1. From clusters of particles to 2D bubble clusters
Edwin Flikkema, Simon Cox
IMAPS, Aberystwyth University, UK

2. Introduction and overview
The minimal perimeter problem for 2D equal area bubble clusters.
Systems of interacting particles
Global optimisation
2D particle clusters to 2D bubble clusters
Voronoi construction
2D particle systems:
-log(r) or 1/rp repulsive potential
Harmonic or polygonal confining potentials
Results

3. 2D bubble clusters Minimal perimeter problem: 2D cluster of N bubbles.
All bubbles have equal area.
Free or confined to the interior of a circle or polygon.
Minimize total perimeter (internal + external).
Objective: apply techniques used in interacting particle clusters to this minimal perimeter problem.

4. Systems of interacting particles
System energy:
Usually:
Example:
Lennard-Jones potential:
LJ13: Ar13
Used in: Molecular Dynamics, Monte Carlo, Energy landscapes

5. Energy landscapes Energy vs coordinate
Stationary points of U: zero net force on each particle
Minima of U correspond to (meta-)stable states.
Global minimum is the most stable state.
Local optimisation (finding a nearby minimum) relatively easy:
Steepest descent, L-BFGS, Powell, etc.
Global optimisation: hard.
Energy vs coordinate
Local optimisation

6. Global optimisation methods
Inspired by simulated annealing:
Basin hopping
Minima hopping
Evolutionary algorithms:
Genetic algorithm
Other:
Covariance matrix adaption
Simply starting from many random geometries

7. 2D particle systems Energy: Repulsive inter-particle potential:
Confining potential: or
harmonic
polygonal

8. 2D particle clusters
Pictures of particle clusters: e.g. N=41, bottom 3 in energy

9. Particles to bubbles Qhull Surface Evolver particle cluster
Voronoi cells
optimized perimeter

10. 2D particle clusters Polygonal confining potential: e.g. triangular
unit vectors
contour lines
discontinuous gradient: smoothing needed?

11. Technical details List of unique 2D geometries produced
Problem: permutational isomers.
Distinguishing by energy U not sufficient:
Spectrum of inter-particle distances compared.
Gradient-based local optimisers have difficulty with polygonal potential due to discontinuous gradient
Smoothing needed?
Use gradient-less optimisers (e.g. Powell)?

12. Results: bubble clusters: Free, circle, hexagon

13. Results: bubble clusters: pentagon, square, triangle
Elec. J. Combinatorics 17:R45 (2010)

14. Conclusions
Optimal geometries of clusters of interacting particles can be used as candidates for the minimal perimeter problem.
Various potentials have been tried. 1/r seems to work slightly better than –log(r).
Using multiple potentials is recommended.
Polygonal potentials have been introduced to represent confinement to a polygon

15. Acknowledgements
Simon Cox
Adil Mughal

16. Energy landscapes Energy vs coordinate
Stationary points of U: zero net force on each particle
Minima of U correspond to (meta-)stable states.
Global minimum is the most stable state.
Saddle points (first order): transition states
Network of minima connected by transition states
Local optimisation (finding a nearby minimum) relatively easy:
L-BFGS, Powell, etc.
Global optimisation: hard.
Local optimisation
Energy vs coordinate

17. 2D clusters: perimeter
is fit to data for free clusters