A MULTISCALE APPROACH TO A MULTISCALE APPROACH TO MATERIALS USING - - PowerPoint PPT Presentation

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Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 169 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801 Email: zabaras@cornell.edu URL: http://www.mae.cornell.edu/zabaras

NICHOLAS ZABARAS NICHOLAS ZABARAS

A MULTISCALE APPROACH TO A MULTISCALE APPROACH TO MATERIALS USING STOCHASTIC MATERIALS USING STOCHASTIC AND COMPUTATIONAL STATISTICS AND COMPUTATIONAL STATISTICS TECHNIQUES TECHNIQUES

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NEED FOR MULTISCALE ANALYSIS NEED FOR MULTISCALE ANALYSIS

Fingering in porous media

Water injector site Porous bed-rock

• Water displaces the oil

layer to the receiving site

• Water accelerates more in

areas with high permeability

• Fingering reduces quality
• f the oil received (polluted

with water) Oil receiver site

• Permeability of bed rock is

inherently stochastic

• Statistics like mean

permeability, correlation structure are usually constant for a given rock type

• Stationary probability

models can be used

• Direct simulation of the effect of uncertainty in permeability on the amount of oil received

requires enormous computational power

• Bed rock length scale – typically of order of kms
• Length scale for permeability variation – typically of order of cms
• Requirement – 10000 blocks for a single dimension (1012 blocks overall)

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NEED FOR MULTISCALE ANALYSIS NEED FOR MULTISCALE ANALYSIS

Transport phenomena in material processes like solidification

Engineering component Microstructural features Formation of dendrites, micro- scale flow structure, heat transfer patterns are highly sensitive to perturbations

• Microstructure is dynamic

and evolves with the materials process

• Uncertainties at the micro-

scale are loosely correlated, however macro-scale features like species concentration, temperature, stresses are highly correlated

• Uncertainty analysis at

micro-scale requires considerable computational effort

• Macro-properties dependant on the dendrite patterns and uncertainty propagation at the micro-

scales

• Uncertainty interactions are no longer satisfy stationarity assumptions – newer probability

models based on image analysis and experimentation needed Length scale ~ meters Length scale ~ 10-4 meters

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STOCHASTIC VARIATIONAL MULTISCALE STOCHASTIC VARIATIONAL MULTISCALE

Physical model

• Statistical variations in

properties are significant

• Discontinuities, loosely

correlated structures in properties

Large scale system Micro scale features

• Statistical variations

are relatively negligible

• Discontinuities get

smoothed out

• Process interactions,

properties become correlated

Uncertainties in boundary and initial conditions

• Discrete

probability distributions to model properties

• Image analysis

to develop the correlation structure Bayesian data analysis interface Green’s functions, RFB type models Explicit subgrid scale model - FEM Experimental, Monte Carlo/ MD models Spectral stochastic/ support-space representation of uncertainty Discretization method like FEM, Spectral, FDM Subgrid scale models Large scale simulation Averaging out the higher statistical features of subgrid scale solutions using Karhunen-loeve/ wavelet filtering Residual Statistical features Large scale solutions

• btained from the

explicit discretization approach

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MULTISCALE TRANSPORT SYSTEMS MULTISCALE TRANSPORT SYSTEMS

Flow past an aerofoil Atmospheric flow in Jupiter Solidification process

Modeling of dendrites at small scale, fluid flow and transport at large scale Large scale turbulent structures, small scale dissipative eddies, surface irregularities Astro-physical flows, effects of gravitational and magnetic fields

• Presence of a variety of spatial and time scales - commonality
• Varied applications – Engineering, Geophysical, Materials
• Boundary conditions, material properties, small scale behavior inherently

are uncertain

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IDEA BEHIND VARIATIONAL MULTISCALE IDEA BEHIND VARIATIONAL MULTISCALE -

• VMS

VMS

Solidification process Physical model

Micro-constitutive laws from experiments, theoretical predictions Subgrid model Resolved model

• Green’s function
• Residual free bubbles
• MsFEM “Hou et al.”
• TLFEM “Hughes et al.”
• FEM
• FDM
• Spectral

Large scale behavior – explicit resolution Small scale behavior – statistical resolution Large scale Residual Subgrid scale solution Where does uncertainty fit in ?

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WHY STOCHASTIC MODELING IN VMS ? WHY STOCHASTIC MODELING IN VMS ?

Model uncertainty Material uncertainty Computational uncertainty

• Imprecise knowledge
• f governing physics
• Models used from

experiments

• Uncertain boundary

conditions

• Inherent initial

perturbations

• Small scale interactions

Surroundings uncertainty Solidification microscale features

• Material properties

fluctuate – only a statistical description possible

• Uncertainty in codes
• Machine precision errors

Not accounted for in analysis here

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SOME PROBABILITY THEORY SOME PROBABILITY THEORY

Probability space – A triplet

• Collection of all basic outcomes of the experiment
• Permutation of the basic outcomes
• Probability associated with the permutations

Ω ( , , ) F P

Ω

F

P :( ) ( , , ) W D T W x t ω × ×Ω →

Sample space Real interval

Ω ξ ω ( )

Random variable – a function Stochastic process – a random function at each space and time point Notations:

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SPECTRAL STOCHASTIC EXPANSIONS SPECTRAL STOCHASTIC EXPANSIONS

1

( , , ) ( , ) ( , ) ( )

i i i

W x t W x t W x t ω ξ ω

∞ =

= +∑

• Covariance kernel required – known only for inputs
• Best possible representation in mean-square sense
• Series representation of stochastic processes with finite second moments

( , ) W x t

( , )

i

W x t ( )

i

ξ ω

• Mean of the stochastic process
• Coefficient dependant on the eigen-pairs of the covariance

kernel of the stochastic process

• Orthogonal random variables

Karhunen-Loeve expansion Generalized polynomial chaos expansion

( , , ) ( , ) (ξ( ))

i i i

W x t W x t ω ψ ω

∞ =

=∑

( , ) (ξ( ))

i i

W x t ψ ω

• Coefficients dependant on chaos-polynomials chosen
• Chaos polynomials chosen from Askey-series (Legendre –

uniform, Jacobi – beta)

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SUPPORT SUPPORT-

• SPACE/STOCHASTIC GALERKIN

SPACE/STOCHASTIC GALERKIN

( ( )) f ξ ω

• Joint probability density function of the inputs

{ ( ) : ( ( )) 0} A f ξ ω ξ ω = >

• The input support-space denotes the regions where input

joint PDF is strictly positive

Triangulation

• f the support-

space Any function can be represented as a piecewise polynomial on the triangulated support-space

• Function to be approximated

( ( )) X ξ ω ( ( ))

h

X ξ ω

• Piecewise polynomial approximation
• ver support-space

( ( ))

h

X ξ ω

L2 convergence – (mean-square)

2 1

( ( ( )) ( ( ))) ( ( ))d

h q A

X X f Ch ξ ω ξ ω ξ ω ξ

+

− ≤

∫

h = mesh diameter for the support-space discretization q = Order of interpolation

Error in approximation is penalized severely in high input joint PDF regions. We use importance based refinement of grid to avoid this

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BOUSSINESQ NATURAL CONVECTION BOUSSINESQ NATURAL CONVECTION

g 2

( )Pr( ) e 2Pr( ) ( ) 1 ( ) [ ( ) ] 2

T

v v v v Ra t v t pI v v v v ω ω θ σ θ θ θ σ ω ε ε ∇ = ∂ + ∇ = − + ∇ ∂ ∂ + ∇ = ∇ ∂ = − + = ∇ + ∇ i i i i

hm

Γ

gm

Γ

gt

Γ

ht

Γ

g

v v =

g

θ θ =

. n q θ ∇ =

. n h σ ∇ =

• Temperature gradients are small
• Constant fluid properties except in the

force term

• viscous dissipation negligible

Momentum equation boundary conditions Energy equation boundary conditions

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DEFINITION OF FUNCTION SPACES DEFINITION OF FUNCTION SPACES

1 2 2 2 2 2 2 1

( ) { : ( ( ) )d } ( ) { : d } ( ) { : d } ( ) { : d }

D D T T

H D v v v x L D v v x L T w w t L T w w t = + ∇ < ∞ = < ∞ = < ∞ = < ∞

∫ ∫ ∫ ∫

D

T

• Spatial domain
• Time interval of simulation [0,tmax]

Function spaces for deterministic quantities Function spaces for stochastic quantities

2 2( )

{ : dP } L ξ ξ

Ω

Ω = < ∞

∫

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DERIVED FUNCTION SPACES DERIVED FUNCTION SPACES

1 d 2 2 1 d 2 2 1 2 2 2 1 2 2 1 2

{ : [ ( ) ( ) ( )] , } { : [ ( ) ( )] , } { : ( ) ( ) ( )} { : ( ) ( )} { : ( ) ( ) ( ), } { : ( ) ( ), }

g gm gm g gt gt

V v v H D L T L v v on V w w H D L w

• n

Q p p L D L T L Q q q L D L E H D L T L

• n

E w w H D L w

• n

θ θ θ θ = ∈ × × Ω = Γ = ∈ × Ω = Γ = ∈ × × Ω = ∈ × Ω = ∈ × × Ω = Γ = ∈ × Ω = Γ

Velocity function spaces

• Uncertainty is incorporated in the function space definition
• Solution velocity, temperature and pressure are in general multiscale quantities (as

Rayleigh number increases) the computational grid capture less and less information Pressure function spaces Temperature function spaces

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WEAK FORMULATION WEAK FORMULATION – – BOUSSINESQ EQNS BOUSSINESQ EQNS

v

( , ) ( . , ) ( , ) ( , )

ht

t

w v w w q w θ θ θ

Γ

∂ + ∇ + ∇ ∇ =

v g

( , ) ( . , ) ( , ( )) ( , ) ( ( )Pr( ) e , ) ( , )

hm

tv w

v v w v h w Ra w v q σ ε ω ω θ

Γ

∂ + ∇ + = − ∇ = i

E θ ∈

w E ∈

Find such that for all , the following holds

[ , ] [ , ] v p V Q ∈ [ , ] [ , ] w q V Q ∈

Find such that for all , the following holds Energy equation – weak form Momentum and continuity equations – weak form

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VARIATIONAL MULTISCALE DECOMPOSITION VARIATIONAL MULTISCALE DECOMPOSITION

', ', ' ', ', ' V V V V V V Q Q Q Q Q Q E E E E E E = ⊕ = ⊕ = ⊕ = ⊕ = ⊕ = ⊕

', ', ' v v v p p p θ θ θ = + = + = +

• Bar denotes large scale/resolved quantity
• Prime denotes subgrid scale/ unresolved quantity

Induced multiscale decomposition for function spaces Interpretation

• Large scale function spaces correspond to finite element spaces – piecewise

polynomial and hence are finite dimensional

• Small scale function spaces are infinite dimensional

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SCALE DESOMPOSED WEAK FORM SCALE DESOMPOSED WEAK FORM -

• ENERGY

ENERGY

v v

( ', ) ( . . ', ) ( ', ) ( , ) ( ', ') ( . . ', ') ( ', ') ( , ')

ht ht

t t t t

w v v w w q w w v v w w q w θ θ θ θ θ θ θ θ θ θ θ θ

Γ Γ

∂ + ∂ + ∇ + ∇ + ∇ + ∇ ∇ = ∂ + ∂ + ∇ + ∇ + ∇ + ∇ ∇ =

Find and such that for all and , the following holds

' ' E θ ∈

E θ ∈

w E ∈

' ' w E ∈

Small scale strong form of equations

2 2

' . ' ' ( . )

t t

v v R θ θ θ θ θ θ ∂ + ∇ −∇ = − ∂ + ∇ −∇ =

2

( ) : . L v θ θ θ = ∇ −∇

1 1

1 ( ), (1 )

t n n n n n n

f f f f f f t

γ

γ γ δ

+ + +

∂ = − = + −

Time discretization rule

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ELEMENT FOURIER TRANSFORM ELEMENT FOURIER TRANSFORM D

Spatial domain

( ) e

D

• ( )

ˆ ˆ exp( ) ( , )d ( , ) ( , )

e

j j j D

k k g k x n i g x i g k i g k x h h h ω ω ω ∂ = − Γ + ≈ ∂

∫

i

( )

ˆ( , ) exp( ) ( , )d

e

D

k x g k i g x x h ω ω = −

∫

i

Subgrid scale solution denotes unresolved part

• f the solution, hence dominated by large wave

number modes!!

Spatial derivative approximation

• Other techniques to solve for an approximate subgrid solution include:
• Residual-free bubbles, Green’s function approach
• Two-level finite element method – explicit evaluation
• Multiscale FEM – Incorporates subgrid features in large scale weighting function

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ALGEBRAIC SUBGRID SCALE MODEL ALGEBRAIC SUBGRID SCALE MODEL

2 2

' . ' ' ( . )

t t

v v R θ θ θ θ θ θ ∂ + ∇ −∇ = − ∂ + ∇ −∇ =

' '

( )

t n n n

L R

γ γ

θ θ +

+

∂ + =

2 ' ' 2

1 1 ˆ ˆ ˆ

n n n

k v k i R t h h t

γ γ

θ θ γδ γδ

+ +

   + +  = +     i

1 2 2 2 ' ' 1 2 2

1 1 1 ,

n t n n t

v R c c t h t h

γ γ

θ τ θ τ γδ γδ

− + +

          ≈ + = + +                

Time discretization Element Fourier transform Parseval’s theorem Mean value theorem

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STABILIZED FINITE ELEMENT EQUATIONS STABILIZED FINITE ELEMENT EQUATIONS

Strong regularity conditions

v 2 2 1 ' 2 1

( , ) ( , ) ( , ) ( , ) ( , ( ) ( , /( ) ) )

ht

t n n n Nel t n n n t e Nel n t t e

w v w w q w v w v w w w t v w w w

γ γ γ γ

θ θ θ θ θ θ τ θ τ γδ τ

+ + Γ + + = =

∂ + ∇ + ∇ ∇ −   + ∂ + ∇ −∇ − + ∇ + ∇     + − ∇ −∇ − =  

∑ ∑

i

• i

i

• i

2 v v

( ', ) ( ', ), ( ', ) ( ', ) v w v w w w θ θ θ θ ∇ = − ∇ ∇ ∇ = − ∇ i i

Stabilized weak formulation

/( ) w w t γδ =

• where

Time integration has a role to play in the stabilization (Codina et al.) Stochastic intrinsic time scale (subgrid scale solution has a stochastic model)

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CONSIDERATIONS FOR MOMENTUM EQUATION CONSIDERATIONS FOR MOMENTUM EQUATION

Picard’s linearization

' v v v v v v ∇ ≈ ∇ + ∇ i i i

Fairly accurate for laminar up to transition (moderate Reynolds number flows) For high Reynolds number flows, the term assumes importance since small scales act as momentum dissipaters

' ' v v ∇ i

Small scale strong form of equations

g

' ' ( ', ') ( )Pr( ) e ( , ) '

t t

v v v v p Ra v v v v p v v σ ω ω θ σ ∂ + ∇ −∇ = − − ∂ − ∇ + ∇ ∇ = −∇ i i i i i i

g

( )Pr( ) e ( , )

mom t n n n n

R Ra v v v v p

γ γ γ

ω ω θ σ

+ + +

= − − ∂ − ∇ + ∇ i i

con n

R v

γ +

= −∇i

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SUBGRID VELOICTY AND PRESSURE SUBGRID VELOICTY AND PRESSURE

2 ' ' ' 2 '

1 1 ˆ ˆ ˆ ˆ Pr( ) ˆ ˆ

n n mom n n con

k v k k i v i p R v t h h h t k v i R h

γ γ γ

ω γδ γδ

+ + +

   + +  + = +     = i i

Element Fourier transform Simultaneous solve Parseval’s theorem Mean-value theorem

1 2 2 2 2 2 ' 1 1 ' 2 ' 1

, Pr( ) ,

n c con c n n m mom m c

c v h h p R c t c v h v R t c

γ γ

τ τ ω γδ τ τ γδ τ

+ +

        ≈ = + +               ≈ + =    

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STABILIZED FINITE ELEMENT EQUATIONS STABILIZED FINITE ELEMENT EQUATIONS

Strong regularity conditions

v v

( ', ) ( ', ), (Pr( ) ( '), ( )) ( ',Pr( ) ( )) v v w v v w v w v w ω ε ε ω ε ∇ = − ∇ = − ∇ i i i

Stabilized weak formulation

/( ) w w t γδ =

• where

( )

' g 1 '

( , ) ( , ) (2Pr( ) ( ), ( )) ( , ) ( )Pr( ) e ( , ) , Pr( ) ( ) ( , ) ( , )

hm

t n n n n Nel n t n n n n m e n n c

v v v w p w v w h w v Ra v v v v p w v w w t v w v w

γ γ γ γ γ γ γ

ω ε ε ω ω θ σ τ ω ε γδ τ

+ + + Γ + + + = +

∂ + ∇ − ∇ + −   + − − ∂ − ∇ + − + ∇ + ∇     − + ∇ ∇ =

∑

i i

• i

i i

• i

i

' g 1

( , ) ( ( )Pr( ) e ( , ) ,

Nel n n t n n n n m e

v v q Ra v v v v p q t

γ γ γ γ

ω ω θ σ τ γδ

+ + + + =

  ∇ + − − ∂ − ∇ + − ∇ =    

∑

i i

Momentum equation Continuity equation

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IMPLEMENTATION ISSUES IMPLEMENTATION ISSUES -

• GPCE

GPCE D

Spatial domain

( ) e

D

nbf 1

( , ) ( ) ( ) f x f N x

α α α

ω ω

=

= ∑

Generic function Random coefficient Galerkin shape function GPCE expansion for random coefficients

( ) (ξ( ))

P i i i

f f

α α

ω ψ ω

=

= ∑

Random coefficient Askey polynomial

• Each node has P+1 degrees of

freedom for each scalar stochastic process

• Interpolation is accomplished by

tensor-product basis functions

• (P+1) times larger than

deterministic problems

• Assume the inputs have been represented in Karhunen-Loeve expansion such that the

input uncertainty is summarized by few random variables

1 n

ξ( ) {ξ ( ), ,ξ ( )} ω ω ω = …

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IMPLEMENTATION ISSUES IMPLEMENTATION ISSUES – – SUPPORT SPACE SUPPORT SPACE D

Spatial domain

A stochastic process can be interpreted as a random variable at each spatial point

( , , ) W x t ω

Two-level grid approach Spatial grid Support-space grid

• Mesh dense in

regions of high input joint PDF

( ) e

D

Element

( , ) x ω

A

( ') e

A

nbf nbf ' nbf ' 1 1 1

( , ) ( ) ( ) ( ) ( )

i i i j j i i j i

f x f N x f N N x ω ω ω

= = =

= =

∑ ∑∑

• There is finite element interpolation at

both spatial and random levels

• Each spatial location handles an

underlying support-space grid

• Highly OOP structure

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NUMERICAL EXAMPLES NUMERICAL EXAMPLES

• Flow past a circular cylinder with uncertain inlet velocity – Transient behavior
• RB convection in square cavity with adiabatic body at the center – uncertainty in

the hot wall temperature (simulation away from critical points)

• Transient behavior
• Simulation using GPCE, validation using deterministic simulation
• RB convection in square cavity – uncertainty in Rayleigh number (simulation

about a critical point)

• Failure of the GPCE approach
• Analysis support-space method
• Comparison of prediction by support-space method with deterministic

simulations In the last example, temperature contours do not convey useful information and hence are ignored

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FLOW PAST A CIRCULAR CYLINDER FLOW PAST A CIRCULAR CYLINDER

X Y

5 10 15 20 2 4 6 8

• Computational details – 2000 bilinear elements for spatial grid, third order Legendre chaos

expansion for velocity and pressure, preconditioned parallel GMRES solver

• Time of simulation

– 180 non- dimensional units

• Inlet velocity –

Uniform random variable between 0.9 and 1.1

• Kinematic viscosity

0.01

• Time stepping –

0.03 non- dimensional units

Inlet Traction free outlet

( ) U ω

No-slip No-slip

Investigations

• Onset of vortex shedding
• Shedding near wake regions, flow statistics

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ONSET OF VORTEX SHEDDING ONSET OF VORTEX SHEDDING

X Y

2 4 6 8 10 12 14 16 18 20 2 4 6 8

• 0.50 -0.42 -0.34 -0.26 -0.18 -0.10 -0.02 0.06

0.14 0.22 0.30 0.39 0.47 0.55 0.63

X Y

2 4 6 8 10 12 14 16 18 20 2 4 6 8

• 0.14 -0.12 -0.10 -0.08 -0.06 -0.04 -0.03 -0.01 0.01 0.03

0.05 0.07 0.09 0.10 0.12

Mean pressure at t = 79.2

• Vortex shedding is just

initiated

• Not in the periodic shedding

mode First order term in Legendre chaos expansion of pressure at t = 79.2

• Vortex shedding is

predominant

• Periodic shedding behavior

noticed

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FULLY DEVELOPED VORTEX SHEDDING FULLY DEVELOPED VORTEX SHEDDING

Mean pressure contours First order term in LCE of pressure contours Second order term in LCE of pressure contours

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VORTEX SHEDDING VORTEX SHEDDING -

• CONTD

CONTD

Frequency Amplitude

0.1 0.2 0.3 0.4 0.5 0.03 0.06 0.09 0.12 0.15

• The FFT of the mean velocity shows a broad spectrum with peak at frequency 0.162
• The spectrum is broad in comparison to deterministic results wherein a sharp shedding

frequency is obtained

• Mean velocity has superimposed frequencies
• Mean velocity has comparatively lower magnitude than the deterministic velocity (Y-

velocities compared at near wake region)

X V

5 8 11 14 17 20

• 0.6
• 0.4
• 0.2

0.2 0.4 0.6

Deterministic Mean

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RB CONVECTION RB CONVECTION -

• CENTRAL ADIABATIC BODY

CENTRAL ADIABATIC BODY

• Computational details – 2048 bilinear elements for spatial grid, third order Legendre chaos

expansion for velocity, pressure and temperature, preconditioned parallel GMRES solver

• Time of simulation –

1.5 non-dimensional units

• Rayleigh number - 104
• Prandtl number – 0.7
• Time stepping – 0.002

non-dimensional units

• Transient behavior of temperature statistics ( Flow results in paper )

X Y

0.25 0.5 0.75 1 0.25 0.5 0.75 1

Cold wall Hot wall Insulated Insulated Adiabatic body

c

θ = [0.95,1.05]

h

U θ =

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TRANSIENT BEHAVIOR TRANSIENT BEHAVIOR – – TEMPERATURE TEMPERATURE

• Mean temperature

contours

• Steady conduction like

state not reached

• Second order term in the

Legendre chaos expansion of temperature

• First order term in the

Legendre chaos expansion of temperature

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CAPTURING UNSTABLE EQUILIBRIUM CAPTURING UNSTABLE EQUILIBRIUM

• Computational details – 1600 bilinear elements for spatial grid
• Time of simulation – 1.5 non-

dimensional units

• Rayleigh number – uniformly

distributed random variable between 1530 and 1870 (10% fluctuation about 1700)

• Prandtl number – 6.95
• Time stepping – 0.002 non-

dimensional units

• Support-space grid – One-

dimensional with ten linear elements

• Simulation about the critical Rayleigh number – conduction below, convection above
• Both GPCE and support-space methods are used separately for addressing the problem
• Failure of Generalized polynomial chaos approach
• Support-space method – evaluation and results against a deterministic simulation

Cold wall Hot wall Insulated Insulated

c

θ = 1

h

θ =

X Y

0.25 0.5 0.75 1 0.25 0.5 0.75 1

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FAILURE OF THE GPCE FAILURE OF THE GPCE

X Y

0.25 0.5 0.75 1 0.25 0.5 0.75 1 9.2E-07 5.7E-06 1.0E-05 1.5E-05 2.0E-05 2.5E-05 3.0E-05 3.4E-05

X Y

0.25 0.5 0.75 1 0.25 0.5 0.75 1

• 6.4E-08

4.3E-07 9.3E-07 1.4E-06 1.9E-06 2.4E-06 2.9E-06 3.4E-0

X Y

0.25 0.5 0.75 1 0.25 0.5 0.75 1

• 5.0E-03 -3.6E-03 -2.1E-03 -7.1E-04

7.1E-04 2.1E-03 3.6E-03 5.0E-03

X Y

0.25 0.5 0.75 1 0.25 0.5 0.75 1

• 3.2E-03 -2.0E-03 -8.0E-04

3.9E-04 1.6E-03 2.8E-03 4.0E-03 5.2E-03

X-vel X-vel Y-vel Y-vel Mean X- and Y- velocities determined by GPCE yields extremely low values !! (Gibbs effect) X- and Y- velocities

• btained from a

deterministic simulation with Ra = 1870 (the upper limit)

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PREDICTION BY SUPPORT PREDICTION BY SUPPORT-

• SPACE METHOD

SPACE METHOD

X Y

0.25 0.5 0.75 1 0.25 0.5 0.75 1

• 5.0E-03 -3.6E-03 -2.1E-03 -7.1E-04

7.1E-04 2.1E-03 3.6E-03 5.0E-03

X Y

0.25 0.5 0.75 1 0.25 0.5 0.75 1

• 3.2E-03 -2.0E-03 -8.0E-04

3.9E-04 1.6E-03 2.8E-03 4.0E-03 5.2E-03

X-vel X-vel Y-vel Y-vel Mean X- and Y- velocities determined by support-space method at a realization Ra=1870 X- and Y- velocities

• btained from a

deterministic simulation with Ra = 1870 (the upper limit)

X Y

0.25 0.5 0.75 1 0.25 0.5 0.75 1

• 5.0E-03 -3.6E-03 -2.1E-03 -7.1E-04

7.1E-04 2.1E-03 3.6E-03 5.0E-03

X Y

0.25 0.5 0.75 1 0.25 0.5 0.75 1

• 3.2E-03 -2.0E-03 -7.4E-04

4.9E-04 1.7E-03 2.9E-03 4.2E-03 5.4E-03

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MICROSTRUCTURE RECONSTRUCTION MICROSTRUCTURE RECONSTRUCTION & CLASSIFICATION WITH & CLASSIFICATION WITH APPLICATIONS IN APPLICATIONS IN MATERIALS MATERIALS-

• BY

BY-

• DESIGN

DESIGN

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

Email: zabaras@cornell.edu URL: http://www.mae.cornell.edu/zabaras/

Veeraraghavan Sundararaghavan and Nicholas Zabaras

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MATERIALS DESIGN FRAMEWORK

Machine learning schemes Microstructure Information library Accelerated Insertion of new materials Optimization of existing materials Tailored application specific material properties Virtual process simulations to evaluate alternate designs Computational process design simulator Virtual materials design framework

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DESIGNING MATERIALS WITH TAILORED PROPERTIES

Micro problem driven by the velocity gradient L Macro problem driven by the macro-design variable β

Bn+1

Ω = Ω (r, t; L)

~

Polycrystal plasticity

x = x(X, t; β) L = L (X, t; β)

L = velocity gradient

Fn+1 B0 Reduced Order Modeling Data mining techniques Database Multi-scale Computation Design variables (β) are macro design variables Processing sequence/parameters Design objectives are micro-scale averaged material/process properties

Process Process parameters Values .. Tension Strain rate, time, velocity gradient 0.56 Forging Forging velocity ,Initial Temperature 2.13

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DATABASE FOR POLYCRYSTAL MATERIALS

Meso-scale database for polycrystalline materials

Machine Learning

Database Feature Extraction Data Organization

Reduced order basis generation

Youngs Modulus

RD TD

Multi-scale microstructure evolution models Process design for desired properties

RD R-value

0.985 0.99 0.995 1 1.005 1.01 1.015 1.02 1.025 1.03 1.035 10 20 30 40 50 60 70 80 90 Angle from rolling direction

Initial Intermediate Optimal Desired

TD

ODF Pole Figures

20 40 60 80 144 144.1 144.2 144.3 144.4 144.5 144.6 144.7

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MOTIVATION MOTIVATION

• 1. Creation of 3D microstructure models for property analysis from

2D images 2. 3D imaging requires time and effort. Need to address real–time methodologies for generating 3D realizations. 3. Make intelligent use of available information from computational models and experiments.

vision Database Pattern recognition Microstructure Analysis 2D Imaging techniques

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PATTERN RECOGNITION (PR) STEPS PATTERN RECOGNITION (PR) STEPS

DATABASE CREATION FEATURE EXTRACTION TRAINING PREDICTION

Datasets: microstructures from experiments or physical models Extraction of statistical features from the database Creation of a microstructure class hierarchy: Classification methods Prediction of 3D reconstruction, process paths, etc. PATTERN RECOGNITION : A DATA-DRIVEN OPTIMIZATION TOOL

• Feature matching for reconstruction of 3D microstructures
• Microstructure representation
• Texture(ODF) classification for process path selection

Real-time

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3D MICROSTRUCTURE RECOGNITION: A TWO 3D MICROSTRUCTURE RECOGNITION: A TWO-

• CLASS PROBLEM

CLASS PROBLEM

Training Features

36.52 36.52 14.08 14.08 52.15 52.15 24.02 24.02

• 1

1 9.52 9.52 20.01 20.01 160.12 160.12 21.30 21.30 1 1 11.30 11.30 25.30 25.30 158.20 158.20 20.10 20.10 1 1 2.52 2.52 23.01 23.01 154.12 154.12 23.32 23.32 1 1

Feature Vector (x) Feature Vector (x) – – single feature type (Grain size feature) single feature type (Grain size feature) Class(y) Class(y) Match lower order features using PR New Feature (From a 2D image) – To which 3D class does this belong?

2.31 2.31 24.10 24.10 153.14 153.14 21.45 21.45

Heyn intercept histogram of a 2D cross-section

Feature Extraction

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MULTIPLE CLASSES MULTIPLE CLASSES

Class-A Class-B

Class-C

A C B A B C

Given a new planar microstructure with its ‘s’ features given by find the class of 3D microstructure (y ) to which it is most likely to belong.

[1,2,3,..., ] p ∈

1 2

1 1 1 2 2 2 1 1 2 2 1 2 1 2

{ , ,...., }, { , ,...., },..., { , ,...., }

s

T T T s s s m m s m

x x x x x x x x x x x x = = =

p = 3

One Against One Method:

• Step 1: Pair-wise

classification, for a p class problem

• Step 2: Given a data point,

select class with maximum votes out of

( 1) 2 p p − ( 1) 2 p p −

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TWO PHASE MICROSTRUCTURE: CLASS HIERARCHY TWO PHASE MICROSTRUCTURE: CLASS HIERARCHY

Class - 1 3D Microstructures

Feature vector : Three point probability function

3D Microstructures Class - 2

Feature:

Autocorrelation function

LEVEL - 1 LEVEL - 2

r µm γ

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STATISTICAL CORRELATION MEASURES STATISTICAL CORRELATION MEASURES

MC Sampling: Computing the three point probability function of a 3D microstructure(40x40x40 mic) S3(r,s,t), r = s = t = 2, 5000 initial points, 4 samples at each initial point. Rotationally invariant probability functions (Si

N ) can be interpreted as

the probability of finding the N vertices of a polyhedron separated by relative distances x1, x2,..,xN in phase i when tossed, without regard to orientation, in the microstructure.

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3D RECONSTRUCTION 3D RECONSTRUCTION

Ag-W composite (Umekawa 1969) A reconstructed 3D microstructure

3 point probability function Autocorrelation function

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ELASTIC PROPERTIES: YOUNGS MODULUS ELASTIC PROPERTIES: YOUNGS MODULUS

170 190 210 230 250 270 290 310 200 400 600 800 1000 Temperature (deg-C) Youngs Modulus (GPa) HS bounds BMMP bounds Experimental FEM

3D image derived through pattern recognition

Experimental image

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MICROSTRUCTURE REPRESENTATION USING SVM & PCA MICROSTRUCTURE REPRESENTATION USING SVM & PCA

COMMON-BASIS FOR MICROSTRUCTURE REPRESENTATION

Does not decay to zero A DYNAMIC LIBRARY APPROACH

• Classify microstructures based
• n lower order descriptors.
• Create a common basis for

representing images in each class at the last level in the class hierarchy.

• Represent 3D microstructures

as coefficients over a reduced basis in the base classes.

• Dynamically update the basis

and the representation for new microstructures

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PCA MICROSTRUCTURE RECONSTRUCTION PCA MICROSTRUCTURE RECONSTRUCTION

Pixel value round-off

Basis Components

X 5.89 X 14.86

+

Project

• nto basis

Reconstruct using two basis components

Representation using just 2 coefficients (5.89,14.86)

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ORIENTATION FIBERS: LOWER ORDER FEATURES OF AN ODF 1 ( 1 . r h y+ (h+y)) h y λ = × +

Points (r) of a (h,y) fiber in the fundamental region

angle

Crystal Axis = h Sample Axis = y

φ

φ

Rotation (R) required to align h with y (invariant to , )

φ φ

Fibers: h{1,2,3}, y || [1,0,1] {1,2,3} Pole Figure Point y (1,0,1)

• h||y

R.h=h, h||y 1 P(h,y) = (P (h,y)+P (-h,y)) 2 1 P(h,y) = 2 Adθ π ∫

Integration is performed over all fibers corresponding to crystal direction h and sample direction y For a particular (h), the pole figure takes values P(h,y) at locations y on a unit sphere.

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LIBRARY FOR TEXTURES

[100] pole figure [110] pole figure

Parameter Feature Vector DATABASE OF ODFs

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SUPERVISED CLASSIFICATION USING SUPPORT VECTOR MACHINES

Given ODF/texture Tension (T) S t a g e 1

LEVEL – 2 CLASSIFICATION Plane strain compression

T+P

LEVEL – I CLASSIFICATION Tension identified

S t a g e 2 Stage 3

Multi-stage classification with each class affiliated with a unique process

Identifies a unique processing sequence: Fails to capture the non-uniqueness in the solution

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UNSUPERVISED CLASSIFICATION

Find the cluster centers {C1,C2,…,Ck} such that the sum of the 2-norm distance squared between each feature xi , i = 1,..,n and its nearest cluster center Ch is minimized.

2 1 2 2 1,.., 1

1 ( , ,.., ) ( ) 2

min

n k h i h k i

J c c c x C

= =

= −

∑

Identify clusters Clusters DATABASE OF ODFs Feature Space Cost function

Each class is affiliated with multiple processes

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PROCESS PARAMETERS LEADING TO DESIRED PROPERTIES Young’s Modulus (GPa) Angle from rolling direction

CLASSIFICATION BASED ON PROPERTIES

Class - 1 Class - 2 Class - 3 Class - 4

0.5 0.25 0.25

• 1.25

0.75 0.5 0.75

• 1.25

Velocity Gradient Different processes, Similar properties Database for ODFs Property Extraction ODF Classification Identify multiple solutions

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A TWO A TWO-

• STAGE PROBLEM

STAGE PROBLEM

Process – 2 Plane strain compression α = 0.3515 Process – 1 Tension α = 0.9539 Initial Conditions: Stage 1 Sensitivity of material property Initial Conditions- stage 2 DATABASE

Reduced Basis φ(1) φ(2) Direct problem Sensitivity problem

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MULTIPLE PROCESS ROUTES MULTIPLE PROCESS ROUTES

10 20 30 40 50 60 70 80 90 144 144.5 145 145.5

Angle from the rolling direction Youngs Modulus (GPa)

Desired Young’s Modulus distribution Magnetic hysteresis loss distribution

10 20 30 40 50 60 70 80 90 1.205 1.21 1.215 1.22 1.225 1.23 1.235 1.24

Magnetic hysteresis loss (W/kg)

Stage: 1 Shear-1 α = 0.9580 Stage: 2 Plane strain compression (α = -0.1597 ) Stage: 1 Shear -1 α = 0.9454 Stage: 2 Rotation-1 (α = -0.2748) Stage 1: Tension α = 0.9495 Stage 2: Shear-1 α = 0.3384 Stage 1: Tension α = 0.9699 Stage 2: Rotation-1 α = -0.2408 Classification

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DESIGN FOR DESIRED ODF: A MULTI STAGE PROBLEM DESIGN FOR DESIRED ODF: A MULTI STAGE PROBLEM

5 10 15 20 25 0.2 0.4 0.6 0.8 1 Iteration Index Normalized objective function

Initial guess, α1 = 0.65, α2 = -0.1 Desired ODF Optimal- Reduced order control Full order ODF based on reduced

• rder control parameters

Stage: 1 Plane strain compression (α1 = 0.9472) Stage: 2 Compression (α2 = -0.2847)

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DESIGN FOR DESIRED MAGNETIC PROPERTY DESIGN FOR DESIRED MAGNETIC PROPERTY

I te ra tio n I n d e x Normalized objective function 5 1 0 1 5 0 .2 0 .4 0 .6 0 .8 1

h

Crystal <100> direction. Easy direction

• f

magnetization – zero power loss External magnetization direction

20 40 60 80 1.21 1.215 1.22 1.225 1.23 1.235 Angle from the rolling direction Magnetic hysteresis loss (W/Kg)

Desired property distribution Optimal (reduced) Initial

Stage: 1 Shear – 1 (α1 = 0.9745) Stage: 2 Tension (α2 = 0.4821)