Tessellation Methods for Modeling the Material Structure

Abstract

Tessellation methods are a relatively new approach for modeling the structure of a material. In this paper, such structures are interpreted as sphere packing models, where molecules and atoms represent spheres of equal or different size. Based on the review of the literature, it is shown that the tessellation approach is a powerful method for modeling and simulating such structures with desirable metric and topological properties. Two basic tessellation methods are considered more in detail: the Delaunay tessellation and the Voronoi diagram in Laguerre geometry, as well as some of their generalizations. The principal concepts of both tessellation methods are briefly explained for a better understanding of the application details. It is noted that packing models created by tessellation methods are not based on the use of the gravity camp effect, which is a difference to numerical and mathematic programming modeling approaches. Therefore, tessellation methods permit the development of structures without taking into account the gravitation, what is important for modeling the structure on the microscopic and nano levels, where the influence of the gravitation is studied insufficiently. A review of the related literature is given, focusing on the details of the tessellation method and the particle size distribution.

Modeling the copper microstructure and elastic anisotropy and studying its impact on reliability in nanoscale interconnects

“…There is a growing interest in the computation community to use Voronoi tessellation to model microstructures and study their impact (Burtseva et al 2015;Itakura et al 2005;Rickman and Barmak 2013;Nabiollahi et al 2015;Rudd and Belak 2002;Musienko et al 2007;Piekoś et al 2008;Gao Guo Jie et al 2013. A Voronoi diagram in simplest case can be defined as follows: Given a finite set of generating points in a plane, their Voronoi diagram divides the plane into convex polygons containing exactly one generating point, such that, the line segments forming the Voronoi diagram are all the points in the plane that are equidistant to the two nearest generating points.…”

Overview and comparison of modelling methods for foams

“…The spheres which are computed with an algorithm of random closed packing of spheres101–103 are the input for the power tessellation 91. Thus, the resulting cell size distribution is similar to the sphere size distribution obtained by the algorithm 95…”

“…The Voronoi tessellation algorithm leads to geometric deviations compared to the real foam geometry. For example, in the foam structure which is obtained by the Poisson-Voronoi point tessellation algorithm the average number of faces per cell is higher than in real foams and the variations of cell sizes is statistically constant due to the placement of the seed points with this algorithm 95. Therefore, variations of the classical Voronoi tessellation were developed to better approximate the geometry of real foams, for example:Voronoi diagram with Laguerre geometry…”

“…This is a weighted Voronoi tessellation, also called radical or power tessellation 89,95–97. The term weighted Voronoi tessellation comes from the fact that every nucleus has a different growing speed (stated as weight) whereas in the classical Voronoi tessellations all the nuclei have the same growing speed/weight 98.…”

“…The term weighted Voronoi tessellation comes from the fact that every nucleus has a different growing speed (stated as weight) whereas in the classical Voronoi tessellations all the nuclei have the same growing speed/weight 98. The higher the weight of the corresponding nuclei, the higher the growing speed and thus the bigger the cell size 95. This results in a pre-determined size distribution which leads to circles (2 D) or spheres (3D) with different radii and thus to a cell size distribution in the model 99.…”