Summer School Utrecht,

1
Homogenization Theory Block 2 Sabine Attinger
2
Lecture Homogenization
3
Elliptic Equation

• Derivation of Homogenized Equations

4
Elliptic Equation1-dimensional
Fine scale problem
with K a Y-periodic field
1. Step
5
Elliptic Equation

• 2. Step
• we equate equal powers of ,
• 3. Step
• require that the equations in orders and
  
• vanish (singular perturbations) and
• disregard all terms of orders higher than 1

6
Elliptic Equation

• 
  (2.1)
• 
  (2.2)
• 
  (2.3)

Result
7
Elliptic Equation

• Solvability condition
• A necessary condition for the existence of
  solutions (2.1) to (2.3) is that the integral
  over the r.h.s. -called -
  vanishes

8
Solvability condition

• Proof

9
Elliptic Equation

• 4. Step Solution to (2.1)
• the only solutions are constants in
• which implies that the homogenized solution is
  independent on the fine scale

10
Elliptic Equation

• Proof

11
Elliptic equation

• 5. Step
• Solution to (2.2) Check on solvability o.k.
• Proof

12
Elliptic equation

• 5. Step
• making use of separation of variables
• yields the cell problem which determines

13
Elliptic equation

• 6. Step
• Solution to (2.3) - solvability condition implies

14
Elliptic equation

• Homogenized elliptic equation
• and is determined by the cell problem

15
Comparison of methods
Local periodicity
In stochastic theory we have the requirement of
local stationarity. This is equivalent to local
periodicity in periodic media.
l
Stationarity
REV
Periodicity
REV
16
Elliptic Equation
Interpretation of the result The conductivity in
the fine scale problem has be replaced by an
effective conductivity
Note the heterogeneous fields have not only been
replaced by it mean values!!
17
Exercise 2

• 1. Derive the homogenized solution for the
  d-dimensional elliptic equation!
• 2. Investigate the properties of the effective
  conductivity tensor!

18
Elliptic Equationd-dimensional
Fine scale problem
1. Step
19
Elliptic Equationd-dimenional

• 2. and 3. Step

20
Elliptic Equationd-dimensional

• 4. Step Solution to (2.1)
• the only solutions are constants in
• which implies that the homogenized solution is
  independent on the fine scale

21
Elliptic equationd-dimensional

• 5. Step Solution to (2.2) Check on solvability-
  o.k
• making use of separation of variables
• yields the cell problem which determines the

22
Elliptic equationd-dimensional

• 6. Step Solution to (2.3)
• solvability condition implies

23
Elliptic equationd-dimensional

• Homogenized elliptic equation
• and is determined by the cell problems