homogenization theory - universiteit utrecht · it is one of the basic assumptions in...

Summer School Utrecht, August 14-25

UFZ Centre for Environmental ResearchLeipzig-Hallein the Helmholtz Association

Friedrich-Schiller University Jena

Homogenization Theory

Sabine Attinger




Lecture: Homogenization

e.g. Advection-Diffusion-Equ.

ExercisesExercises

HT of otherEquations

Elliptic Equations:HT in comparison with

other Upscaling Methods

Elliptic EquationsDerivation of

Homogenized Equations

LectureBlock 2

EllipticEquations

NumericalHomogenization

Elliptic EquationsCalculation of Effective

Coefficients

MotivationBasic Ideas

LectureBlock 1

ThursdayAugust 17

WednesdayAugust 16

TuesdayAugust 15




A DefinitionHomogenization Theory is concerned with the analysis of

Partial Differential Equations (PDEs) with rapidlyoscillating coefficients

(1.1)

wherea differential operatorthe solutiona nondimensional parameter associatedwith the oscillations

fu =Α εε

εΑεuε




Example

Steady flow through a saturated porous medium

(1.2)

with

pressure headconductivitysource/sink term

( ) ( ) fK =∇∇− xx φ

( )xφ( )xKf

lL




Example• 2 scales: observation scale and conductivity

oscillations on scale

• rewriting equation (1.2) in dimensionlessvariables

ε1ˆˆ x

lL

Lx

lx

Lxx ≡=⇒≡

( ) ( )xxxxx ˆˆˆ,ˆˆˆˆ fAK ==⎟

⎠⎞

⎜⎝⎛∇⎟

⎠⎞

⎜⎝⎛∇− εεφ

εφ

ε

Ll




Other examples

ρψ

±∇=∂

∂q

tC p ( ) ( )zpp +∇−= ψψKq

pressure/head

capacity

Darcy flow

sink terms

conductivity

3D Richards/Pressure Equation 3D Darcy Eqaution




Other examples

( ),...iiiit cccc ρθ ±∇∇+∇−=∂ Dq

concentrationTransport velocityporosityMolecular diffusionMachanical dispersionChemical reactions...




What is the problem?Heterogeneities may cause large computational

problems (3D)

ml 1≈lh

51

≈

mxmxm 10100100

810≈

Area

Typical aquifer heterogeneitiesNumerical resolution

Total number of grid cells

Is it possible to reduce the computationalresolution with tolerable errors?




Convergence estimates for the Method of Finite Elements yield

What is the problem?

εhCuu h ≤−

Heterogeneities may cause ill-conditioned (stiff)numerical problems

Is it possible to formulate a well-conditionednumerical problem?




Basic IdeasIs there an equivalent homogeneous aquifer?

InteractiveGroundWater (IGW), by Dr. Li

http://www.egr.msu.edu/igw/

Flow and Transport through a Complex Aquifer System




Basic IdeasLarge Scale Flow Model with effective conductivity

fine

grid

mod

ellarge grid

model

( ) f=∇+⋅∇− )()(~ xxKK εε φ ( ) f=∇⋅∇− )(0eff xK φ

K x( )r 0K




We want to derive an effective, homogeneous model, where the heterogeneity is no longer seen

Consider the limit

Basic Ideas

l

L0lL

ε = →

0→ε




Basic IdeasA heterogeneous medium is “similar” to a periodic field.

Two distinct length scales:l

L

Unit cell:




Basic IdeasWe want to derive an effective, homogeneous model, where the heterogeneity is no longer seen

Consider the limit

...




Questions1. Convergence to a limit=homogenized solution:

• Is there a limit , as ? • In which sense should we understand the

convergence (i.e. in which norm)? • What is the convergence rate?

u 0→ε




Questions

2. Derivation of the homogenized solution:

• What kind of equation does the limit satisfy? Suppose that the limiting equation is of the followingform

• Is the operator of the same type as ?

u

fAu =A εA

( ) f=∇+⋅∇− )()(~ xxKK εε φ( ) f=∇⋅∇− )(0eff xK φExample:




Questions3. Calculation and properties of :

• How can we compute homogenized operators / effectivecoefficients?

• How do the properties of the homogenized equationcompare with those of the fine scale problem?

• How do the effective coefficients depend on the fine scale problem?

A

( ) f=∇⋅∇− )(0eff xK φExample:




Questions

4. Comparison with other upscaling methods

How does Homogenization compare e.g. to - Stochastic Modelling (Ensemble Averaging)- Volume Averaging?




Questions

5. Numerical Homogenization:

• Can we design and implement efficientalgorithms for problem (1.1) based on themethod of homogenization?

• Can we calculate the homogenized equation in a computationally efficient manner?




Idea: is a small parameter in (1.1), thus it isnatural to expand in a power series in

•all terms depend explicitly on both and

Two-scale Expansion

εεu

ε

x̂ ε/x̂

( ) ...,,, 22

10 +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

εε

εε

εε xxuxxuxxuxu




•assumption of x and y as independent variables

(only for problems with scale separation

or for y-periodic problems)

Two-scale Expansion

Ll <<

( ) ( ) ( ) ( ) ...,,, 22

10 +++= yxuyxuyxuxu εεε

It is one of the basic assumptions in homogenization theory, that the solution can be expanded like this. The convergence of the expansion has in principle to be proved!!

( ) ( )xuxu 00⎯⎯ →⎯ →εε




Two-Scale ExpansionCounterexample for scale separation:

L

l ????




Two-Scale Expansion

L

L

L

Example: Propagation of a wave with wavelength in a heterogeneous porous medium

Fluctuations due to the heterogeneity

has to be large compared to lL




Two-Scale ExpansionExample:




Exercise 1How do spatial derivatives of a two-scale

function look like?

...,... +∇→∇+∂∂

→∂∂

xxx

Example:




Two-Scale Expansion• Partial derivatives then become

• Procedure:1.Insert the two-scale ansatz into the fine scale problem (1.1)2.Group the terms in orders of3.Take the limit4.Solve resulting equations for

yxyxx∇+∇→∇

∂∂

+∂∂

→∂∂

εε1,1

ε0→ε

,..., 10 uu




Comparison with Volume Averaging

Averaging volume

Averaging volume

Averaging Volume=REV: small compared to the macroscopic Volume large enough to contain all information about heterogeneities




Volume Averaging

• Introduction of a Filter Function (= SpatialAverage Function) as moving average

( )∫ −≡ )(1)( yyxx φφ Vd

VFyd

V

V




Comparison withVolume Averaging

• Starting Point

• Volume Average

?

( ) ( ) ( )xxx fK =∇∇− φ

( ) ( ) ( ) ( ) ( )xxxxx φφ ∇−∇≠=∇∇− KfKV

( ) ( )xxK f=∇∇− φVolAve

V




Comparison withStochastic Theories

Assumptions:• Spatially heterogeneous medium properties are modelled

as random space function or stochastic process• governing differential equations with dependent

variables become stochastic PDE’s.

Procedure:• Specific aquifer is considered as one

realisation out of the ensemble of all possible realisations.

• Average over all realizations




Comparison withStochastic Theories

• Averages over the ensemble first of all describestatistical properties of the formation.

• Their predictive value with respect to a particular(deterministic) geologic formation might be very limited.

• If deviations from the mean are small the mean value ischaracteristic or predictive also for a single deterministicrealisation (ergodicity assumption).




Example: Stochastic Theory

500 1000 1500x_1 [m]

200

700

1200

x_2

[m]

Stauffer et al., WRR, 2002

Different realizations of a catchment zone

Risk Assessment

Mean catchment zone




Comparison withStochastic Theory

• Starting Point

• Ensemble Average

( ) ( ) ( )xxx fK =∇∇− φ

( ) ( ) ( ) ( ) ( )xxxxx φφ KfK −≠=∇∇−

( ) ( )xx fK =∇∇− φens




Volume Averaging and ensemble Averaging becomes equivalent if ergodicity holds

Comparison of methods

∫∫ ==REV

dVfdffPf )(

Ensemble Average

Ergodicity:REV

l

Volume Average




In stochastic theory we have the requirement of local stationarity. This is equivalent to local periodicity in periodic media.

Comparison of methods

( ) ( )lxfxff +==

∫∫+

=lREVREV

dVfdVf

Local periodicity Stationarity

Stationarity:

Periodicity:REV

l

REV

( ) ( )lxfxf +=




Numerical HomogenizationIntroduction of mathematical norms to measure if e.g. errors

•numerical errors

•Homogenization errors…

are limited by an upper finite bound




Two-Scale ExpansionExample:




Numerical Homogenization

Example:




Summary - Block 1

• What is the problem with two-scale equations like

?• Introduction of the main questions:

1. Derivation of homogenized equations2. Calculation of coefficients3. Comparison to other upscaling methods4. Numerical Homogenization

( ) ( )xxxxx ˆˆˆ,ˆˆˆˆ fAK ==⎟

⎠⎞

⎜⎝⎛∇⎟

⎠⎞

⎜⎝⎛∇− εεφ

εφ

ε

homogenization theory - universiteit utrecht · it is one of the basic assumptions in...

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