June 2015 review bhattacharya - Carnegie Mellon...

K. Bha'acharya Review 2015: #1

Compact representa=on of micromechanical fields

Kaushik Bha'acharya

Gal Shmuel Jin Yang

Chun-‐Jen Hsueh Md. Zubaer Hossain Dingyi Sun

Acknowledge: Mary Comer, Charles Bouman, G. Ravichandran


Mul=scale understanding of materials

Brinson et al.

James and Chu

Schryvers

Sun

Rapid growth of digital experimental techniques and computa=onal power has given unprecedented level of data about materials

How can we harness this for understanding and ul=mately designing?

How do we represent micro-‐mechanical fields?

Mabe


Topics Mul=scale Phase-‐field Simula=on Digital Image Correla=on

Fracture of heterogeneous materials Coarse-‐grained Density Func=onal Theory

Axial strain using 10% of Symlet5 wavelet detail coefficients

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05Exact axial strain

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05


Phase transforma=on


Mul=scale phase field method

= min⌫,µ

Z

⌦1

✓Z

RW (�, x)d⌫

x

(�) +1

2

Z

RT (', x) · LT (', x)d⌫

x

(�)

�1

2

Z

S

2

A

�1pi

(↵)↵q

L

pqrs

↵

b

L

ibcd

dµ

rscd

x

(↵)

◆dx.

lim⌘!0

min�,u

E

⌘(�, u) = inf�,u

Z

⌦1

✓W (', x) +

1

2(ru� T (', x)) · L(ru� T (', x))

◆dx

E

↵L

3=

Z

⌦1

✓1

2

�

↵L

2|r'|2 +W (', x) +

1

2(ru� T (', x)) · L(ru� T (', x))

◆dx

E =

Z

⌦

✓�

2|r'|2 +W (', x) +

1

2(ru� T (', x)) · L(ru� T (', x))

◆dx

Energy

Scaling: Length scale L, Energy scale α x 7! 1

L

x, W 7! 1

↵

W, L 7! 1

↵

L

E⌘ ⌘2

Large body limit

= E


Measures and elas=c energy

Young measure (Ball, Tartar) νx describes the one-‐point sta=s=cs near the point x

•  Local volume frac=on •  Pole figures

H measure (Tartar) µx describes the part of the two-‐point sta=s=cs that is related to the induced elas=c energy. Closely related to

•  Localiza=on tensor •  Eshelby tensor •  (Structure factor)

E = min⌫

✓minµ

. . .

◆

⇡ minu

min�i

Z

⌦

0

@X

wi�i +X

i,j

�imij�j +1

2

e(u)�

X

i

�iTi(x)

!· L

e(u)�

X

i

�iTi(x)

!1

Adx

New transla=on bounds

Postulate

�̇ = K(��E)Phase field model for volume frac=ons


Superelasticity"

Initiation. Well oriented grains Stress concentrations

Progress. Autocatalyic strips Some reorienta=on

Saturation. Incomplete transformation Reorientation

Reversal. Last on, first off Slight differences from (B)

Richards et al. (2013)

Is there a compact representa=on for these

transforma=on strain fields that retains the essen=al physics


Wavelets Generate a nested sequence of subspaces {0} ⇢ ...Vj ⇢ Vj+1 ⇢ ... ⇢ L2

(R):

Define Vj = {�j,k (x) , k 2 Z},and note Vj ⇢ Vj+1 = Vj �Wj where Wj = { j,k (x) , k 2 Z}

Approximate a function at scale j0 as

f(x) =

X

k2Zak'j0,k(x) +

X

j>j0,k2Zdj,k j,k(x)

• Scaling function: '(x) satisfying

Z|'|dx = 1

• Mother wavelet: (x) satisfying

Z dx = 0

• (x) =X

n2Zbn'(2x� n)

Introduce

Space

Freq

uency

-1.0-0.5 0.5 1.0 1.5 2.0

-1.0-0.5

0.51.0

-1.0-0.5 0.5 1.0 1.5 2.0

-1.0-0.5

0.51.0

Haar

-1.0-0.5 0.5 1.0 1.5 2.0-1.0-0.5

0.51.01.5

-1.0-0.5 0.5 1.0 1.5 2.0-1.0-0.5

0.51.01.5

Daubechies

Consider their translations and dilations

j,k (x) := 2j/2 �2jx� k

�

'j,k (x) := 2j/2'�2jx� k

�j, k 2 Z

Compression Adaptively choose basis


Compressing the results using wavelets

0 1 2 3 40

100

200

300

400

500

600

700

!̄x (%)

!̄x(M

Pa)

A B

C

ED

•  Simulate using FFT •  At each =me step, take a (Symlet 5) wavelet transform of the result •  Keep only 1 % of the coefficients •  Compute the macroscopic stress-‐strain curve1


Wavelets also provide new insight

0 1 2 3 40

100

200

300

400

500

600

700

800

900

1000

!̄x(%)

!̄x(M

Pa)

•  Rela=ve change in # of ac=ve wavelets: ~4% •  Ac=ve wavelets are predictors of stress and strain

intensi=es •  Dominant wavelets are basis for model reduc=on •  Current ac=ve set are predictors of transforma=on •  Ac=ve wavelets carry informa=on about inter-‐

granular compa=bility

Ac=ve

Inac=ve

Total


Experiments with full-‐field strain

Daly, Ravichandran and Bha'acharya, Acta Mater. (2007)


Axial strain using 10% of Symlet5 wavelet detail coefficients

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045


0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045


0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05Axial strain using 10% of Symlet5 wavelet detail coefficients

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045


0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05Axial strain using 10% of Symlet5 wavelet detail coefficients

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Representa=on of Daly et al. Experiments

99.3% 99.9% 99.8% ||"approxyy ||||"yy||

=

Compression using 1% coefficients

1.93% 2.94% 1.34% Total strain


Digital image correla=on


Digital image correla=on

Su'on et al., Image Correla=on …, Springer Hild and Roux, in Op#cal Methods for Solid Mechanics, Wiley

y x

y(x)

f(x) g(y) miny:⌦!Rn

Z

⌦|f(x)� g(y(x))|2 dx

() max

y:⌦!Rn

Z

⌦f(x)g(y(x))dx

Can be done locally •  Very computa=onally efficient

But, •  Need high frequency random pa'ern … large data sets

•  Results can be very noisy

Can we use filtered/ compressed images for DIC?


DIC with compressed images

Original 20 % DCT min

y:⌦!Rn

Z

⌦|f(x)� g(y(x))|2 dx

() max

y:⌦!Rn

Z

⌦f(x)g(y(x))dx

Since the method is local, •  we need high frequency random images •  filtered and lost during compression

Generally gives significant error


Need to introduce global informa=on

miny:⌦!Rn

Z

⌦

⇣|f(x)� g(y(x))|2 + ↵ |ry|2

⌘dx

•  Global Method and Regulariza=on (Hild and Roux)

Reduces noise, but expensive

Use finite element basis for y

•  Local method with global constraints

max

�,µmin

{yi},{Fi},y

Z

⌦

8<

:X

i

�i|f(x)� g(yi + Fix)|2 + �

��y �X

i

�iyi

��

2

+ µ

��ry �X

i

�iFi

��

29=

; dx

max

�,µmin

{yi},{Fi},y

Z

⌦

8<

:X

i

�i|f(x)� g(yi + Fix)|2 + �

��y �X

i

�iyi

��

2

+ µ

��ry �X

i

�iFi

��

29=

; dx

Par==on of unity Related interpola=on

-  Computa=onally efficient (mul=scale approach)

-  Consistent with compression


Results with global + compression

100%

20%

Original

Global Local

20% compressed

Avg Eyy = 1.89 % Avg Eyy = -‐1.09 %

Global 0.0071 % 0.044 %

Local 0.49 % 4.61 %


Use of priors: Fracture

Seek the stress intensity, but the displacements are extremely small.

minr,R,c,KI ,x0

Z

⌦r,R

|f(x)� g(KI

U(x� x0) + c)|2 dx


Impact/Future •  Personnel

–  Gal Shmuel, Post-‐doc (currently Asst. Prof. Technion, Israel) –  Md. Zubaer Hossain, Post-‐doc (par=al, soon Asst. Prof. U Delaware) –  Jin Yang, Graduate student –  Chun-‐Jen Hsueh, Graduate student (par=al) –  Dingyi Sun, Graduate student (NDSEG)

•  Publica=ons –  G. Shmuel, A.T. Thorgeirsson, and K.Bha'acharya. 2014. “Wavelet Analysis of

Microscale Strains.” Acta Materialia 76: 118–126. –  J. Yang, G. Ravichandran and K. Bha'acharya. 2015. “Data compression for digital

image correla=on”. In prepara=on for submission to Experimental Mechanics –  4 others

•  Provisional Patent –  C-‐J Hsueh, G. Ravichandran, K. Bha'acharya. 2015. “A Novel device of measuring

the fracture toughness of heterogeneous materials”

•  Future direc=ons –  Digital Image Correla=on with compression (Comer) –  Scale-‐dependent Young measure, H-‐measures (James) –  Mechanical proper=es of Al-‐Si (Voorhees/Kalidinidi)

June 2015 review bhattacharya - Carnegie Mellon...

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