Tailored design of centroidal Voronoi tessellations - Franco, Juan Sebastián Castaño (2025)
The Voronoi diagram has been the subject of study in many fields of science because of the range of applications that it can have. One of special interest is the centroidal Voronoi tessellation for its applications in optimal mesh generation, data compression, optimal quadrature rules for computing integrals, among others. In this work, our objective is to show how to build centroidal Voronoi tessellations using the theory developed by E. G. Birgin, A. Laurain and T. C. Menezes in their paper ``Sensitivity analysis and tailored design of minimization diagrams\" using optimization algorithms. After defining the objective functions, calculating the gradients and obtaining the centroidal Voronoi tessellations, several numerical experiments are presented to generate these diagrams with some specific properties. The first one consists of building centroidal Voronoi tessellations with cells of equal volume, two others are for building centroidal Voronoi tessellations with cells satisfying conditions on the relative size of their edges or their internal angles, and the last one is to construct centroidal Voronoi tessellations with cells of varying sizes governed by a given function. The experiments show that it is possible to construct customized centroidal Voronoi tessellations using off-the-shelf-well-established optimization algorithms.