Quantifying Uncertainty
Lucor, D., Su, C.-H., and Karniadakis, G.E.. "Generalized Polynomial Chaos and Random Oscillators" 60
(3).
2004.
pp. 571--596.
We present a new approach to obtain solutions for general random oscillators using a broad class of polynomial chaos expansions, which are more efficient than the classical Wiener Hermite expansions. The approach is general but here we present results for linear oscillators only with random forcing or random coefficients. In this context, we are able to obtain relatively sharp error estimates in the representation of the stochastic input as well as the solution. We have also performed computational comparisons with Monte Carlo simulations which show that the new approach can be orders of magnitude faster. especially for compact distributions. Copyright (C) 2004 John Wiley Sons, Ltd.
Lucor, D. and Karniadakis, G.E.. "Adaptive Generalized Polynomial Chaos for Nonlinear Random Oscillators" SIAM J. Sci. Comput.. 26
(2).
2005.
pp. 720--735.
The solution of nonlinear random oscillators subject to stochastic forcing is investigated numerically. In particular, solutions to the random Duffing oscillator with random Gaussian and non-Gaussian excitations are obtained by means of the generalized polynomial chaos (GPC). Adaptive procedures are proposed to lower the increased computational cost of the GPC approach in large-dimensional spaces. Adaptive schemes combined with the use of an enriched representation of the system improve the accuracy of the GPC approach by reordering the random modes according to their magnification by the system.
Lucor, D., Xiu, D., Su, C.-H., and Karniadakis, G.E.. "Predictability and uncertainty in CFD" International Journal for Numerical Methods in Fluids. 43
(5).
OCT 20 2003.
pp. 483--505.
CFD has reached some degree of maturity today, but the new question is how to construct simulation error bars that reflect uncertainties of the physical problem, in addition to the usual numerical inaccuracies. We present a fast Polynomial Chaos algorithm to model the input uncertainty and its propagation in incompressible flow simulations. The stochastic input is represented spectrally by Wiener-Hermite functionals, and the governing equations are formulated by employing Galerkin projections. The resulted system is deterministic, and therefore existing solvers can be used in this new context of stochastic simulations. The algorithm is applied to a second-order oscillator and to a flow-structure interaction problems. Open issues and extensions to general random distributions are presented. Copyright (C) 2003 John Wiley Sons, Ltd.
Lucor, D. and Karniadakis, G.E.. "Predictability and uncertainty in flow-structure interactions" European Journal of Mechanics B-Fluids. 23
(1).
JAN-FEB 2004.
pp. 41--49.
Direct numerical simulation advances in the field of flow-structure interactions are reviewed both from a deterministic and stochastic point of view. First, results of complex wake flows resulting from vibrating cylindrical bluff bodies in linear and exponential sheared flows are presented. On the structural side, non-linear modeling of cable structures with variable tension is derived and applied to the problem of a catenary riser of complex shape. Finally, a direct approach using Polynomial Chaos to modeling uncertainty associated with flow-structure interaction is also described. The method is applied to the two-dimensional flow-structure interaction case of an elastically mounted cylinder with random structural parameters subject to vortex-induced vibrations. (C) 2003 Elsevier SAS. All rights reserved.
Lucor, D., Meyers, J., and Sagaut, P.. "Sensitivity analysis of large-eddy simulations to subgrid-scale-model parametric uncertainty using polynomial chaos" Journal of Fluid Mechanics.
vol. 585.
AUG 25 2007.
pp. 255--279.
We address the sensitivity of large-eddy simulations (LES) to parametric uncertainty in the subgrid-scale model. More specifically, we investigate the sensitivity of the LES statistical moments of decaying homogeneous isotropic turbulence to the uncertainty in the Smagorinsky model free parameter C-s (i.e. the Smagorinsky constant). Our sensitivity methodology relies on the non-intrusive approach of the generalized Polynomial Chaos (gPC) method; the gPC is a spectral non-statistical numerical method well-suited to representing random processes not restricted to Gaussian fields. The analysis is carried out at Re-lambda=100 and for different grid resolutions and C-s distributions. Numerical predictions are also compared to direct numerical simulation evidence. We have shown that the different turbulent scales of the LES solution respond differently to the variability in C-s. In particular, the study of the relative turbulent kinetic energy distributions for different C-s distributions indicates that small scales are mainly affected by changes in the subgrid-model parametric uncertainty.