Hessam Babaee, PhD

Assistant Professor

1102 Benedum Hall | 3700 O’Hara Street | Pittsburgh, PA 15261 P: 412-383-0560

h.babaee@pitt.edu

AStochasticFrameworkforComputationofSensitivitiesinChaoticFlows

The work of Dr. Babaee’s research group is at the intersection of high performance computing, uncertainty quantification and predictive modeling. Advances in the field of machine learning provide a rigorous mathematical framework that have begun to allow combining data with variable fidelities from different sources to generate or improve predictive models. Furthermore, the advent of efficient numerical techniques, particularly in the field of stochastic differential equations and high performance computing have paved the way for quantifying uncertainty in complex engineering systems. The combination of these components is just starting to bear fruit and unprecedented opportunities exist for constructing predictive models endowed with rigorous certificates of fidelity in multi-physics/multi-scale engineering systems. This is the scope of Dr. Babaee’s research group.

Uncertainty Quantification and Stochastic Modeling

Complex thermo-fluid systems are often poorly understood due to the uncertainties in the models, parameters, experimental measurements and numerical simulations, and operating conditions. Propagating and managing uncertainty in these systems is a daunting task. These systems are often governed by multi-physics/multiscale phenomenon – described by highdimensional coupled differential/integral equations – such as heat transfer in gas turbines or turbulent two-phase flows. Our group develops stochastic reduced order models (ROM) by exploiting the correlations among the different realizations of a physical system. ROM has orders of magnitude smaller dimension than the full-dimensional model and it serves two main purposes: (1) ROM is a diagnostic tool that can be used to characterize and analyze the behavior of complex dynamical systems; (2) ROM can be used instead of the full-dimensional model to significantly reduce the computational burden of propagating uncertainty in high-dimensional dynamical systems.

Multi-Fidelity Modeling for Design and Optimization

Engineering designs and resilient systems require management of data from a variety of sources, efficient allocation of computational resources, and, most importantly, quantification of uncertainty inherent in multi-physics models of variable fidelity and operating conditions, as well as utilization of such information in risk-averse decision making. Even for classical engineering systems that can be described at various levels of fidelity and for which experimental data may exist, currently there are no mathematically rigorous methods to combine these disparate information sources into a viable framework for the purpose of design and optimization. In our group we develop multi-fidelity framework by employing modern elements of machine learning – such as non-parametric Bayesian regression – that are capable of blending information from sources of different fidelity, and can

Low-dimensional Attractor

(a) (b) Figure2: PIBabaee’sresearchonaminimizationprincipleforbuildingtime-dependentmodes: (a)FeaturedontheCoverPageof ProceedingsoftheRoyalSocietyofLondonA:Mathematical, PhysicalandEngineeringSciences [33];(b)Smoke:volumerenderingofDNS;Iso-surface:optimally time-dependent(OTD)modes.

Time-Dependent Basis

Figure 1

increasestheinfluenceoftheunresolvedfluctuationsbecomesmoresignificant. Insuchsituations, thenon-linearityofdynamicalsystemresultsingrowthoftheunresolvedfluctuations.Foraccurate predictionoftheflowstructure,itisessentialtoaccountforthee↵ not,thealiasingerrorswillsurelyleadtonon-physicalresults. Thisissuehasbeenattheheart ofcompressibleflowresearchforover60yeasnow[34–38]. Thecruxoftheproblemisduetothe interactionofpressureandvorticitymodes,withescalatedcomplexityastheflowbecomesturbulent. Wewouldliketoexamineseveralmeansofaccountingforthee↵ectsoftheunresolvedmodesin highReynoldsnumberflows.Werecognizethattherearecertainsimilaritiesbetweentheloworder modaldecompositionandthefilteringoperationasemployedinLES[39]. Therefore,wespeculate thatsomeoftheideasissubgridscale(SGS)modelingmaybeusefulforourpurpose. Essentially, theissueisclosureofthecorrelation: Tij = E[uiuj ] − E[ui]E[uj ]. TheliteratureofLESdescribes manyofsuchclosures[39–41].Someofthemappearpromisingforourpurposeandwillbetriedhere. Figure 2 provide an information-fusion framework, in which prediction First,wewouldliketoexaminethe scale-similaritymodel (SSM)[42–44]asthismodelappearsvery for quantities of interest and their associated uncertainties naturalforourpurpose.Withthismodel,simulationswillbeconductedwith(atleast)twosetsof are determined. resolutions.Thiscanbeachievedbyincreasing r.Theseresultswillbeusedtoprovideamodelfor theunresolvedfluctuations.Anotherpossibilitywouldbethe mixedmodel basedonthecombinationoftheSSMandthegeneralSGSviscositytypemodel.Again,twosetsof 114 DEPARTMENT OF MECHANICAL ENGINEERING AND MATERIALS SCIENCE simulationswithvaryingresolutionsareneededtodeterminethee↵ectsoftheunresolvedfluctuations. Wewillalsoconsiderastochasticmethodology.Sincethismethodisnew,abitofmoredescription