Inverse Problems in Microstructural Evolution of Polycrystalline Materials

工程系统中使用的绝大多数固体材料是多晶的。它们由许多单晶体组成，这些单晶体通过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由晶界网络的结构决定。这种材料的例子是纳米孪晶铜，其强度由相干孪晶界决定;微电子中的精细尺度互连，其电阻率由晶界决定。虽然这些材料具有非常吸引人的特性，但它们的基本机制却知之甚少，因此对进一步开发提出了挑战。 为了迎接这一挑战，在实验、理论、建模和仿真方面都出现了新的研究方向。这项工作的目标是开发数学工具及其软件实现，并提供一个工具箱，用于根据实验数据提取这些材料特性。 研究人员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）-来实现这些目标，以确定在应用中普遍使用的材料中的能量和流动性。定义在五维空间上的晶界能和迁移率是控制热载荷下晶界演变的物理参数，其中最可接受的模型是Mullins方程。HEDM技术的出现为探测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。这与由穆林斯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运输，偏微分方程的数值解和优化技术解决反问题。 这项工作涉及新的数学和计算方法，以满足材料科学的重要挑战。特别是，了解机械和热负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。我们将开发技术，以准确地识别这些性质的材料的工程应用。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。 这些研究将为多晶材料热机械载荷响应的研究铺平道路。这些结果的应用将在发生机械载荷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中产生更广泛的影响。

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.