theory of homogenization - colgate university

Theory of Homogenization

Silvia Jimenez Bolanos

September 28 2007

Silvia Jimenez Bolanos Theory of Homogenization

Outline

Acknowledgement

Introduction

Hypotheses (BVP and Homogenized BVP)

Asymptotic Theory for Averages

Motivation for the Linear Case

Limit of Product of Weakly Converging Gradients

Behavior of gradients of solutions to PDEs with highlyoscillatory coefficients

Motivation for Nonlinear Case

Behavior of gradients of solutions to nonlinear PDEs withhighly oscillatory coefficients

Bibliography

Outline

Acknowledgement

Introduction

Bibliography

Outline

Acknowledgement

Introduction

Bibliography

Outline

Acknowledgement

Introduction

Bibliography

Outline

Acknowledgement

Introduction

Bibliography

Outline

Acknowledgement

Introduction

Bibliography

Outline

Acknowledgement

Introduction

Bibliography

Outline

Acknowledgement

Introduction

Bibliography

Outline

Acknowledgement

Introduction

Bibliography

Outline

Acknowledgement

Introduction

Bibliography

Outline

Acknowledgement

Introduction

Bibliography

Acknowledgement

I would like to thank Prof. Robert Lipton for his help and patience.

Introduction-Definition of Composite Materials

Composites are materials that are combinations of two or moreorganic or inorganic components.

Fiberglass is the most common composite material, and consists ofglass fibers embedded in a resin matrix.

Composites are materials that are combinations of two or moreorganic or inorganic components.Fiberglass is the most common composite material, and consists ofglass fibers embedded in a resin matrix.

Introduction-Examples of Composite Materials:Sail

Introduction-Examples of Composite Materials:Bone

Introduction-Composite Materials

Important in many branches of Mechanics, Physics, Chemistryand Engineering.

Contain features that are different at different length scales.

Figure: Heterogeneous Material

Physical parameters (conductivity, elasticity coefficients, etc)are discontinuous and change values between components(across a small length scale ε).

When components are intimately mixed, the physicalparameters oscillate rapidly and microscopic structurebecomes complicated.

Introduction-Homogenization Theory

An important problem is the determination of the macroscopicproperties of heterogeneous materials.

A good approximation to the macroscopic behavior of suchmaterials is obtained through a suitable asymptotic theorycalled Homogenization Theory.

Homogenization Theory provides an accurate description ofthe macroscopic properties as the length scale ε tends to zeroin the equations describing phenomena such as heatconduction or elasticity.

Introduction

A good model for the study of physical behaviour of heterogeneousmaterial is given by

(τε)

−div

xε ,Duε

))= f on Ω,

uε ∈W1,p0 (Ω).

Ω is a bounded open set in Rn (piece of heterogeneousmaterial)

f given source term

ε > 0 (length scale)

uε interpreted as the electric potential, magnetic potential,temperature.

A describes the physical properties of different materials in thebody.

Introduction

(τε)

−div

xε ,Duε

))= f on Ω,

uε ∈W1,p0 (Ω).

f given source term

Introduction

(τε)

−div

xε ,Duε

))= f on Ω,

uε ∈W1,p0 (Ω).

f given source term

Introduction

(τε)

−div

xε ,Duε

))= f on Ω,

uε ∈W1,p0 (Ω).

f given source term

Introduction

(τε)

−div

xε ,Duε

))= f on Ω,

uε ∈W1,p0 (Ω).

f given source term

Introduction

(τε)

−div

xε ,Duε

))= f on Ω,

uε ∈W1,p0 (Ω).

f given source term

Introduction

If ε is really small, a direct numerical approximation to thesolution of (τε) may be expensive or even impossible.

Then homogenization gives an alternative way byapproximating these solutions by a function uH which solves

−div

(b(DuH

))= f on Ω,

uH ∈W1,p0 (Ω),

The ”homogenized” b: Physical parameters of a homogeneousbody, whose behaviour is ”equivalent” to the behaviour of thematerial with the given microstructure (Effective parameters).

Introduction

−div

(b(DuH

))= f on Ω,

uH ∈W1,p0 (Ω),

Introduction

−div

(b(DuH

))= f on Ω,

uH ∈W1,p0 (Ω),

Notation

Y := (0, 1)n is the unit cube in Rn.

Lpn(Ω) =

f : Ω→ Rn :

∫Ω|f (x)|p dx <∞

‖f ‖Lpn(Ω) =

(∫Ω|f (x)|p dx

If f ∈ L1n,loc and E ⊂ Rn is a bounded set of positive measure,

we define:

〈f 〉E =1

f (x)dx .

For p > 1 and q such that 1p + 1

q = 1,

W1,pper(Y)

u(y) ∈W1,p(Y) : u has the same trace on the opposite faces of Y

Notation

Lpn(Ω) =

f : Ω→ Rn :

∫Ω|f (x)|p dx <∞

‖f ‖Lpn(Ω) =

(∫Ω|f (x)|p dx

we define:

〈f 〉E =1

f (x)dx .

q = 1,

W1,pper(Y)

Notation

Lpn(Ω) =

f : Ω→ Rn :

∫Ω|f (x)|p dx <∞

‖f ‖Lpn(Ω) =

(∫Ω|f (x)|p dx

we define:

〈f 〉E =1

f (x)dx .

q = 1,

W1,pper(Y)

Notation

Lpn(Ω) =

f : Ω→ Rn :

∫Ω|f (x)|p dx <∞

‖f ‖Lpn(Ω) =

(∫Ω|f (x)|p dx

we define:

〈f 〉E =1

f (x)dx .

q = 1,

W1,pper(Y)

Notation

Lpn(Ω) =

f : Ω→ Rn :

∫Ω|f (x)|p dx <∞

‖f ‖Lpn(Ω) =

(∫Ω|f (x)|p dx

we define:

〈f 〉E =1

f (x)dx .

q = 1,

W1,pper(Y)

Notation

Lpn(Ω) =

f : Ω→ Rn :

∫Ω|f (x)|p dx <∞

‖f ‖Lpn(Ω) =

(∫Ω|f (x)|p dx

we define:

〈f 〉E =1

f (x)dx .

q = 1,

W1,pper(Y)

Notation

Lpn(Ω) =

f : Ω→ Rn :

∫Ω|f (x)|p dx <∞

‖f ‖Lpn(Ω) =

(∫Ω|f (x)|p dx

we define:

〈f 〉E =1

f (x)dx .

q = 1,

W1,pper(Y)

Hypotheses on A

A (x , ξ) satisfies the following conditions:

H1 A is Y-periodic and measurable with respect to the firstvariable.

H2 For any x ∈ Rn a.e, and ξ1, ξ2 ∈ Rn, (p ≥ 2):

i |A (x , ξ1)− A (x , ξ2)| ≤ β(|ξ1|+ |ξ2|)p−2 |ξ1 − ξ2|;ii (A (x , ξ1)− A (x , ξ2) , ξ1 − ξ2) ≥ α |ξ1 − ξ2|2 (|ξ1|+ |ξ2|)p−2;

H3 A (x , 0) ∈ Lqn(Ω).

Hypotheses on A

H3 A (x , 0) ∈ Lqn(Ω).

Hypotheses on A

H3 A (x , 0) ∈ Lqn(Ω).

Hypotheses on A

H3 A (x , 0) ∈ Lqn(Ω).

Boundary Value Problem and Homogenized BoundaryValue Problem

(τε)

−div

xε ,Duε

))= f on Ω,

uε ∈W1,p0 (Ω).

Ω is a bounded open set in Rn

f ∈ Lq(Ω)

ε > 0

Boundary Value Problem and Homogenized BoundaryValue Problem

−div

(b(DuH

))= f on Ω,

uH ∈W1,p0 (Ω),

b (ξ) =

A (y ,Dv(y)) dy ;

and v(y) is solution of the problem:∫Y(A (y ,Dv(y)) · Dϕ(y))dy = 0, ∀ϕ ∈W1,p

per (Y)

v(y) ∈ ξ · y + W1,pper (Y).

Homogenization Theorem

ASYMPTOTIC THEORY FOR AVERAGES

If A(x , ξ) satisfies the structure conditions H1, H2 and H3, thenfor any f ∈W−1,q, if uε is the solution of (τε) and uH is thesolution of (τ0), we have that:

uε uH in W1,p(Ω),

ε,Duε

) b(DuH) in Lq

n(Ω).

Given ω ⊂⊂ Ω, we have

〈uε〉ω −→⟨uH⟩ω

.⟨A( x

ε ,Duε)⟩ω−→

⟨b(DuH)

uε uH in W1,p(Ω),

ε,Duε

) b(DuH) in Lq

n(Ω).

.⟨A( x

⟨b(DuH)

uε uH in W1,p(Ω),

ε,Duε

) b(DuH) in Lq

n(Ω).

.⟨A( x

⟨b(DuH)

uε uH in W1,p(Ω),

ε,Duε

) b(DuH) in Lq

n(Ω).

.⟨A( x

⟨b(DuH)

uε uH in W1,p(Ω),

ε,Duε

) b(DuH) in Lq

n(Ω).

⟨A( x

⟨b(DuH)

uε uH in W1,p(Ω),

ε,Duε

) b(DuH) in Lq

n(Ω).

.⟨A( x

⟨b(DuH)

Compensated Compactness and Convergence of Energies

Div-CurlLet 1 < p < +∞. Let uh and gh be two sequences such that:

uh u, in W1,p(Ω),

gh g , in Lqn(Ω),

-div(gh) converges to -div(g) strongly in W−1,q(Ω).

Then, ∫Ω

(gh,Duh)ϕdx −→∫

Ω(g ,Du)ϕdx ,

for all ϕ ∈ C∞0 (Ω).

uh u, in W1,p(Ω),

gh g , in Lqn(Ω),

Then, ∫Ω

Ω(g ,Du)ϕdx ,

uh u, in W1,p(Ω),

gh g , in Lqn(Ω),

Then, ∫Ω

Ω(g ,Du)ϕdx ,

uh u, in W1,p(Ω),

gh g , in Lqn(Ω),

Then, ∫Ω

Ω(g ,Du)ϕdx ,

Using Convergence Theorem and CompensatedCompactness we get Convergence of Energies

uε uH in W1,p(Ω),

xε ,Duε

) b(DuH) in Lq

n(Ω),

−div(A(

xε ,Duε

)) = f = −div(b(DuH)).

Then∫Ω

ε,Duε

)· Duε

)ϕdx −→

DuH)· DuH

)ϕdx ,

uε uH in W1,p(Ω),

xε ,Duε

) b(DuH) in Lq

n(Ω),

−div(A(

xε ,Duε

)) = f = −div(b(DuH)).

Then∫Ω

ε,Duε

)· Duε

)ϕdx −→

DuH)· DuH

)ϕdx ,

uε uH in W1,p(Ω),

xε ,Duε

) b(DuH) in Lq

n(Ω),

−div(A(

xε ,Duε

)) = f = −div(b(DuH)).

Then∫Ω

ε,Duε

)· Duε

)ϕdx −→

DuH)· DuH

)ϕdx ,

uε uH in W1,p(Ω),

xε ,Duε

) b(DuH) in Lq

n(Ω),

−div(A(

xε ,Duε

)) = f = −div(b(DuH)).

Then∫Ω

ε,Duε

)· Duε

)ϕdx −→

DuH)· DuH

)ϕdx ,

Good model for the study of the physical behavior of aheterogeneous body with a fine periodic structure.For example, in:

Electrostatics

Magnetostatics

Stationary heat diffusion

Electrostatics

Magnetostatics

Electrostatics

Magnetostatics

Electrostatics

Magnetostatics

Linear Case

−div

)Duε(X )

)= f in Ω,

uε ∈ H1,p0 (Ω),

whereA(x

)αi ,

and p = 2.

Figure: Two phase material: A(

)= α1χ1

)+ α2χ2

Linear Case

−div

)Duε(X )

)= f in Ω,

uε ∈ H1,p0 (Ω),

whereA(x

)αi ,

and p = 2.

Figure: Two phase material: A(

)= α1χ1

)+ α2χ2

Linear case

The div-curl lemma applies in this case:∫Ω

)Duε(x) · Duε(x)dx

AHDuH(x) · DuH(x)dx ,

AHij =

A(y)(Dw i (y) + e i

)·(Dw j(y) + e j

and the functions w i (y) satisfy

−div(A(y)(Dy w i (y) + e i )) = 0, in Y.

Linear case

AHij =

A(y)(Dw i (y) + e i

)·(Dw j(y) + e j

Linear case

AHij =

A(y)(Dw i (y) + e i

)·(Dw j(y) + e j

Corrector Theory

Corrector Matrix

Pij(y) = ∂yi wj(y) + e j

where e ji = δij and w i periodic functions that satisfy

div(A(y)(Dw i (y) + e i )) = 0, y in Y.

Natural Question:Can we find an explicit formula for the limit of the product∫

Ω χεi (x)Duε(x) · Duε(x)dx?

∫Ωχεi (x)Duε(x) · Duε(x)dx −→

∫Ω∂αi A

HDuH(x) · DuH(x)dx

∫Ω∂αi A

HDuH(x) · DuH(x)dx

∫Ω∂αi A

HDuH(x) · DuH(x)dx

For example, if p = 2 and

A(x) = α1χ1(x) + α2χ2(x),

we have(∂αi A

∫Yχi (y)

(Dwk(y) + ek

Dw l(y) + e l)

For example, if p = 2 and

A(x) = α1χ1(x) + α2χ2(x),

we have(∂αi A

∫Yχi (y)

(Dwk(y) + ek

Dw l(y) + e l)

Motivation

Let α1 < α2 and

)= Aε(x) = α1χ

ε1(x) + α2χ

ε2(x) = α1χ1

)+ α2χ2

AεDuε · Duε =

(χε1 +

α1χε2

)Duε · Duε ≥

∫Ω|Duε|2

taking limits as ε tends to 0, we obtain:

AHDuH(x) · DuH(x)dx ≥ lim supε→0

∫Ω|Duε(x)|2 dx .

Motivation

Let α1 < α2 and

)= Aε(x) = α1χ

ε1(x) + α2χ

ε2(x) = α1χ1

)+ α2χ2

AεDuε · Duε =

(χε1 +

α1χε2

)Duε · Duε ≥

∫Ω|Duε|2

Motivation

Let α1 < α2 and

)= Aε(x) = α1χ

ε1(x) + α2χ

ε2(x) = α1χ1

)+ α2χ2

AεDuε · Duε =

(χε1 +

α1χε2

)Duε · Duε ≥

∫Ω|Duε|2

Motivation

Let α1 < α2 and

)= Aε(x) = α1χ

ε1(x) + α2χ

ε2(x) = α1χ1

)+ α2χ2

AεDuε · Duε =

(χε1 +

α1χε2

)Duε · Duε ≥

∫Ω|Duε|2

Motivation

Result:

limε→0

∫Ω|Duε|2 =

α1AH − (α2 − α1) ∂α2AH

)DuH · DuH

α1AHDuH · DuH −

((α2 − α1) ∂α2AH

)DuH · DuH

∂α2AH =

∫Yχ2(y)(Dw i (y) + e i ) · (Dw j(y) + e j)dy .

Motivation

Result:

limε→0

∫Ω|Duε|2 =

α1AH − (α2 − α1) ∂α2AH

)DuH · DuH

α1AHDuH · DuH −

((α2 − α1) ∂α2AH

)DuH · DuH

∂α2AH =

Motivation

Result:

limε→0

∫Ω|Duε|2 =

α1AH − (α2 − α1) ∂α2AH

)DuH · DuH

α1AHDuH · DuH −

((α2 − α1) ∂α2AH

)DuH · DuH

∂α2AH =

Motivation

Result:

limε→0

∫Ω|Duε|2 =

α1AH − (α2 − α1) ∂α2AH

)DuH · DuH

α1AHDuH · DuH −

((α2 − α1) ∂α2AH

)DuH · DuH

∂α2AH =

Motivation for the Nonlinear Case

Good model for:

Nonlinear Conducting Media

Power Law Plasticity

Good model for:

Let W 1,pper (Y ) be the set of all functions u ∈W1,p(Y ) with

mean value zero which have the same trace on the oppositefaces of Y .

We consider N-phase materials. The characteristic function

for the i-th material χi (y) is Y -periodic andN∑

χi (y) = 1.

Let A : Rn × Rn → Rn be defined by

A(y , λ) =N∑

χi (y)ai |λ|p−2 λ, with ai ≥ 0.

χi (y) = 1.

A(y , λ) =N∑

χi (y) = 1.

A(y , λ) =N∑

A satisfies:

For every λ ∈ Rn, A(·, λ) is Y -periodic and Lebesguemeasurable.

Have |A(y , 0)| = 0 for all y ∈ Rn.

Continuity

|A(y , λ1)− A(y , λ2)| ≤ C1 |λ1 − λ2| (|λ1|+ |λ2|+ 1)p−2 .

Monotoniciy

(A(y , λ1)− A(y , λ2), λ1 − λ2) ≥ C2 |λ1 − λ2|p

A satisfies:

Continuity

|A(y , λ1)− A(y , λ2)| ≤ C1 |λ1 − λ2| (|λ1|+ |λ2|+ 1)p−2 .

Monotoniciy

(A(y , λ1)− A(y , λ2), λ1 − λ2) ≥ C2 |λ1 − λ2|p

A satisfies:

Continuity

|A(y , λ1)− A(y , λ2)| ≤ C1 |λ1 − λ2| (|λ1|+ |λ2|+ 1)p−2 .

Monotoniciy

(A(y , λ1)− A(y , λ2), λ1 − λ2) ≥ C2 |λ1 − λ2|p

A satisfies:

Continuity

|A(y , λ1)− A(y , λ2)| ≤ C1 |λ1 − λ2| (|λ1|+ |λ2|+ 1)p−2 .

Monotoniciy

(A(y , λ1)− A(y , λ2), λ1 − λ2) ≥ C2 |λ1 − λ2|p

In this particular case, we have∫D

∣∣∣P(y ,∇uH(x))∣∣∣p dydx ≤ lim inf

k→∞

∫D|∇uε(x)|p dx .

What is the next step?

We want to find the same kind of bound for

A (x , ξ) = α1χ1 (x) |ξ|α1−2 ξ + α2χ2 (x) |ξ|α2−2 ξ

with α2 ≥ α1 > 2.

What is the next step?We want to find the same kind of bound for

A (x , ξ) = α1χ1 (x) |ξ|α1−2 ξ + α2χ2 (x) |ξ|α2−2 ξ

with α2 ≥ α1 > 2.

Dal Maso, Gianni and Defranceschi, Anneliese,Correctors forthe homogenization of Monotone Operators ,Differential andIntegral Equations, Volume 3, Number 6, November 1990,1151–1166.

Defranceschi, A., An Introduction to Homogenization andG-convergence, Lecture Notes, School on Homogenization,ICTP, Trieste, 1993.

Evans, Lawrence C., Partial Differential Equations, GraduateStudies in Mathematics, 19, 1998.

Fusco, N. and Moscariello, G., On the homogenization ofquasilinear divergence structure operators, Ann. Mat. PuraAppl. (4), Annali di Matematica Pura ed Applicata. SerieQuarta, 146, 1987, 1–13.

Lipton, Robert P., Assessment of the local stress state throughmacroscopic variables,R. Soc. Lond. Philos. Trans. Ser. AMath. Phys. Eng. Sci.,The Royal Society of London.Philosophical Transactions.Series A. Mathematical, Physicaland Engineering Sciences, 361, 2003, 1806, 921–946.

Lipton, Robert P., Homogenization of the product of weaklyconverging sequences of gradients,2000.

Lipton, Robert P.,Homogenization and fiel concentrations inheterogeneous media ,SIAM Journal on MathematicalAnalysis, 38, 2006, 1048–1059.

Pedregal,Pablo,Parametrized measures and variationalprinciples , Birkhauser Verlag, Boston, 1997.

Svanstedt, N., G-convergence and Homogenization ofSequences of Linear and Nonlinear Partial DifferentialOperators (PhD. Thesis).

Trudinger, N., Elliptic Partial Differential Equations of SecondOrder ; Springer, 2001.

theory of homogenization - colgate university

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