Parametric representation and stochastic 3D modeling of grain microstructures in polycrystalline materials using random marked tessellations

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In the proposed project, we will develop a flexible platform for the stochastic analysis, modeling and simulation of 3D grain microstructures in particulate materials, using tools from stochastic geometry. This is motivated by the fact that, in many cases, the 3D microstructure of materials significantly influences their physical properties. A special goal of the present German-Czech research project is the application of the developed mathematical tools in order to better understand the behavior of the 3D microstructure of polycrystalline metallic alloys under repeated thermal and mechanical treatments. Therefore, methods from stochastic geometry are used to develop parametric 3D models which are based on random closed sets, or, more precisely, on spatial point processes and random tessellations. We will closely cooperate with the group of Prof. Viktor Benes at Charles University in Prague. Both the Schmidt and Benes groups cooperate with experimental physicists and material scientists at their respective institutions (Prof. Carl E. Krill III, Ulm University, Institute of Micro and Nanomaterials, and Dr. Ales Jäger, Academy of Science of the Czech Republic, Institute of Physics). They will contribute to the project with their expertise in performing physical experiments with polycrystalline materials and they will provide the tomographic data base. In the proposed project, we will represent the particulate microstructures observed in experimental image data by (deterministic) tessellations, where one cell corresponds to exactly one grain region. Generalizations of Laguerre tessellations, so-called generalized balanced power diagrams (GBPD), are of special interest. They are able to model curved grain boundaries and non-convex cells. In order to fit GBPD models to data, high-dimensional optimization problems need to be solved. As a result, modified versions of stochastic optimization techniques, such as the cross-entropy method, will be developed. The next step is the development of stochastic GBPD models for the 3D microstructure of particulate materials. This will entail both theoretical and empirical investigation of the properties of these models, which are not yet well studied. Where available, the crystallographic grain orientations will be included into the stochastic model as random marks of the cells. By fitting random marked tessellations to data we will gain a detailed understanding of spatial dependencies between grain volumes, shapes and orientations. In addition to these 3D models, we will describe grain coarsening dynamics using Markov chains. The Markov transition kernels will help to understand the grain coarsening behavior in dependence of the local grain neighborhoods. To validate our stochastic grain-tracking models, we will use experimental 4D image data of polycrystalline alloys, provided by the Krill group.