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受约束团簇的联系来推进描述多晶体的几何图形结构的理论和计算，从而实现新材料的发现。新材料发现的基本挑战之一是了解和控制金属、合金、陶瓷和其他多晶体中的颗粒结构的形成。在实践中，颗粒结构可作为颗粒图访问，即，作为由成像过程产生的大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的性质至关重要。到目前为止，由于处理中涉及大量的像素/体素，详细的时间分辨研究是不可能的。然而，颗粒结构可以用多边形来近似，或者更好地，由具有分段二次边界的对象来近似。如果这些表示只涉及少量参数，则在此几何级别上执行的计算可以显著减少计算时间。我们在2015年已经证明，通过聚类技术计算的所谓广义平衡功率图，与测量的谷物结构产生了前所未有的良好拟合。正如我们最近可以展示的那样，集群技术可以被优化为只涉及少量的像素/体素，现在是时候将这些和类似的颗粒图开发成新材料发现的变革性工具，首次实现时间分辨的高分辨率研究。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高效的算法：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样的图，为以前用目前的方法无法实现的颗粒形成过程的数据分析和模拟奠定基础。对于通过表面成像获得的给定粒子图，确定“最佳拟合”的粒子图。这将第一次展示无噪声情况下的结果如何传递到相关的噪声情况下，从而产生一种分析颗粒地图的概念上的新方法。通过颗粒图对动态颗粒结构进行建模和分析。这一分析新的动态镶嵌模型的科学基础将使我们能够对特定实验条件下的纹理形成机制获得新的见解或观点。该项目将结合和推进数据科学(约束聚类)、优化和凸几何的方法。真实的数据将由该项目的合作者提供。总而言之，这些研究的结果将提供严格的模型、算法和数学分析，允许根据测量的参数首次表征大型静态和动态颗粒图。