Geometric Self-Assembly of Rigid Shapes

AbstractSelf-assembly of shapes from spheres to nonsmooth and possibly nonconvex shapes is pervasive throughoutthe sciences. These arrangements arise in biology for animal flocking and herding, in condensed matterphysics with molecular and nano self-assembly, and in control theory for coordinated motion problems. Whileidealizing these often nonconvex objects as points or spheres aids in analysis, the effects of shapecurvature and convexity are often dramatic, especially for short-range interactions. In this paper, wedevelop a general-purpose model for arranging rigid shapes in Euclidean domains and on flat tori. The shapesare arranged optimally with respect to minimization of a geometric Voronoi-based cost function whichgeneralizes the notion of a centroidal Voronoi tessellation from point sources to general rigid shapes.Building upon our previous work in [L. J. Larsson, R. Choksi, and J.-C. Nave, SIAM J. Sci.Comput., 36 (2014), pp. A792--A827], we present an efficient and fast algorithm for the minimizationof this nonlocal, albeit finite-dimensional variational problem. The algorithm applies in any space dimensionand can be used to generate self-assemblies of any collection of nonconvex, piecewise smooth shapes. We alsoprovide a result which supports the intuition that self-assembled shapes should be centered in and alignedwith their Voronoi regions.