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.

The description of distributions related to grain microstructure helps physicists to understand the processes in materials and their properties. This paper presents a general statistical methodology for the analysis of crystallographic orientations of grains in a 3D Laguerre tessellation dataset which represents the microstructure of a polycrystalline material. We introduce complex stochastic models which may substitute expensive laboratory experiments: conditional on the Laguerre tessellation, we suggest interaction models for the distribution of cubic crystal lattice orientations, where the interaction is between pairs of orientations for neighbouring grains in the tessellation. We discuss parameter estimation and model comparison methods based on maximum pseudolikelihood as well as graphical procedures for model checking using simulations. Our methodology is applied for analysing a dataset representing a nickel-titanium shape memory alloy.

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.

We study a particular marked three-dimensional point process sample that represents a Laguerre tessellation. It comes from a polycrystalline sample of aluminium alloy material. The 'points' are the cell generators while the 'marks' are radius marks that control the size and shape of the tessellation cells. Our statistical mark correlation analyses show that the marks of the sample are in clear and plausible spatial correlation: the marks of generators close together tend to be small and similar and the form of the correlation functions does not justify geostatistical marking. We show that a simplified modelling of tessellations by Laguerre tessellations with independent radius marks may lead to wrong results. When we started from the aluminium alloy data and generated random marks by random permutation we obtained tessellations with characteristics quite different from the original ones. We observed similar behaviour for simulated Laguerre tessellations. This fact, which seems to be natural for the given data type, makes fitting of models to empirical Laguerre tessellations quite difficult: the generator points and radius marks have to be modelled simultaneously. This may imply that the reconstruction methods are more efficient than point-process modelling if only samples of similar Laguerre tessellations are needed. We also found that literature recipes for bandwidth choice for estimating correlation functions should be used with care.

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

A comparison study of grain segmentation in 2D and 3D electron backscatter diffraction (EBSD) measurements of polycrystalline materials, based on a misorientation threshold, is presented. Using Laguerre tessellations with orientation marks, a simulation study that demonstrates a different sensitivity of segmentation in the case of low angle and high angle grain boundaries is involved. Then two metallic materials measured by 3D EBSD are investigated concerning segmentation, and there are given some recommendations for the misorientation thresholds. Computational tools such as DREAM 3D software enable significant data evaluation that is completed by a five-parameter analysis of grain boundaries.

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 aU-statistic of the process. Under the assumption of a suitable lower bound on the variance, this implies a central limit theorem for suchU-statistics of the Gibbs particle process. A by-product of our approach is a new uniqueness result for Gibbs particle processes.

In this paper, the grain microstructure and strain partitioning in a polycrystalline NiTi wire subjected to tensile loading was reconstructed from an experimental 3D-XRD dataset. The reconstruction of a volume containing more than 8000 stressed grains involved optimization with respect to both the geometrical features and material elastic properties. The geometrical features of the microstructure were reconstructed using Laguerre tessellations based on the experimental 3D-XRD dataset. Two different algorithms fitting Laguerre tessellations were applied in order to assess the sensitivity of the reconstruction to the choice of the algorithm. The material properties in terms of elastic anisotropy were refined from an initial published value to minimize the mismatch between experiment and simulation using an optimization algorithm based on linear elasticity simulations. As a result of this, we constructed a numerical microstructure model that statistically matches the experimentally probed material in terms of positions and sizes of grains as well as partitioning of elastic strain and stress in the microstructure (average elastic properties and standard deviations of piecewise constant components of elastic strain and stress tensors in grains).

The aim of this paper is to evaluate an ambitious imaging experiment and to contribute to the methodology of statistical inference of the three-dimensional microstructure of polycrystalline materials. The microstructure of the considered Al-3Mg-0.2Sc alloy was investigated by three-dimensional electron backscattered diffraction (3D-EBSD), i.e., tomographic imaging with xenon plasma focused ion beam (Xe-FIB) alongside EBSD. The samples were subjected to severe plastic deformations by equal channel angular pressing (ECAP) and annealed subsequently prior to the mapping. First we compared the misorientation level needed for a reliable segmentation of grains distinguishing between conventional evaluation of two-dimensional cuts and the 3D data set. Then, using methods of descriptive spatial statistics, various morphological characteristics of a large number of grains were analyzed, as well as the crystallographic texture and the spatial distribution of grain boundaries. According to the results stated so far in the literature, an even microstructure was expected, nevertheless local inhomogeneities in grains and grain boundaries with regard to their size, texture and spatial distribution were observed and justified.

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 data sets, the extent of edge effects needs to be controlled. This is illustrated using a model for a multiphase material that is incorporated in solid oxide fuel cells.

Far-field three-dimensional X-ray diffraction microscopy allows for quick measurement of the centers of mass and volumes of a large number of grains in a polycrystalline material, along with their crystal lattice orientations and internal stresses. However, the grain boundaries-and, therefore, individual grain shapes-are not observed directly. The present paper aims to overcome this shortcoming by reconstructing grain shapes based only on the incomplete morphological data described above. To this end, cross-entropy (CE) optimization is employed to find a Laguerre tessellation that minimizes the discrepancy between its centers of mass and cell sizes and those of the measured grain data. The proposed algorithm is highly parallel and is thus capable of handling many grains (>8,000). The validity and stability of the CE approach are verified on simulated and experimental datasets.

In the present work, the long-term stability of ultrafine-grained (UFG) copper at room temperature was investigated. The pure copper specimen was processed by 10 revolutions of high-pressure torsion (HPT) at room temperature. This procedure imposes an equivalent strain of about 300 to the material sample. In the region of these large strains, a saturation in grain size refinement occurs. UFG copper, deformed up to the region of microstructure saturation, was subsequently annealed at room temperature for 6 years. Microstructure changes of HPT-processed copper were investigated by means of 2D and 3D electron back scatter diffraction (EBSD) and also by transmission electron microscopy. It was found that the UFG microstructure of copper with saturated HPT-grain sizes coarsens significantly during long-term storage at room temperature. The analysis of grain volumes showed that the boundaries of coarse grains often contain flat segments with the coincidence site lattices (CSL) Sigma 3 and Sigma 9. The misorientation distributions revealed that most boundaries in the annealed microstructure are low energy grain boundaries of these kinds. However, groups of fine grains that are surrounded by random boundaries can also be found in the microstructure. Furthermore, 3D EBSD data were analysed in order to obtain a statistical microstructural information. The microstructure contains a high number of fine grains, but they form only a minority of the investigated volume. Quantitative geometrical characteristics of grain boundaries including CSL were described and interpreted.

The paper deals with planar segment processes given by a density with respect to the Poisson process. Parametric models involve reference distributions of directions and/or lengths of segments. These distributions generally do not coincide with the corresponding observed distributions. Statistical methods are presented which first estimate scalar parameters by known approaches and then the reference distribution is estimated non-parametrically. Besides a general theory we offer two models, first a Gibbs type segment process with reference directional distribution and secondly an inhomogeneous process with reference length distribution. The estimation is demonstrated in simulation studies where the variability of estimators is presented graphically.

In the paper asymptotic properties of functionals of stationary Gibbs particle processes are derived. Two known techniques from the point process theory in the Euclidean space R^d are extended to the space of compact sets on R^d equipped by the Hausdorff metric. First, conditions for the existence of the stationary Gibbs point process with given conditional intensity have been simplified recently. Secondly, the Malliavin-Stein calculus was applied to the estimation of Wasserstein distance between the Gibbs input and standard Gaussian distribution. We transform these theories to the space of compact sets and use them to derive a Gaussian approximation for functionals of a planar Gibbs segment process.

It is shown how the central limit theorem for U-statistics of spatial Poisson point processes can help to derive the central limit theorem for U-statistics of a Gibbs facet process from stochastic geometry. A full-dimensional submodel enables a simpler approach to the investigation. Finally the general situation is studied and the asymptotics with increasing intensity is described.

A main task of weather services is the issuing of warnings for potentially harmful weather events. Automated warning guidances can be derived, e.g., from statistical post-processing of numerical weather prediction using meteorological observations. These statistical methods commonly estimate the probability of an event (e.g. precipitation) occurring at a fixed location (a point probability). However, there are no operationally applicable techniques for estimating the probability of precipitation occurring anywhere in a geographical region (an area probability). We present an approach to the estimation of area probabilities for the occurrence of precipitation exceeding given thresholds. This approach is based on a spatial stochastic model for precipitation cells and precipitation amounts. The basic modeling component is a non-stationary germ-grain model with circular grains for the representation of precipitation cells. Then, we assign a randomly scaled response function to each precipitation cell and sum these functions up to obtain precipitation amounts. We derive formulas for expectations and variances of point precipitation amounts and use these formulas to compute further model characteristics based on available sequences of point probabilities. Area probabilities for arbitrary areas and thresholds can be estimated by repeated Monte Carlo simulation of the fitted precipitation model. Finally, we verify the proposed model by comparing the generated area probabilities with independent rain gauge adjusted radar data. The novelty of the presented approach is that, for the first time, a widely applicable estimation of area probabilities is possible, which is based solely on predicted point probabilities (i.e., neither precipitation observations nor further input of the forecaster are necessary). Therefore, this method can be applied for operational weather predictions.

In the paper asymptotic properties of functionals of stationary Gibbs particle processes are derived. Two known techniques from the point process theory in the Euclidean space R^d are extended to the space of compact sets on R^d equipped by the Hausdorff metric. First, conditions for the existence of the stationary Gibbs point process with given conditional intensity have been simplified recently. Secondly, the Malliavin-Stein calculus was applied to the estimation of Wasserstein distance between the Gibbs input and standard Gaussian distribution. We transform these theories to the space of compact sets and use them to derive a central limit theorem for functionals of a planar Gibbs segment process.

In the paper asymptotic properties of functionals of stationary Gibbs particle processes are derived. Two known techniques from the point process theory in the Euclidean space R^d are extended to the space of compact sets on R^d equipped by the Hausdorff metric. First, conditions for the existence of the stationary Gibbs point process with given conditional intensity have been simplified recently. Secondly, the Malliavin-Stein calculus was applied to the estimation of Wasserstein distance between the Gibbs input and standard Gaussian distribution. We transform these theories to the space of compact sets and use them to derive a central limit theorem for functionals of a planar Gibbs segment process.