Collaborative Research: Hybrid Finite-Element Level-Set Methods for Stress-Driven Interface Dynamics

随着材料系统尺寸的减小，其界面原子和体原子的数量之比增加。因此，在通常具有非凡物理力量的小系统中，界面性质可以成为主导。例如，正如文献所记载的那样，具有高度有序量子点、线或环组装的纳米级界面系统表现出显著的光电、磁性和机械性能，具有广泛的应用范围。对于纳米尺度的界面，基本理论是无效的，必须重新审视和修正。该项目开发了一大类复杂界面的计算技术和数学理论，这些界面由纳米尺度下的潜在异质系统的机械特性控制，旨在理解这些界面在材料行为中的关键作用。它将新颖的有限元方法和强大的水平集技术结合到新一代多尺度模拟程序中，并直接应用于许多纳米尺度材料过程。这些过程包括金属和合金的微观结构演变，外延薄膜中应变岛的聚集，以及应力固体中普遍的形态不稳定。在严谨的数学理论的支持下，本研究将产生一套经过良好测试的、适应学术和工业需求的仿真技术。同时，建立数学与纳米科学之间的联系，培养学生的跨学科研究能力。

As the size of a material system decreases, the ratio of the amount of its interfacial and bulk atoms increases. Consequently, interfacial properties can become dominant in small systems that often possess extraordinary physical powers. As is well documented, for instance, nanoscale interfacial systems with highly ordered assemblies of quantum dots, wires, or rings exhibit remarkable optoelectronic, magnetic, and mechanical properties that have a wide range of applications. Generally invalid for nanoscale interfaces, fundamental theories for interfaces of bulk materials must be examined and reworked. This project develops computational techniques and mathematical theories for a large class of complex interfaces governed by the mechanical properties of underlying heterogeneous systems at nanoscale, aiming at the understanding of the crucial role of such interfaces in material behavior. It combines the novel finite-element methodology and the powerful level-set technique into a new generation multiscale simulation program, and applies directly to a host of nanoscale material processes. Such processes include microstructural evolution in metals and alloys, aggregation of strained islands in epitaxial thin films, and general morphological instability in stressed solids. Powered by rigorous mathematical theories, a set of well tested simulation technologies that are adapted to academic and industrial needs should result from this research. Meanwhile, connections between mathematics and nanoscience should be established, and students will be trained in this interdisciplinary research.