比较了计算流体动力学不确定度分析中的验证与确认方法和不确定度量化方法之间的区别.介绍了一种最新的不确定度量化方法——非侵入式混沌多项式法,并应用该方法和传统的蒙特卡洛法分别对二维随机阻曳流进行不确定度量化,其中蒙特卡洛法选用了随机采样法和拉丁超立方采样法两种方法.研究了当进出口压力为服从均匀分布的随机变量时所引起的流动的不确定性.研究结果表明,应用蒙特卡洛法进行不确定度量化时,随机采样法和拉丁超立方采样法效果差别不大;混沌多项式方法相较于蒙特卡洛法可以更高效地模拟不确定性在流场中的传播.
In this paper, verification & validation and uncertainty quantification in uncertainty analysis for computational fluid dynamics (CFD) simulation are compared. A state-of-the-art method for uncertainty quantification problems, i.e., the non-intrusive polynomial chaos (NIPC) method, is introduced and applied to quantifying the uncertainty of two-dimensional stochastic drag flow, together with the Monte-Carlo (MC) method. For the MC method, the random sampling (RS) method and the Latin hypercube sampling (LHS) method are adopted. The uncertainty of the stochastic drag flow induced by the inlet and outlet pressure boundaries is studied, with the boundaries treated as stochastic variables with uniform distribution. It is shown that there is no big difference between LHS and RS, and the NIPC method can simulate the uncertainty propagation better.
[1]MARCELLO A, SPINELLA S, RINAUDO S. Stochastic response surface method and tolerance analysis in microelectronics[J]. COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, 2003, 22(2): 314-327.
[2]WIENER N. The homogeneous chaos[J]. American Journal of Mathematics, 1938, 60(4): 897-936.
[3]GHANEM R G, SPANOS P D. Stochastic finite elements: A spectral approach[M]. New York: Springer-Verlag, 1991: 113-185.
[4]XIU D, KARNIADAKIS G E. The Wiener-Askey polynomial chaos for stochastic differential equations[J]. SIAM Journal on Scientific Computing, 2002, 24(2): 619-644.
[5]XIU D, KARNIADAKIS G E. Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos[J]. Computer Methods in Applied Mechanics and Engineering, 2002, 191(43): 4927-4948.
[6]XIU D, KARNIADAKIS G E. Modeling uncertainty in flow simulations via generalized polynomial chaos[J]. Journal of Computational Physics, 2003, 187(1): 137-167.
[7]LOEVEN G J A, WITTEVEEN J A S, BIJL H. Probabilistic collocation: An efficient non-intrusive approach for arbitrarily distributed parametric uncertainties[C]∥Proceedings of 45th Aerospace Sciences Meeting and Exhibit. Reno, Nevada, USA: AIAA, 2007: No.2007-317.
[8]LOEVEN G J A. Efficient uncertainty quantification in computational fluid dynamics[D]. Delft: Delft University of Technology, 2010.
[9]MARGHERI L, SAGAUT P. A hybrid anchored-ANOVA-POD/Kriging method for uncertainty quantification in unsteady high-fidelity CFD simulations[J]. Journal of Computational Physics, 2016, 324: 137-173.
[10]MATHELIN L, HUSSAINI M Y. Uncertainty quantification in CFD simulations: A stochastic spectral approach[M]. ARMFIELD S W, MORGAN P, SRINIVAS K. Computational Fluid Dynamics 2002. Berlin: Springer, 2003: 65-70.
[11]MATHELIN L, HUSSAINI M Y, ZANG T A. Stochastic approaches to uncertainty quantification in CFD simulations[J]. Numerical Algorithms, 2005, 38(1-3): 209-236.
[12]LACOR C, DINESCU C, HIRSCH C, et al. Implementation of intrusive polynomial chaos in CFD codes and application to 3D Navier-Stokes[M]. BIJL H, LUCOR D, MISHRA S, et al. Uncertainty Quantification in Computational Fluid Dynamics. Heidelberg: Springer, 2013: 193-223.
[13]HOSDER S, WALTERS R, PEREZ R. A non-intrusive polynomial chaos method for uncertainty propagation in CFD simulations[C]∥Proceedings of 44th Aerospace Sciences Meeting and Exhibit. Reno, Nevada, USA: AIAA, 2006: No.2006-891.
[14]HOSDER S, WALTERS R, BALCH M. Efficient sampling for non-intrusive polynomial chaos applications with multiple uncertain input variables[C]∥Proceedings of 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. Honolulu, Hawaii, USA: AIAA, 2007: No.2007-1939.
[15]HOSDER S, WALTERS R. Non-intrusive polynomial chaos methods for uncertainty quantification in fluid dynamics[C]∥Proceedings of 48th Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition. Honolulu, Hawaii, USA: AIAA, 2010: No.2010-129.
[16]SALEHI S, RAISEE M, CERVANTES M J, et al. Efficient uncertainty quantification of stochastic CFD problems using sparse polynomial chaos and compressed sensing[J]. Computers & Fluids, 2017, 154: 296-321.
[17]SALEHI S, RAISEE M, CERVANTES M J, et al. The effects of inflow uncertainties on the characteristics of developing turbulent flow in rectangular pipe and asymmetric diffuser[J]. Journal of Fluids Engineering, 2017, 139(4): 041402.
[18]WANG X D, KANG S. Application of polynomial chaos on numerical simulation of stochastic cavity flow[J]. Science China Technological Sciences, 2010, 53(10): 2853-2861.
[19]HE W, DIEZ M, CAMPANA E F, et al. A one-dimensional polynomial chaos method in CFD-based uncertainty quantification for ship hydrodynamic performance[J]. Journal of Hydrodynamics, 2013, 25(5): 655-662.
[20]DIEZ M, HE W, CAMPANA E F, et al. Uncertainty quantification of Delft catamaran resistance, sinkage and trim for variable Froude number and geometry using metamodels, quadrature and Karhunen-Loève expansion[J]. Journal of Marine Science and Technology, 2014, 19(2): 143-169.
[21]STERN F, VOLPI S, GAUL N J, et al. Development and assessment of uncertainty quantification methods for ship hydrodynamics[C]∥Proceedings of 55th Aerospace Sciences Meeting. Grapevine, Texas, USA: AIAA, 2017: No.2017-1654.
[22]毕超. 计算流体力学有限元方法及其编程详解[M]. 北京: 机械工业出版社, 2013: 37-72.
BI Chao. Finite element method in computational fluid dynamics and its detailed programming[M]. Beijing: China Machine Press, 2013: 37-72.
[23]CELIK I B, GHIA U, ROACHE P J. Procedure for estimation and reporting of uncertainty due to discretization in CFD applications[J]. Journal of Fluids Engineering, 2008, 130(7): 0780011-0780014.