Higher Order Macro Coefficients in Periodic Homogenization

M.VanninathanTIFR - Centre for Applicable Mathematics, Bangalore

Collaborators: Conca, San Martin (Chile)Smaranda (Romania)

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Heat Conduction Problem

Conductivity matrix: a(y) = (akl(y)), y ∈ IRN

Symmetric matrixBoundedness |akl(y)| ≤ MPositive definiteness a(y) ≥ a0I ∀y with a0 > 0Periodicity: akl(y) is Y− periodicExample ( Two phase conductor with phases (α0, α1))

akl(y) = a(y)δkl , a(y) = α0χT c (y) + α1χT (y)

Volume Proportion is (1− γ, γ) where γ = |T ||Y |

χT constitutes microstructure

Hear Conduction Problem

Basic operator A ≡ −div(a(y)∇) ≡ − ∂∂yk

{akl(y) ∂

∂yl

}Aε ≡ − ∂

∂xk

{akl

(xε

) ∂

∂xl

}We consider the equation Aεuε = f in IRN

uε is temperature

f is Heat source

Classical Homogenization Problem

Restrictions on oscillations

Assume uεH1

⇀ u

Aεuε ≡ − ∂

∂xk

{akl

(xε

) ∂uε∂xl

}= f in IRN

Question: What is the limit of the constitutive relation?

akl

(xε

) ∂uε∂xl

L2⇀??

How to take the average 〈akl( xε )∂uε

∂xl〉?

What is the equation satisfied by u?

Homogenization Theorem

There are two steps:

(a) Existence of q:Fourier conduction law is preserved.

(b) Evaluation of q using a cell test problem.

(a) There exist homogenized coefficients/matrix q = (qkl)

akl(x

ε)∂uε

∂xl

L2⇀ qkl

∂u

∂xl∀k = 1 . . .N

Qu ≡ − ∂

∂xk

{qkl

∂u

∂xl

}= f in IRN

(b)

qkl =1

|Y |

∫Y

amn(y)∂

∂yn(χk + yk)

∂

∂ym(χl + yl)

Cell Test functions :

{Aχk = ∂akl

∂ylχk is Y − periodic

Note the dependence on the microstructure

Above result establishes the existence of mixtures of materials withconductivities α0, α1 . . . in volume proportions γ0, γ1 . . . by varyingmicrostructures in a periodic manner.

Conductivities of the mixture are given by the eigenvalues of thematrix qExample (N = 1, 1 d Homogenization)Mixture of 2 phases (α0, α1) in proportion (1− γ, γ)

Homogenized Coefficient: q =

(1− γα0

+γ

α1

)−1(Harmonic Mean)

No dependence on the microstructure (represented by χT )

Bloch Wave Approach

How to go beyond the usual homogenized coefficients?Homogeneous medium (IRN− translation invariance) :Plane waves e iη.y

Periodic medium ( ZN - translation invariance) : Bloch WavesNew condition is (η − Y ) periodicity with η ∈ Y ′ =]− 1

2 ,12 [N

Periodic cell Y =]0, 2π[N

Dual cell Y ′ =]− 1

2,

1

2[N

We consider the eigenvalue problem for (λ(η), ψ(y , η))

Aψ ≡ − ∂

∂yk

{akl(y)

∂ψ

∂yl

}= λ(η)ψ in IRN

ψ is (η − Y ) periodic

i.e,ψ(y + 2πp) = ψ(y)e2πip.η ∀p ∈ ZN

change of variables ψ(y) = φ(y)e iη.y

Parametrized eigenvalue problems

A(η)φ = λ(η)φ, φ is Y − periodic

Shifted operator A(η) ≡ −(

∂∂yk

+ iηk

){akl(y)

(∂∂yl

+ iηl

)}Eigenvalues: 0 ≤ λ1(η) ≤ λ2(η) . . .→∞Eigenvalues: φ1(y , η)φ2(y , η) . . . (0.n basis)

Connection with Homogenization

At η = 0, λ1(0) = 0 is a simple eigenvalue of the operatorA(0) = A with eigenvector = constant.Regular perturbation theory then implies that

η 7−→ (λ1(η), φ1(y , η))

is analytic when |η| is small

λ1(η) = λ1(0) + λ′1(0)η + 12!λ

21(0)η2 + 1

3!λ(3)1 (0)η3 + 1

4!λ(4)1 (0)η4 + . . .

φ1(η) = φ1(0) + ∂φ1

∂ηk(0)ηk + . . .

Results λ1(0) = 0,All odd order derivatives of λ1(η) at η = 0 vanish

12!λ

(2)1 (0) = q, homogenized matrix

∂φ1∂ηk

(y , 0) = iχk(y), cell test functions

Interpretation of Homogenization: Lowest energy level (m = 1)Largest scale (η ≈ 0)

There are two possibilities to go further.

(i) High energy homogenization ( at level m > 1)

(ii) Keep m = 1, decrease scale i.e., |η| is not very small)

In case (ii), we can see the importance of higher order macro

coefficients d = 14!λ

(4)1 (0) . . . ( apart from q = 1

2!λ(2)1 (0))

Various derivatives of φ1(y , η) at η = 0 yield new cell testfunctions.

Aim Study properties of d , compare with q. In particular, seekproperties independent of microstructures. They are useful inOptimal Design Problems

High Energy Homogenization (level m)

Structural Conditions

(a) minη∈Y ′

λm(η) is attained at η = 0

It is the unique minimizer

(b) λm(0) is a simple eigenvalue of A(0) = A

(c) λ(2)m (0) is a positive definite matrix.

The corresponding homogenized matrix is 12!λ

(2)m (0).

Acoustic Wave Propagation

v εtt + Aεv ε = 0

v ε(x , 0) = φ1

(xε, εξ)v0(x)

v εt (x , 0) = φ1

(xε, εξ)v1(x).

λε1(ξ) =1

2!λ(2)1 (0)ξ2 +

1

4!ε2λ

(4)1 (0)ξ4 + O(ε4|ξ|6)

If |ξ| = o(ε−12 ) then ε2|ξ|4 = o(1).

In this case, q is relevant.Homogenized Medium is good enough:

Aε ≈ −q d2

dx2

For shorter waves |ξ| = o(ε−23 ) then ε2|ξ|4 = o(ε−

23 ) and

ε4|ξ|6 = o(1)In this case, both (q, d) are relevant: Homogenized Medium alone

is not good enough. Aε ≈ −q d2

dx2+ ε2d d4

dx4

With the decrease of scales, short waves undergo dispersion.

Physical Space Representation

We already have (energy - momentum) space representation

q =1

2!λ(2)1 (0), d =

1

4!λ(4)1 (0)

Basic operator A ≡ − ∂∂yk

{akl(y) ∂

∂yl

}Bilinear Form a(φ, ψ) =

∫Y

akl(y) ∂φ∂yl∂ψ∂yk

Cell test functions (χ(1), χ(2)) Y− periodic

Aχ(1) = ηk∂akl∂yl

Note χ(1) = ηkχk

Aχ(2) = (akl − qkl)ηkηl − ηkCkχ(1),

Ckφ ≡ −akl ∂φ∂yl −∂∂yl

(aklφ)

Physical space Representation

qη.η = 1|Y |a(χ(1) + η.y , χ(1) + η.y)

dη4 = − 1|Y |a(χ(2) − 1

2(χ(1))2, χ(2) − 12(χ(1))2)

Consequences

q is positive qη.η ≥ c|η|2

d is negative dη4 ≤ 0

These hold universally, in particular independent of microstructureMicrostructure contributes to q through χ(1)

Microstructure contributes to d through (χ(1), χ(2))In d , we see interaction of microstructures, including a self-interaction represented by (χ(1))2.

Question We would like to exploit negativity of d .How negative d can be?What is the minimum value of d when we vary microstructure?What is the microstructure at which this min value is attained ?Is it classical ( i.e., given by characteristic function ) ?Is it generalized / relaxed?

Results in one dimension

Two phases with conductivities (α0, α1) in proportion (1− γ, γ)

1. Value of q = (1−γα0+ γ

α1)−1

q is insensitive to microstructure

2. The values of d over classical microstructures range over theentire interval

− 1

12|Y |2γ2(1− γ)2q3(

1

α1− 1

α0)2 ≤ d < 0

3. Max value for d is attained at the followinggeneralized/relaxed microstructure ( not a characteristicfunction)

θ∗max(y) ≡ γ (local density of α1 − phase)

4. Min value for d is attained at the following classicalmicrostructure

θ∗min(y) = χ[0,γ|Y |](y)

5. As we introduce more and more interfaces between phases,the value of d increases.At min value, we have just one interface.At max value , there is a continuum of interfaces.

Remark Taken together the values of (q, d) provide Harmonicmean and a correction to it at the next scale.What about Arithmetic Mean ?

Murat-Tartar Theorem (α0 > α1)

µ−(γ) ≤ µj ≤ µ+(γ),

N∑j=1

1

µj − α1≤ 1

µ−(γ)− α1+

N − 1

µ+(γ)− α1,

N∑j=1

1

α0 − µj≤ 1

α0 − µ−(γ)+

N − 1

α0 − µ+(γ).

Here, we have used the following notations for the Arithmetic andHarmonic Means:

µ+(γ) = (1− γ)α0 + γα1,

1

µ−(γ)=

1− γα0

+γ

α1.

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Case of Laminates

As microstructure varies among classical laminates, we have

1. qη.η = µ−(γ)η21 + µ+(γ)|η̃|2 ( fixed value)

2. dη4 fills up the following interval

− 1

12|Y |2γ2(1−γ)2µ−(γ)−1

{µ−(γ))2η21(

1

α1− 1

α0) + |η̃|2(α1 − α0)

}2

≤ dη4 < 0

New feature: With the decrease of scale, we are able to seeinteraction between longitudinal and transverse modes.

Legendre- Hadamard Condition

Even though dη4 ≤ 0, d does not satisfy L-H condition.There are vectors η̃ and η such that d(η ⊗ η̃) · (η ⊗ η̃) > 0. Moreprecisely, we obtain the spectral decomposition of the map

d : SymN → SymN , (Mmn) 7→ (dklmnMmn)

Case : N = 2. Above map has two negative eigenvalues λ1 and λ2

and a third eigenvalue λ3 which is positive.They are given by

λ1,2 = −A2 + B2

2±

√(A2 − B2

2

)2

+1

9A2B2.

λ3 =2

3AB.

A =1

m2(1/α)

(1

α1− 1

α0

)√m(1/α)(m(b2T )−m2(bT )) > 0,

B = (α0 − α1)√m(1/α)(m(b2T )−m2(bT )) > 0.

bT :]0, 2π[→ IR is defined by

bT (y1) =

y1∫0

(χT1(s)− γ)ds.

T1 ⊂ [0, 2π]

T = T1 × [0, 2π]N−1

γ =|T1|2π

=|T ||Y |

.

Blossoming Principle

For any multivariate polynomial pn : IRm → IR of total degree n,there exists a unique symmetric n-affine functionPn : IRm × . . . IRm → IR satisfying

Pn(x , . . . , x) = pn(x) for all x ∈ IRm.

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Homogenization of Hashin Structures

Hashin - Shtrikman method/Formula

qH − α1

qH + (N − 1)α1= (1− γ)

(α0 − α1)

α0 + (N − 1)α1

q = qH is the ( scalar) homogenized coefficient

Questions What is minimum value of d over all Hashin structures?What is the minimizer?

We applied the principle from one - dimension:

d minimizes interface area(By duality, we maximize volume)

Our Guess Hashin structure with minimum number of Hashininclusions: APOLLONIAN - HASHIN structureNumerics confirm that our guess is correct.

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