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Homogenization Theory
Sabine Attinger
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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
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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
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Example
Steady flow through a saturated porous medium
(1.2)
with
pressure headconductivitysource/sink term
( ) ( ) fK =∇∇− xx φ
( )xφ( )xKf
lL
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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
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Other examples
ρψ
±∇=∂
∂q
tC p ( ) ( )zpp +∇−= ψψKq
pressure/head
capacity
Darcy flow
sink terms
conductivity
3D Richards/Pressure Equation 3D Darcy Eqaution
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Other examples
( ),...iiiit cccc ρθ ±∇∇+∇−=∂ Dq
concentrationTransport velocityporosityMolecular diffusionMachanical dispersionChemical reactions...
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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?
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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?
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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
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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
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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
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Basic IdeasA heterogeneous medium is “similar” to a periodic field.
Two distinct length scales:l
L
Unit cell:
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Basic IdeasWe want to derive an effective, homogeneous model, where the heterogeneity is no longer seen
Consider the limit
...
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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
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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:
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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:
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Questions
4. Comparison with other upscaling methods
How does Homogenization compare e.g. to - Stochastic Modelling (Ensemble Averaging)- Volume Averaging?
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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?
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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
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•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
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Two-Scale ExpansionCounterexample for scale separation:
L
l ????
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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
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Two-Scale ExpansionExample:
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Exercise 1How do spatial derivatives of a two-scale
function look like?
...,... +∇→∇+∂∂
→∂∂
xxx
Example:
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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
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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
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Volume Averaging
• Introduction of a Filter Function (= SpatialAverage Function) as moving average
( )∫ −≡ )(1)( yyxx φφ Vd
VFyd
V
V
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Comparison withVolume Averaging
• Starting Point
• Volume Average
?
( ) ( ) ( )xxx fK =∇∇− φ
( ) ( ) ( ) ( ) ( )xxxxx φφ ∇−∇≠=∇∇− KfKV
( ) ( )xxK f=∇∇− φVolAve
V
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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
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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).
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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
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Comparison withStochastic Theory
• Starting Point
• Ensemble Average
( ) ( ) ( )xxx fK =∇∇− φ
( ) ( ) ( ) ( ) ( )xxxxx φφ KfK −≠=∇∇−
( ) ( )xx fK =∇∇− φens
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Volume Averaging and ensemble Averaging becomes equivalent if ergodicity holds
Comparison of methods
∫∫ ==REV
dVfdffPf )(
Ensemble Average
Ergodicity:REV
l
Volume Average
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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 +=
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Numerical HomogenizationIntroduction of mathematical norms to measure if e.g. errors
•numerical errors
•Homogenization errors…
are limited by an upper finite bound
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Two-Scale ExpansionExample:
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Numerical Homogenization
Example:
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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