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Identificación de Propiedades de Materiales por Métodos Inversos 1

MODELAGEM MULTIESCALA: HOMOGENEIZAÇÃO

Daniel Alves Castello

Red Sudamericana de Identificación de Propiedades de Materiales por Métodos

Inversos – Mar del Plata

Identificación de Propiedades de Materiales por Métodos Inversos 2

OUTLINE

• Heterogeneous Materials

• Homogenization

• Homogenization Applied to the HeatConduction Problem (HCP)

• Homogenization Applied to the ElasticityProblem (EP)

Identificación de Propiedades de Materiales por Métodos Inversos 3

HETEROGENEOUS MATERIALS

• Hystory Maxwell, J.C.(1873)

Effective conductivity of a dispersion of espheresthat is exact for dilute sphere concentrations.

Rayleigh, L.(1892) Effective conductivity of regular arrays of espheres

(used to this day).

Einstein, A.(1906) Effective viscosity of a dilute suspension of

espheres.

Identificación de Propiedades de Materiales por Métodos Inversos 4

HETEROGENEOUS MATERIALS

• Definition A heterogeneous material is comprised of

domains composed of different materials, such as a composite, or these domains caneven be composed of the same material butin different states, such as a polycristal.

Identificación de Propiedades de Materiales por Métodos Inversos 5

HETEROGENEOUS MATERIALS

• Examples: synthetic products Gels

Foams

Concrete

Cellular solids

Coloids

Composites

...

Identificación de Propiedades de Materiales por Métodos Inversos 6

HETEROGENEOUS MATERIALS

• Examples: in nature Soils Sandstone Wood Bone Lungs Blood Animal and Plant Tissue ...

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HETEROGENEOUS MATERIALS

• Examples of natural random heterogeneousmaterials.(Extracted from S.Torquato, Random Heterogeneous Materials, Springer Verlag, 2001.)

(Red blood cells) (Celullar Structure of bone) (Fontainebleau sandstone)

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HETEROGENEOUS MATERIALS

• Examples of synthetic random heterogeousmaterials. (Extracted from S.Torquato, Random Heterogeneous Materials, Springer Verlag, 2001.)

Aluminum (white) and anotherceramic phase (gray)

Fiber-reinforce cermet Colloidal system of hardSpheres.

Identificación de Propiedades de Materiales por Métodos Inversos 9

HETEROGENEOUS MATERIALS

• P1: Given K1, K2 and F can we determine the displacement field u(x) of this body ?

K1 and K2 : elastic properties

Identificación de Propiedades de Materiales por Métodos Inversos 10

HETEROGENEOUS MATERIALS

• P1: How can we solve problem P1 ?

Analytical solution (Depending on the shapeof the inclusion)

Finite Element Method

...?

Identificación de Propiedades de Materiales por Métodos Inversos 11

HETEROGENEOUS MATERIALS

• P2: Given K1, K2 and F can we determine the displacement field u(x) of this body ?

K1 and K2 : elastic properties

Identificación de Propiedades de Materiales por Métodos Inversos 12

HETEROGENEOUS MATERIALS

• P2: How can we solve problem P2 ? Analytical Solution

Finite Element Method

Black Box Approach : u(xm

) = g(F,α1,...,αN)

Phenomenological Approach: u(x) = g(F,p1,...,pN)

Homogenizationu(x) = g(F,K1,...,KN,? )

Identificación de Propiedades de Materiales por Métodos Inversos 13

HETEROGENEOUS MATERIALS

• P2: Phenomenological Approach

Hypothesis:

There is an equivalent macroscopic description.

Parameters:

Through experiments one can obtain theequivalent physical parameters p1, ..., pN.

Identificación de Propiedades de Materiales por Métodos Inversos 14

HETEROGENEOUS MATERIALS

• P2: Homogenization

Hypothesis: There is an equivalent macroscopic description.

Governing Equations:Build the macroscopic governing equations

starting at the heterogeneity scale (microscopic).

Identificación de Propiedades de Materiales por Métodos Inversos 15

HOMOGENIZATION

• DEFINITION

The homogenization theory is aimed at determining the macroscopic governing partialdifferential equations of a heterogeneousmedium when the characteristic length of theheterogeneities is much times lower than thecharacteristic length of the medium.

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HOMOGENIZATION• METHODS (Bensoussan et. al)

Based on the construction of asymptotic expansionsusing multiple scales.

Based on energy estimates. Based on probabilistic arguments and works

whenever the problem admits probabilistic formulationor has a probabilistic origin.

Based on the spectral decomposition of operatorswith periodic coefficients, the so-called expansionBloch waves (problems involving high frequencywave propagation in rapidly varying periodic media) .

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HOMOGENIZATION

• MACROSCOPIC DESCRIPTION ? What are we really looking for when we say

that we are searching the macroscopicgoverning equations for the medium ?

R.: Ariault states that we are looking for anequivalent boundary value problem, i.e., weare looking for the relations between themacroscopic variables of the problem and itsassociated effective properties.

Identificación de Propiedades de Materiales por Métodos Inversos 18

HOMOGENIZATION

• WHAT DO WE MEAN BY MACRO AND MICRO ?

The physical phenomena under analysisoccur on the microscopic length scales thatspan from tens of nanometers in the case of gels to meters in the case of geologicalmedia. (Extracted from S.Torquato, Random HeterogeneousMaterials, Springer Verlag, 2001.)

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HOMOGENIZATION

• EFFECTIVE PROPERTIES ?

What do we mean by effective properties ?

R.: The effective properties of heterogeneousmaterials are determined by suitableaverages of local fields derived fromappropriate governing continuum equations of the problem under analysis.

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HOMOGENIZATION

• EFFECTIVE PROPERTIES (Key-Questions)

Given a medium composed of phases such thateach phase is characterized by its propertiesdenoted by K1 , ..., KN and its volumetric fractionsdenoted by , φ1, ..., φN, respectively, how are itseffective properties mathematically defined ?

Do we have to take the microstructure intoaccount ?

Identificación de Propiedades de Materiales por Métodos Inversos 21

HOMOGENIZATION• MICROSTRUCTURE AND EFFECTIVE

PROPERTIES

In order to evaluate if the analysis of themicrostructure is important for the effective calculationof the effective properties lets consider a simpleexample.

The example consists of a heterogeneous mediumcomposed of two phases with very diffrent elasticproperties but with φ1=φ2= 0.5.

Identificación de Propiedades de Materiales por Métodos Inversos 22

HOMOGENIZATION• EXAMPLE

Lets consider the blue phase highly stiffcompared to the yellow one.

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HOMOGENIZATION

• QUESTION

Which one possesses the higher effectivestiffness ?

R.: The right one.

Even though φ1 = φ2 for both composites, theireffective properties will be extremely different, therefore

1 1( )e N N Yf … …φ φ= , , , , , ,ΩK K K

Identificación de Propiedades de Materiales por Métodos Inversos 24

HOMOGENIZATION• TARGET ?

• HOW ?

Identificación de Propiedades de Materiales por Métodos Inversos 25

HOMOGENIZATION-HCP

• Heat Conduction Problem: PeriodicStructure

Ω: Ω: Ω: Ω: Macroscopic Domain

Y:Microscopic Domain

ε: ε: ε: ε: Scale parameter

L:Characteristic lenght of

the macroscopic scale

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HOMOGENIZATION-HCP

• Governing Equations

• Boundary Conditions

( )( ) ( )

ij

i j

TK g

x x

εε

∂ ∂− = , ∀ ∈Ω

∂ ∂

xx x x

1 1T C

ε = , ∀ ∈Γx

2 2ij i

j

TK n C

x

εε ∂

− = , ∀ ∈Γ∂

x

1 2Γ Γ = ∂ΩU

Kε: Conductivity tensor

Tε:Temperature field

Identificación de Propiedades de Materiales por Métodos Inversos 27

HOMOGENIZATION-HCP

• Continuity Conditions (Perfect contact)

• Hypothesis : Separation of Scales is coherent.

( ) ( )T T Y+ −= ,∀ ∈∂x x x

( ) ( )( ) ( )

ij ij

i j i j

T TK K Y

x x x x

ε εε ε

+ − ∂ ∂ ∂ ∂− = − ,∀, ∈∂

∂ ∂ ∂ ∂

x xx x x

Identificación de Propiedades de Materiales por Métodos Inversos 28

HOMOGENIZATION-HCP

• SEPARATION OF SCALES ? Lets consider the property Pε(x) described

below:

Pe(x) suggests a possibleuse of two spatial scales !

Identificación de Propiedades de Materiales por Métodos Inversos 29

HOMOGENIZATION-HCP• HOW CAN WE MATHEMACALLY

PERFORM THE SEPARATION OF SCALES ?

Concerning the separation of scales, Joseph B. Keller in [6] states that in order to allow a variable to vary in a macroscale(O(L)) and also in a microscale (O(l)) we can represent it as a function of the macro/slow variable x and of the micro/rapidvariable y. HOW ?

1ε −=y xl

Lε = (Scale parameter)

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HOMOGENIZATION-HCP

• WHY y IS CALLED FAST VARIABLE ? Lets consider a function g

1

( ) ( ) ( / )

( )j j

g g g

g g

x y

ε ε

ε −

= =

∂ ∂=

∂ ∂

x y x

y

Identificación de Propiedades de Materiales por Métodos Inversos 31

HOMOGENIZATION-HCP

• AFTER ADOPTING THE HYPOTHESIS OF TWO SEPARATE SCALES HOW CAN WE TAKE THIS INTO ACCOUNT TO FIND THE SOLUTION Tε(x) ?

The presence of two disparate scales motivates themethod of asymptotic expansions using multiplescales.

How is the general dependence of Tε(x) with thesescales ?

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HOMOGENIZATION-HCP

• WHAT IS THE GENERAL FORM OF Tε(x) ?

• J.B. Keller also mentions that the fact that thefast variable was considered explicitly in v is that he believes the dependence of Tε(x) in εεεε is due to y and moreover, he also believes that vdepends regularly on ε, viz.

( ) ( )T vε ε= , ,x x y

(0) (1) 2 (2)( ) ( ) ( ) ( ) ( )T v T T T …

ε ε ε ε= , , = , + , + , +x x y x y x y x y

Identificación de Propiedades de Materiales por Métodos Inversos 33

HOMOGENIZATION-HCP

• ASPECTS OF THE PROPOSED ASYMPTOTIC FORM OF Tε(x) ?

Now the unknown fields become T(0)(x,y), T(1)(x,y), T(2)(x,y), ...

It is shown in Bensoussan et al [2] that Tε(x) converges weakly to T(0)(x,y) as ε tends to zero

Identificación de Propiedades de Materiales por Métodos Inversos 34

HOMOGENIZATION-HCP

• HOW DO WE PERFORM THE SPATIAL DERIVATIVES ?

As we have adopted the hypothesis of separation of scales we must first treat x and yas independent variables and subsequentlyreplace y by ε-1x .

Example ?

Identificación de Propiedades de Materiales por Métodos Inversos 35

HOMOGENIZATION-HCP

• EXAMPLE OF SPATIAL DERIVATIVE WITH A FUNCTION WHICH DEPENDS ON TWO SCALES

If fε(x) = f(x,y), then

1i

i i i i i i

yf f f f f

x x y x x y

ε

ε

∂∂ ∂ ∂ ∂ ∂= + = +

∂ ∂ ∂ ∂ ∂ ∂

1x x y

f f fε

ε∇ = ∇ + ∇

Identificación de Propiedades de Materiales por Métodos Inversos 36

HOMOGENIZATION-HCP

• GOVERNING EQUATIONS OF THE HCP

( )(0) (1) 2 (2)1 1ij

i i i i

K T T T … gx y x y

ε ε εε ε

∂ ∂ ∂ ∂+ − + + + + =

∂ ∂ ∂ ∂

( )( ) ( )ij

i j

TK g

x x

εε

∂ ∂− = , ∀ ∈Ω

∂ ∂

xx x x

Identificación de Propiedades de Materiales por Métodos Inversos 37

HOMOGENIZATION-HCP

• REARRANGING THE GOVERNING

EQUATIONS IN POWERS OF ε:(0) (1) (1) (2)

0

(0) (0) (1)

1

( ) ij ij

i j j i j j

ij ij

i j i j j

T T T TO K K

x x y y x y

T T TK K

x y y x y

ε ε

ε ε

ε ε

ε

ε

−

∂ ∂ ∂ ∂ ∂ ∂ + − + + − + + ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ + − + − + + ∂ ∂ ∂ ∂ ∂

+(0)

2

ij

i j

TK g

y y

ε− ∂ ∂ − = ∂ ∂

Identificación de Propiedades de Materiales por Métodos Inversos 38

HOMOGENIZATION-HCP

• DIFFERENTIAL OPERATORS (Kε periodic)

0( )

ij

i j

A Ky y

ε ∂ ∂

= − ∂ ∂

y

1( ) ( )

ij ij

i j i j

A K Kx y y x

ε ε ∂ ∂ ∂ ∂

= − − ∂ ∂ ∂ ∂

y y

2( )ij

i j

A Kx x

ε ∂ ∂

= − ∂ ∂

y

(0)

00A T =

(1) (0)

0 10A T A T+ =

(2) (1) (0)

0 1 2A T A T A T g+ + =

Identificación de Propiedades de Materiales por Métodos Inversos 39

HOMOGENIZATION-HCP

• HOW DO WE DEFFINE A SPATIAL AVERAGE OF A Y-PERIODIC FUNCTION ?

The spatial average of a Y-periodic function F(y) over a unit cell denoted by <F(y)> is defined as follows

where |Y| is a d-dimensional volume (measure) of Y . The quantity may represent a tensor of any order.

1( ) ( )

Yd

Y | |= ∫F y F y y

Identificación de Propiedades de Materiales por Métodos Inversos 40

HOMOGENIZATION-HCP

• THEOREM 1

Let F(y) be a Y-periodic function that is squareintegrable. For the boundary value problem

where Φ(y) is Y-periodic the following hold:

(i) There exists a solution if and only if

(ii) If a solution exists, it is unique up to an additiveconstant.

0( ) ( )A YΦ = , ∀ ∈y F y y

Identificación de Propiedades de Materiales por Métodos Inversos 41

HOMOGENIZATION-HCP

• HOW DO WE SOLVE A0T(0)=0 ?

The first step is simply to note that equation thisequation fulfills the conditions specified in Theorem 1 and hence there exists a solution T(0).

The second step is to note that the operator A0

involves only derivatives with respect to y whatmeans that in this equation the variable x workssimply as a parameter.

Identificación de Propiedades de Materiales por Métodos Inversos 42

HOMOGENIZATION-HCP

• SOLUTION ?

Among a myriad of possible solutions are the constant ones, i.e., those solutions which depend only on x. J.B. Keller states that A0 is such that these constants are the only solutions which are defined in the entire Y-pace and are bounded, therefore

(0) (0)( )T T= x

Identificación de Propiedades de Materiales por Métodos Inversos 43

HOMOGENIZATION-HCP

• DOMAIN OF INTEGRATION ?

As we are considering spatial periodic properties we perform integrations over the domain of a periodic cell.

Note: Another interesting comment related to the periodicity hypothesis is that we may consider only solutions that satisfy the same periodicity !

Identificación de Propiedades de Materiales por Métodos Inversos 44

HOMOGENIZATION-HCP

• HOW DO WE SOLVE A0T(1) + A1T(0) =0 ?

First question : Does T(1) exist ?

As A0 involves only derivatives with respect to y, in right side of the equation above x acts as a parameter.

(0)

(1) (0)

0 1

( ) ( )ij

i j

K TA T A T

y x

∂ ∂= − =

∂ ∂

y x

Identificación de Propiedades de Materiales por Métodos Inversos 45

HOMOGENIZATION-HCP

• ADJOINT PROBLEM

Lets consider the following adjoint problem defined in the cell domain

We can prove that this auxiliary problem has solution if it fulfills the requirements states in Theorem 1.

0( )

ij

j

i

KA Y

yχ

∂= , ∀ ∈

∂y y

Identificación de Propiedades de Materiales por Métodos Inversos 46

HOMOGENIZATION-HCP

• ADJOINT PROBLEM From the divergence theorem one gets

Due to the Y-periodicity of Kij the right side of this equation is zero. Therefore,

And χj(y) exists.

ij

ij jY Y

i

Kd K n dS

y ∂

∂=

∂∫ ∫y

ij

i

Kis Y periodic

y

∂−

∂

Identificación de Propiedades de Materiales por Métodos Inversos 47

HOMOGENIZATION-HCP

• SOLUTION OF T(1)(x,y)

(0)

(1)( ) ( ) ( )

j

j

TT u

xχ

∂, = +

∂x y y x

Identificación de Propiedades de Materiales por Métodos Inversos 48

HOMOGENIZATION-HCP

• HOW DO WE SOLVE A0T(2)+A1T(1)+A2T(0) = g ?

Which conditions should be fulfilled to assure that T(2)

exists ?

What do we need to to apply Theorem 1 ?

(2) (1) (0)

0 1 2A T g AT A T= − −

(1) (0)

1 20g AT A T− − =

Identificación de Propiedades de Materiales por Métodos Inversos 49

HOMOGENIZATION-HCP

• Y-PERIODICITY CONDITIONS

( )2 (0)

2 (0) 2 (0)

1

10

ij

ij qY

i j q i j

q

ijY

j i q i j

KT ug K d

Y y x x y x

T TK d

Y y x x x x

χ

χ

∂∂ ∂ ∂+ +

| | ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂+ = | | ∂ ∂ ∂ ∂ ∂

+∫

∫

y

y

Simplifications

Identificación de Propiedades de Materiales por Métodos Inversos 50

HOMOGENIZATION-HCP

• SIMPLIFICATIONS

( ) ( )1 ( ) 1 ( )0

| | | |

ij ij

Y Yi j j i

K Ku ud d

Y y x Y x y

∂ ∂∂ ∂= =

∂ ∂ ∂ ∂∫ ∫y yx x

y y

2 (0) 2 (0)1 1 ( )

( ) ( ( ) ( )) 0ij q ij qY Y

i j q j q i

T TK d K d

Y y x x Y x x yχ χ

∂ ∂ ∂ ∂= =

| | ∂ ∂ ∂ | | ∂ ∂ ∂∫ ∫x

y y y y

Identificación de Propiedades de Materiales por Métodos Inversos 51

HOMOGENIZATION-HCP

• RESULTING EQUATIONS

2 (0)( )1 1 ( )( ) ( ) 0

j

ij irY Y

r i j

Tg d K K d

Y Y y x x

χ∂ ∂+ + =

| | | | ∂ ∂ ∂ ∫ ∫

y xy y y y

CONSTANT

FUNCTION WHICH DEPENDS ONLY ON THE MACRO SCALE

Identificación de Propiedades de Materiales por Métodos Inversos 52

HOMOGENIZATION-HCP

• STANDARD FASHION

2 (0)

( )eq

ij

i j

TK g

x x

∂− =

∂ ∂x

jeq

ij ij ir

r

K K Ky

χ∂= +

∂

Identificación de Propiedades de Materiales por Métodos Inversos 53

HOMOGENIZATION-EP

• Elasticity Problem: Periodic Structure

Governing Equation

Boundary Conditions

] [1,0),(. ∈=

− xxf

dx

duE

dx

d εε

0)1()0( == εεuu

Identificación de Propiedades de Materiales por Métodos Inversos 54

HOMOGENIZATION-EP

• Periodic Structure

Hypothesis: It is possible to adopt a separation of scales

)()( lxExE += εε

...),(),(),()()2(2)1()0( +++= yxyxyxx uuuu εεε

Identificación de Propiedades de Materiales por Métodos Inversos 55

HOMOGENIZATION-EP

• Homogenization

∫

∫

=

=

=−

l

l

eq

eq

dyfl

f

dzZElE

fdx

udE

0

0

2

)0(2

1

)(

111

Identificación de Propiedades de Materiales por Métodos Inversos 56

HOMOGENIZATION-EP

• This one-dimensional problem has an analytic solution

∫ ∫

∫ ∫∫

+−=

=

x z

z

dzcdttfzE

xu

dzdttfzE

dzzE

c

0

0

0

1

0 0

1

0

0

)()/(

1)(

)()/(

1

)/(

1

ε

εε

ε

Identificación de Propiedades de Materiales por Métodos Inversos 57

HOMOGENIZATION-EP

• Parameters

2

5

2

1

))2sin(1)((2

1)(

1)(

==

++−=

=

βα

απαβ zzE

xf

Identificación de Propiedades de Materiales por Métodos Inversos 58

HOMOGENIZATION-EP

• Solution ε =1

Identificación de Propiedades de Materiales por Métodos Inversos 59

HOMOGENIZATION-EP

• Solution ε =1/2

Identificación de Propiedades de Materiales por Métodos Inversos 60

HOMOGENIZATION-EP

• Solution ε =1/4

Identificación de Propiedades de Materiales por Métodos Inversos 61

HOMOGENIZATION-EP

• Solution ε =1/8

Identificación de Propiedades de Materiales por Métodos Inversos 62

HOMOGENIZATION-EP

• Solution ε =1/16

Identificación de Propiedades de Materiales por Métodos Inversos 63

HOMOGENIZATION-EP

• Solution ε =1/32

Identificación de Propiedades de Materiales por Métodos Inversos 64

HOMOGENIZATION-EP

• Solution ε =1/256

Identificación de Propiedades de Materiales por Métodos Inversos 65

MODELAGEM MULTIESCALA: HOMOGENEIZAÇÃO

• Heterogeneous Materials

• Separation of Scales

• Homogenized Solution

• Examples

Identificación de Propiedades de Materiales por Métodos Inversos 66

MODELAGEM MULTIESCALA: HOMOGENEIZAÇÃO

¡ Muchas Gracias !