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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 ...
Identificación de Propiedades de Materiales por Métodos Inversos 7
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)
Identificación de Propiedades de Materiales por Métodos Inversos 8
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.
Identificación de Propiedades de Materiales por Métodos Inversos 16
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) .
Identificación de Propiedades de Materiales por Métodos Inversos 17
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.)
Identificación de Propiedades de Materiales por Métodos Inversos 19
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.
Identificación de Propiedades de Materiales por Métodos Inversos 20
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.
Identificación de Propiedades de Materiales por Métodos Inversos 23
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
Identificación de Propiedades de Materiales por Métodos Inversos 26
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)
Identificación de Propiedades de Materiales por Métodos Inversos 30
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 ?
Identificación de Propiedades de Materiales por Métodos Inversos 32
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 !