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

在拟议的项目中，我们将开发一个灵活的平台，用于随机分析，建模和模拟颗粒材料中的三维颗粒微观结构，使用随机几何工具。这是因为在许多情况下，材料的三维微观结构会显著影响其物理性质。目前德国-捷克研究项目的一个特殊目标是应用开发的数学工具，以便更好地了解多晶金属合金在反复热处理和机械处理下的三维微观结构的行为。因此，随机几何的方法被用于开发基于随机闭集的参数化三维模型，或者更准确地说，基于空间点过程和随机镶嵌。我们将与布拉格查尔斯大学维克托·贝内斯教授小组密切合作。Schmidt和Benes小组都与各自机构的实验物理学家和材料科学家合作（乌尔姆大学微观和纳米材料研究所Carl E. Krill III教授和捷克共和国科学院物理研究所Ales博士Jäger）。他们将以他们在多晶材料物理实验方面的专业知识为该项目做出贡献，他们将提供层析数据库。在提议的项目中，我们将通过（确定性）镶嵌来表示实验图像数据中观察到的颗粒微观结构，其中一个细胞对应于一个颗粒区域。拉盖尔镶嵌的推广，所谓的广义平衡幂图（GBPD），是特别有趣的。他们能够模拟弯曲的晶界和非凸细胞。为了使GBPD模型与数据拟合，需要解决高维优化问题。因此，随机优化技术的改进版本，如交叉熵法，将得到发展。下一步是为颗粒材料的三维微观结构开发随机GBPD模型。这将需要对这些模型的特性进行理论和实证研究，这些特性尚未得到很好的研究。在可用的情况下，晶体学的晶粒方向将作为细胞的随机标记包含在随机模型中。通过将随机标记的镶嵌拟合到数据中，我们将详细了解颗粒体积，形状和方向之间的空间依赖关系。除了这些3D模型外，我们还将使用马尔可夫链描述晶粒粗化动力学。马尔可夫转移核将有助于理解依赖于局部颗粒邻域的颗粒粗化行为。为了验证我们的随机晶粒跟踪模型，我们将使用Krill团队提供的多晶合金的实验4D图像数据。

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.