Mesh generation for FEM based on centroidal Voronoi tessellations

Centroidal Voronoi tessellations (CVTs) are very useful in a variety of applications, which can be used in triangular or tetrahedral mesh generations. There are several algorithms for determining CVTs, including MacQueen's method, Lloyd's method, and generalized probabilistic Lloyd's method. The latter is a combination of MacQueen's method and Lloyd's method, which is thought to be one of the most efficient methods to determine high-quality CVTs without the need to explicitly construct Voronoi diagrams. However, the convergence of these methods is difficult to achieve, since they are inclined to be trapped at local minima of cost functional. In this paper, simulated annealing (SA) is introduced to overcome this problem, which is applied to make mesh generation in domains including convex domains, a concaved domain, a multi-connected domain, and a circular domain. The efficiency of this method, and 2-D and 3-D mesh generations are successfully verified through examples.