QUIZ

1
QUIZ
Which of these convection patterns is non-Boussine
sq?
2
Homological Characterization Of Convection
Patterns

• Kapilanjan Krishan Marcio Gameiro
• Michael Schatz
  Konstantin Mischaikow
• School of Physics School of
  Mathematics
• Georgia Institute of Technology

Supported by DOE, DARPA, NSF
3
Patterns and Drug Delivery
Caffeine in Polyurethane Matrix
D. Saylor et al., (U.S. Food and Drug
Administration)
4
Patterns and Strength of Materials
Maximal Principal Stresses in Alumina
E. Fuller et al., (NIST)
5
Patterns and Convection
Light Source
Camera
Reduced Rayleigh number e(T-Tc)/ Tc
Convection cell
e0.125
6
Spiral Defect Chaos
7
Homology

• Using algebra to determine topology

Representations
Simplicial Cubical
8
Elementary Cubes and Chains

0-cube
2-cube
1-cube
2-chain
1-chain
0-chain
9
Boundary Operator
e5
e1

e6
f
e2
e8
e4
e7
e3

dimension of Loops enclosing holes
of
homology group H1
10
Homology Summary

• Patterns are described by
• Dimension of , the ith
  Betti number
• Homology Computable topology

11
Reduction to Binary Representation
Cold flow
Experiment image
Hot flow
12
Number of Components
Zeroth Betti number 34
13
Hot flows vs. Cold flows
Hot flow
Cold flow
14
Spiral Defect Chaos
e2
e1
15
Number of distinct components
Hot flow vs. Cold flow
Time 103 tn
Time 103 tn
e1
e2
16
Number of holes
First Betti number 13
17
Number of distinct holes
Hot flow vs. Cold flow
Time 103 tn
Time 103 tn
e1
e2
18
Betti numbers vs Epsilon
Hot flow and Cold flow
b0
Betti numbers
b1
b0
b1
e
Asymmetry between hot and cold regions Non-Boussin
esq effects  ?
19
Which of these convection patterns is non-Boussine
sq?
20
Simulations (SF6 )
(Madruga and Riecke)
Boussinesq
Non-Boussinesq
e1.4
e1.4, Q4.5
21
Boussinesq Simulations (SF6 )
Time Series
Components
Holes
b0
b0
b1
b1
22
Non-Boussinesq Simulations (SF6 )
Time Series
Components
Holes
b0
b1
b0
b1
23
Simulations (CO2 ) at Experimental Conditions
e2, Q0.7
Components
Holes
b0
b1
b0
b1
24
Boundary Influence
25
Number of connected components
Hot flow vs. Cold flow
Time 103 tn
Time 103 tn
e1
e2
26
Percentage of connected components
Hot flow vs. Cold flow
Time 103 tn
Time 103 tn
e1
e2
27
Convergence to Attractor
Frequency of occurrence
bcold0 Number of cold flow components ( e1 )
28
Entropy
Joint Probability P(bhot0 ,bcold0 , bhot1 ,
bcold1)
Entropy (-S Pi log Pi)
e
29
Entropy vs epsilon
e1.00
e1.13
e1.25

Entropy8.3
Entropy7.9
Entropy8.9
30
Space-Time Topology
Space
1-D Gray-Scott model
Time
Time SeriesFirst Betti number Exhbits Chaos
31
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