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3D晶粒微观结构，使用随机几何工具。这是因为在许多情况下，材料的3D微观结构会显著影响其物理性能。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3D微观结构的行为。因此，随机几何的方法被用来开发基于随机闭集的参数化3D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的Viktor Benes教授小组密切合作。施密特和贝奈斯的研究小组都与各自机构的实验物理学家和材料科学家合作（卡尔·E·施密特教授）。Krill III，乌尔姆大学，微纳米材料研究所，和Ales Jäger博士，捷克共和国科学院，物理研究所）。他们将利用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，并将提供断层数据库。在拟议的项目中，我们将代表实验图像数据中观察到的颗粒微观结构（确定性）镶嵌，其中一个细胞对应于一个晶粒区域。广义拉盖尔镶嵌，所谓的广义平衡功率图（GBPD），是特别感兴趣的。他们能够模拟弯曲的晶界和非凸细胞。为了使GBPD模型适合数据，需要解决高维优化问题。因此，将开发改进的随机优化技术，如交叉熵方法。下一步是颗粒材料的3D微观结构的随机GBPD模型的开发。 这将需要对这些模型的性质进行理论和实证研究，这些模型尚未得到很好的研究。在可用的情况下，晶粒取向将作为细胞的随机标记被包括到随机模型中。通过将随机标记的镶嵌拟合到数据，我们将详细了解颗粒体积，形状和方向之间的空间依赖性。除了这些3D模型，我们将使用马尔可夫链描述晶粒粗化动态。马尔可夫转移核将有助于理解依赖于局部晶粒邻域的晶粒粗化行为。为了验证我们的随机颗粒跟踪模型，我们将使用由Krill组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.