Reduced-order modeling of stochastic transport processes

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


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/




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




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




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.




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




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




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




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

Materials Process Design and Control Laboratory



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




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




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




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




Natural convection in stochastic domainNatural convection in stochastic domain

Governing Equations

Boundary Conditions

Initial Conditions




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)




Concurrent model reduction applied to natural convectionConcurrent model reduction applied to natural convection

Momentum

Energy




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



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))




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





SSFEM

CRM

SD Velocity - x SD Velocity - y SD Temperature





SSFEM

CRM






SSFEM

CRM





Uncertainty in Rayleigh number – MC comparisonsUncertainty in Rayleigh number – MC comparisons

Final centroidal velocity

MC results from 2000 samples generated using Latin Hypercube Sampling




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



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




Uncertainty in porosity - results t = 0.2Uncertainty in porosity - results t = 0.2

SSFEM

CRM






SSFEM

CRM






SSFEM

CRM






SSFEM

CRM





•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




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

Documents

Reduced-order modeling of stochastic transport processes