Inverse Problems in Microstructural Evolution of Polycrystalline Materials

工程系统中使用的绝大多数固体材料都是多晶材料。它们由许多单晶组成，由称为晶界的内部界面的三维网络连接在一起。在许多情况下，材料的性能和完整性是由晶界网络的结构决定的。这种材料的例子是纳米孪晶铜，其强度主要由相干孪晶边界决定；微电子学中电阻率受晶界支配的细尺度互连。虽然这些材料呈现出极具吸引力的特性，但其背后的基本机制尚不清楚，因此对进一步开发提出了挑战。迎接这一挑战的努力已经在实验、理论、建模和仿真方面带来了新的研究方向。本工作的目标是开发数学工具及其软件实现，并提供一个基于实验数据提取这些材料属性的工具箱。研究者和他的同事将使用反问题公式结合最近开发的实验技术-高能衍射显微镜(HEDM) -来确定应用中普遍使用的材料的能量和迁移率，以实现这些目标。晶界能和迁移率是在五维空间上定义的，是控制热载荷作用下晶界演化的物理参数，其中最可接受的模型是Mullins方程。HEDM技术的出现为探索多晶材料的演变开辟了新的可能性，并将使我们能够直接计算晶界速度。这与由Mullins方程控制的逆问题相结合，将使我们准确地估计能量和流动性。我们在这项工作中使用的数学技术包括最优传输、偏微分方程的数值解和求解逆问题的优化技术。这项工作涉及新的数学和计算方法，以满足材料科学中的重要挑战。特别是了解在机械和热载荷下的微观结构演变。微观结构影响材料的可靠性和失效，并强烈依赖于物理参数-能量和流动性。我们将开发在工程应用材料中准确识别这些特性的技术。这对于电子、结构和燃烧部件的更好设计和可靠性至关重要。这些研究将为研究多晶材料对热机械载荷的响应铺平道路。这些结果的应用将对机械载荷发生的所有其他工程分支（桥梁，汽车，飞机，MEMS设备，假肢）以及地质材料的研究产生更广泛的影响。

The vast majority of the solid materials used in engineered systems are polycrystalline. They are composed of many single crystals joined together by a three dimensional network of internal interfaces called grain boundaries. In many cases, the performance and integrity of a material are determined by the structure of the grain-boundary network. Examples of such materials are nano-twinned copper whose strength is dominated by coherent twin boundaries; fine-scale interconnects in microelectronics whose resistivity is dominated by grain boundaries. While these material present extremely attractive properties, the fundamental mechanisms that underlie them are poorly-understood thus presenting a challenge to further development. Efforts to meet this challenge have led to new directions of research in experiment, theory, modeling and simulation. The goal of this work is to develop the mathematical tools and its software implementation and to provide a toolbox for extracting these material properties based on experimental data. The investigator and his colleagues will achieve these goals using inverse problem formulations in conjunction with recently developed experimental technique - The High Energy Diffraction Microscopy (HEDM) - to identify energy and mobility in materials ubiquitously used in applications. Grain boundary energy and mobility, which are defined on a five dimensional space, are the physical parameters that govern the evolution of grain boundaries under thermal loading where the most acceptable model is the Mullins equation. The advent of the HEDM technique opens new possibilities for probing the evolution of polycrystalline materials and will enable us the direct calculation of grain boundary velocities. This in conjunction with inverse problems governed by Mullins equation will lead us to accurate estimates of both energy and mobility. The mathematical techniques that we use in this work include Optimal Transport, Numerical solution of PDEs and Optimization Techniques for solving the inverse problems. This work deals with novel mathematical and computational approaches for meeting important challenges in materials science. In particular, understanding microstructure evolution under mechanical and thermal loading. Microstructure affects materials reliability and failure and is strongly dependent on physical parameters - the energy and mobility. We will develop techniques for accurate identification of these properties in materials of engineering applications. This is pivotal for better design and reliability of electronic, structures and combustion components. These studies will pave the road to deal with response to thermo-mechanical loading in polycrystalline materials. The applications of these results will have broader impact in essentially all the other branches of engineering where mechanical loads occur (bridges, cars, planes, MEMS devices, prostheses), as well as the study of geological materials.