Optimizing Voronoi Diagrams for Polygonal Finite Element Computations

Abstract

Summary. We present a 2D mesh improvement technique that optimizes Voronoi diagrams for their use in polygonal finite element computations. Starting from a centroidal Voronoi tessellation of the simulation domain we optimize the mesh by minimizing a carefully designed energy functional that effectively removes the major reason for numerical instabilities-short edges in the Voronoi diagram. We evaluate our method on a 2D Poisson problem and demonstrate that our simple but effective optimization achieves a significant improvement of the stiffness matrix condition number.

Trefftz polygonal finite element for linear elasticity: convergence, accuracy, and properties

“…To improve the quality of the Voronoï tessellation, the generating point of each Voronoï cell can be used as its center of mass, leading to a special type of Voronoï diagram, called the centroidal Vornoï tessellation (CVT) [40]. Sieger et al [41] presented an optimizing technique to improve the Voronoï diagrams for use in FE computations.…”

The finite cell method for polygonal meshes: poly-FCM

“…Therefore, Sieger et al employ a direct meshing approach [17]. They developed an algorithm to improve the quality of Voronoï diagrams for the use in FE computations [17]. The basic idea is to employ a simple variational mesh optimization procedure which is able to remove short edges from a centroidal Voronoï diagram (CVT) input mesh by minimizing a suitable energy functional [17].…”

“…Both methods, however, rely on Voronoï diagrams and their properties [16]. In the direct approach, the Voronoï tessellation is used to generate a suitable polytope mesh [17] while in the indirect approach, the duality transform is exploited [2]. That is to say, in a first step a primal triangular (tetrahedral) mesh is created.…”

“…In the computational mechanics community it is widely known that a single "bad" element may be sufficient to destroy the numerical conditioning of the system matrices. Therefore, Sieger et al employ a direct meshing approach [17]. They developed an algorithm to improve the quality of Voronoï diagrams for the use in FE computations [17].…”

Integrating Spatial and Attribute Characteristics of Extended Voronoi Diagrams in Spatial Patterning Research: A Case Study of Wuhan City in China

“…The Voronoi diagram is an important spatial interpolation method because of its geometric structure. Indeed, it can be used to determine the value of any unknown point based on the nearest known point's value [66][67][68]. The concept of distance is central to Voronoi diagrams.…”