QUIZ

Which of these convection patternsis

non-Boussinesq?

Homological Characterization Of

Convection Patterns Kapilanjan Krishan Marcio Gameiro

Michael Schatz Konstantin Mischaikow

School of Physics School of Mathematics

Georgia Institute of TechnologySupported by:

DOE, DARPA, NSF

Patterns and Drug Delivery

Caffeine in Polyurethane Matrix

D. Saylor et al., (U.S. Food and Drug Administration)

Patterns and Strength of Materials

Maximal Principal Stresses in Alumina

E. Fuller et al., (NIST)

Patterns and Convection

Camera

Light Source

Reduced Rayleigh

number=(T-Tc)/ Tc

=0.125

Convection cell

Spiral Defect Chaos

Homology

Using algebra to determine topology

Simplicial Cubical

Representations

Elementary Cubes and Chains

0-cube 1-cube 2-cube

1-chain 2-chain

v

0-chain

e f

Boundary Operator

f

e1

e2

e3

e4

e5

e6

e7

e8

1 2 3 4

1 2 3 4

ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ( ) 0

f e e e e

e e e e

5 6 7 8ˆ ˆ ˆ ˆ( ) 0e e e e

dimension# of Loops enclosing holes = of

homology group H1

Homology Summary

• Patterns are described by

• Dimension of = , the ith Betti number

• Homology: Computable topology

iH

iiH

Reduction to Binary Representation

Hot flow Cold flowExperiment image

Number of Components

Zeroth Betti number = 34

Hot flows vs. Cold flows

Hot flow Cold flow

Spiral Defect Chaos

Time ~ 103 Time ~ 103

Number of distinct components

Hot flow vs. Cold flow

Number of holes

First Betti number = 13

Time ~ 103 Time ~ 103

Number of distinct holes

Hot flow vs. Cold flow

Betti numbers vs EpsilonHot flow and Cold flow

Asymmetry between hot and cold regions

Non-Boussinesq effects  ?

Bettinumbers

Which of these convection patternsis

non-Boussinesq?

Simulations (SF6 )

Boussinesq Non-Boussinesq

(Madruga and Riecke)

Q=4.5

Boussinesq Simulations (SF6 )Time Series

Components Holes

Non-Boussinesq Simulations (SF6 )

Components Holes

Time Series

Simulations (CO2 ) at Experimental Conditions

Components Holes

Q=0.7

Boundary Influence

Time ~ 103 Time ~ 103

Number of connected components

Hot flow vs. Cold flow

Time ~ 103 Time ~ 103

Percentage of connected components

Hot flow vs. Cold flow

Convergence to Attractor

Frequency of

occurrence

cold0

: Number of cold

flow components ( ~1 )

EntropyJoint Probability

P(hot0 ,cold

0 , hot1 , cold

1)

Entropy(-Pi log

Pi)

Bifurcations?

Entropy vs epsilon

Entropy=8.3 Entropy=7.9 Entropy=8.9

Space-Time Topology

1-D Gray-Scott model

Space

Time

Time Series—First Betti number

Exhbits Chaos

Summary

• Homology characterizes complex patterns

• Underlying symmetries detected in data

• Alternative measure of boundary effects

• Detects transitions between complex states

• Space-time topology may reveal new insights

Homology source codes available at:

http://www.math.gatech.edu/~chomp