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