arXiv subjects

Viktor Beneš

Publications and source records attributed to Viktor Beneš.

4 recordsLinked to original sources

arXiv2025

Random marked nested tessellations applied to the modelling of deformation twinning in polycrystalline materials

Stochastic geometry provides a powerful framework for modelling complex random structures, with applications in physics, materials science, biology, and other fields. The three-dimensional microstructure of polycrystalline materials is usually modeled by a randomly marked tessellation, where the marks correspond to crystallographic orientations. The purpose of this study is to extend the modelling approach to a finer scale, focusing on the subcells that emerge when a material specimen is exposed to mechanical loading. Specifically, the deformation twinning gives rise to nested tessellation, where the subcells are parallel twin lamellae and their complement is embedded within the original mother cells. The aim of this study is to develop a parametric mathematical model of marked nested tessellation and to realize it using stochastic simulations. We were able to deal with this model using computational tools. The sensitivity of the model to selected key parameters was investigated using statistical methods. As an application, a numerical simulation of the stress and strain fields resulting from deformation twinning is provided, and the contribution of the subcells to the total strain energy density under varying initial conditions was evaluated. This study highlights the dynamic capabilities of stochastic geometry in modelling a phenomenon that changes the microstructure.

math-ph↗

arXiv2022

Fitting three-dimensional Laguerre tessellations by hierarchical marked point process models

We present a general statistical methodology for analysing a Laguerre tessellation data set viewed as a realization of a marked point process model. In the first step, for the points we use a nested sequence of multiscale processes which constitute a flexible parametric class of pairwise interaction point process models. In the second step, for the marks/radii conditioned on the points we consider various exponential family models where the canonical sufficient statistic is based on tessellation characteristics. For each step parameter estimation based on maximum pseudolikelihood methods is tractable. Model checking is performed using global envelopes and corresponding tests in the first step and by comparing observed and simulated tessellation characteristics in the second step. We apply our methodology for a 3D Laguerre tessellation data set representing the microstructure of a polycrystalline metallic material, where simulations under a fitted model may substitute expensive laboratory experiments.

stat.ME↗

arXiv2020

Decorrelation of a class of Gibbs particle processes and asymptotic properties of U-statistics

We study a stationary Gibbs particle process with deterministically bounded particles on Euclidean space defined in terms of an activity parameter and non-negative interaction potentials of finite range. Using disagreement percolation we prove exponential decay of the correlation functions, provided a dominating Boolean model is subcritical. We also prove this property for the weighted moments of a U-statistic of the process. Under the assumption of a suitable lower bound on the variance, this implies a central limit theorem for such U-statistics of the Gibbs particle process. A byproduct of our approach is a new uniqueness result for Gibbs particle processes.

math.PR↗

arXiv2018

Estimation of geodesic tortuosity and constrictivity in stationary random closed sets

We investigate the problem of estimating geodesic tortuosity and constrictivity as two structural characteristics of stationary random closed sets. They are of central importance for the analysis of effective transport properties in porous or composite materials. Loosely speaking, geodesic tortuosity measures the windedness of paths whereas the notion of constrictivity captures the appearance of bottlenecks resulting from narrow passages within a given materials phase. We first provide mathematically precise definitions of these quantities and introduce appropriate estimators. Then, we show strong consistency of these estimators for unboundedly growing sampling windows. In order to apply our estimators to real datasets, the extent of edge effects needs to be controlled. This is illustrated using a model for a multi-phase material that is incorporated in solid oxid fuel cells.

math.ST↗