Turbulent Combustion Modeling pp 381-407 | Cite as
Uncertainty Quantification in Fluid Flow
Chapter
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.
References
- 1.Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. Dover, New York (1970) Google Scholar
- 2.Agarwal, N., Aluru, N.: A domain adaptive stochastic collocation approach for analysis of MEMS under uncertainties. J. Comput. Phys. 228, 7662–7688 (2009) MATHMathSciNetCrossRefGoogle Scholar
- 3.Askey, R., Wilson, J.: Some basic hypergeometric polynomials that generalize jacobi polynomials. Memoirs Amer. Math. Soc. 319, 1–55 (1985) MathSciNetGoogle Scholar
- 4.Asokan, B., Zabaras, N.: Using stochastic analysis to capture unstable equilibrium in natural convection. J. Comput. Phys. 208, 134–153 (2005) MATHMathSciNetCrossRefGoogle Scholar
- 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) MATHMathSciNetCrossRefGoogle Scholar
- 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) MATHMathSciNetCrossRefGoogle Scholar
- 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) MATHCrossRefGoogle Scholar
- 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.Barthelmann, V., Novak, E., Ritter, K.: High-dimensional polynomial interpolation on sparse grids. Adv. Compu. Math. 12, 273–288 (2000) MATHMathSciNetCrossRefGoogle Scholar
- 10.Beran, P.S., Pettit, C.L., Millman, D.R.: Uncertainty quantification of limit-cycle oscillations. J. Comput. Phys. 217, 217–247 (2006). MATHCrossRefGoogle Scholar
- 11.Bieri, M., Schwab, C.: Sparse high order FEM for elliptic sPDEs. CMAME 198, 1149–1170 (2009) MATHMathSciNetGoogle Scholar
- 12.Boyd, J.: The rate of convergence of Hermite function series. Math. Comput. 35, 1309–1316 (1980) MATHCrossRefGoogle Scholar
- 13.Cacuci, D. (ed.): Sensitivity and Uncertainty Analysis Theory, vol. 1. Chapman & Hall/CRC, Boca Raton, FL (2003) Google Scholar
- 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.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.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.Canavan, G.: Some properties of a Lagrangian Wiener-Hermite expansion. J. Fluid Mech. 41, 405–412 (1970) CrossRefGoogle Scholar
- 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) MATHCrossRefGoogle Scholar
- 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) MATHMathSciNetCrossRefGoogle Scholar
- 20.Chorin, A.: A numerical method for solving incompressible viscous flow problems. J. Comput. Phys. 2, 12–26 (1967) MATHCrossRefGoogle Scholar
- 21.Chorin, A.: Gaussian fields and random flow. J. Fluid Mech. 63, 21–32 (1974) MATHMathSciNetCrossRefGoogle Scholar
- 22.de Cooman, G., Ruan, D., Kerre, E. (eds.): Foundations and Applications of Possibility Theory. World Scientific Publishing, Singapore (1995) MATHGoogle Scholar
- 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.Crow, S., Canavan, G.: Relationship between a Wiener-Hermite expansion and an energy cascade. J. Fluid Mech. 41, 387–403 (1970) MATHCrossRefGoogle Scholar
- 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) MATHMathSciNetCrossRefGoogle Scholar
- 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.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) MATHMathSciNetCrossRefGoogle Scholar
- 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.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.Frauenfelder, P., Schwab, C., Todor, R.: Finite elements for elliptic problems with stochastic coefficients. Comput. Methods Appl. Mech. Engrg. 194, 205–228 (2005) MATHMathSciNetCrossRefGoogle Scholar
- 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.Ganapathysubramanian, B., Zabaras, N.: Sparse grid collocation schemes for stochastic natural convection problems. J. Comput. Phys. 225, 652–685 (2007) MATHMathSciNetCrossRefGoogle Scholar
- 33.Gautschi, W.: On generating orthogonal polynomials. SIAM J. Sci. Stat. Comput. 3, 289–317 (1982) MATHMathSciNetCrossRefGoogle Scholar
- 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) MATHMathSciNetCrossRefGoogle Scholar
- 35.Gerstner, T., Griebel, M.: Dimension adaptive tensor product quadrature. Computing 71, 2003 (2003) MathSciNetCrossRefGoogle Scholar
- 36.Ghanem, R.: Probabilistic characterization of transport in heterogeneous media. Comput. Methods Appl. Mech. Engrg. 158, 199–220 (1998) MATHCrossRefGoogle Scholar
- 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.Ghanem, R., Spanos, P.: Stochastic Finite Elements: A Spectral Approach. Springer Verlag, New York (1991) MATHGoogle Scholar
- 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) MATHMathSciNetCrossRefGoogle Scholar
- 40.Guan, J., Bell, D.: Evidence Theory and its Applications, vol. I. Elsevier Science Publishers, Amsterdam, The Netherlands (1991) Google Scholar
- 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.Hemsch, M.: Statistical Analysis of Computational Fluid Dynamics Solutions from the Drag Prediction Workshop. J. Aircraft 41, 95–103 (2004) CrossRefGoogle Scholar
- 43.Hoeting, J., Madigan, D., Raftery, A., Volinsky, C.: Bayesian Model Averaging: A Tutorial. Stat. Sci. 14, 382–417 (1999) MATHMathSciNetCrossRefGoogle Scholar
- 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.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.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) MATHMathSciNetCrossRefGoogle Scholar
- 47.Jaynes, E.: Probability Theory: The Logic of Science, G.L. Bretthorst, Ed. Cambridge University Press, Cambridge, UK (2003) CrossRefGoogle Scholar
- 48.Kaipio, J., Somersalo, E.: Statistical and Computational Inverse Problems. Springer (2005) MATHGoogle Scholar
- 49.Kennedy, M., O’Hagan, A.: Predicting the output from a complex computer code when fast approximations are available. Biometrika 87, 1–13 (2000) MATHMathSciNetCrossRefGoogle Scholar
- 50.Kennedy, M.C., O’Hagan, A.: Bayesian calibration of computer models. J. Royal Stat. Soc.: Series B 63, 425–464 (2001) MATHMathSciNetCrossRefGoogle Scholar
- 51.Kim, J., Moin, P.: Application of a fractional-step method to incompressible Navier-Stokes equations. J. Comput. Phys. 59, 308–323 (1985) MATHMathSciNetCrossRefGoogle Scholar
- 52.Knio, O., Le Maître, O.: Uncertainty propagation in cfd using polynomial chaos decomposition. Fluid Dyn. Res. 38, 616–640 (2006) MATHCrossRefGoogle Scholar
- 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.Kozine, I.: Imprecise Probabilities Relating to Prior Reliability Assessments. In: 1st Int. Symp. on Imprecise Probabilities and their Applications. Belgium (1999) Google Scholar
- 55.Kraichnan, R.: Direct-interaction approximation for a system of several interacting simple shear waves. Phys. Fluids 6, 1603 (1963) MATHCrossRefGoogle Scholar
- 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.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.Le Maître, O., Ghanem, R., Knio, O., Najm, H.: Uncertainty propagation using Wiener-Haar expansions. J. Comput. Phys. 197, 28–57 (2004) MATHMathSciNetCrossRefGoogle Scholar
- 59.Le Maître, O., Knio, O.: A stochastic particle-mesh scheme for uncertainty propagation in vortical flows. J. Comput. Phys. 226, 645–671 (2007) MATHMathSciNetCrossRefGoogle Scholar
- 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) MATHMathSciNetCrossRefGoogle Scholar
- 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) MATHMathSciNetCrossRefGoogle Scholar
- 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) MATHMathSciNetCrossRefGoogle Scholar
- 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) MATHMathSciNetCrossRefGoogle Scholar
- 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) MATHMathSciNetCrossRefGoogle Scholar
- 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.Lin, G., Su, C.H., Karniadakis, G.: Predicting shock dynamics in the presence of uncertainties. J. Comput. Phys. 217, 260–276 (2006) MATHMathSciNetCrossRefGoogle Scholar
- 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.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) MATHMathSciNetCrossRefGoogle Scholar
- 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.Lucor, D., Xiu, D., Su, C., Karniadakis, G.: Predictability and uncertainty in cfd. Int. J. Num. Meth. Fluids 43, 483–505 (2003) MATHMathSciNetGoogle Scholar
- 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) MATHMathSciNetCrossRefGoogle Scholar
- 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) MATHMathSciNetCrossRefGoogle Scholar
- 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) MATHMathSciNetCrossRefGoogle Scholar
- 74.Mathelin, L., Hussaini, M., Zang, T.: Stochastic Approaches to Uncertainty Quantification in CFD Simulations. Num. Algorithms 38, 209–236 (2005) MATHMathSciNetGoogle Scholar
- 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.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.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) MATHMathSciNetCrossRefGoogle Scholar
- 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.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.Najm, H.: Uncertainty quantification and plynomial chaos techniques in computational fluid dynamics. Ann. Rev. Fluid Mech. 41, 35–52 (2009) MathSciNetCrossRefGoogle Scholar
- 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) MATHCrossRefGoogle Scholar
- 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) MATHMathSciNetCrossRefGoogle Scholar
- 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) MATHMathSciNetCrossRefGoogle Scholar
- 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) MATHMathSciNetCrossRefGoogle Scholar
- 85.Novak, E., Ritter, K.: High dimensional integration of smooth functions over cubes. Numerische Mathematik 75, 79–97 (1996) MATHMathSciNetCrossRefGoogle Scholar
- 86.Novak, E., Ritter, K.: Simple cubature formulas with high polynomial exactness. Constructive Approximation 15, 499–522 (1999) MATHMathSciNetCrossRefGoogle Scholar
- 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) MATHMathSciNetCrossRefGoogle Scholar
- 88.Oakley, J., O’Hagan, A.: Bayesian inference for the uncertainty distribution of computer model outputs. Biometrika 89, 769–784 (2002) CrossRefGoogle Scholar
- 89.Oakley, J., O’Hagan, A.: Probabilistic sensitivity analysis of complex models: a Bayesian approach. J.R. Statist. Soc. B 66, 751–769 (2004) MATHMathSciNetCrossRefGoogle Scholar
- 90.Oberkampf, W., Barone, M.: Measures of agreement between computation and experiment: Validation metrics. J. Comput. Phys. 217, 5–36 (2006) MATHCrossRefGoogle Scholar
- 91.Oberkampf, W., Blottner, F.: Issues in computational fluid dynamics code verification and validation. AIAA J. 36, 687–695 (1998) CrossRefGoogle Scholar
- 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.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.Oberkampf, W., Trucano, T.: Validation methodology in computational fluid dynamics. AIAA-2000-2549 (2000) Google Scholar
- 95.Ogura, H.: Orthogonal Functionals of the Poisson Process. IEEE Trans. Info. Theory 18, 473–481 (1972) MATHMathSciNetCrossRefGoogle Scholar
- 96.Orszag, S., Bissonnette, L.: Dynamical properties of truncated Wiener-Hermite expansions. Phys. Fluids 10, 2603–2613 (1967) MATHCrossRefGoogle Scholar
- 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.Petras, K.: On the smolyak cubature error for analytic functions. Advances in Computational Mathematics 12, 71–93 (2000) MATHMathSciNetCrossRefGoogle Scholar
- 99.Petras, K.: Fast calculation of coefficients in the smolyak algorithm. Numerical Algorithms 26, 93–109 (2001) MATHMathSciNetCrossRefGoogle Scholar
- 100.Petras, K.: Smolyak cubature of given polynomial degree with few nodes for increasing dimension. Numerische Mathematik 93, 729–753 (2003) MATHMathSciNetCrossRefGoogle Scholar
- 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.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.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.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.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.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.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.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.Rebba, R., Mahadevan, S.: Model Predictive Capability Assessment Under Uncertainty. AIAA J. 44, 2376–2384 (2006) CrossRefGoogle Scholar
- 110.Roache, P.: Code verification by the method of manufactured solutions. ASME J. Fluids Eng. 124, 4–10 (2002) CrossRefGoogle Scholar
- 111.Rosenblatt, M.: Remarks on a multivariate transformation. Annals Math. Stat. 23, 470–472 (1952) MATHMathSciNetCrossRefGoogle Scholar
- 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.Sagaut, P., Deck, S.: Large eddy simulation for aerodynamics: status and perspectives. Phil. Trans. Royal Soc. A 367, 2849–2860 (2009) MATHCrossRefGoogle Scholar
- 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.Schoutens, W.: Stochastic Processes and Orthogonal Polynomials. Springer (2000) MATHGoogle Scholar
- 116.Schwab, C., Todor, R.: Sparse finite elements for stochastic elliptic problems. Numer. Math. 95, 707–734 (2003) MATHMathSciNetCrossRefGoogle Scholar
- 117.Schwab, C., Todor, R.: Sparse finite elements for stochastic elliptic problems – higher order moments. Computing 71, 43–63 (2003) MATHMathSciNetCrossRefGoogle Scholar
- 118.Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton, NJ (1976) MATHGoogle Scholar
- 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.Soize, C., Ghanem, R.: Physical systems with random uncertainties: Chaos representations with arbitrary probability measure. SIAM J. Sci. Comput. 26, 395–410 (2004) MATHMathSciNetCrossRefGoogle Scholar
- 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) MATHCrossRefGoogle Scholar
- 122.Tartakovsky, D., Xiu, D.: Stochastic analysis of transport in tubes with rough walls. J. Comput. Phys. 217, 248–259 (2006) MATHMathSciNetCrossRefGoogle Scholar
- 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) MATHMathSciNetCrossRefGoogle Scholar
- 124.Unwin, S.: A fuzzy set theoretic foundation for vagueness in uncertainty analysis. Risk Analysis 6, 27–34 (1986) CrossRefGoogle Scholar
- 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.Wan, X., Karniadakis, G.: Long-term behavior of polynomial chaos in stochastic flow simulations. Comput. Meth. Appl. Mech. Eng. 195, 5582–5596 (2006) MATHMathSciNetCrossRefGoogle Scholar
- 127.Wan, X., Karniadakis, G.: Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures. SIAM J. Sci. Comput. 28, 901–928 (2006) MATHMathSciNetCrossRefGoogle Scholar
- 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) MATHMathSciNetCrossRefGoogle Scholar
- 129.Warnatz, J.: Resolution of gas phase and surface combustion chemistry into elementary reactions. Proc. Combust. Inst. 24, 553–579 (1992) Google Scholar
- 130.Wiener, N.: The homogeneous chaos. Am. J. Math. 60, 897–936 (1938) MathSciNetCrossRefGoogle Scholar
- 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.Wiener, N.: Nonlinear Problems in Random Theory. MIT Press, Cambridge, MA (1958) MATHGoogle Scholar
- 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.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.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) MATHCrossRefGoogle Scholar
- 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.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.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.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.Xiu, D., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27, 1118–1139 (2005) MATHMathSciNetCrossRefGoogle Scholar
- 141.Xiu, D., Karniadakis, G.: The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24, 619–644 (2002) MATHMathSciNetCrossRefGoogle Scholar
- 142.Xiu, D., Karniadakis, G.: Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187, 137–167 (2003) MATHMathSciNetCrossRefGoogle Scholar
- 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.Xiu, D., Tartakovsky, D.: Numerical methods for differential equations in random domains. SIAM J. Sci. Comput. 28, 1167–1185 (2006) MATHMathSciNetCrossRefGoogle Scholar
- 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