Geometric Self-Assembly of Rigid Shapes: A Simple Voronoi Approach

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 minimiza...

[1]  G. Whitesides,et al.  Molecular self-assembly and nanochemistry: a chemical strategy for the synthesis of nanostructures. , 1991, Science.

[2]  N. Sloane,et al.  The Optimal Lattice Quantizer in Three Dimensions , 1983 .

[3]  Oliver Deussen,et al.  Beyond Stippling 
— Methods for Distributing Objects on the Plane , 2003, Comput. Graph. Forum.

[4]  Mark A. Peletier,et al.  Small Volume-Fraction Limit of the Diblock Copolymer Problem: II. Diffuse-Interface Functional , 2010, SIAM J. Math. Anal..

[5]  Ronald Fedkiw,et al.  Level set methods and dynamic implicit surfaces , 2002, Applied mathematical sciences.

[6]  Kazuo Murota,et al.  A fast Voronoi-diagram algorithm with applications to geographical optimization problems , 1984 .

[7]  K. Kawasaki,et al.  Equilibrium morphology of block copolymer melts , 1986 .

[8]  Felipe Cucker,et al.  Emergent Behavior in Flocks , 2007, IEEE Transactions on Automatic Control.

[9]  Sharon C Glotzer,et al.  Calculation of partition functions for the self-assembly of patchy particles. , 2011, The journal of physical chemistry. B.

[10]  I. Couzin.  Collective minds , 2007, Nature.

[11]  S. Nash,et al.  Truncated Newton-Based Multigrid Algorithm for Centroidal Voronoi Diagram Calculation , 2012 .

[12]  F. Stillinger,et al.  Improving the Density of Jammed Disordered Packings Using Ellipsoids , 2004, Science.

[13]  R. Choksi,et al.  On global minimizers for a variational problem with long-range interactions , 2012 .

[14]  Michael L Klein,et al.  Self-assembly and properties of diblock copolymers by coarse-grain molecular dynamics , 2004, Nature materials.

[15]  David J. Thuente,et al.  Line search algorithms with guaranteed sufficient decrease , 1994, TOMS.

[16]  J. Nocedal.  Updating Quasi-Newton Matrices With Limited Storage , 1980 .

[17]  Xin Zhao,et al.  Area-Preservation Mapping using Optimal Mass Transport , 2013, IEEE Transactions on Visualization and Computer Graphics.

[18]  Marco Dorigo,et al.  Self-Organized Coordinated Motion in Groups of Physically Connected Robots , 2007, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[19]  Qiang Du,et al.  Probabilistic methods for centroidal Voronoi tessellations and their parallel implementations , 2002, Parallel Comput..

[20]  O. Deussen,et al.  Capacity-constrained point distributions: a variant of Lloyd's method , 2009, SIGGRAPH 2009.

[21]  Andrew J. Bernoff,et al.  A Primer of Swarm Equilibria , 2010, SIAM J. Appl. Dyn. Syst..

[22]  Mathieu Desbrun,et al.  Blue noise through optimal transport , 2012, ACM Trans. Graph..

[23]  Mark A. Peletier,et al.  Small Volume Fraction Limit of the Diblock Copolymer Problem: I. Sharp-Interface Functional , 2009, SIAM J. Math. Anal..

[24]  Mathieu Lewin,et al.  The Crystallization Conjecture: A Review , 2015, 1504.01153.

[25]  J. MacQueen.  Some methods for classification and analysis of multivariate observations , 1967 .

[26]  J. Vermant,et al.  Directed self-assembly of nanoparticles. , 2010, ACS nano.

[27]  Wenping Wang,et al.  GPU-Assisted Computation of Centroidal Voronoi Tessellation , 2011, IEEE Transactions on Visualization and Computer Graphics.

[28]  Y. Brenier.  Polar Factorization and Monotone Rearrangement of Vector-Valued Functions , 1991 .

[29]  Jorge Nocedal,et al.  On the limited memory BFGS method for large scale optimization , 1989, Math. Program..

[30]  Mark A. Peletier,et al.  Hexagonal Patterns in a Simplified Model for Block Copolymers , 2013, SIAM J. Appl. Math..

[31]  Qiang Du,et al.  Centroidal Voronoi Tessellations: Applications and Algorithms , 1999, SIAM Rev..

[32]  Xiaofeng Ren,et al.  Evolution in off-critical diblock copolymer melts , 2008, Networks Heterog. Media.

[33]  Hongkai Zhao,et al.  A fast sweeping method for Eikonal equations , 2004, Math. Comput..

[34]  Thomas C. Hales.  Sphere packings, I , 1997, Discret. Comput. Geom..

[35]  L. Tóth.  Lagerungen in der Ebene auf der Kugel und im Raum , 1953 .

[36]  Nestor Guillen,et al.  Five lectures on optimal transportation: Geometry, regularity and applications , 2010, 1011.2911.

[37]  C. Villani.  Optimal Transport: Old and New , 2008 .

[38]  Robert V. Kohn,et al.  Energy-Driven Pattern Formation , 2006 .

[39]  Irad Yavneh,et al.  Adaptive Multiscale Redistribution for Vector Quantization , 2005, SIAM J. Sci. Comput..

[40]  Franz Aurenhammer,et al.  Minkowski-Type Theorems and Least-Squares Clustering , 1998, Algorithmica.

[41]  Marcelo Gattass,et al.  Lloyd's Algorithm on GPU , 2008, ISVC.

[42]  Quentin Mérigot,et al.  A Multiscale Approach to Optimal Transport , 2011, Comput. Graph. Forum.

[43]  Jean-Christophe Nave,et al.  An Iterative Algorithm for Computing Measures of Generalized Voronoi Regions , 2014, SIAM J. Sci. Comput..

[44]  Qiang Du,et al.  Uniform Convergence of a Nonlinear Energy-Based Multilevel Quantization Scheme , 2008, SIAM J. Numer. Anal..

[45]  Stanley Osher,et al.  Fast Sweeping Algorithms for a Class of Hamilton-Jacobi Equations , 2003, SIAM J. Numer. Anal..

[46]  Q. Du,et al.  The optimal centroidal Voronoi tessellations and the gersho's conjecture in the three-dimensional space , 2005 .

[47]  G. Fredrickson.  The equilibrium theory of inhomogeneous polymers , 2005 .

[48]  G. Fredrickson,et al.  Block Copolymers—Designer Soft Materials , 1999 .

[49]  Jiwon Kim,et al.  Self-assembly: from crystals to cells , 2009 .

[50]  Henry Cohn,et al.  New upper bounds on sphere packings I , 2001, math/0110009.

[51]  G. Whitesides,et al.  Self-Assembly at All Scales , 2002, Science.

[52]  Donald J. Newman,et al.  The hexagon theorem , 1982, IEEE Trans. Inf. Theory.

[53]  Sharon C. Glotzer,et al.  Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra , 2009, Nature.

[54]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[55]  Allen Gersho,et al.  Asymptotically optimal block quantization , 1979, IEEE Trans. Inf. Theory.

[56]  Lin Lu,et al.  Centroidal Voronoi Tessellation of Line Segments and Graphs , 2012, Comput. Graph. Forum.

[57]  Rustum Choksi,et al.  Coarsening and self-organization in dilute diblock copolymer melts and mixtures , 2009 .

[58]  Wolfgang Kreitmeier,et al.  Optimal vector quantization in terms of Wasserstein distance , 2011, J. Multivar. Anal..

[59]  James A. Sethian,et al.  Level Set Methods and Fast Marching Methods , 1999 .

[60]  Irad Yavneh,et al.  A multigrid approach to the scalar quantization problem , 2005, IEEE Transactions on Information Theory.

[61]  M. Peletier,et al.  Optimality of the Triangular Lattice for a Particle System with Wasserstein Interaction , 2012, 1212.6973.

[62]  Chenglei Yang,et al.  On centroidal voronoi tessellation—energy smoothness and fast computation , 2009, TOGS.

[63]  X. Ren,et al.  On the Derivation of a Density Functional Theory for Microphase Separation of Diblock Copolymers , 2003 .

[64]  Francesco Bullo,et al.  Esaim: Control, Optimisation and Calculus of Variations Spatially-distributed Coverage Optimization and Control with Limited-range Interactions , 2022 .

[65]  S. P. Lloyd,et al.  Least squares quantization in PCM , 1982, IEEE Trans. Inf. Theory.

[66]  Qiang Du,et al.  Convergence of the Lloyd Algorithm for Computing Centroidal Voronoi Tessellations , 2006, SIAM J. Numer. Anal..

[67]  Atsuyuki Okabe,et al.  Spatial Tessellations: Concepts and Applications of Voronoi Diagrams , 1992, Wiley Series in Probability and Mathematical Statistics.

[68]  M. Seul,et al.  Domain Shapes and Patterns: The Phenomenology of Modulated Phases , 1995, Science.

[69]  Xiaofeng Ren,et al.  Spherical Solutions to a Nonlocal Free Boundary Problem from Diblock Copolymer Morphology , 2008, SIAM J. Math. Anal..

[70]  Gunzburger,et al.  Advances in Studies and Applications of Centroidal Voronoi Tessellations , 2010 .

[71]  Emelianenko.  Fast Multilevel CVT-Based Adaptive Data Visualization Algorithm , 2010 .