Accurate representation of the distributions of the 3D Poisson-Voronoi typical cell geometrical features

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Understanding the intricate and complex materials microstructure and how it is related to materials properties is an important problem in the Materials Science field. For a full comprehension of this relation, it is fundamental to be able to describe the main characteristics of the 3-dimensional microstructure. The most basic model used for approximating steel microstructure is the Poisson-Voronoi diagram. Poisson-Voronoi diagrams have interesting mathematical properties, and they are used as a good model for single-phase materials. In this paper we exploit the scaling property of the underlying Poisson process to derive the distribution of the main geometrical features of the grains for every value of the intensity parameter. Moreover, we use a sophisticated simulation program to construct a close Monte Carlo based approximation for the distributions of interest. Using this, we determine the closest approximating distributions within the mentioned frequently used parametric classes of distributions and conclude that these representations can be quite accurate. Finally we consider a 3D volume dataset and compare the real volume distribution to what is to be expected under the Poisson-Voronoi model.