Reduced-order modeling of stochastic Reduced-order modeling of stochastic transport processes transport processes
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Swagato Acharjee
and
Nicholas Zabaras
Materials Process Design and Control LaboratorySibley School of Mechanical and Aerospace Engineering
188 Frank H. T. Rhodes HallCornell University
Ithaca, NY 14853-3801
Email: [email protected]: http://mpdc.mae.cornell.edu/
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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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
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Outline of PresentationOutline 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 ?
X Y
Z
Cval0.0980.0960.0940.0920.0900.0880.0850.0830.081
X Y
Z
Cval0.1590.1430.1350.1270.1190.1020.0940.0860.0780.0620.054
(b)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.
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Overview of stochastic PDEs – Heat diffusion equationOverview of stochastic PDEs – Heat diffusion equation
Deterministic PDE Stochastic PDE
( , ) 2( , )
T x tk T x t x
t
( , 0) ( )T x T xo
( , )( , )
T x tk q x t x qn
( , ) ( , ) T x t T x t xe e
( , , )( ) ( , , )
T x tk q x t x qn
( , , ) ( , , ) T x t T x t xe e
( , , ) 2( ) ( , , )
T x tk T x t x
t
Primary variables and coefficients have space and time dimensionality
θ = random dimension
Primary variables and coefficients have space time and random dimensionality – stochastic process
( , 0, ) ( , )T x T xo
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Spatial model reductionSpatial 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
1
( , )n
i iT x t
1
( )m
i ix
1
ˆ( , ) ( ) ( )m
j jj
T x t T t x
2
| ( , ) |max
|| ||
T
C
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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 snapshotsMethod 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
1
nj
j i ij
u T
CU U
1ij i jC TT d
n
1
( )m
j ix
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Model reduction along the random dimensionModel 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
0
( , , ) ( , ) ( )q
i ii
A x t A x t
0
( )q
i ix
0
( , , ) ( , ) ( )i ii
A x t A x t
Generalized Polynomial chaos expansion (Weiner, Karniadakis)
Hypergeometric orthogonal polynomials from the Askey series
0
1
22
( ) 1
( ) ( )
( ) ( ) 1
Basis functions in terms of Hermite polynomials
Orthogonality relation
i j ijd R
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Generalized polynomial chaos expansion - overview
αn
iii
n txWtxW0
)( )(),(~
),,(
Stochastic Stochastic processprocess
Chaos Chaos polynomialspolynomials
(random (random variables)variables)Reduced order representation of a stochastic processes.
Subspace spanned by orthogonal basis functions from the askey series.
Chaos polynomialChaos polynomial Support spaceSupport space Random variableRandom variable
LegendreLegendre [[]] UniformUniform
JacobiJacobi [[]] BetaBeta
HermiteHermite [-[-∞,∞]∞,∞] Normal, LogNormalNormal, LogNormal
LaguerreLaguerre [0, [0, ∞]∞] GammaGamma
Number of chaos polynomials used to represent output uncertainty depends on Number of chaos polynomials used to represent output uncertainty depends on
- Type of uncertainty in input- Type of uncertainty in input - Distribution of input uncertainty- Distribution of input uncertainty- Number of terms in KLE of input - Degree of uncertainty propagation desired- Number of terms in KLE of input - Degree of uncertainty propagation desired
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Reduced order subspacesReduced order subspaces
, j
i R
Random dimension
Space dimension
R R
Basis functions
Basis functions
Inner product
Inner product
, ( )a b ab d R
R
, ( )a b ab d
,a bR
,a b
- Generated using POD
- Generated using truncated GPCE
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Concurrent Reduced order problem formulationConcurrent Reduced order problem formulation
Expansion along random dimension
Subsequent Expansion in a POD basis
, i R R R
Фij corresponds to the jth basis function in the expansion of the ith GPCE coefficient
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Analogy of the reduced models with FEMAnalogy of the reduced models with FEM
FEM Spatial Reduced Random reduced
Interpolation
Method of generating
basis
Domain discretization into
elements
POD GPCE
Trial function
Test function
0
( , , ) ( , ) ( )q
i ii
A x t A x t
1
ˆ( , ) ( ) ( )m
j jj
T x t T t x
1
( , ) ( ) ( )m
j jj
T x t T t N x
(local) (global) (global)
( )jN x ( )j x ( )j
( )jN x ( )j x ( )j
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Natural convection in stochastic domainNatural convection in stochastic domain
Governing Equations
Boundary Conditions
Initial Conditions
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Natural convection in stochastic domainNatural 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 convectionConcurrent model reduction applied to natural convection
Momentum
Energy
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Example problem 1 – Uncertainty in Rayleigh numberExample problem 1 – Uncertainty in Rayleigh number
[0,0.4]t
l=1 Ra(θ)
l=1vx = vy = 0
vx = 0
vy = 0
vx = vy = 0
vx = 0
vy = 0
q = 2.5t
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
Basis infoBasis info
Other parameters
Darcy number 7:812e-6
Porosity = 1.0
Diffusivity = 1.0
Grid size – 50x50
DOFs in SSFEM energy equation – 10404
DOFs in SSFEM momentum equation - 31212
DOFs in CRM energy equation – 16
DOFs in CRM momentum equation - 32
=(θ)Ra 1e4(1+ 0.05ξ(θ))
Functional form for Ra(θ)
=(θ)Ra 5e4(1+ 0.07ξ(θ))
=(θ)Ra 5e5(1+ 0.05ξ(θ))
=(θ)Ra 1e5(1+ 0.05ξ(θ))
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Uncertainty in Rayleigh number - results t = 0.2Uncertainty in Rayleigh number - results t = 0.2
SSFEM
CRM
Mean Velocity - x Mean Velocity - y Mean Temperature
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Uncertainty in Rayleigh number - results t = 0.2Uncertainty in Rayleigh number - results t = 0.2
SSFEM
CRM
SD Velocity - x SD Velocity - y SD Temperature
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Uncertainty in Rayleigh number - results t = 0.4Uncertainty in Rayleigh number - results t = 0.4
SSFEM
CRM
Mean Velocity - x Mean Velocity - y Mean Temperature
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Uncertainty in Rayleigh number - results t = 0.4Uncertainty in Rayleigh number - results t = 0.4
SSFEM
CRM
SD Velocity - x SD Velocity - y SD Temperature
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Uncertainty in Rayleigh number – MC comparisonsUncertainty 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 porosityExample problem 2 – Uncertainty in porosity
[0,0.4]t
l=1 ε(θ)
l=1vx = vy = 0
vx = 0
vy = 0
vx = vy = 0
vx = 0
vy = 0
q = 2.5t
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
Basis infoBasis info
Other parameters
Darcy number 7:812e-6
Rayleigh Number = 1e4
Diffusivity = 1.0
Grid size – 50x50
DOFs in SSFEM energy equation – 26010
DOFs in SSFEM momentum equation - 78030
DOFs in CRM energy equation – 50
DOFs in CRM momentum equation - 100
2
0 i n ii=1
=(θ, p) (θ)ε ε (1+ ξ λ f (p) )
KL expansion for ε(θ)
ε0 = 0.8, σ=0.05 , b=10
1 2-r2
= exp( )b
(p p )C ,
Exponential covariance kernel
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Uncertainty in porosity - results t = 0.2Uncertainty in porosity - results t = 0.2
SSFEM
CRM
Mean Velocity - x Mean Velocity - y Mean Temperature
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
Uncertainty in porosity - results t = 0.2Uncertainty in porosity - results t = 0.2
SSFEM
CRM
SD Velocity - x SD Velocity - y SD Temperature
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
Uncertainty in porosity - results t = 0.4Uncertainty in porosity - results t = 0.4
SSFEM
CRM
Mean Velocity - x Mean Velocity - y Mean Temperature
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
Uncertainty in porosity - results t = 0.4Uncertainty in porosity - results t = 0.4
SSFEM
CRM
SD Velocity - x SD Velocity - y SD Temperature
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•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.
SummarySummary
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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More complicated input uncertainties, higher degree of randomness.
Other stochastic PDEs .
Application to stochastic Inverse problems.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15
Iterations
0.985
0.99
0.995
1
1.005
1.01
1.015
1.02
1.025
1.03
1.035
0 10 20 30 40 50 60 70 80 90
Angle from rolling direction
InitialIntermediateOptimalDesired
Nor
mal
ized
hy
ster
esis
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)
No
n-d
imen
sio
nal
mea
nflu
x
0.5 1 1.5 2
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
SSFEMAnalytical
Non-dimensional time
Flu
xst
and
ard
dev
iatio
n
0 0.5 1 1.5 2
-0.4
-0.3
-0.2
-0.1
0
Reconstructed heat flux with comparisons to analytical meanNon-dimensional time
Tem
per
ature
confid
ence
inte
rval
0 0.5 1 1.5 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
SSFEM meanC.I upper limitC.I lower limit
Required design temperature readings
Unknown flux
Temperature sensor readings
PotentialPotential