Quantifying Material Microstructures with Quaternions

用四元数量化材料微观结构

• 批准号: 0855402
• 负责人: Christopher Schuh
• 金额: $ 31.5万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-15 至 2012-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0855402&HistoricalAwards=false
• 关键词: Quantifying; Material; Microstructures; Quaternions

This Award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).TECHNICAL SUMMARY:This project seeks to develop a new mathematical framework for the description of crystal orientations, as well as misorientations between crystals. The approach is built upon four-dimensional vectors called quaternions; a quaternion can represent the axis and angle of a rotation in a simple way, and distributions of quaternions can be used to describe populations of crystal orientations (i.e., texture information), as well as populations of grain boundary misorientations. The intellectual merit of the present project lies in three technical objectives. The first of these is the development of the mathematical tools required to express orientation distributions in the quaternion framework, inclusive of relevant crystal and sample symmetries. For the case of grain boundaries dual symmetries reflective of the two adjacent crystallites are simultaneously applied. The second technical objective of the project is to provide a sound mathematical connection between the new quaternion framework and the more conventional Euler angle-based framework in wide use in the field of texture analysis today; this connection permits forward-translation of decades of prior research into the new framework. The final technical objective of the project is to apply the new quaternion-based framework to address some pressing problems in the crystallography of grain boundaries and grain boundary junctions in polycrystalline materials; in particular the local correlations in grain boundary misorientations induced by triple junctions and quadruple nodes are assessed mathematically using this framework. The project also has significant broader impacts for the scientific community. In addition to student training in materials microstructures, texture analysis, and advanced characterization techniques, the project develops open-source software that is freely disseminated on the Internet. The software renders all of the mathematical manipulations developed under the project accessible to the community in the form of simple algorithmic tools. These tools permit researchers to evaluate crystallographic texture, intuitively represent grain boundary statistics, and map grain boundary misorientations overlaid upon conventional micrographs.NON-TECHNICAL SUMMARY:The discipline of materials science and engineering has reached a point where it is possible to design a material at the level of assembling individual crystals into a solid. Among the important considerations in materials design are the orientations of individual crystals in the material, and the interfaces or boundaries between crystals in the structure. The specific crystal orientations and interface structures formed in a material dictate to a very large extent the material?s ability to resist damage, to exhibit high strength and toughness, and to display functional properties. The present project develops the mathematical tools necessary to design materials microstructure, with specific emphasis on crystal orientations and misorientations between crystals. The project specifically develops methods based on so-called quaternions, four-dimensional vectors that can describe a crystal orientation. The use of quaternions in the field of material microstructure greatly simplifies the description of many important problems, and is conducive to the development of fast algorithms to characterize, analyze, and design new materials microstructures. The present project, in addition to developing these mathematical tools, also seeks a broader impact by disseminating computer algorithms free for use by the entire scientific community. These computer codes allow researchers to more intuitively visualize materials structure, and to perform the calculations required to optimize them as well.

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。技术摘要：该项目旨在开发一个新的数学框架，用于描述晶体取向，以及晶体之间的取向差。该方法建立在称为四元数的四维向量上;四元数可以以简单的方式表示旋转的轴和角度，并且四元数的分布可以用于描述晶体取向的群体（即，纹理信息），以及晶界取向不良的群体。本项目的智力价值在于三个技术目标。 第一个是发展所需的数学工具来表达取向分布的四元数框架，包括相关的晶体和样品的对称性。对于晶界的情况下，双对称性反射的两个相邻的微晶同时应用。该项目的第二个技术目标是在新的四元数框架和当今纹理分析领域广泛使用的更传统的基于欧拉角的框架之间提供良好的数学联系;这种联系允许将数十年的先前研究向前转化为新的框架。该项目的最终技术目标是应用新的基于四元数的框架来解决多晶材料中晶界和晶界结的晶体学中的一些紧迫问题;特别是使用该框架对由三重结和四重结引起的晶界取向差的局部相关性进行数学评估。该项目还对科学界产生了广泛的影响。除了对学生进行材料微观结构、纹理分析和先进表征技术方面的培训外，该项目还开发了在互联网上免费传播的开源软件。该软件以简单算法工具的形式向社区提供在该项目下开发的所有数学运算。这些工具使研究人员能够评估晶体学纹理，直观地表示晶界统计，并绘制覆盖在传统显微照片上的晶界取向差。非技术摘要：材料科学与工程学科已经达到了一个可以在将单个晶体组装成固体的水平上设计材料的程度。材料设计中的重要考虑因素包括材料中单个晶体的取向，以及结构中晶体之间的界面或边界。材料中形成的特定晶体取向和界面结构在很大程度上决定了材料的性能。具有抗损伤能力，表现出高强度和韧性，并显示出功能特性。本项目开发设计材料微观结构所需的数学工具，特别强调晶体取向和晶体之间的取向差。该项目专门开发了基于所谓的四元数的方法，四元数是可以描述晶体取向的四维向量。四元数在材料微结构领域的应用，大大简化了许多重要问题的描述，有利于发展快速算法来表征、分析和设计新材料微结构。本项目除了开发这些数学工具外，还通过免费传播计算机算法供整个科学界使用，寻求产生更广泛的影响。这些计算机代码使研究人员能够更直观地可视化材料结构，并进行优化所需的计算。