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International Journal for Uncertainty Quantification
IF: 0.967 5-Year IF: 1.301 SJR: 0.531 SNIP: 0.8 CiteScore™: 1.52

ISSN Print: 2152-5080
ISSN Online: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2012003594
pages 371-395


S. Maysam Mousaviraad
IIHR-Hydroscience and Engineering, The University of Iowa, Iowa City, Iowa 52242, USA
Wei He
Visiting scholar from NAOCE, Shanghai Jiao Tong University, Shanghai, China
Matteo Diez
Visiting scholar from CNR-INSEAN, Via di Vallerano 139, 00128 Rome, Italy
Frederick Stern
IIHR-Hydroscience and Engineering, The University of Iowa, Iowa City, Iowa 52242, USA


A framework is described for convergence and validation of nonintrusive uncertainty quantification (UQ) methods; the relationship between deterministic verification and validation (V&V) and stochastic UQ is studied, and an example is provided for a unit problem. Convergence procedures are developed for Monte Carlo (MC) without and with metamodels, showing that in addition to the usual user-defined acceptable confidence intervals, convergence studies with systematic refinement ratio are required. A UQ validation procedure is developed using the benchmark UQ results and defining the comparison error and its uncertainty to evaluate validation. A stochastic influence factor is defined to evaluate the effects of input variability on the performance expectation and four possibilities are identified in making design decisions. The unit problem studies a two-dimensional airfoil with variable Re and normal distribution using high-fidelity Reynolds-averaged Navier-Stokes (RANS) simulations. Deterministic V&V studies achieve monotonic grid convergence and validation at the validation uncertainty interval of 2.2% D, averaged between lift and drag, with an average error of 0.25% D. For MC with Latin hypercube sampling the converged results are obtained with 400 computational fluid dynamics (CFD) simulations and are used as validation benchmark in the absence of experimental UQ. The stochastic influence factor is small such that the output expected value is not distinguishable from the deterministic solution. The output uncertainty is one order of magnitude smaller for lift than drag, implying that lift is only weakly dependent on Re. Several metamodels are used with MC, reducing the number of CFD simulations to a minimum of 4. The results are converged and validated at the average intervals of 0.1% for expected value (EV) and 10.7% for standard deviation (SD). The Gauss quadrature and the polynomial chaos (PC) method are validated using 8 and 7 CFD simulations, respectively, at the average intervals of 0.08% for EV and 7.5% for SD. The error values are smallest for the metamodels, followed by the PC method and then the Gauss quadratures.