Preface Introduction

Homogenization Theory and some of itsapplications

John Fabricius Peter Wall

Department of MathematicsLuleå University of Technology

November 19, 2008

Preface Introduction

Homogenization of heterogeneous materials

Finding the macroscopic properties of a material that hasinhomogenities on the microscopic scale

or more generally

Homogenization of a heterogeneous material is a processleading to its macroscopic characterization with fewerparameters than those needed for a full description of theobject.

Preface Introduction

Some historical remarks

Dispersion – a system of homogeneous particles that are evenlydistributed in a homogeneous medium.

Poisson (1824) – A theory of induced magnetism forheterogeneous media based on a dispersion ofspherical and ellipsoidal particles.

Maxwell (1873) – Calculating the effective heat conductivity ofa dilute dispersion.

Einstein (1905) – Calculating the macroscopic viscosity of adilute suspension using the viscosity of thesolvent and the volume fraction of the dissolved(spherical) particles.

Bruggeman (1935) – Calculating the elasticity constants (bulkand shear moduli) of a dilute solid dispersion.

Preface Introduction

Some well known homogenization results for binarymixtures

Heat propagation

• Maxwell’s formula• Clausius–Mossotti relation (dielectric constant)• Maxwell Garnett mixing rule (permettivity,dielectric

constant??)• Lorenz–Lorentz equation (refractive index)

• The Bergman formula• Hashin–Shtrikman’s estimates (two-phase isotropic

composites)• The Reuss–Voigt–Wiener bounds• The Beran bounds

Preface Introduction

. . . the basic and classical ideas are old and, typically for anyclassic, have been rediscovered many times.

Preface Introduction

Homogenization

A mathematical theory for studying• differential operators with rapidly oscillating coefficients,• boundary value problems with rapidly changing

boundary conditions• equations in perforated domains.

The first results in homogenization theory were obtained byDe Giorgi and Spagnolo around 1970.

Homogenization techniques multiscale convergence,asymptotic expansion method, compensatedcompactness, Γ-convergence, G-convergence,H-convergence, p-connectedness, Youngmeasures, periodic unfolding method, stochastichomogenization.

Preface Introduction

Stationary heat problem

{−∇ · (κε(x)∇uε) = f in Ωuε = 0 on ∂Ω.

(1)

κ1 – conductivity of the surronding materialκ2 – conductivity of the spherical inclusionsε – parameter related to the size of the inclusionsκε – conductivity of the dispersionWhat happens as ε → 0?

Preface Introduction

Homogenized equation (ε → 0)

The appropriate macroscopic equation is

−∇ · (b∇u) = f

where b is a constant matrix. Moreover, as ε → 0,

κε(x)∇uε → b∇u weakly in L2(Ω).

The macroscopic material properties are contained in the(possibly isotropic) matrix b.

Preface Introduction

Optimal structures

The Hashin-Shtrikman bounds are valid for any two-phasecomposite and describe the possible eigenvalues of the matrixb.

Preface Introduction

Elasticity theory

• Hashin–Shtrikman bounds for the effective bulk modulusof a two-phase composite.

Fluid mechanics

• Darcy’s law for flow in porous media• Macroscopic properties of liquid dispersions and

emulsions

Preface Introduction

Iterated homogenization

{−∇ · a

( xε ,

xε2

,∇uε)

= f in Ωuε = 0 on ∂Ω

Preface Introduction

Iterated structures

Rank-two laminate

ε2→0→ ε→0→

Iterated honeycomb

Preface Introduction

Tribology

• The science of interacting surfaces in relative motion.• The word is derived from the Greek tribos which means

‘rubbing’.• An interdisciplinary science dealing with such diverse

phenomena as• friction• wear• lubrication• contact mechanics.

Preface Introduction

Reynolds’ equation

Lubrication theory is based on Reynolds’ equation

∇ ·(h3∇p

)= 6µv

∂h∂x1

in Ω.

For hydrodynamic lubrication the boundary condition is p = 0on ∂Ω.A 2D approximation for 3D viscuous thin-film flow that is usedto calculate the pressure distribution p = p(x1, x2) in fluid filmbearings.

Preface Introduction

Surface roughness

Roughness effects are usually ne-glected for laminar flow. But for thinfilm flows even small roughness be-comes significant.

• Technical surfaces are never perfectly smooth due tolimitations in the manufacturing process.

• Almost smooth surfaces are very expensive to produce.• Smoothening of a surface may lead to a decrease in

hydrodynamic performance.

Preface Introduction

Two kinds of roughness

ε

h(x, y, z) = h0(x) + hT(y) + hR(z)

ε2

h(x, y, z) = h0(x) + hT(y) + hR(z)

Preface Introduction

Homogenization of Reynolds’ equation

The effects of surface roughness can be analyzed byhomogenization theory. To this end, given ε > 0 we form thefunction

hε(x) = h0(x) + h1(x

ε

),

where h0 : Ω → R (continuous) and h1 : R2 → R (continuousand periodic). Leads to a Reynolds equation with rapidlyoscillating coefficients

∇ ·(h3ε∇pε

)= 6µv

∂hε∂x1

. (2)

In reality ε is very small, suggesting that we should let ε → 0.

Preface Introduction

The homogenized Reynolds equation

By homogenization theory one can prove that as ε → 0, thesolutions pε of (2) converge to a function p that solves

2

∑i,j=1

∂

∂xi

(aij

∂p∂xj

)= 6µv

∂h0∂x1

−2

∑i=1

∂bi∂xi

. (3)

The coefficents of the homogenized equation (3) are computedby the “averaging formulae”(

a11 a12 b1a21 a22 b2

)=

1|Y|

∫Y

h3(

1 + ∂v1∂y1∂v2∂y1

∂v0∂y1

∂v1∂y2

1 + ∂v2∂y2∂v0∂y2

)dy, (4)

where. . .

Preface Introduction

. . . v0, v1 and v2, also know as “flow factors”, are solutions ofthe “cell problems” (periodic boundary conditions)

∂

∂y1

(h3

∂v0∂y1

)+

∂

∂y2

(h3

∂v0∂y2

)= 6µv

∂h∂y1

(5a)

∂

∂y1

(h3

∂vi∂y1

)+

∂

∂y2

(h3

∂vi∂y2

)= −∂h

3

∂yi(i = 1, 2), (5b)

Here Y denotes the cell of periodicity, usually [0, 1)N, and hdenotes the function h(x, y) = h0(x) + h1(y).

Preface Introduction

An illustration of the convergence

ε→0−→

pε (deterministic) p (homogenized)

PrefaceIntroduction