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
Edwin Flikkema, ICMS, Edinburgh, March 2012
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
Edwin Flikkema, ICMS, Edinburgh, March 2012
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.
Edwin Flikkema, ICMS, Edinburgh, March 2012
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
Edwin Flikkema, ICMS, Edinburgh, March 2012
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
Edwin Flikkema, ICMS, Edinburgh, March 2012
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
Edwin Flikkema, ICMS, Edinburgh, March 2012
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
Edwin Flikkema, ICMS, Edinburgh, March 2012
2D particle clusters2D particle clusters• Pictures of particle clusters: e.g. N=41, bottom 3 in energy
-945.421319 -945.419781 -945.419508
Edwin Flikkema, ICMS, Edinburgh, March 2012
Particles to bubblesParticles to bubbles
particle cluster Voronoi cells optimized perimeter
Qhull Surface Evolver
Edwin Flikkema, ICMS, Edinburgh, March 2012
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
Edwin Flikkema, ICMS, Edinburgh, March 2012
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)?
Edwin Flikkema, ICMS, Edinburgh, March 2012
N=31-37
Results: bubble clusters: Results: bubble clusters: Free, circle, hexagonFree, circle, hexagon
Edwin Flikkema, ICMS, Edinburgh, March 2012
N=31-37
Results: bubble clusters: Results: bubble clusters: pentagon, square, trianglepentagon, square, triangle
Elec. J. Combinatorics 17:R45 (2010)
Edwin Flikkema, ICMS, Edinburgh, March 2012
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
Edwin Flikkema, ICMS, Edinburgh, March 2012
AcknowledgementsAcknowledgements• Simon Cox
• Adil Mughal
Edwin Flikkema, ICMS, Edinburgh, March 2012
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
Edwin Flikkema, ICMS, Edinburgh, March 2012
2D clusters: perimeter
is fit to data for free clusters