Materials Process Design and Control Laboratory A STOCHASTIC VARIATIONAL MULTISCALE METHOD FOR DIFFUSION…

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

CCOORRNNEELLLL U N I V E R S I T Y

A STOCHASTIC VARIATIONAL MULTISCALE METHOD FOR DIFFUSION IN HETEROGENEOUS

RANDOM MEDIA

N. ZABARAS AND B. VELAMUR ASOKANMaterials Process Design and Control Laboratory

Sibley School of Mechanical and Aerospace Engineering188 Frank H. T. Rhodes Hall

Cornell University Ithaca, NY 14853-3801

Email: [email protected]

URL: http://mpdc.mae.cornell.edu/



WHY UNCERTAINTY AND MULTISCALING ?

MacroMesoMicro

Uncertainties introduced across various length scales have a non-trivial interaction

Current sophistications – resolve macro uncertainties

Use micro averaged models for resolving physical scales

Imprecise boundary conditions Initial perturbations

Physical properties, structure follow a statistical description



EXAMPLE 2: DIFFUSION IN A RANDOM MICROSTRUCTURE

DIFFUSION COEFFICIENTS OF INDIVIDUAL CONSTITUENTS NOT KNOWN EXACTLY

– A MIXTURE MODEL IS USED

* *0 *1( ) ( )k k k

( , ) ( ( ) ( )) ( ) ( )k x k k I x k

THE INTENSITY OF THE GRAY-SCALE IMAGE IS MAPPED TO THE CONCENTRATIONS

DARKEST DENOTES PHASE

LIGHTEST DENOTES PHASE

(x,0, ) 0u ( 1) ( 0) ( 0,1)| 0, | 1, | 0x x yuu u kn



OUTLINE Motivation: coupling multiscaling and uncertainty analysis

Mathematical representation of uncertainty

Variational multiscale method (VMS)

Application of VMS with explicit subgrid model

Stochastic multiscale diffusion equation

Increasing efficiency in uncertainty modeling techniques

Sparse grid quadrature, support-space method

Future directions



UNCERTAINTY ANALYSIS TECHNIQUES Monte-Carlo : Simple to implement, computationally expensive

Perturbation, Neumann expansions : Limited to small fluctuations, tedious for higher order statistics

Sensitivity analysis, method of moments : Probabilistic information is indirect, small fluctuations

Spectral stochastic uncertainty representation

Basis in probability and functional analysis

Can address second order stochastic processes

Can handle large fluctuations, derivations are general



RANDOM VARIABLES = FUNCTIONS ? Math: Probability space (, F, P)

Sample space Sigma-algebra Probability measure

F

W : Random variableW

: ( )W

Random variable

A stochastic process is a random field with variations across space and time

: ( , , )X x t



SPECTRAL STOCHASTIC REPRESENTATION

: ( , , )X x t

A stochastic process = spatially, temporally varying random function

CHOOSE APPROPRIATE BASIS FOR THE

PROBABILITY SPACE

HYPERGEOMETRIC ASKEY POLYNOMIALS

PIECEWISE POLYNOMIALS (FE TYPE)

SPECTRAL DECOMPOSITION

COLLOCATION, MC (DELTA FUNCTIONS)

GENERALIZED POLYNOMIAL CHAOS EXPANSION

SUPPORT-SPACE REPRESENTATION

KARHUNEN-LOÈVE EXPANSION

SMOLYAK QUADRATURE, CUBATURE, LH



KARHUNEN-LOEVE EXPANSION

1

( , , ) ( , ) ( , ) ( )i ii

X x t X x t X x t

Stochastic process

Mean function

ON random variablesDeterministic functions

Deterministic functions ~ eigen-values , eigenvectors of the covariance function

Orthonormal random variables ~ type of stochastic process

In practice, we truncate (KL) to first N terms

1( , , ) fn( , , , , )NX x t x t



GENERALIZED POLYNOMIAL CHAOS Generalized polynomial chaos expansion is used to represent the stochastic output in terms of the input

0

( , , ) ( , ) (ξ( ))i ii

Z x t Z x t

Stochastic output

Askey polynomials in inputDeterministic functions

Stochastic input

1( , , ) fn( , , , , )NX x t x t

Askey polynomials ~ type of input stochastic process

Usually, Hermite, Legendre, Jacobi etc.



SUPPORT-SPACE REPRESENTATION Any function of the inputs, thus can be represented as a function defined over the support-space

1( , , ) : ( ) 0NA f ξ ξ

JOINT PDF OF A TWO RANDOM

VARIABLE INPUT

FINITE ELEMENT GRID REFINED IN HIGH-DENSITY REGIONS

2

2

1

ˆ

ˆ( ( ) ( )) ( )d

L

A

q

X X

X X f

Ch

ξ ξ ξ ξ

OUTPUT REPRESENTED ALONG SPECIAL COLLOCATION POINTS

– SMOLYAK QUADRATURE

– IMPORTANCE MONTE CARLO



NEED FOR SUPPORT-SPACE APPROACH GPCE and Karhunen-Loeve are Fourier like expansions

Gibb’s effect in describing highly nonlinear, discontinuous uncertainty propagation

Onset of natural convection

[Zabaras JCP 208(1)] – Using support-space method

[Ghanem JCP 197(1)] – Using Wiener-Haar wavelets Finite element representation of stochastic processes [stochastic Galerkin method: Babuska et al]

Incorporation of importance based meshing concept for improving accuracy [support space method]



CURSE OF DIMENSIONALITY Both GPCE and support-space method are fraught with the curse of dimensionality

As the number of random input orthonormal variables increase, computation time increases exponentially

Support-space grid is usually in a higher-dimensional manifold (if the number of inputs is > 3), we need special tensor product techniques for generation of the support-space

Parallel implementations are currently performed using PETSc (Parallel scientific extensible toolkit )



VARIATIONAL MULTISCALE METHOD

h

Subgrid scale

solution

Coarse scale

solution

Actual solution

Hypothesis Exact solution = Coarse resolved part + Subgrid part [Hughes, 95, CMAME]

Induced function space

Solution function space = Coarse function space + Subgrid function space

Idea

Model the projection of weak form onto the subgrid function space, calculate an approximate subgrid solution

Use the subgrid solution to solve for coarse solution



VARIATIONAL MULTISCALE BASICS

DERIVE THE WEAK FORMULATION FOR

THE GOVERNING EQUATIONS

PROJECT THE WEAK FORMULATION ON COARSE AND FINE

SCALES

COARSE WEAK FORM

FINE (SUBGRID) WEAK FORM

ALGEBRAIC SUBGRID MODELS

COMPUTATIONAL SUBGRID MODELS

APPROXIMATE SUBGRID SOLUTION

REMOVE SUBGRID EFFECTS IN THE COARSE

WEAK FORM USING STATIC CONDENSATION

MODIFIED MULTISCALE COARSE WEAK FORM INCLUDING SUBGRID EFFECTS

SOLUTION FUNCTION SPACES ARE NOW

STOCHASTIC FUNCTION SPACES

NEED TECHNIQUES TO SOLVE STOCHASTIC

PDEs



We will restrict ourselves to computational subgrid models



MODEL MULTISCALE HEAT EQUATION

( )u K u ft

Dgu u

in

on

D

0( ,0) ( )u x u x in DTHE DIFFUSION COEFFICIENT K IS HETEROGENEOUS AND POSSESSES RAPID

RANDOM VARIATIONS IN SPACE

FLOW IN HETEROGENEOUS POROUS MEDIA

INHERENTLY STATISTICAL

DIFFUSION IN MICROSTRUCTURES

OTHER APPLICATIONS

– DIFFUSION IN COMPOSITES

– FUNCTIONALLY GRADED MATERIALS

DEFINE PROBLEM

DERIVE WEAK FORM

VMS HYPOTHESIS

COARSE-TO-SUBGRID MAP

FINAL COARSE FORMULATION

AFFINE CORRECTION



STOCHASTIC WEAK FORM

u U : Find such that, for all v V

,( , ) ( , ) ( , ); ( , ) : ( , )tu v a u v f v a u v K u v

12

12

: ( ; ( )),

: ( ; ( )), 0

gU u u L H D u u

V v v L H D v

Weak formulation

VMS hypothesisC Fu u u

Exact solution Coarse solution Subgrid solutionC FU U U C FV V V

DEFINE PROBLEM

DERIVE WEAK FORM

VMS HYPOTHESIS



AFFINE CORRECTION



EXPLICIT SUBGRID MODELLING: IDEA

DERIVE THE WEAK FORMULATION FOR

THE GOVERNING EQUATIONS

PROJECT THE WEAK FORMULATION ON COARSE AND FINE

SCALES

COARSE WEAK FORM

FINE (SUBGRID) WEAK FORM

COARSE-TO-SUBGRID MAP EFFECT OF

COARSE SOLUTION ON SUBGRID SOLUTION

AFFINE CORRECTION SUBGRID DYNAMICS

THAT ARE INDEPENDENT OF THE COARSE SCALE

LOCALIZATION, SOLUTION OF SUBGRID EQUATIONS NUMERICALLY

FINAL COARSE WEAK FORMULATION THAT ACCOUNTS FOR THE SUBGRID SCALE EFFECTS



SCALE PROJECTION OF WEAK FORM

, ,( , ) ( , ) ( , ) ( , ) ( , )C C F C C C F C Ct tu v u v a u v a u v f v

C Cu UFind such that, for all C Cv VF Fu Uandand F Fv V Projection of weak form on coarse scale

Projection of weak form on subgrid scale

, ,( , ) ( , ) ( , ) ( , ) ( , )C F F F C F F F Ft tu v u v a u v a u v f v

DEFINE PROBLEM

DERIVE WEAK FORM

VMS HYPOTHESIS



AFFINE CORRECTION

0ˆF F Fu u u

EXACT SUBGRID SOLUTION


SUBGRID AFFINE CORRECTION



SPLITTING THE SUBGRID SCALE WEAK FORM

Coarse-to-subgrid map

Subgrid affine correction

, ,ˆ ˆ( , ) ( , ) ( , ) ( , ) 0C F F F C F F Ft tu v u v a u v a u v

0 0,( , ) ( , ) ( , )F F F F Ftu v a u v f v

Subgrid scale weak form

, ,( , ) ( , ) ( , ) ( , ) ( , )C F F F C F F F Ft tu v u v a u v a u v f v



NATURE OF MULTISCALE DYNAMICS

ũC ūC

ûF

( )A t

1 1

t t

Coarse solution field at start of time step

Coarse solution field at end of time step

( )B ttt

ASSUMPTIONS:

NUMERICAL ALGORITHM FOR SOLUTION OF THE MULTISCALE PDE

COARSE TIME STEP

SUBGRID TIME STEP



REPRESENTING COARSE SOLUTION

( )eD

COARSE MESH ELEMENT

1 0( ) ( ) ( )Cnbf P C

s ssu t x

( , , )Cu x t

1( , ) ( )nbf Cu t x

RANDOM FIELD DEFINED OVER THE ELEMENT

FINITE ELEMENT PIECEWISE POLYNOMIAL REPRESENTATION

USE GPCE TO REPRESENT THE RANDOM COEFFICIENTS

Given the coefficients , the coarse scale solution is completely defined

( )Csu t




( )eD

COARSE MESH ELEMENT ANY INFORMATION FROM COARSE TO SUBGRID SOLUTION CAN BE PASSED ONLY THROUGH

( )Csu t

1 0ˆ ( , , ) ( ) ( , , )Cnbf PF C F

s ssu x t u t x t

COARSE-TO-

SUBGRID MAP

INFORMATION FROM COARSE

SCALE

BASIS FUNCTIONS THAT ACCOUNT FOR FINE SCALE

EFFECTS

DEFINE PROBLEM

DERIVE WEAK FORM

VMS HYPOTHESIS



AFFINE CORRECTION



SOLVING FOR THE COARSE-TO-SUBGRID MAP

( ) , , ( ) , 0C F F C F Fs s s t s s su v K u v

, ,ˆ ˆ( , ) ( , ) ( , ) ( , ) 0C F F F C F F Ft tu v u v a u v a u v

START WITH THE WEAK FORM

APPLY THE MODELS FOR COARSE SOLUTION AND THE C2S MAP

AFTER SOME ASSUMPTIONS ON TIME STEPPING

( ), , ( ), 0F F F Fs s t s sv K v

t t0

THIS IS DEFINED OVER EACH ELEMENT, IN EACH COARSE TIME STEP



BCs FOR THE COARSE-TO-SUBGRID MAP

( ), , ( ), 0F F F Fs s t s sv K v

Fs s s

INTRODUCE A SUBSTITUTION

, , , 0F Fs t sv K v

1x

2x

3x 4x

s s

CONSIDER AN ELEMENT

, ,

,

, ,

, , 0

( , ', ) ( )

( ,0, ) ( , , )

F Fs t s

s t s

s t s t n

v K v

x t

x x t



SOLVING FOR SUBGRID AFFINE CORRECTION

START WITH THE WEAK FORM

0 0,( , ) ( , ) ( , )F F F F Ftu v a u v f v

1x

2x

3x 4xCONSIDER AN ELEMENT

0 0Fu

WHAT DOES AFFINE CORRECTION MODEL?

– EFFECTS OF SOURCES ON SUBGRID SCALE

– EFFECTS OF INITIAL CONDITIONS

IN A DIFFUSIVE EQUATION, THE EFFECT OF INITIAL CONDITIONS DECAY WITH TIME. WE CHOOSE A CUT-OFF

To reduce cut-off effects and to increase efficiency, we can use the quasistatic subgrid assumption

0, , 0Fs t tu

DEFINE PROBLEM

DERIVE WEAK FORM

VMS HYPOTHESIS



AFFINE CORRECTION



COMPUTATIONAL ISSUES Based on the indices in the C2S map and the affine correction, we need to solve (P+1)(nbf) problems in each coarse element

At a closer look we can find that

This implies, we just need to solve (nbf) problems in each coarse element (one for each index s)

0 ( )s s



MODIFIED COARSE SCALE FORMULATION We can substitute the subgrid results in the coarse scale variational formulation to obtain the following

We notice that the affine correction term appears as an anti-diffusive correction

Often, the last term involves computations at fine scale time steps and hence is ignored

0 0

, , , , ,

( , ) , , ,

C C C C C Cs t s s s t s s

C F C F Ct

u v u v u K v

f v K u v u v

DEFINE PROBLEM

DERIVE WEAK FORM

VMS HYPOTHESIS



AFFINE CORRECTION



NUMERICAL EXAMPLES Stochastic investigations

Example 1: Decay of a sine hill in a medium with random diffusion coefficient

The diffusion coefficient has scale separation and periodicity

Example 2: Planar diffusion in microstructures

The diffusion coefficient is computed from a microstructure image

The properties of microstructure phases are not known precisely [source of uncertainty]

Future issues



EXAMPLE 10 1

1

0 00

0 0

1

1 11

1 1

0 1

( )

(2 sin(2 / )) (2 sin(2 / ))1(2 cos(2 / )) (2 sin(2 / ))

(2 sin(2 / )) (2 sin(2 / ))(2 cos(2 / )) (2 sin(2 / ))

1.8, 0.08, 0.04

K K K

P x yKP y P x

P x yKP y P x

P

(x,0, ) sin( )sin( )u x y 0u 0u

0u

0u



EXAMPLE 1 [RESULTS AT T=0.05]












QUASISTATIC SEEMS BETTER There are two important modeling considerations that were neglected for the dynamic subgrid model

Effect of the subgrid component of the initial conditions on the evolution of the reconstructed fine scale solution

Better models for the initial condition specified for the C2S map (currently, at time zero, the C2S map is identically equal to zero implying a completely coarse scale formulation)

In order to avoid the effects of C2S map, we only store the subgrid basis functions beyond a particular time cut-off (referred to herein as the burn-in time)

These modeling issues need to be resolved for increasing the accuracy of the dynamic subgrid model



EXAMPLE 2: DIFFUSION IN A RANDOM MICROSTRUCTURE

DIFFUSION COEFFICIENTS OF INDIVIDUAL CONSTITUENTS NOT KNOWN EXACTLY

– A MIXTURE MODEL IS USED

* *0 *1( ) ( )k k k

( , ) ( ( ) ( )) ( ) ( )k x k k I x k

THE INTENSITY OF THE GRAY-SCALE IMAGE IS MAPPED TO THE CONCENTRATIONS

DARKEST DENOTES PHASE

LIGHTEST DENOTES PHASE

(x,0, ) 0u ( 1) ( 0) ( 0,1)| 0, | 1, | 0x x yuu u kn



RESULTS AT TIME = 0.05MEAN FIRST ORDER GPCE

COEFFSECOND ORDER GPCE

COEFF

RE

CO

NS

TRU

CTE

D F

INE

S

CA

LE S

OLU

TIO

N (V

MS

)FU

LLY

RE

SO

LVE

D

GP

CE

SIM

ULA

TIO

N



RESULTS AT TIME = 0.2MEAN FIRST ORDER GPCE

COEFFSECOND ORDER GPCE

COEFF

RE

CO

NS

TRU

CTE

D F

INE

S

CA

LE S

OLU

TIO

N (V

MS

)FU

LLY

RE

SO

LVE

D

GP

CE

SIM

ULA

TIO

N



HIGHER ORDER TERMS AT TIME = 0.2FOURTH ORDER GPCE

COEFFFIFTH ORDER GPCE

COEFF

RE

CO

NS

TRU

CTE

D F

INE

S

CA

LE S

OLU

TIO

N (V

MS

)FU

LLY

RE

SO

LVE

D

GP

CE

SIM

ULA

TIO

NTHIRD ORDER GPCE

COEFF



UNCERTAINTY RELATED

THE EXAMPLES USED ASSUME A CORRELATION FUNCTION FOR INPUTS, USE KARHUNEN-LOEVE EXPANSION GPCE (OR) SUPPORT-SPACE

– PHYSICAL ASPECTS OF AN UNCERTAINTY MODEL, DERIVATION OF CORRELATION, DISCTRIBUTIONS USING EXPERIMENTS AND SIMULATION

ROUGHNESS PERMEABILITY – AVAILABLE GAPPY DATA

– BAYESIAN INFERENCE

– WHAT ABOUT THE MULTISCALE NATURE ?

BOTH GPCE AND SUPPORT-SPACE ARE SUCCEPTIBLE TO CURSE OF DIMENSIONALITY

– USE OF SPARSE GRID QUADRATURE SCHEMES FOR HIGHER DIMENSIONS (SMOLYAK, GESSLER, XIU)

– FOR VERY HIGH DIMENSIONAL INPUT, USING MC ADAPTED WITH SUPPORT-SPACE, GPCE TECHNIQUES



SPARSE GRID QUADRATURE If the number of random inputs is large (dimension D ~ 10 or higher), the number of grid points to represent an output on the support-space mesh increases exponentially

GPCE for very high dimensions yields highly coupled equations and ill-conditioned systems (relative magnitude of coefficients can be drastically different)

Instead of relying on piecewise interpolation, series representations, can we choose collocation points that still ensure accurate interpolations of the output (solution)



SMOLYAK ALGORITHM

LET OUR BASIC 1D INTERPOLATION SCHEME BE SUMMARIZED AS

IN MULTIPLE DIMENSIONS, THIS CAN BE WRITTEN AS

( ) ( )ii i

i ix

x X

U f a f x

1 11

1 1

( )( ) ( ) ( , , )d di id

i i i id d

i ii ix x

x X x X

U U f a a f x x

TO REDUCE THE NUMBER OF SUPPORT NODES WHILE MAINTAINING ACCURACY WITHIN A LOGARITHMIC FACTOR, WE USE SMOLYAK METHOD

1

0 11

, 1,

0, ,

( ) ( ) ( )( )d

i i id

iiq d q d

i q

U U U i i i

A f A f f

IDEA IS TO CONSTRUCT AN EXPANDING SUBSPACE OF COLLOCATION POINTS THAT CAN REPRESENT PROGRESSIVELY HIGHER ORDER POLYNOMIALS IN MULTIPLE DIMENSIONS

A FEW FAMOUS SPARSE QUADRATURE SCHEMES ARE AS FOLLOWS: CLENSHAW CURTIS SCHEME, MAXIMUM-NORM BASED SPARSE GRID AND CHEBYSHEV-GAUSS SCHEME



SPARSE GRID COLLOCATION METHOD

PREPROCESSINGCompute list of collocation points based on number of stochastic dimensions, N and level of interpolation, q

Compute the weighted integrals of all the interpolations functions across the stochastic space (w i)

Solve the deterministic problem defined by each set of collocated points

POSTPROCESSINGCompute moments and other statistics with simple operations of the deterministic data at the collocated points and the preprocessed list of weights

Solution Methodology

Use any validated deterministic solution procedure.

Completely non intrusive

0.3010.2600.2200.1800.1400.1000.0600.020

0.3010.2600.2200.1800.1400.1000.0600.020

Std deviation of temperature: Natural convection



USING THE COLLOCATION METHOD FOR HIGHER DIMENSIONS

1. Flow through heterogeneous random media

Alloy solidification, thermal insulation, petroleum prospecting

Look at natural convection through a realistic sample of heterogeneous material

Square cavity with free fluid in the middle part of the domain. The porosity of the material is taken from experimental data1

Left wall kept heated, right wall cooled

Numerical solution procedure for the deterministic procedure is a fractional time stepping method

1. Reconstruction of random media using Monte Carlo methods, Manwat and Hilfer, Physical Review E. 59 (1999)



0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Index

Eige

nval

ue

5 10 15 200

5

10

15

FLOW THROUGH HETEROGENEOUS RANDOM MEDIA

Experimental correlation for the porosity of the sandstone.

Eigen spectrum is peaked. Requires large dimensions to accurately represent the stochastic space

Simulated with N= 20

Number of collocation points is 11561 (level 4 interpolation)

Material: Sandstone

Numerically computed

Eigen spectrum



0.90.80.70.60.40.30.20.1

0.90.80.70.60.40.30.20.1

0.90.80.70.60.40.30.20.1

9.16.43.71.0

-1.7-4.4-7.1-9.8

14.210.1

6.12.0

-2.0-6.1

-10.1-14.1

12.18.65.11.7

-1.8-5.3-8.7

-12.2

Snapshots at a few collocation points

Temperature y-Velocity0.940.810.690.560.440.310.190.06

7.04.41.8

-0.8-3.4-6.0-8.6

-11.2

FIRST MOMENT

0.0970.0840.0710.0580.0450.0320.0190.006

5.0564.3823.7083.0342.3591.6851.0110.337

SECOND MOMENT

FLOW THROUGH HETEROGENEOUS RANDOM MEDIA

Temperature

Temperature Y velocity

Y velocity

Streamlines



USING THE COLLOCATION METHOD FOR HIGHER DIMENSIONS

2. Flow over rough surfaces

Thermal transport across rough surfaces, heat exchangers

Look at natural convection through a realistic roughness profile

Rectangular cavity filled with fluid.

Lower surface is rough. Roughness auto correlation function from experimental data2

Lower surface maintained at a higher temperature

Rayleigh-Benard instability causes convection

Numerical solution procedure for the deterministic procedure is a fractional time stepping method

2. H. Li, K. E. Torrance, An experimental study of the correlation between surface roughness and light scattering for rough metallic surfaces, Advanced Characterization Techniques for Optics, Semiconductors, and Nanotechnologies II,

T (y) = 0.5

T (y) = -0.5

y = f(x,ω)



NATURAL CONVECTION ON ROUGH SURFACES

V1

V2

0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

IndexEi

genv

alue

5 10 15 200

4

8

12

16

Experimental ACF

0.440.310.190.06

-0.06-0.19-0.31-0.44

0.440.310.190.06

-0.06-0.19-0.31-0.44

0.440.310.190.06

-0.06-0.19-0.31-0.44

Sample realizations of temperature at collocation points

Experimental correlation for the surface roughness

Eigen spectrum is peaked. Requires large dimensions to accurately represent the stochastic space

Simulated with N= 20 (Represents 94% of the spectrum)

Number of collocation points is 11561 (level 4 interpolation)

Numerically computed Eigen spectrum

0.440.310.190.06

-0.06-0.19-0.31-0.44



NATURAL CONVECTION ON ROUGH SURFACES

0.440.310.190.06

-0.06-0.19-0.31-0.44

FIRST MOMENT

Temperature

Streamlines

0.170.140.120.100.080.060.030.01

7.636.625.604.583.562.541.530.51

Temperature

Y Velocity

SECOND MOMENT

Roughness causes improved thermal transport due to enhanced nonlinearities

Results in thermal plumes

Can look to tailor material surfaces to achieve specific thermal transport



PROBLEM DEFINITION

We have a class of microstructures which share certain features between each other. We want to compute statistical variability of certain diffusion related fields, such as temperature, within this class of microstructures. The variability in the microstructural class is due to variations in grain sizes.

SUB PROBLEMS

1. How do you compute the class of microstructures?

- MaxEnt

2. How do you interrogate this class of microstructures for diffusion problems?

- Stochastic collocation schemes

3. How do you compute microstructures at collocation points?

- POD method



DETAILED SOLUTION METHODOLOGY

Microstructure samples computed using MaxEnt

Index

Eige

nval

ue

5 10 15 200

5

10

15

Use POD technique to compute most energetic eigen

microstructures

Compute collocation points

based on N=20(99.9% energy)

Compute microstructures at collocation points as a linear

combination of the eigen microstructures

+ ++

+ …Each of the collocation samples

is interrogated for computing the required statistical field such as temperature. The

statistics account for topological variability within

the class of microstructures



Limited set of input microstructures computed using

phase field technique

Statistical samples of microstructure at computed using maximum entropy

technique. Collocation point microstructures are derived from this

Diffusion on the class of microstructures computed at collocation points

DIFFUSION IN MICROSTRUCTURES INDUCED BY TOPOLOGICAL UNCERTAINTY

Temperature

Pro

babi

lityVariability of

temperature at a fixed point across the entire class of microstructures

http://www.ime.auc.dk/research/materials/labs/image/foam.jpg





STOCHASTIC MULTISCALING: Open problems

A TYPICAL MULTISCALE PROCESS IS CHARACTERIZED BY PHYSICS AT VARIOUS LENGTH SCALES

– VMS IS ESSENTIALLY A SINGLE GOVERNING EQUATION MODEL

– HOW TO COMBINE VMS WITH OTHER COARSE-GRAINING TYPE, MULTISCALE METHODS

– HOW TO ADAPTIVELY SELECT MULTISCALE REGIONS : POSTERIORI ERROR MEASURES, CONTROL THEORY

IN COUPLING MULTIPLE EQUATION MODELS, STATISTICS MUST BE CONSISTENT

TRANSFERRING DATA, STATISTICS ACROSS LENGTH SCALES USING INFORMATION THEORY

http://www.oja-services.nl/iea-pvps/ar02/images/dnk03.jpg

Materials Process Design and Control Laboratory A STOCHASTIC VARIATIONAL MULTISCALE METHOD FOR DIFFUSION…

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