The mineralogical composition of a particle system is characterised based on a total of six different characteristics describing size and shape of the particles, which are obtained through X-ray micro tomography measurements [1]. Since these six characteristics are highly correlated, it is difficult to estimate their joint probability distribution directly. Therefore, the marginal distributions of each characteristic are estimated separately, and their correlation structure is modelled by an Archimedean copula.

[1] O. Furat, T. Leißner, K. Bachmann, J. Gutzmer, U. Peuker and V. Schmidt, Stochastic modeling of multidimensional particle characteristics using parametric copulas. Microscopy and Microanalysis 25 (2019), 720-734

Nonwoven fiber-based material have important applications in various areas, such as fuel cell technology, filtration or hygiene products. In this work, a single-fiber model is proposed by describing a fiber as a sequence of bond and torsion angles. The whole fiber is then modelled by a k-th order Markov chain, where an R-vine copula is used to model the transition kernel of the Markov chain [2]. Simulating a set of fibers in this way generates a structure that exhibits similar properties to those observed in experimental data.

[2] M. Weber, A. Grießer, E. Glatt, A. Wiegmann and V. Schmidt, Modeling curved fibers by fitting R-vine copulas to their Frenet representations. Microscopy and Microanalysis (in print).

The Boolean model is a powerful, but simple model in stochastic geometry. It is used as a baseline of many more complex models. The Boolean model is defined as union of independent and identically distributed random compact sets (called grains or particles), the random locations of which are given by a Poisson point process [4]. In this talk, basic properties of the Boolean model and a simulation algorithm will be presented.

[4] S.N. Chiu, D. Stoyan, W.S. Kendall, J. Mecke. Stochastic Geometry and its Applications, Chapter 3. J. Wiley & Sons (2013).

By sampling a set of vertices from a random point process and connecting them in a deterministic way (according to the rule of beta-skeletons) we result in a  random graph in the sense that the vertices are generated at random. This type of model can be used to model two- or three-phase materials, where connectivity properties of all phases can be nicely adjusted by parameters of the underlying graph. It has been used to model the microstructure of mesoporous silica [9] as well as solid oxide fuel cell electrodes [10].

[9] B. Prifling, M. Neumann, D. Hlushkou, C. Kübel, U. Tallarek and V. Schmidt, Generating digital twins of mesoporous silica by graph-based stochastic microstructure modeling. Computational Materials Science 187 (2021), 109934.

[10] M. Neumann, J. Stanek, O. Pecho, L. Holzer, V. Benes and V. Schmidt, Stochastic 3D modeling of complex three-phase microstructures in SOFC-electrodes with completely connected phases. Computational Materials Science 118 (2016), 353-364

Markov random fields (MRFs) are graph models, the vertices of which are random variables. The dependency structure of these random variables is tractable by the Markov property. By chosing a regular grid as the underlying graph, the realizations of the MRF can be considered to be images, i.e., MRFs can be used as random image models. Thus, have a broad range of applications ranging from image and texture synthesis, image compression and restoration, image segmentation and super-resolution to information retrieval [11]. Especially, using MRFs for image and texture synthesis makes them interesting as stochastic geometry models. In this talk basic properties of MRFs are given and an application for generating synthetic image data of polycrystalline materials is shown [12].

[11] L.Y. Wei, S. Lefebvre, V. Kwatra, G. Turk. State of the art in example-based texture synthesis. In Eurographics 2009, State of the Art Report, EG-STAR. Eurographics Association (2009), pp. 93-117.

[12] A. Kumar, L. Nguyen, M. DeGraef, V. Sundararaghavan, Markov random field approach for microstructure synthesis. Modelling and Simulation in Materials Science and Engineering (2016), 24(3), 035015.

Generative adversial networks (GANs) which are usually models for unsupervised learning have, for example, various applications in texture and image synthesis [13]. After training with a given data set of images, these networks can generate new images which are statistically similar to those of the training data set. Therefore, GANs can be considered to be spatial stochastic models. For example, in [14] a GAN was trained with with 3D image data depicting the three-phased microstructure of lithium-ion battery cathodes and solid oxide fuell cell anodes. Then, similar to a stochastic geometry model, the trained network is able to generate virtual, but realistic 3D image data.

[13] I.J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, Y. Bengio. Generative adversarial networks (2014). arXiv preprint arXiv:1406.2661.

[14] A. Gayon-Lombardo, L. Mosser, N.P. Brandon, S.J. Cooper. Pores for thought: generative adversarial networks for stochastic reconstruction of 3D multi-phase electrode microstructures with periodic boundaries. npj Computational Materials (2020), 6(1), 1-11.

Stochastic geometry models for the outer shell of individual particles are the basis for more complex models which consider the spatial positions of particles for the generation of virtual cathode microstructures. For modeling the shape of the outer shells of such particles often random fields on the unit sphere are used. In the Gaussian case the parameters of these models can be estimated using analytical formulas [7]. However, for more complex models (e.g, mixtures of Gaussian random fields) a direct estimation of model parameters is difficult. In this case machine learning libraries can be utilized for a fast optimization of model parameters [8].

[7] J. Feinauer, T. Brereton, A. Spettl, M. Weber, I. Manke and V. Schmidt, Stochastic 3D modeling of the microstructure of lithium-ion battery anodes via Gaussian random fields on the sphere. Computational Materials Science 109 (2015), 137-146.

[8] O. Furat, L. Petrich, D. Finegan, D. Diercks, F. Usseglio-Viretta, K. Smith and V. Schmidt, Artificial generation of representative single Li-ion electrode particle architectures from microscopy data. npj Computational Materials 7 (2021), 105.