Funding NSF DMS-0510799

A Multi-element Generalized Polynomial Chaos Method for Modeling Uncertainty in Flow Simulations

With the computational fluid dynamics field reaching now some degree of maturity, we pose the more general question in this proposal of how to model and propagate uncertainty in flow systems, and how to formulate efficient and robust algorithms for the corresponding stochastic incompressible and compressible Navier-Stokes equations. To this end, we propose a multi-element generalized Polynomial Chaos (ME-gPC) method that can overcome the difficulties associated with long-term integration, stochastic bifurcations and strong nonlinearities typically encountered in unsteady fluid dynamics problems. To evaluate this approach we consider three representative applications involving biological flows, transitional flows, and shock dynamics that reflect uncertainty in constitutive laws, sensitivity to the free stream random fluctuations, and random roughness.

Specifically, we will extend the pioneering ideas of Norbert Wiener on polynomial chaos and our previous work on Wiener-Askey expansions in order to handle stochastic inputs with arbitrary probability distributions, either continuous or discrete. This fundamental development, in turn, will allow us to introduce a subdomain decomposition of the random space, thus resolving adaptively large levels of localized random fluctuations and even stochastic discontinuities. In fact, discretizing the random space with N elements within which we employ generalized polynomial chaos (gPC) expansion of order p results in a very fast convergence of the form N^(-2p+2). This possibility for dual path of convergence, either increasing N or p, is similar to the h-p convergence of spectral element methods that the PI has been developing in the last twenty years for deterministic problems. Moreover, the ability to handle arbitrary PDFs will open the possibility of "racking" he PDF of the solution output - instead of the stochastic input as we currently do - thus updating the gPC trial basis on-the-°y and correspondingly reducing significantly errors associated with long-term integration due to strong flow nonlinearities.

The proposed work will have significant and broad impact as it will set the foundations of data assimilation and rigorous sensitivity analysis in computational fluid dynamics. It will establish, for first-time, a composite error bar in flow simulations that goes beyond numerical accuracy and includes uncertainties in operating conditions, the physical parameters, and the domain. The proposed approach will affect fundamentally the way we design new experiments and the type of questions that we can address, while the interaction between simulation and experiment will become more meaningful and more dynamic. This, in turn, will find its way in the design of flow systems equipment and will provide a rigorous reliability framework.

On the education front, the new knowledge will contribute towards understanding nonlinear systems subject to noise, fundamentals in stochastic dynamics, data assimilation, and design under uncertainty. Currently, there is very little in our curricula on such subjects where the "deterministic" thinking prevails. We plan to incorporate these new ideas in engineering and applied mathematics courses we teach at Brown. Outside of the classroom, undergraduate students, through Brown's UTRA (Undergraduate Teaching and Research Assistantships) program, will be involved in the research projects, either during the academic year or the summer. For pre-college students, we will expand the Artemis program at Brown University which is designed to engage young women from the Providence area in computer and computational sciences. We will use immersive flow visualizations at Brown's CAVE as an opportunity to educate students about simulation, predictability, and other issues of computational science and applied mathematics.

Publications

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