A MULTISCALE APPROACH TO MATERIALS USING ...paulino.ce.gatech.edu/.../TUESDAY/T_Morning_II/Zabaras.pdfmodel - FEM Experimental, Monte Carlo/ MD models Spectral stochastic/ support-space

Materials Process Design and Control LaboratoryMaterials Process Design and Control LaboratoryCCOORRNNEELLLL 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 LaboratorySibley School of Mechanical and Aerospace Engineering

169 Frank H. T. Rhodes HallCornell University

Ithaca, NY 14853-3801

Email: [email protected]: http://www.mae.cornell.edu/zabaras

NICHOLAS ZABARASNICHOLAS ZABARAS

A MULTISCALE APPROACH TO A MULTISCALE APPROACH TO MATERIALS USING STOCHASTIC MATERIALS USING STOCHASTIC

AND COMPUTATIONAL STATISTICS AND COMPUTATIONAL STATISTICS TECHNIQUESTECHNIQUES


NEED FOR MULTISCALE ANALYSISNEED FOR MULTISCALE ANALYSISFingering 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 of 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)


NEED FOR MULTISCALE ANALYSISNEED FOR MULTISCALE ANALYSISTransport 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


STOCHASTIC VARIATIONAL MULTISCALESTOCHASTIC 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 obtained from the

explicit discretization approach


MULTISCALE TRANSPORT SYSTEMSMULTISCALE 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


IDEA BEHIND VARIATIONAL MULTISCALE IDEA BEHIND VARIATIONAL MULTISCALE -- VMSVMS

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

?


WHY STOCHASTIC MODELING IN VMS ?WHY STOCHASTIC MODELING IN VMS ?Model uncertainty

Material uncertainty Computational uncertainty

• Imprecise knowledge of 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


SOME PROBABILITY THEORYSOME 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 PΩFP

: ( )( , , )

W D TW 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:


SPECTRAL STOCHASTIC EXPANSIONSSPECTRAL STOCHASTIC EXPANSIONS

1( , , ) ( , ) ( , ) ( )i i

iW x t W x t W x tω 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( , )iW x t

( )iξ 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 0( , , ) ( , ) (ξ( ))i i

iW x t W x tω ψ ω

∞

=

=∑( , )(ξ( ))

i

i

W x tψ ω

- Coefficients dependant on chaos-polynomials chosen

- Chaos polynomials chosen from Askey-series (Legendre –uniform, Jacobi – beta)


SUPPORTSUPPORT--SPACE/STOCHASTIC GALERKINSPACE/STOCHASTIC GALERKIN( ( ))f ξ ω - Joint probability density function of the inputs

{ ( ) : ( ( )) 0}A fξ ω ξ ω= > - The input support-space denotes the regions where input joint PDF is strictly positive

Triangulation of the support-space

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

- Function to be approximated( ( ))X ξ ω

( ( ))hX ξ ω - Piecewise polynomial approximation over support-space

( ( ))hX ξ ω

L2 convergence – (mean-square)2 1( ( ( )) ( ( ))) ( ( ))dh q

A

X X f Chξ 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


BOUSSINESQ NATURAL CONVECTIONBOUSSINESQ NATURAL CONVECTION

g

2

0

( ) Pr( ) e

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

T

vv v v Rat

vt

pI v

v v v

ω ω θ σ

θ θ θ

σ ω ε

ε

∇ =∂

+ ∇ = − +∇∂

∂+ ∇ = ∇

∂= − +

= ∇ + ∇

i

i i

i

hmΓ

gmΓ

gtΓ

htΓ

gv v=

gθ θ=

0.n qθ∇ =

.n hσ∇ =

• Temperature gradients are small

• Constant fluid properties except in the force term

• viscous dissipation negligible

Momentum equation boundary conditions

Energy equation boundary conditions


DEFINITION OF FUNCTION SPACESDEFINITION OF FUNCTION SPACES

1 2 2

22

22

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

= + ∇ < ∞

= < ∞

= < ∞

= < ∞

∫

∫

∫

∫DT

- Spatial domain

- Time interval of simulation [0,tmax]

Function spaces for deterministic quantities

Function spaces for stochastic quantities

22 ( ) { : dP }L ξ ξ

Ω

Ω = < ∞∫


DERIVED FUNCTION SPACESDERIVED FUNCTION SPACES

1 d2 2

1 d0 2

2 1 2

0 2 21

2 2

10 2

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

{ : [ ( ) ( )] , 0 }

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

{ : ( ) ( ) ( ), }

{ : ( ) ( ), 0 }

g gm

gm

g gt

gt

V v v H D L T L v v on

V w w H D L w on

Q p p L D L T LQ q q L D L

E H D L T L on

E w w H D L w on

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


WEAK FORMULATION WEAK FORMULATION –– BOUSSINESQ EQNSBOUSSINESQ EQNS

v 0( , ) ( . , ) ( , ) ( , ) htt w v w w q wθ θ θ Γ∂ + ∇ + ∇ ∇ =

v g( , ) ( . , ) ( , ( )) ( , ) ( ( ) Pr( ) e , )

( , ) 0hmt

v w v v w v h w Ra w

v q

σ ε ω ω θΓ∂ + ∇ + = −

∇ =i

Eθ ∈ 0w E∈Find such that for all , the following holds

[ , ] [ , ]v p V Q∈ 0 0[ , ] [ , ]w q V Q∈Find such that for all , the following holds

Energy equation – weak form

Momentum and continuity equations – weak form


VARIATIONAL MULTISCALE DECOMPOSITIONVARIATIONAL MULTISCALE DECOMPOSITION

0 0 0

0 0 0 0 0 0

', ', '

', ', '

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


SCALE DESOMPOSED WEAK FORM SCALE DESOMPOSED WEAK FORM -- ENERGYENERGY

v 0

v 0

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

( ', ') ( . . ', ') ( ', ') ( , ')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 θΓ

Γ

∂ + ∂ + ∇ + ∇ + ∇ +∇ ∇ =

∂ + ∂ + ∇ + ∇ + ∇ +∇ ∇ =

Find and such that for all and , the following holds

' 'Eθ ∈Eθ ∈ 0w E∈ 0' 'w E∈

Small scale strong form of equations

2 2' . ' ' ( . )t tv v Rθ θ θ θ θ θ∂ + ∇ −∇ = − ∂ + ∇ −∇ =2( ) : .L vθ θ θ= ∇ −∇

1 11 ( ), (1 )t n n n n n nf f f f f ft γ

γ γδ + + +

∂ = − = + −Time discretization rule


ELEMENT FOURIER TRANSFORMELEMENT FOURIER TRANSFORM

DSpatial domain

( )eD

( )

ˆ ˆexp( ) ( , )d ( , ) ( , )e

j jj

D

k kg k xn i g x i g k i g kx h h h

ω ω ω∂ = − Γ + ≈∂ ∫

i

( )

ˆ ( , ) exp( ) ( , )deD

k xg k i g x xh

ω ω= −∫i

Subgrid scale solution denotes unresolved part of 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


ALGEBRAIC SUBGRID SCALE MODELALGEBRAIC SUBGRID SCALE MODEL2 2' . ' ' ( . )t tv v Rθ θ θ θ θ θ∂ + ∇ −∇ = − ∂ + ∇ −∇ =

' '( )t n n nL Rγ γθ θ + +∂ + =

2' '

2

1 1ˆ ˆˆn n n

kv ki Rt h h tγ γ

θ θγδ γδ+ + + + = +

i

122 2

' '1 22

1 1 1,n t n n tv

R c ct h t hγ γ

θ τ θ τγδ γδ

−

+ +

≈ + = + +

Time discretization

Element Fourier transform

Parseval’s theorem

Mean value theorem


STABILIZED FINITE ELEMENT EQUATIONSSTABILIZED FINITE ELEMENT EQUATIONSStrong regularity conditions

v 0

2 2

1

' 2

1

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

( , ( )

( , /( ) ) ) 0

htt n n n

Nel

t n n n te

Nel

n t te

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

2v v( ', ) ( ', ), ( ', ) ( ', )v w v w w wθ θ θ θ∇ = − ∇ ∇ ∇ = − ∇i i

Stabilized weak formulation

/( )w w tγδ=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)


CONSIDERATIONS FOR MOMENTUM EQUATIONCONSIDERATIONS FOR MOMENTUM EQUATIONPicard’s linearization 'v v v v v v∇ ≈ ∇ + ∇i i iFairly 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 tv v v v p Ra v v v v p

v v

σ ω ω θ σ∂ + ∇ −∇ = − −∂ − ∇ +∇

∇ = −∇

i i i ii i

g( ) Pr( ) e ( , )mom t n n n nR Ra v v v v pγ γ γω ω θ σ+ + += − − ∂ − ∇ +∇i i

con nR v γ+= −∇i


SUBGRID VELOICTY AND PRESSURESUBGRID VELOICTY AND PRESSURE2

' ' '2

'

1 1ˆˆ ˆ ˆPr( )

ˆ ˆ

n n mom n

ncon

k v k ki v i p R vt h h h t

k vi R

h

γ γ

γ

ωγδ γδ+ +

+

+ + + = +

=

i

i

Element Fourier transform

Simultaneous solve

Parseval’s theorem

Mean-value theorem1

22 222'

1 1

' 2'

1

, Pr( )

,

n c con c

nn m mom m

c

c v hhp Rc t c

v hv Rt c

γ

γ

τ τ ωγδ

τ τγδ τ

+

+

≈ = + +

≈ + =


STABILIZED FINITE ELEMENT EQUATIONSSTABILIZED FINITE ELEMENT EQUATIONSStrong regularity conditions

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

Stabilized weak formulation

/( )w w tγδ=where

( )'

g1

'

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

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

( , ) ( , ) 0

hmt n n n n

Neln

t n n n n me

n n c

v v v w p w v w h w

vRa v v v v p w v w wt

v w v w

γ γ γ

γ γ γ

γ

ω ε ε

ω ω θ σ τ ω εγδ

τ

+ + + Γ

+ + +=

+

∂ + ∇ − ∇ + −

+ − −∂ − ∇ + − + ∇ + ∇

− + ∇ ∇ =

∑

i i

i i i

i i

'

g1

( , ) ( ( ) Pr( ) e ( , ) , 0Nel

nn t n n n n m

e

vv q Ra v v v v p qtγ γ γ γ

ω ω θ σ τγδ+ + + +=

∇ + − −∂ − ∇ + − ∇ =

∑i i

Momentum equation

Continuity equation


IMPLEMENTATION ISSUES IMPLEMENTATION ISSUES -- GPCEGPCE

DSpatial domain

( )eD

nbf

1

( , ) ( ) ( )f x f N xα αα

ω ω=

=∑

Generic function

Random coefficient

Galerkin shape

function

GPCE expansion for random coefficients

0

( ) (ξ( ))P

i ii

f fα αω 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 nξ( ) {ξ ( ), ,ξ ( )}ω ω ω= …


IMPLEMENTATION ISSUES IMPLEMENTATION ISSUES –– SUPPORT SPACESUPPORT SPACE

DSpatial 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

( )eD

Element

( , )x ω

A

( ')eA

nbf nbf ' nbf'

1 1 1( , ) ( ) ( ) ( ) ( )i i i j j i

i j if x f N x f N N xω 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


NUMERICAL EXAMPLESNUMERICAL 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


FLOW PAST A CIRCULAR CYLINDERFLOW PAST A CIRCULAR CYLINDER

X

Y

0 5 10 15 200

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


ONSET OF VORTEX SHEDDINGONSET OF VORTEX SHEDDING

X

Y

0 2 4 6 8 10 12 14 16 18 200

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

0 2 4 6 8 10 12 14 16 18 200

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


FULLY DEVELOPED VORTEX SHEDDINGFULLY DEVELOPED VORTEX SHEDDING

Mean pressure contours

First order term in LCE of pressure contours

Second order term in LCE of pressure contours


VORTEX SHEDDING VORTEX SHEDDING -- CONTDCONTD

Frequency

Ampl

itude

0.1 0.2 0.3 0.4 0.50

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

0.2

0.4

0.6

DeterministicMean


RB CONVECTION RB CONVECTION -- CENTRAL ADIABATIC BODYCENTRAL 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 0.25 0.5 0.75 10

0.25

0.5

0.75

1Cold wall

Hot wall

Insulated InsulatedAdiabatic

body

0cθ =

[0.95,1.05]h Uθ =


TRANSIENT BEHAVIOR TRANSIENT BEHAVIOR –– TEMPERATURETEMPERATURE

• 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


CAPTURING UNSTABLE EQUILIBRIUMCAPTURING 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

0cθ =

1hθ =X

Y

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1


FAILURE OF THE GPCEFAILURE OF THE GPCE

X

Y

0 0.25 0.5 0.75 10

0.25

0.5

0.75

19.2E-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 0.25 0.5 0.75 10

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 0.25 0.5 0.75 10

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

XY

0 0.25 0.5 0.75 10

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 obtained from a deterministic simulation with Ra = 1870 (the upper limit)


PREDICTION BY SUPPORTPREDICTION BY SUPPORT--SPACE METHODSPACE METHOD

X

Y

0 0.25 0.5 0.75 10

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

XY

0 0.25 0.5 0.75 10

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 obtained from a deterministic simulation with Ra = 1870 (the upper limit)

X

Y

0 0.25 0.5 0.75 10

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 0.25 0.5 0.75 10

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


MICROSTRUCTURE RECONSTRUCTION MICROSTRUCTURE RECONSTRUCTION & CLASSIFICATION WITH & CLASSIFICATION WITH

APPLICATIONS IN APPLICATIONS IN MATERIALSMATERIALS--BYBY--DESIGNDESIGN

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://www.mae.cornell.edu/zabaras/

Veeraraghavan Sundararaghavan and Nicholas Zabaras


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


DESIGNING MATERIALS WITH TAILORED PROPERTIES

Micro problem driven by the velocity gradient L

Macro problem driven by the macro-design

variable βBn+1

Ω = Ω (r, t; L)~Polycrystal

plasticityx = x(X, t; β)L = L (X, t; β)

ODF: 1 2 3 4 5 6 7

L = velocity gradient

Fn+1

B0

Reduced Order Modeling

Data mining techniques

Database

Multi-scale Computation

Design variables (β) are macrodesign variables Processing sequence/parameters

Design objectives are micro-scaleaveraged material/process

properties

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


DATABASE FOR POLYCRYSTAL MATERIALS

Meso-scale database for polycrystalline materials Machine Learning

Database

Feature Extraction

Data Organization

Reduced order basisgeneration

You

ngsM

odul

us

RD TD

Multi-scale microstructure

evolution models

Process design for desired properties

RD

R-v

alue

0.9850.99

0.9951

1.0051.01

1.0151.02

1.0251.03

1.035

0 10 20 30 40 50 60 70 80 90

Angle from rolling direction

InitialIntermediateOptimalDesired

TD

ODF

Pole Figures

0 20 40 60 80144

144.1

144.2

144.3

144.4

144.5

144.6

144.7


MOTIVATIONMOTIVATION1. 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

MicrostructureAnalysis

2D Imaging techniques


PATTERN RECOGNITION (PR) STEPSPATTERN RECOGNITION (PR) STEPS

DATABASE CREATION

FEATURE EXTRACTION

TRAINING

PREDICTION

Datasets: microstructures from experiments or physical models

Extraction of statistical features from the databaseCreation of a microstructure class hierarchy: Classification methodsPrediction 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


3D MICROSTRUCTURE RECOGNITION: A TWO3D MICROSTRUCTURE RECOGNITION: A TWO--CLASS PROBLEMCLASS PROBLEM

Training Features

36.5236.5214.0814.0852.1552.1524.0224.02--11

9.529.5220.0120.01160.12160.1221.3021.3011

11.3011.3025.3025.30158.20158.2020.1020.1011

2.522.5223.0123.01154.12154.1223.3223.3211

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?2.312.3124.1024.10153.14153.1421.4521.45

Heyn intercept histogram of a 2D cross-section

Feature Extraction


MULTIPLE CLASSESMULTIPLE CLASSES

Class-AClass-B

Class-CA

CB

AB

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 21 1 2 2 1 2 1 2{ , , ...., }, { , , ...., }, ..., { , , ...., }sT T T s s s

m m s mx x x x x x x x x x x x= = =

p = 3One 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)

2p p −

( 1)2

p p −


TWO PHASE MICROSTRUCTURE: CLASS HIERARCHYTWO PHASE MICROSTRUCTURE: CLASS HIERARCHY

Class - 1

3D MicrostructuresFeature vector : Three point probability

function

3D Microstructures

Class - 2

Feature:Autocorrelation

function

LEVEL - 1 LEVEL - 2

r µm

γ


STATISTICAL CORRELATION MEASURESSTATISTICAL 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 (SiN ) 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.


3D RECONSTRUCTION3D RECONSTRUCTION

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

3 point probability function

Autocorrelation function


ELASTIC PROPERTIES: YOUNGS MODULUSELASTIC PROPERTIES: YOUNGS MODULUS

170

190

210

230

250

270

290

310

0 200 400 600 800 1000Temperature (deg-C)

You

ngs

Mod

ulus

(GP

a)

HS boundsBMMP boundsExperimentalFEM

3D image derived through pattern

recognition Experimental image


MICROSTRUCTURE REPRESENTATION USING SVM & PCAMICROSTRUCTURE REPRESENTATION USING SVM & PCA

COMMON-BASIS FOR MICROSTRUCTURE REPRESENTATION

Does not decay to zero

A DYNAMIC LIBRARY APPROACH

•Classify microstructures based on 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


PCA MICROSTRUCTURE RECONSTRUCTIONPCA MICROSTRUCTURE RECONSTRUCTION

Pixel value round-off

Basis Components

X 5.89

X 14.86

+

Project

onto basis

Reconstruct using two basis components

Representation using just 2 coefficients (5.89,14.86)


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 FigurePoint y (1,0,1)

0 0

h ||y

R .h=h , h||y1P (h,y) = (P (h,y)+P (-h,y))21P (h,y) =

2Adθ

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.


LIBRARY FOR TEXTURES

[100] pole figure

[110] pole figure

Parameter Feature Vector DATABASE OF ODFs


SUPERVISED CLASSIFICATION USING SUPPORT VECTOR MACHINES

Given ODF/texture

Tension (T)

Stage 1

LEVEL – 2 CLASSIFICATIONPlane strain compression

T+P

LEVEL – I CLASSIFICATIONTension identified

Stag

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


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.

21 2

21,..,1

1( , ,.., ) ( )2min

nk h

ih ki

J c c c x C==

= −∑

Identify clusters

Clusters

DATABASE OF ODFs

Feature Space

Cost function Each class is affiliated with multiple processes


PROCESS PARAMETERS LEADING TO DESIRED PROPERTIESY

oung

’s M

odul

us (G

Pa)

Angle from rolling direction

CLASSIFICATION BASED ON PROPERTIES

Class - 1 Class - 2

Class - 3Class - 4 0.5 0.25 00.25 -1.25 0

0 0 0.75

0.5 0 00 0.75 00 0 -1.25

Velocity Gradient

Different processes, Similar properties

Database for ODFs

Property Extraction

ODF Classification

Identify multiple solutions


A TWOA TWO--STAGE PROBLEMSTAGE PROBLEM

Process – 2 Plane strain compression α = 0.3515

Process – 1 Tension α = 0.9539Initial Conditions:

Stage 1

Sensitivity of material property

Initial Conditions- stage 2

DATABASE Reduced Basis

φ(1) φ(2)

Direct problem

Sensitivity problem


MULTIPLE PROCESS ROUTESMULTIPLE PROCESS ROUTES

0 10 20 30 40 50 60 70 80 90144

144.5

145

145.5

Angle from the rolling direction

You

ngs

Mod

ulus

(GP

a)

Desired Young’s Modulus distribution

Magnetic hysteresis loss distribution

0 10 20 30 40 50 60 70 80 901.205

1.21

1.215

1.22

1.225

1.23

1.235

1.24

Mag

netic

hys

tere

sis

loss

(W/k

g) 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


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

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

Iteration Index

Nor

mal

ized

obj

ectiv

e fu

nctio

nInitial guess, α1 = 0.65,

α2 = -0.1

Desired ODF Optimal- Reduced order control

Full order ODF based on reduced order control parameters

Stage: 1 Plane strain compression (α1 = 0.9472)

Stage: 2 Compression (α2 = -0.2847)


DESIGN FOR DESIRED MAGNETIC PROPERTYDESIGN FOR DESIRED MAGNETIC PROPERTY

I te ra t io n I n d e x

Nor

mal

ized

obj

ectiv

e fu

nctio

n

5 1 0 1 50

0 .2

0 .4

0 .6

0 .8

1

h

Crystal direction.

Easy direction of

magnetization – zero power

loss

External magnetization direction

0 20 40 60 80

1.21

1.215

1.22

1.225

1.23

1.235

Angle from the rolling direction

Mag

netic

hys

tere

sis

loss

(W/K

g)

Desired property distributionOptimal (reduced)Initial

Stage: 1 Shear – 1 (α1 = 0.9745)

Stage: 2 Tension

(α2 = 0.4821)

Documents

A MULTISCALE APPROACH TO MATERIALS USING ...paulino.ce.gatech.edu/.../TUESDAY/T_Morning_II/Zabaras.pdfmodel - FEM Experimental, Monte Carlo/ MD models Spectral stochastic/ support-space