A parametric stochastic 3D model for the description of complex three-phase microstructures is developed. Such materials occur for example in anodes of solid oxide fuel cells (SOFC) which consist of pores, nickel (Ni) and yttriastabilized zirconia (YSZ). The model is constructed using tools from stochastic geometry. More precisely, we model the backbones of the three phases by a certain class of random geometric graphs called beta-skeletons. This allows us to reproduce complete connectivity of all three phases as observed in experimental image data of a pristine Ni-YSZ anode as well as the prediction of volume fractions by model parameters. Finally a slightly generalized version of this model enables a good fit to experimental image data with respect to transport relevant microstructure characteristics and the length of triple phase boundary. Model validation is performed by comparing effective transport properties from finite element (FE) simulations based on 3D-data from the stochastic model and from tomography of real Ni-YSZ anodes. Moreover, the virtual, but realistic Ni-YSZ microstructures can be used for investigating the quantitative influence of microstructure characteristics on various physical properties and consequently on the performance of the

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.

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.

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.