Abstract
Self-assembly of shapes from spheres to nonsmooth and possibly nonconvex shapes is pervasive throughout the sciences. These arrangements arise in biology for animal flocking and herding, in condensed matter physics with molecular and nano self-assembly, and in control theory for coordinated motion problems. While idealizing these often nonconvex objects as points or spheres aids in analysis, the effects of shape curvature and convexity are often dramatic, especially for short-range interactions. In this paper, we develop a general-purpose model for arranging rigid shapes in Euclidean domains and on flat tori. The shapes are arranged optimally with respect to minimization of a geometric Voronoi-based cost function which generalizes 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 minimization of this nonlocal, albeit finite-dimensional variational problem. The algorithm applies in any space dimension and can be used to generate self-assemblies of any collection of nonconvex, piecewise smooth shapes. We also provide a result which supports the intuition that self-assembled shapes should be centered in and aligned with their Voronoi regions.
Submitted Version (
Free)
Published Version
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call