Exploration of Gibbs-Laguerre Tessellations for Three-Dimensional Stochastic Modeling

Abstract

Random tessellations are well suited for the probabilistic modeling of three-dimensional (3D) grain microstructure of polycrystalline metals. The present paper deals with so-called Gibbs-Laguerre tessellations where the generators of a Laguerre tessellation form a Gibbs point process. The goal is to construct an energy function of the Gibbs point process from a suitable set of potentials, such that the resulting Gibbs-Laguerre tessellation matches some desired geometrical properties. Since the model is analytically hardly tractable, our main tool of analysis are stochastic simulations based on Markov chain Monte Carlo. These enable us to investigate the properties of the models, and, in the next step, to apply the thus gained knowledge to do a statistical reconstruction of an aluminum alloy based on 3D tomographic image data.

Existence and simulation of Gibbs-Delaunay-Laguerre tessellations

“…Note that the bounds (12) and (13) are not optimal. Both are highly sensitive to the value of K 0 , forcing it to be zero in practice, and especially for the model (LD, ϕ S ).…”

“…In [13] we presented an algorithm for the simulations of Gibbs-Laguerre tessellations in 3D. We were motivated by [4], where Gibbs Voronoi and Gibbs Delaunay tessellations in 2D were investigated.…”

“…For a general version of the algorithm see [8]. The detailed algorithm for simulations of Gibbs-Laguerre tessellations is available in [13], including the proposal distributions used here. Several parameters have to be specified, in our case W = 0.15 (upper bound of the space of marks) and z = 1000 (activity).…”

“…In [13] an algorithm for the simulation of stationary Gibbs Laguerre tessellations in R 3 was developed, inspired by 2D Voronoi case from [4]. One has to approximate the random tessellation in a bounded window and make use of finite volume Gibbs models.…”

Comparison of segmentation of 2D and 3D EBSD measurements in polycrystalline materials

“…2. The Markov chain Monte Carlo method of simulation of a Gibbs-Laguerre tessellation [16] is extended by adding the Euler angles to the state space. For this model of a tessellation structure, two different scenarios of orien- tations of cubic crystal grid of grains were simulated, namely: (A) the misorientation between neighbouring grains is low, (see Fig.…”