Reduced-order modeling of stochastic transport processes Materials Process Design and Control Laboratory Swagato Acharjee and Nicholas Zabaras Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY URL:

Materials Process Design and Control Laboratory Research Sponsors U.S. AIR FORCE PARTNERS Materials Process Design Branch, AFRL Computational Mathematics Program, AFOSR CORNELL THEORY CENTER ARMY RESEARCH OFFICE Mechanical Behavior of Materials Program NATIONAL SCIENCE FOUNDATION (NSF) Design and Integration Engineering Program

Materials Process Design and Control Laboratory Outline of Presentation Motivation – why lower dimension models in transport processes Stochastic PDEs – overview Model reduction in spatial domain Model reduction in stochastic domain Concurrent model reduction applied to stochastic PDEs – Natural Convection Example problems Conclusions and Discussion

Materials Process Design and Control Laboratory Why Lower Dimension Models ? Solute concentrations (a) without any magnetic field (b) under the influence of a magnetic field. (Zabaras,Samanta 2004) (a) (b) Transport problems that involve partial differential equations are formidable problems to solve. Binary Alloy Solidification Mean Higher order statistics Flow past a cylinder (Stochastic Simulation) (Badri Narayanan, Zabaras 2004) Probabilistic modeling and control are all the more daunting. Need to come up with efficient solution methods without losing out on accuracy or physics.

Materials Process Design and Control Laboratory Overview of stochastic PDEs – Heat diffusion equation Deterministic PDE Stochastic PDE Primary variables and coefficients have space and time dimensionality θ = random dimension Primary variables and coefficients have space time and random dimensionality – stochastic process

Materials Process Design and Control Laboratory Spatial model reduction Suppose we had an ensemble of data (from experiments or simulations) : such that it can represent the variable as: Is it possible to identify a basis POD technique (Lumley) Maximize the projection of the data on the basis. Leads to the eigenvalue problem C – full p x p matrix: leads to a large eigenvalue problem with p the number of grid points Introduce method of snapshots

Materials Process Design and Control Laboratory Method of snapshots (Lumley, Ly, Ravindran.) Eigenvalue problem where C – n x n matrix n – ensemble size Leads to the basis which is optimal for the ensemble data Method of snapshots Other features Generated basis can be used in the interpolatory as well as the extrapolatory mode First few basis vectors enough to represent the ensemble data

Materials Process Design and Control Laboratory Model reduction along the random dimension Fourier type expansion along the random dimension such that it can represent the variable as: Is it possible to identify an optimal basis Generalized Polynomial chaos expansion (Weiner, Karniadakis) Hypergeometric orthogonal polynomials from the Askey series Basis functions in terms of Hermite polynomials Orthogonality relation

Materials Process Design and Control Laboratory Generalized polynomial chaos expansion - overview   α  n i ii n txWtxW 0 )( )(),( ~ ),,(   Stochastic process Chaos polynomials (random variables) Reduced order representation of a stochastic processes. Subspace spanned by orthogonal basis functions from the askey series. Chaos polynomial Support space Random variable Legendre [  ] Uniform Jacobi Beta Hermite [-∞,∞] Normal, LogNormal Laguerre [0, ∞] Gamma Number of chaos polynomials used to represent output uncertainty depends on - Type of uncertainty in input- Distribution of input uncertainty - Number of terms in KLE of input - Degree of uncertainty propagation desired

Materials Process Design and Control Laboratory Reduced order subspaces Random dimension Space dimension Basis functions Inner product - Generated using POD - Generated using truncated GPCE

Materials Process Design and Control Laboratory Concurrent Reduced order problem formulation Expansion along random dimension Subsequent Expansion in a POD basis Ф ij corresponds to the j th basis function in the expansion of the i th GPCE coefficient

Materials Process Design and Control Laboratory Analogy of the reduced models with FEM FEMSpatial ReducedRandom reduced Interpolation Method of generating basis Domain discretization into elements PODGPCE Trial function Test function (local)(global)

Materials Process Design and Control Laboratory Natural convection in stochastic domain Governing Equations Boundary Conditions Initial Conditions

Materials Process Design and Control Laboratory Natural convection in stochastic domain Governing Equations for GPCE formulation Solution scheme based on a SUPG/PSPG Stabilized FEM technique for the analogous deterministic problem (Zabaras,2004, Heinridge, 1998)

Materials Process Design and Control Laboratory Concurrent model reduction applied to natural convection Momentum Energy

Materials Process Design and Control Laboratory Example problem 1 – Uncertainty in Rayleigh number l=1 Ra(θ) l=1 v x = v y = 0 v x = 0 v y = 0 v x = v y = 0 v x = 0 v y = 0 q = 2.5t Total 90 snapshots from third-order SSFEM simulations 30 snapshots at equal intervals with Using 4 out of a possible 90 basis vectors for the energy and momentum equations. 1D order 3 GPCE used for random discretization Basis info Other parameters Darcy number 7:812e-6 Porosity = 1.0 Diffusivity = 1.0 Grid size – 50x50 DOFs in SSFEM energy equation – DOFs in SSFEM momentum equation DOFs in CRM energy equation – 16 DOFs in CRM momentum equation - 32 Functional form for Ra(θ)

Materials Process Design and Control Laboratory Uncertainty in Rayleigh number - results t = 0.2 SSFEM CRM Mean Velocity - x Mean Velocity - yMean Temperature

Materials Process Design and Control Laboratory Uncertainty in Rayleigh number - results t = 0.2 SSFEM CRM SD Velocity - x SD Velocity - ySD Temperature

Materials Process Design and Control Laboratory Uncertainty in Rayleigh number - results t = 0.4 SSFEM CRM Mean Velocity - x Mean Velocity - yMean Temperature

Materials Process Design and Control Laboratory Uncertainty in Rayleigh number - results t = 0.4 SSFEM CRM SD Velocity - x SD Velocity - ySD Temperature

Materials Process Design and Control Laboratory Uncertainty in Rayleigh number – MC comparisons Final centroidal velocity MC results from 2000 samples generated using Latin Hypercube Sampling

Materials Process Design and Control Laboratory Example problem 2 – Uncertainty in porosity l=1 ε(θ)ε(θ) v x = v y = 0 v x = 0 v y = 0 v x = v y = 0 v x = 0 v y = 0 q = 2.5t Total 90 snapshots from third-order SSFEM simulations 30 snapshots at equal intervals with ε 0 = 0.5; σ = snapshots at equal intervals with ε 0 = 0.6; σ = snapshots at equal intervals with ε 0 = 0.7; σ = 0.02 Using 5 out of a possible 90 POD basis vectors for the energy and momentum equations. 2D order 3 basis used for random dimension Basis info Other parameters Darcy number 7:812e-6 Rayleigh Number = 1e4 Diffusivity = 1.0 Grid size – 50x50 DOFs in SSFEM energy equation – DOFs in SSFEM momentum equation DOFs in CRM energy equation – 50 DOFs in CRM momentum equation KL expansion for ε(θ) ε 0 = 0.8, σ=0.05, b=10 Exponential covariance kernel

Materials Process Design and Control Laboratory Uncertainty in porosity - results t = 0.2 SSFEM CRM Mean Velocity - x Mean Velocity - yMean Temperature

Materials Process Design and Control Laboratory Uncertainty in porosity - results t = 0.2 SSFEM CRM SD Velocity - x SD Velocity - ySD Temperature

Materials Process Design and Control Laboratory Uncertainty in porosity - results t = 0.4 SSFEM CRM Mean Velocity - x Mean Velocity - yMean Temperature

Materials Process Design and Control Laboratory Uncertainty in porosity - results t = 0.4 SSFEM CRM SD Velocity - x SD Velocity - ySD Temperature

Materials Process Design and Control Laboratory Concurrent Model reduction applied to thermal transport. GPCE in the random domain, POD in the spatial domain. Captures all the essential physics of the problem without signicant loss of accuracy Quite generic – applies to other PDEs also. Useful tool for fast solution of complex SPDEs especially when previous simulation data is available. Speed up of several orders of magnitude compared to full model MC sampling.Summary Relevant Publication "A concurrent model reduction approach on spatial and random domains for stochastic PDEs", International Journal for Numerical Methods in Engineering, in press

Materials Process Design and Control Laboratory More complicated input uncertainties, higher degree of randomness. Other stochastic PDEs. Application to stochastic Inverse problems. Normalized hysteresis loss Objective function Inverse problem - POD based control of texture for desired properties (Acharjee, Zabaras 2003) GPCE based Stochastic inverse heat conduction (Badri Narayanan, Zabaras 2004) Reconstructed heat flux with comparisons to analytical mean Required design temperature readings Unknown flux Temperature sensor readingsPotential