Optimization of centroidal Voronoi tessellations

TL;DR

This paper develops an optimization framework for centroidal Voronoi tessellations considering geometric constraints, using shape calculus and sensitivity analysis, with numerical experiments demonstrating its effectiveness.

Contribution

It introduces a novel optimization approach for CVTs that incorporates geometric constraints and computes derivatives via shape calculus.

Findings

Effective optimization of CVTs with geometric constraints

Numerical experiments validate the approach

Applicable to density-based distributions

Abstract

In this paper, we investigate the optimization of Centroidal Voronoi Tessellations (CVT) under geometric constraints. For this purpose, we minimize a linear combination of the standard CVT energy functional with terms involving geometric attributes such as area and perimeter. The derivative of the objective functional with respect to the position of the generators is computed using techniques of shape calculus and sensitivity analysis of minimization diagrams. Several numerical experiments are presented to explore the geometric constraints of cells with identical areas, cells without small edges, and density-based distributions of cells.