TY - JOUR
T1 - Periodic Centroidal Voronoi Tessellations
AU - Zhang , J.
AU - Emelianenko , M.
AU - Du , Q.
JO - International Journal of Numerical Analysis and Modeling
VL - 4
SP - 950
EP - 969
PY - 2012
DA - 2012/09
SN - 9
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/667.html
KW - Centroidal Voronoi Tessellations, Periodic Centroidal Voronoi Tessellations, Clustering energy, Critical point, Saddle point, Lloyd's algorithm, Shrinking dimer dynamics, Complex energy landscape, Optimal material design.
AB - <p style="text-align: justify;">Centroidal Voronoi tessellations (CVTs) are Voronoi tessellations whose generators
coincide with the mass centroids of the respective Voronoi regions. CVTs have become useful tools
in many application domains of arts, sciences and engineering. In this work, for the first time
the concept of the periodic centroidal Voronoi tessellations (PCVTs) — CVTs that exhibit certain
periodicity properties in the Euclidean space — is introduced and given a rigorous treatment. We
discuss the basic mathematical structures of the PCVTs and show how they are related to the so-called CVT clustering energy. We demonstrate by means of a concrete example that the clustering
energy can lose smoothness at degenerate points which disproves earlier conjectures about the
CVT energy being globally $C^2$-smooth. We discuss a number of algorithms for the computation
of PCVTs, including modifications of the celebrated Lloyd algorithm and a recently developed
algorithm based on the shrinking dimer dynamics for saddle point search. As an application, we
present a catalog of numerically computed PCVT patterns for the two dimensional case with a
constant density and a square unit cell. Examples are given to demonstrate that our algorithms
are capable of effectively probing the energy surface and produce improved patterns that may be
used for optimal materials design. The numerical results also illustrate the intrinsic complexity
associated with the CVT energy landscape and the rich geometry and symmetry represented by
the underlying PCVTs.</p>