A MULTISCALE APPROACH TO A MULTISCALE APPROACH TO MATERIALS USING STOCHASTIC MATERIALS USING STOCHASTIC AND COMPUTATIONAL STATISTICS AND COMPUTATIONAL STATISTICS TECHNIQUES TECHNIQUES NICHOLAS ZABARAS NICHOLAS ZABARAS Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 169 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801 Email: zabaras@cornell.edu URL: http://www.mae.cornell.edu/zabaras CO OR RN NE EL LL L C CO OR RN NE EL LL L C Materials Process Design and Control Laboratory Materials Process Design and Control Laboratory U N I V E R S I T Y U N I V E R S I T Y
NEED FOR MULTISCALE ANALYSIS NEED FOR MULTISCALE ANALYSIS Fingering in porous media Oil receiver site • Water displaces the oil • Permeability of bed rock is layer to the receiving site inherently stochastic • Water accelerates more in • Statistics like mean areas with high permeability permeability, correlation Porous structure are usually • Fingering reduces quality bed-rock constant for a given rock of the oil received (polluted type with water) • Stationary probability models can be used Water injector site • Direct simulation of the effect of uncertainty in permeability on the amount of oil received requires enormous computational power • Bed rock length scale – typically of order of kms • Length scale for permeability variation – typically of order of cms • Requirement – 10000 blocks for a single dimension (10 12 blocks overall) CO OR RN NE EL LL L C CO OR RN NE EL LL L C Materials Process Design and Control Laboratory Materials Process Design and Control Laboratory U N I V E R S I T Y U N I V E R S I T Y
NEED FOR MULTISCALE ANALYSIS NEED FOR MULTISCALE ANALYSIS Transport phenomena in material processes like solidification Microstructural features • Microstructure is dynamic and evolves with the materials process Engineering • Uncertainties at the micro- component Formation of scale are loosely correlated, dendrites, micro- however macro-scale features scale flow structure, like species concentration, Length scale heat transfer temperature, stresses are ~ 10 -4 meters patterns are highly highly correlated sensitive to • Uncertainty analysis at perturbations micro-scale requires considerable computational Length scale effort ~ meters • Macro-properties dependant on the dendrite patterns and uncertainty propagation at the micro- scales • Uncertainty interactions are no longer satisfy stationarity assumptions – newer probability models based on image analysis and experimentation needed CO OR RN NE EL LL L C CO OR RN NE EL LL L C Materials Process Design and Control Laboratory Materials Process Design and Control Laboratory U N I V E R S I T Y U N I V E R S I T Y
STOCHASTIC VARIATIONAL MULTISCALE STOCHASTIC VARIATIONAL MULTISCALE Bayesian data Subgrid scale • Statistical variations in analysis interface models properties are significant Averaging out the higher Green’s • Discontinuities, loosely statistical features of functions, RFB correlated structures in subgrid scale solutions type models properties using Karhunen-loeve/ • Discrete wavelet filtering probability Explicit distributions to subgrid scale Micro scale model properties model - FEM features • Image analysis to develop the Experimental, correlation Monte Carlo/ Residual structure MD models Statistical features Spectral Physical model stochastic/ Large scale support-space system representation of uncertainty • Statistical variations Discretization are relatively negligible Large scale solutions method like Uncertainties obtained from the • Discontinuities get FEM, Spectral, explicit discretization in boundary smoothed out FDM approach and initial • Process interactions, conditions Large scale properties become simulation correlated CO OR RN NE EL LL L C CO OR RN NE EL LL L C Materials Process Design and Control Laboratory Materials Process Design and Control Laboratory U N I V E R S I T Y U N I V E R S I T Y
MULTISCALE TRANSPORT SYSTEMS MULTISCALE TRANSPORT SYSTEMS Flow past an aerofoil Atmospheric flow in Solidification process Jupiter Modeling of dendrites at Large scale turbulent Astro-physical flows, small scale, fluid flow and structures, small scale effects of gravitational transport at large scale and magnetic fields dissipative eddies, surface irregularities • Presence of a variety of spatial and time scales - commonality • Varied applications – Engineering, Geophysical, Materials • Boundary conditions, material properties, small scale behavior inherently are uncertain CO OR RN NE EL LL L C CO OR RN NE EL LL L C Materials Process Design and Control Laboratory Materials Process Design and Control Laboratory U N I V E R S I T Y U N I V E R S I T Y
IDEA BEHIND VARIATIONAL MULTISCALE - - VMS VMS IDEA BEHIND VARIATIONAL MULTISCALE Subgrid model • Green’s function Small scale Micro-constitutive laws • Residual free bubbles behavior – from experiments, statistical • MsFEM “Hou et al.” theoretical predictions resolution • TLFEM “Hughes et al.” Large Subgrid Physical model scale scale Residual solution Solidification process • FEM Large scale behavior – explicit • FDM Where does resolution uncertainty fit in • Spectral ? Resolved model CO OR RN NE EL LL L C CO OR RN NE EL LL L C Materials Process Design and Control Laboratory Materials Process Design and Control Laboratory U N I V E R S I T Y U N I V E R S I T Y
WHY STOCHASTIC MODELING IN VMS ? WHY STOCHASTIC MODELING IN VMS ? Surroundings uncertainty Model uncertainty • Uncertain boundary • Imprecise knowledge conditions of governing physics • Inherent initial perturbations • Models used from experiments • Small scale interactions Solidification microscale • Uncertainty in codes features • Material properties • Machine precision errors fluctuate – only a statistical description Not accounted for in possible analysis here Material uncertainty Computational uncertainty CO OR RN NE EL LL L C CO OR RN NE EL LL L C Materials Process Design and Control Laboratory Materials Process Design and Control Laboratory U N I V E R S I T Y U N I V E R S I T Y
SOME PROBABILITY THEORY SOME PROBABILITY THEORY Ω Probability space – A triplet ( , , F P ) Ω - - Collection of all basic outcomes of the experiment F - - Permutation of the basic outcomes P - - Probability associated with the permutations ξ ω ( ) Random variable – a function Ω Sample space Real interval Stochastic process – a random function at each space and time point × ×Ω → � W :( D T ) Notations: W x t ω ( , , ) CO OR RN NE EL LL L C CO OR RN NE EL LL L C Materials Process Design and Control Laboratory Materials Process Design and Control Laboratory U N I V E R S I T Y U N I V E R S I T Y
SPECTRAL STOCHASTIC EXPANSIONS SPECTRAL STOCHASTIC EXPANSIONS • Series representation of stochastic processes with finite second moments ∞ + ∑ ω = ξ ω Karhunen-Loeve W x t ( , , ) W x t ( , ) W x t ( , ) ( ) i i expansion = i 1 W x t ( , ) - Mean of the stochastic process - Coefficient dependant on the eigen-pairs of the covariance W x t ( , ) i kernel of the stochastic process ξ ω ( ) - Orthogonal random variables i • Covariance kernel required – known only for inputs • Best possible representation in mean-square sense ∞ Generalized = ∑ ω ψ ω W x t ( , , ) W x t ( , ) ( ξ ( )) polynomial chaos i i expansion = i 0 W x t ( , ) - Coefficients dependant on chaos-polynomials chosen i ψ ω ( ξ ( )) - Chaos polynomials chosen from Askey-series (Legendre – i uniform, Jacobi – beta) CO OR RN NE EL LL L C CO OR RN NE EL LL L C Materials Process Design and Control Laboratory Materials Process Design and Control Laboratory U N I V E R S I T Y U N I V E R S I T Y
SUPPORT- -SPACE/STOCHASTIC GALERKIN SPACE/STOCHASTIC GALERKIN SUPPORT f ξ ω ( ( )) - Joint probability density function of the inputs = ξ ω ξ ω > A { ( ) : f ( ( )) 0} - The input support-space denotes the regions where input joint PDF is strictly positive Triangulation h ξ ω X ( ( )) Any function can be represented as a of the support- piecewise polynomial on the triangulated support-space space X ξ ω ( ( )) - Function to be approximated ξ ω h X ( ( )) - Piecewise polynomial approximation over support-space L 2 convergence – (mean-square) ∫ + ξ ω − ξ ω ξ ω ξ ≤ h 2 q 1 ( X ( ( )) X ( ( ))) f ( ( ))d Ch A Error in approximation is penalized h = mesh diameter for the support-space severely in high input joint PDF regions. discretization We use importance based refinement of grid to avoid this q = Order of interpolation CO OR RN NE EL LL L C CO OR RN NE EL LL L C Materials Process Design and Control Laboratory Materials Process Design and Control Laboratory U N I V E R S I T Y U N I V E R S I T Y
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