Binary Stochastic Fields: Theory and Application to Modeling

of Two-Phase Random Media Steve Koutsourelakis

University of InnsbruckGeorge Deodatis

Columbia University

Presented at “Probability and Materials: From Nano- to Macro-Scale,” Johns Hopkins

University, Baltimore, MD. January 5-7, 2005

Effects of Random Heterogeneity of Soil Properties on Bearing Capacity

Radu Popescu and Arash NobaharMemorial University

George DeodatisColumbia University

What is a two-phase medium ?

A continuum which consists of two materials (phases) that have different properties.

What is a random two-phase medium ?

A two-phase medium in which the distribution of the two phases is so intricate that it can only

be characterized statistically.

Examples:

Synthetic: fiber composites, colloids, particulate

composites, concrete.

Natural: soils, sandstone, wood, bone, tumors.

Characterization of Two-Phase Random Media Through Binary Fields

( ) 1 if is in phase ( )

0 otherwisej j

I

xx

black : phase 1

white : phase 2

Complimentarity Condition:

(1) (2)( ) ( ) 1 I I x x x

Binary fields assumed statistically homogeneous

Only one of two phases used to describe medium

j = 1 or 2

Random Fields Description

First Order Moments – Volume Fraction

[ ( )] Pr[ is in phase 1]E I x x

Pr[ is in phase 2] 1 x

Second Order Moments – Autocorrelation

( ) [ ( ) ( )]

Pr[ and are in phase 1]

R E I I

z x x z

x x z

Properties of the Autocorrelation ( )R z

• 2( ) [ ( )] Pr[ ( ) 1]R E I I 0 x x

• If no long range correlation exists:

• Positive Definite (Bochner’s Theorem)

2

lim lim [ ]

R E I I

E I E I

z zz x x z

x x z

Simulation of Homogeneous Binary Fields based on 1st and 2nd order

information

Available Methods:

1) Memoryless transformation of homogeneous Gaussian fields (translation fields) (Berk 1991, Grigoriu 1988 & 1995, Roberts 1995)

Advantage : Computationally Efficient

Disadvantage : Limited Applicability

Simulation of Homogeneous Binary Fields based on 1st and 2nd order

information

Available Methods:

2) Yeong and Torquato 1996

Using a stochastic optimization algorithm, one sample at a time can be generated whose spatial averages match the target.

Advantage : Able to incorporate higher

order probabilistic information

Disadvantage : Computationally costly when

a large number of samples

needs to be generated

medium

Gaussian sequence

Binary sequence

Modeling the Two-Phase Random Medium in 1D Using Zero Crossings

are equidistant values of a stationary, Gaussian stochastic process Y(x) with zero mean, unit variance and autocorrelation

iY

[ ]i j j iE Y Y

iY

iI

medium

0 1

Modeling the Two-Phase Random Medium in 1D

-11 if 0

0 otherwisei i

i

Y YI

is also statistically homogeneous with autocorrelation

iI

jR

0 1

0 0(2)

1 1 1

0 0

1

[ ] Pr[ 1 ] Pr[ 0 ]

( ) ( , ; )

1 cos( )

i i i i

i i i i

R E I I Y Y

f y y dy dy

Arc

2nd order joint

Gaussian p.d.f

Observe that:

1

1

1

1 1

10 21 0

Modeling the Two-Phase Random Medium in 1D

1 1 1

0 0 0(3)

1 1 1 2

0 0 0

1 1

2 1

[ ] Pr[ 1 and 1]

( ) ( , , ; , )

1 1 ( sin( ) 2 sin( ))

4 2

i i i i

i i i

i i i

R E I I I I

f y y y

dy dy dy

Arc Arc

1 0

2 0 1

cos( ) cos( )

cos[2 ( )]

R

R R

1 2

1

1

. 1

1

sym

(3)Γ

For any pair , the correlation matrix :0 1 and R R

is always positive definite

Observe that:

Modeling the Two-Phase Random Medium in 1D

-1 -1

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

(4)1 1 1 1, , 1

[ ] Pr[ 1 and 1]

Pr[ 0 and 0]

(

)

( , , , ; , )

j i i j i i j

i i i j i j

i i i j i j j j j

R E I I I I

Y Y Y Y

f y y y y

1 1 i i i j i jdy dy dy dy

1 1, , 1( , )j j j jR H

The function H doesn’t have an explicit form, except for special cases. It can be calculated numerically with great computational efficiency (Genz 1992).

4th order joint

Gaussian p.d.f

Three Gaussian Autocorrelations

Corresponding Binary Autocorrelations

Sample Realizations of Three Cases with Different Clustering (but same )

case 1 – strong clustering

case 2– medium clustering

case 3 – weak clustering

0.1

Simulation: Inversion Algorithm1 1, , 1( , ) i i i iR H i

For simulation purposes, the inverse path has to be followed.

The goal is to find a Gaussian autocorrelation that

can produce the target binary autocorrelation

i

targetiR

Questions:

Existence of for arbitrary

Uniqueness of

i

i

targetiR

Approximate solutions – Optimization Formulation

Find Gaussian autocorrelation that produces a binary autocorrelation which minimizes the error with :

i

iRtargetiR

targetmax i ii

R R

Iterative Inversion Algorithm – Basic Concept

Step 1: Start with an arbitrary Gaussian

autocorrelation such that and

. Calculate the binary

autocorrelation

and the error

i 0 1

target1 0cos R

1 1, , 1( , )i i i iR H targetmax i i

ie R R

Step 2: Perturb the values of by small

amounts, keeping and the same.

Calculate the new and the new error e.

If the error is smaller, then keep the

changes in otherwise reject them.

i

10

iR

Step 3: Repeat Step 2 until the error e becomes

smaller than a prescribed tolerance of if a

large number of iterations do not further

reduce the error e.

i

Example – Known Gaussian Autocorrelation

Bin

ary

auto

corr

elat

ion

Gau

ssia

n au

toco

rrel

atio

n

Observe stability of the mapping

Example – Debye Medium

target 20(1 )exp( / )R z z z

Bin

ary

auto

corr

elat

ion

0.1

Example – Debye MediumG

auss

ian

auto

corr

elat

ion

Gau

ssia

n S

pect

ral D

ensi

ty

Fun

ctio

n0.1

Example – Debye MediumP

rogr

essi

on o

f E

rror

Sample Realization

0.1

Advantage of the Method Proposed:

The inversion procedure has to be performed only once.

Once the underlying Gaussian autocorrelation is determined, samples of the corresponding Gaussian process can be generated very efficiently using the Spectral Representation Method (Shinozuka & Deodatis 1991).

These Gaussian samples are then mapped according to:

Simulation: Inversion Algorithm

-11 if 0

0 otherwisei i

i

Y YI

in order to produce the samples of the binarysequence.

Example – Anisotropic MediumT

arge

t0.5

Inve

rsio

ntarget 2

1 2 1 01 2 02( , ) (1 )exp( / / )R z z z z z z

01 024, 1z z

Gau

ssia

n au

toco

rrel

atio

nG

auss

ian

Spe

ctra

l Den

sity

F

unct

ion

0.5 Example – Anisotropic Medium

0.5 Example – Anisotropic Medium

Sample Realization

Example – Fontainebleau SandstoneT

arge

tIn

vers

ion

Gau

ssia

n au

toco

rrel

atio

nG

auss

ian

Spe

ctra

l Den

sity

F

unct

ion

Example – Fontainebleau Sandstone

Example – Fontainebleau Sandstone

Actual Image Simulated Image

Generalized Formulation

Consider a homogeneous, zero mean, unit variance, Gaussian random field with autocorrelation:

Y x

E Y Y z x x + z 1 0

1 if 0

0 otherwise

Y YI

x δ xx

will also be homogeneous I x

1 1 sin 2 2 sin

4 2

R E I I

Arc Arc

δ x x δ

δ δ

Generalized Formulation

Autocorrelation R z

, , ,

R E I I

H

z x x z

δ z δ z z δ

In general:

1 cos

R E I

Arc

0 x

δ

Generalized Formulation: Discretization in 1D

Properties of the Autocorrelation

• For we recover the previous formulation1

• depend on: 1

0 parameters

M

i iR M

1

00 1 parameters since =1

1 parameter

M

i iM

Surplus of parameters

Surplus of parameters makes the method more flexible and able to describe a wider range of

binary autocorrelation functions.

Conclusions

It takes advantage of existing methods for the generation of Gaussian samples and requires minimum computational cost especially when a large number of samples is needed.

The method proposed is shown capable of generating samples of a wide range of binary fields using nonlinear transformations of Gaussian fields.

Extension to higher order probabilistic information.

Generalized formulation increases the range of binary fields that can be modeled.

Extension to more than two phases.

Extension to three dimensions.