences Meeting and Exhibit. Reno, NV (2005) Google Scholar
  98. 98.
    Petras, K.: On the smolyak cubature error for analytic functions. Advances in Computational Mathematics 12, 71–93 (2000) zbMATHMathSciNetCrossRefGoogle Scholar
  99. 99.
    Petras, K.: Fast calculation of coefficients in the smolyak algorithm. Numerical Algorithms 26, 93–109 (2001) zbMATHMathSciNetCrossRefGoogle Scholar
  100. 100.
    Petras, K.: Smolyak cubature of given polynomial degree with few nodes for increasing dimension. Numerische Mathematik 93, 729–753 (2003) zbMATHMathSciNetCrossRefGoogle Scholar
  101. 101.
    Pettit, C., Beran, P.: Polynomial chaos expansions applied to airfoil limit cycle oscillations. 45th AIAA/ASME/ASCE/AHS/ASC Structures, Struct. Dyn. and Mat. Conf., Palm Springs, CA, 2004 AIAA-2004-1691 (2004) Google Scholar
  102. 102.
    Pettit, C., Hajj, M., Beran, P.: Gust loads with uncertainty due to imprecise gust velocity spectra. In: AIAA-2007-1965, 48th AIAA/ASME/ASCE/AHS/ASC Structures, Struct. Dyn., and Mat. Conf. Honolulu, Hawaii (2007) Google Scholar
  103. 103.
    Pettit, C.L., Beran, P.S.: Spectral and multiresolution wiener expansions of oscillatory stochastic processes. J. Sound Vibr. 294, 752–779 (2006). CrossRefGoogle Scholar
  104. 104.
    Phenix, B., Dinaro, J., Tatang, M., Tester, J., Howard, J., McRae, G.: Incorporation of parametric uncertainty into complex kinetic mechanisms: Application to hydrogen oxidation in supercritical water. Combust. Flame 112, 132–146 (1998) CrossRefGoogle Scholar
  105. 105.
    Putko, M., Taylor III, A., Newman, P., Green, L.: Approach for uncertainty propagation and robust design in CFD using sensitivity derivatives. ASME J. Fluids Eng. 124, 60–69 (2002) CrossRefGoogle Scholar
  106. 106.
    Reagan, M., Najm, H., Debusschere, B., Le Maître, O., Knio, O., Ghanem, R.: Spectral stochastic uncertainty quantification in chemical systems. Combust. Theory Model. 8, 607–632 (2004) CrossRefGoogle Scholar
  107. 107.
    Reagan, M., Najm, H., Ghanem, R., Knio, O.: Uncertainty quantification in reacting flow simulations through non-intrusive spectral projection. Combust. Flame 132, 545–555 (2003) CrossRefGoogle Scholar
  108. 108.
    Reagan, M., Najm, H., Pébay, P., Knio, O., Ghanem, R.: Quantifying uncertainty in chemical systems modeling. Int. J. Chem. Kin. 37, 368–382 (2005) CrossRefGoogle Scholar
  109. 109.
    Rebba, R., Mahadevan, S.: Model Predictive Capability Assessment Under Uncertainty. AIAA J. 44, 2376–2384 (2006) CrossRefGoogle Scholar
  110. 110.
    Roache, P.: Code verification by the method of manufactured solutions. ASME J. Fluids Eng. 124, 4–10 (2002) CrossRefGoogle Scholar
  111. 111.
    Rosenblatt, M.: Remarks on a multivariate transformation. Annals Math. Stat. 23, 470–472 (1952) zbMATHMathSciNetCrossRefGoogle Scholar
  112. 112.
    Ruscic, B., Pinzon, R., Morton, M., von Laszewski, G., Bittner, S., Nijsure, S., Amin, K., Minkoff, M., Wagner, A.: Introduction to active thermochemical tables: Several “key” enthalpies of formation revisited. J. Phys. Chem. A 108, 9979–9997 (2004) CrossRefGoogle Scholar
  113. 113.
    Sagaut, P., Deck, S.: Large eddy simulation for aerodynamics: status and perspectives. Phil. Trans. Royal Soc. A 367, 2849–2860 (2009) zbMATHCrossRefGoogle Scholar
  114. 114.
    Sagaut, P., Meyers, J., Lucor, D.: Uncertainty modeling, error charts and improvement of subgrid models. In: S.-H. Peng and W. Haase (Eds.): Adv. in Hybrid RANS-LES Modelling, NNFM 97. Springer-Verlag, Berlin (2008) CrossRefGoogle Scholar
  115. 115.
    Schoutens, W.: Stochastic Processes and Orthogonal Polynomials. Springer (2000) zbMATHGoogle Scholar
  116. 116.
    Schwab, C., Todor, R.: Sparse finite elements for stochastic elliptic problems. Numer. Math. 95, 707–734 (2003) zbMATHMathSciNetCrossRefGoogle Scholar
  117. 117.
    Schwab, C., Todor, R.: Sparse finite elements for stochastic elliptic problems – higher order moments. Computing 71, 43–63 (2003) zbMATHMathSciNetCrossRefGoogle Scholar
  118. 118.
    Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton, NJ (1976) zbMATHGoogle Scholar
  119. 119.
    Smolyak, S.A.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Soviet Math. Dokl. 4, 240–243 (1963) Google Scholar
  120. 120.
    Soize, C., Ghanem, R.: Physical systems with random uncertainties: Chaos representations with arbitrary probability measure. SIAM J. Sci. Comput. 26, 395–410 (2004) zbMATHMathSciNetCrossRefGoogle Scholar
  121. 121.
    Stern, F., Wilson, R., Shao, J.: Quantitative V&V of CFD simulations and certification of CFD codes. Int. J. Numer. Meth. Fluids 50, 1335–1355 (2006) zbMATHCrossRefGoogle Scholar
  122. 122.
    Tartakovsky, D., Xiu, D.: Stochastic analysis of transport in tubes with rough walls. J. Comput. Phys. 217, 248–259 (2006) zbMATHMathSciNetCrossRefGoogle Scholar
  123. 123.
    Todor, R., Schwab, C.: Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. IMA Journal of Numerical Analysis 27, 232–261 (2007) zbMATHMathSciNetCrossRefGoogle Scholar
  124. 124.
    Unwin, S.: A fuzzy set theoretic foundation for vagueness in uncertainty analysis. Risk Analysis 6, 27–34 (1986) CrossRefGoogle Scholar
  125. 125.
    Walters, R., Huyse, L.: Uncertainty analysis for fluid mechanics with applications. Tech. rep., ICASE Report No. 2002-1; NASA/CR-2002-211449 (2002) Google Scholar
  126. 126.
    Wan, X., Karniadakis, G.: Long-term behavior of polynomial chaos in stochastic flow simulations. Comput. Meth. Appl. Mech. Eng. 195, 5582–5596 (2006) zbMATHMathSciNetCrossRefGoogle Scholar
  127. 127.
    Wan, X., Karniadakis, G.: Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures. SIAM J. Sci. Comput. 28, 901–928 (2006) zbMATHMathSciNetCrossRefGoogle Scholar
  128. 128.
    Wan, X., Karniadakis, G.E.: An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. J. Comput. Phys. 209, 617–642 (2005) zbMATHMathSciNetCrossRefGoogle Scholar
  129. 129.
    Warnatz, J.: Resolution of gas phase and surface combustion chemistry into elementary reactions. Proc. Combust. Inst. 24, 553–579 (1992) Google Scholar
  130. 130.
    Wiener, N.: The homogeneous chaos. Am. J. Math. 60, 897–936 (1938) MathSciNetCrossRefGoogle Scholar
  131. 131.
    Wiener, N.: The use of statistical theory in the study of turbulence. In: Fifth International Congress for Applied Mechanics. Wiley, New York, NY (1939) Google Scholar
  132. 132.
    Wiener, N.: Nonlinear Problems in Random Theory. MIT Press, Cambridge, MA (1958) zbMATHGoogle Scholar
  133. 133.
    Witteveen, J., Bijl, H.: An alternative unsteady adaptive stochastic finite elements formulation based on interpolation at constant phase. Comput. Meth. Appl. Mech. Eng. 198, 578–591 (2008) CrossRefGoogle Scholar
  134. 134.
    Witteveen, J., Bijl, H.: An unsteady adaptive stochastic finite elements formulation for rigid-body fluid-structure interaction. Comp. Struct. 86, 2123–2140 (2008) CrossRefGoogle Scholar
  135. 135.
    Witteveen, J., Bijl, H.: Effect of randomness on multi-frequency aeroelastic responses resolved by Unsteady Adaptive Stochastic Finite Elements. J. Comput. Phys. 228, 7025–7045 (2009) zbMATHCrossRefGoogle Scholar
  136. 136.
    Witteveen, J., Bijl, H.: Higher period stochastic bifurcation of nonlinear airfoil fluid-structure interaction. Math. Prob. Eng. 394387, 1–26 (2009) Google Scholar
  137. 137.
    Witteveen, J., Loeven, A., Bijl, H.: An adaptive stochastic finite elements approach based on Newton-Cotes quadrature in simplex elements. Comput. Fluids 38, 1270–1288 (2009) MathSciNetCrossRefGoogle Scholar
  138. 138.
    Witteveen, J., Loeven, A., Sarkar, S., Bijl, H.: Probabilistic collocation for period-1 limit cycle oscillations. J. of Sound Vib. 311, 421–439 (2008) CrossRefGoogle Scholar
  139. 139.
    Witteveen, J., Sarkar, S., Bijl, H.: Modeling physical uncertainties in dynamic stall induced fluid-structure interaction of turbine blades using arbitrary polynomial chaos. Comput. Struct. 85, 866–878 (2007) CrossRefGoogle Scholar
  140. 140.
    Xiu, D., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27, 1118–1139 (2005) zbMATHMathSciNetCrossRefGoogle Scholar
  141. 141.
    Xiu, D., Karniadakis, G.: The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24, 619–644 (2002) zbMATHMathSciNetCrossRefGoogle Scholar
  142. 142.
    Xiu, D., Karniadakis, G.: Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187, 137–167 (2003) zbMATHMathSciNetCrossRefGoogle Scholar
  143. 143.
    Xiu, D., Lucor, D., Su, C.H., Karniadakis, G.: Stochastic modeling of flow-structure interactions using generalized polynomial chaos. ASME J. Fluids Eng. 124, 51–59 (2002) CrossRefGoogle Scholar
  144. 144.
    Xiu, D., Tartakovsky, D.: Numerical methods for differential equations in random domains. SIAM J. Sci. Comput. 28, 1167–1185 (2006) zbMATHMathSciNetCrossRefGoogle Scholar
  145. 145.
    Zang, T., Hemsch, M., Hilburger, M., Kenny, S., Luckring, J., Maghani, P., Padula, S., Stroud, W.: Needs and opportunities for uncertainty-based multidisciplinary design methods for aerospace vehicles. Tech. rep., NASA/TM-2002-211462 (2002) Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Sandia National LaboratoriesLivermoreUSA

Personalised recommendations