Reducedorder modeling of stochastic transport processes

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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_at_cornell.edu URL
http//mpdc.mae.cornell.edu/
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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
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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
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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
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Overview of stochastic PDEs Heat diffusion
equation
Deterministic PDE
Stochastic PDE
? random dimension Primary variables and
coefficients have space time and random
dimensionality stochastic process
Primary variables and coefficients have space and
time dimensionality
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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
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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 n x n matrix n ensemble size
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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
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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
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Reduced order subspaces
Random dimension
- Generated using truncated GPCE
Basis functions
Inner product
Space dimension
- Generated using POD
Basis functions
Inner product
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Concurrent Reduced order problem formulation
Expansion along random dimension
Subsequent Expansion in a POD basis
?ij corresponds to the jth basis function in the
expansion of the ith GPCE coefficient
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Analogy of the reduced models with FEM
(local)
(global)
(global)
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Natural convection in stochastic domain
Governing Equations
Initial Conditions
Boundary Conditions
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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)
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Concurrent model reduction applied to natural
convection
Momentum
Energy
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Example problem 1 Uncertainty in Rayleigh number
vx vy 0
Functional form for Ra(?)
q 2.5t
vx 0 vy 0
l1
Ra(?)
vx 0 vy 0
Other parameters Darcy number 7812e-6 Porosity
1.0 Diffusivity 1.0 Grid size 50x50
l1
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
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Uncertainty in Rayleigh number - results t 0.2
Mean Velocity - x
Mean Velocity - y
Mean Temperature
SSFEM
CRM
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Uncertainty in Rayleigh number - results t 0.2
SD Velocity - x
SD Velocity - y
SD Temperature
SSFEM
CRM
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Uncertainty in Rayleigh number - results t 0.4
Mean Velocity - x
Mean Velocity - y
Mean Temperature
SSFEM
CRM
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Uncertainty in Rayleigh number - results t 0.4
SD Velocity - x
SD Velocity - y
SD Temperature
SSFEM
CRM
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Uncertainty in Rayleigh number MC comparisons
Final centroidal velocity
MC results from 2000 samples generated using
Latin Hypercube Sampling
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Example problem 2 Uncertainty in porosity
vx vy 0
KL expansion for e(?)
Exponential covariance kernel
q 2.5t
vx 0 vy 0
l1
e(?)
vx 0 vy 0
e0 0.8, s0.05 , b10
l1
vx vy 0
Other parameters Darcy number 7812e-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 e0 0.5 s
  0.05
• 30 snapshots at equal intervals with e0 0.6 s
  0.03
• 30 snapshots at equal intervals with e0 0.7 s
  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
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Uncertainty in porosity - results t 0.2
Mean Velocity - x
Mean Velocity - y
Mean Temperature
SSFEM
CRM
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Uncertainty in porosity - results t 0.2
SD Velocity - x
SD Velocity - y
SD Temperature
SSFEM
CRM
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Uncertainty in porosity - results t 0.4
Mean Velocity - x
Mean Velocity - y
Mean Temperature
SSFEM
CRM
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Uncertainty in porosity - results t 0.4
SD Velocity - x
SD Velocity - y
SD Temperature
SSFEM
CRM
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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
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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 Stochastic inverse 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
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