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Reduced-order modeling of stochastic transport processes

Reduced-order modeling of stochastic transport processes. 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 14853-3801

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Reduced-order modeling of stochastic transport processes

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  1. Reduced-order modeling of stochastic transport processes Swagato Acharjee and Nicholas Zabaras Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801 Email: zabaras@cornell.edu URL: http://mpdc.mae.cornell.edu/ Materials Process Design and Control Laboratory

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

  3. 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

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

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

  6. Spatial model reduction Suppose we had an ensemble of data (from experiments or simulations) : 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 Is it possible to identify a basis such that it can represent the variable as: Introduce method of snapshots Materials Process Design and Control Laboratory

  7. Method of snapshots Method of snapshots (Lumley, Ly, Ravindran.) Leads to the basis Eigenvalue problem which is optimal for the ensemble data where • 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 C – nx nmatrix n – ensemble size Materials Process Design and Control Laboratory

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

  9. n ~ å = α ( ) n W ( x , t , ) W ( x , t ) ( ) q q i i = 0 i Generalized polynomial chaos expansion - overview Chaos polynomials (random variables) Stochastic process Reduced order representation of a stochastic processes. Subspace spanned by orthogonal basis functions from the askey series. 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

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

  11. Concurrent Reduced order problem formulation Expansion along random dimension Subsequent Expansion in a POD basis Фijcorresponds to the jth basis function in the expansion of the ith GPCE coefficient Materials Process Design and Control Laboratory

  12. Analogy of the reduced models with FEM (local) (global) (global) Materials Process Design and Control Laboratory

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

  14. 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

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

  16. Example problem 1 – Uncertainty in Rayleigh number vx = vy = 0 Functional form for Ra(θ) q = 2.5t vx = 0 vy = 0 l=1 Ra(θ) vx = 0 vy = 0 Other parameters Darcy number 7:812e-6 Porosity = 1.0 Diffusivity = 1.0 Grid size – 50x50 l=1 vx = vy = 0 Basis info • Total 90 snapshots from third-order SSFEM simulations • 30 snapshots at equal intervals with • 30 snapshots at equal intervals with • 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 DOFs in SSFEM energy equation – 10404 DOFs in SSFEM momentum equation - 31212 DOFs in CRM energy equation – 16 DOFs in CRM momentum equation - 32 Materials Process Design and Control Laboratory

  17. Uncertainty in Rayleigh number - results t = 0.2 MeanVelocity - y MeanTemperature Mean Velocity - x SSFEM CRM Materials Process Design and Control Laboratory

  18. Uncertainty in Rayleigh number - results t = 0.2 SD Velocity - x SDVelocity - y SDTemperature SSFEM CRM Materials Process Design and Control Laboratory

  19. Uncertainty in Rayleigh number - results t = 0.4 Mean Velocity - x MeanVelocity - y MeanTemperature SSFEM CRM Materials Process Design and Control Laboratory

  20. Uncertainty in Rayleigh number - results t = 0.4 SD Velocity - x SDVelocity - y SDTemperature SSFEM CRM Materials Process Design and Control Laboratory

  21. 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

  22. Example problem 2 – Uncertainty in porosity vx = vy = 0 KL expansion for ε(θ) Exponential covariance kernel q = 2.5t vx = 0 vy = 0 l=1 ε(θ) vx = 0 vy = 0 ε0 = 0.8, σ=0.05 , b=10 l=1 vx = vy = 0 Other parameters Darcy number 7:812e-6 Rayleigh Number = 1e4 Diffusivity = 1.0 Grid size – 50x50 Basis info • Total 90 snapshots from third-order SSFEM simulations • 30 snapshots at equal intervals with ε0 = 0.5; σ = 0.05 • 30 snapshots at equal intervals with ε0 = 0.6; σ = 0.03 • 30 snapshots at equal intervals with ε0 = 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 DOFs in SSFEM energy equation – 26010 DOFs in SSFEM momentum equation - 78030 DOFs in CRM energy equation – 50 DOFs in CRM momentum equation - 100 Materials Process Design and Control Laboratory

  23. Uncertainty in porosity - results t = 0.2 Mean Velocity - x MeanVelocity - y MeanTemperature SSFEM CRM Materials Process Design and Control Laboratory

  24. Uncertainty in porosity - results t = 0.2 SD Velocity - x SDVelocity - y SDTemperature SSFEM CRM Materials Process Design and Control Laboratory

  25. Uncertainty in porosity - results t = 0.4 Mean Velocity - x MeanVelocity - y MeanTemperature SSFEM CRM Materials Process Design and Control Laboratory

  26. Uncertainty in porosity - results t = 0.4 SD Velocity - x SDVelocity - y SDTemperature SSFEM CRM Materials Process Design and Control Laboratory

  27. Summary • 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. 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

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

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