Theory of Homogenization with Applications to ... - Suraj @ LUMS

exhaustedcrumΜηχανική

24 Οκτ 2013 (πριν από 3 χρόνια και 11 μήνες)

88 εμφανίσεις

Adnan

Khan

Department of Mathematics

Lahore University of Management Sciences


Introduction



Theory of Periodic Homogenization



The Advection Diffusion Equation


Eulerian
and Lagrangian Pictures



Non Standard Homogenization Theory



Summary

International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7
-
8 2010

2


Many physical systems involve more than one
time/space scales



Usually interested in studying the system at
the large scale



Multiscale techniques have been developed for
this purpose



We would like to capture the information at the
fast/small scales in some statistical sense

International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7
-
8 2010

3


Heterogeneous Porous Media


Bhattacharya et.al,
Asymptotics

of solute dispersion in periodic porous media,
SIAM J. APPL. MATH 49(1):86
-
98, 1989



Plasma Physics


Soward

et.al, Large Magnetic
Reynold

number
dynmo

action in spatially periodic
flow with mean motion, Proc. Royal Soc.
Lond
. A 33:649
-
733






Ocean Atmospheric Science


Cushman
-
Roisin

et.al, Interactions between mean flow and finite amplitude
mesoscale

eddies in a
baratropic

ocean
Geophys
.
Astrpophys
.
Fkuid

Dynamics
29:333
-
353, 1984



Astrophysics


Knobloch

et.al, Enhancement of diffusive transport in Oscillatory Flows,
Astroph
.
J., 401:196
-
205, 1992



Fully Developed Turbulence


Lesieur
. M., Turbulence in Fluids, Fluid Mechanics and its Applications 1,
Kluwer
,
Dordrecht, 1990

International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7
-
8 2010

4

International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7
-
8 2010

5


To smooth out small scale heterogeneities



Assume periodicity at small scales for
mathematical simplification



Capture the behavior of the small scales in
some ‘effective parameter’



Obtain course grained ‘homogenized’ equation
at large scale





International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7
-
8 2010

6


As a ‘toy’ problem consider the following
Dirichlet Problem












D is periodic in the second ‘fast’ argument





International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7
-
8 2010

7


Using the ‘
ansatz







Where are periodic functions



We obtain





International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7
-
8 2010

8


Collecting terms with like powers of
ε

we
obtain the following asymptotic hierarchy



O(1):



O(
ε
):



O(
ε
2
):



International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7
-
8 2010

9


Applying periodicity and zero mean conditions



O(1)



O(
ε
)

where → The ‘Cell Problem’




O(
ε
2
) on Homogenized








on



Equation




International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7
-
8 2010

10


We have obtained an ‘homogenized’ equation



The effective diffusivity is given by



Where the average over a period is




a is obtained by solving the ‘cell problem’




International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7
-
8 2010

11


Transport is governed by the following


non dimensionalized
Advection Diffusion
Equation





There are different
distinguished limits





Weak Mean Flow



Equal Strength Mean Flow




Strong Mean Flow




International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7
-
8 2010

12


We study the
simplest case
of
two scales
with
periodic fluctuations and a mean flow



The case of
weak

and
equal strength
mean
flows has been
well studied



For the
strong mean flow
case standard
homogenization theory
seems to
break down


International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7
-
8 2010

13


For the first two cases we obtain a coarse
grained
effective equation





is the effective diffusivity given by




is the solution to the ‘cell problem’



The
goal

is to try an obtain a similar
effective
equation

for the strong mean flow case

International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7
-
8 2010

14


We study the transport using
Monte Carlo
Simulations

for tracer trajectories



We
compare

our
MC

results to numerics
obtained by
extrapolating homogenization
code



We develop a
non standard
homogenization
theory to explain our results


International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7
-
8 2010

15


We use
Monte Carlo Simulations
for the
particle paths to study the problem



The
equations of motion
are given by






The
enhanced diffusivity
is given by







International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7
-
8 2010

16


Some MC runs with Constant Mean Flow & CS
fluctuations



MC

and
homogenization

results
agree











Need a
modified
Homogenization

theory




International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7
-
8 2010

17


We consider one
distinguished limit
where we
take



We develop a
Multiple Scales

calculation for
the strong mean flow case in this limit



We get a
hierarchy

of equations (as in standard
Multiple Scales Expansion) of the form




is the
advection operator
, is a smooth function
with mean zero over a cell


International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7
-
8 2010

18


We develop the correct
solvability condition
for this case



We want to see if becomes large on time scales






This is equivalent to
estimating
the following
integral





The magnitude of this integral will determine the
solvability condition

International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7
-
8 2010

19



has
mean zero
over a ‘
cell




Two cases


Low

order rational
ratio


High

Order rational
ratio



Magnitude of Integral in


both these cases







International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7
-
8 2010

20

International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7
-
8 2010

21


Change to
coordinates

‘s’ & ‘t’
along

and
perpendicular

to the characteristics









Estimate
magnitude

of the integral in
traversing

the
cell

over the
characteristics







Analysis of the integral gives the following




Hence the
magnitude

of the
integral

depends on
the
ratio

of and



For
low order
rational

ratio
the integral gets
in time



For
higher order
rational
ratio

the integral stays
small over time


International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7
-
8 2010

22


We develop the asymptotic expansion in both the
cases



We have the following
multiple scales hierarchy







We derive the
effective equation
for the quantity

International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7
-
8 2010

23


For the
low order rational

case we get







For the
high order rational
ratio case we get





International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7
-
8 2010

24


Brief exposition of Periodic Homogenization



Toy Problem to illustrate the process



Advection Diffusion Equation


Eulerian Approach


Homogenization


Lagrangian Approach


Monte Carlo Simulation



Non Standard Homogenization Theory





International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7
-
8 2010

25