Optimal Grain Diagrams: Mathematical Analysis and Algorithms

最佳晶粒图：数学分析和算法

• 批准号: EP/X035883/1
• 负责人: Andreas Alpers
• 金额: $ 29.82万
• 依托单位: University of Liverpool
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FX035883%2F1
• 关键词: Optimal; Grain; Diagrams; Mathematical; Analysis

The project aims to enable new materials discovery by advancing the theory and computation of geometric diagram structures describing polycrystals by fully exploiting, for the first time, the recent link to constrained clusterings.One of the fundamental challenges for new materials discovery is to understand and control the forming of grain structures in metals, alloys, ceramics, and other polycrystals. In practice, grain structures are accessible as grain maps, i.e., as large 2D or 3D images resulting from an imaging process. Analysing geometric features in these data sets (e.g., grain boundaries) is of crucial importance for understanding and predicting materials properties. Detailed time-resolved studies have so far not been possible due the the large number of pixels/voxels involved in the processing.Grains structures, however, can be approximated by polygons or, even better, by objects with piecewise quadratic boundaries. The computations performed on this geometric level can drastically decrease computation times if these representations involve only small numbers of parameters. We have shown in 2015 that so-called generalised balanced power diagrams, computed by a clustering technique, yield unprecedented good fits to measured grain structures. As we could show recently that the clustering technique can be optimised to involve only a small number of pixes/voxels, it is now timely to develop these and similar grain diagrams into a transformative tool for new materials discovery, enabling, for the first time, time-resolved high-resolution studies.In particular, the aim of this project is to provide an in-depth mathematical analysis and efficient algorithms for the following three tasks/objectives:1. For a given noise-free grain map, determine a 'best fitting' grain diagram. This will allow us to represent such maps by few or physically meaningful parameters, laying the foundation for data analysis and simulations of grain forming processes, which was formerly out of reach with present methods.2. For a given grain map obtained by surface imaging, determine a 'best fitting' grain diagram. This will demonstrate, for the first time, how results from the noise-free case carry over to a relevant noisy case yielding a conceptionally novel method for analysing grain maps.3. Model and analyse dynamic grain structures via grain diagrams. This scientific groundwork for analysing new dynamic tessellation models will allow us to gain new insights or perspectives on grain forming mechanisms under specific experimental conditions.The project will combine and advance methods from data science (constrained clustering), optimisation, and convex geometry. Real data will be provided by collaborators on this project. Together, the results from these studies will provide rigorous models, algorithms, and a mathematical analysis that allows to characterize large static and, for the first time, dynamic grain maps from measured parameters.

该项目旨在通过首次充分利用最近与约束簇的联系，推进描述多晶的几何图结构的理论和计算，从而实现新材料的发现。新材料发现的基本挑战之一是理解和控制金属、合金、陶瓷和其他多晶中晶粒结构的形成。在实践中，颗粒结构可以作为颗粒图来访问，即作为成像过程产生的大 2D 或 3D 图像。分析这些数据集中的几何特征（例如晶界）对于理解和预测材料特性至关重要。由于处理过程中涉及大量像素/体素，迄今为止还无法进行详细的时间分辨研究。然而，颗粒结构可以通过多边形来近似，或者更好的是通过具有分段二次边界的对象来近似。如果这些表示仅涉及少量参数，则在此几何级别上执行的计算可以大大减少计算时间。我们在 2015 年证明，通过聚类技术计算的所谓广义平衡功率图可以对测量的晶粒结构产生前所未有的良好拟合。正如我们最近表明的那样，聚类技术可以优化为仅涉及少量像素/体素，现在是将这些和类似的晶粒图开发为新材料发现的变革性工具，从而首次实现时间分辨高分辨率研究。特别是，该项目的目的是为以下三个任务/目标提供深入的数学分析和有效的算法：1。对于给定的无噪声颗粒图，确定“最佳拟合”颗粒图。这将使我们能够用很少的或有物理意义的参数来表示这些图，为晶粒形成过程的数据分析和模拟奠定基础，而这在以前是现有方法无法实现的。2．对于通过表面成像获得的给定颗粒图，确定“最佳拟合”颗粒图。这将首次证明无噪声情况的结果如何转移到相关的噪声情况，从而产生一种概念上新颖的颗粒图分析方法。3．通过晶粒图对动态晶粒结构进行建模和分析。这一用于分析新的动态镶嵌模型的科学基础将使我们能够获得关于特定实验条件下晶粒形成机制的新见解或观点。该项目将结合并推进数据科学（约束聚类）、优化和凸几何的方法。真实数据将由该项目的合作者提供。总之，这些研究的结果将提供严格的模型、算法和数学分析，从而可以根据测量的参数来表征大型静态和动态颗粒图。