From clusters of particles to 2D bubble clusters

Edwin Flikkema, ICMS, Edinburgh, March 2012

From clusters of particles to 2D bubble From clusters of particles to 2D bubble clustersclusters

Edwin Flikkema, Simon Cox IMAPS, Aberystwyth University, UK

http://www.aber.ac.uk/


Introduction and overviewIntroduction and overview

Introduction:The minimal perimeter problem for 2D equal area bubble

clusters.Systems of interacting particlesGlobal optimisation

2D particle clusters to 2D bubble clustersVoronoi construction

2D particle systems: -log(r) or 1/rp repulsive potentialHarmonic or polygonal confining potentials

Results



2D bubble clusters2D 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.



Systems of interacting particlesSystems of interacting particles● System energy:● Usually:● Example:

...,,,

kjikjiijkji

jiij

iii rrrVrrVrVU

||||, jijiij rrVrrV

612

11rr

rV

Lennard-Jones potential:

LJ13: Ar13

● Used in: Molecular Dynamics, Monte Carlo, Energy landscapes



Energy landscapesEnergy landscapes● 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



Global optimisation methodsGlobal optimisation methods• Inspired by simulated annealing:

Basin hopping Minima hopping

• Evolutionary algorithms: Genetic algorithm

• Other: Covariance matrix adaption Simply starting from many random geometries



2D particle systems2D particle systems• Energy:

• Repulsive inter-particle potential:

• Confining potential:

i

iconfi ij

jirep rVrrVU

)log(rrVrep or prep rrV 1

2

21 rKrVconf harmonic

polygonal 2

max21

kkconf urKrV

,...2,1p



2D particle clusters2D particle clusters• Pictures of particle clusters: e.g. N=41, bottom 3 in energy

-945.421319 -945.419781 -945.419508



Particles to bubblesParticles to bubbles

particle cluster Voronoi cells optimized perimeter

Qhull Surface Evolver



2D particle clusters2D particle clusters• Polygonal confining potential: e.g. triangular

1u2u

3u

2

max21

kkconf urKrV contour lines

discontinuous gradient: smoothing needed?

unit vectors



Technical detailsTechnical 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)?



N=31-37

Results: bubble clusters: Results: bubble clusters: Free, circle, hexagonFree, circle, hexagon



N=31-37

Results: bubble clusters: Results: bubble clusters: pentagon, square, trianglepentagon, square, triangle

Elec. J. Combinatorics 17:R45 (2010)



ConclusionsConclusions• 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



AcknowledgementsAcknowledgements• Simon Cox

• Adil Mughal



Energy landscapesEnergy landscapes● 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.

Energy vs coordinate

Local optimisation



2D clusters: perimeter

is fit to data for free clusters


From clusters of particles to 2D bubble clusters

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