Computational Methods for Heteroepitaxial Growth, Grain Boundary Motion, and High Frequency Wave Propagation

这项建议涉及三个项目。第一个是用动力学蒙特卡罗模拟和有效模拟异质外延生长，它将建立在以前的NSF支持下，从而发展了一种傅立叶多重网格法，用于快速求解复杂几何形状的离散弹性方程。这项工作将扩展到开发获得廉价的上键的方法，使用弹性方程的局部计算，以及包含多个物种的混合。第二个项目涉及使用最近发展的多相变分水平集框架来模拟二维和三维的晶界运动，该框架允许系统地推导曲率流动下的颗粒网络的水平集方程。我们计划扩展这个公式，允许只使用几个级别集函数来模拟数千个种子。第三个项目是高频波传播的有效计算和薛定谔方程的半经典极限。所提出的算法是基于这样的观察，即在大多数情况下，在这些限制区域中，解非常局限于波数域。这可以通过使用快速的局部卷积来求解该域中的方程来利用。计划通过使用Krylov子空间方法计算矩阵指数来更新解决方案。每个拟议的项目都有可能对基本和技术上重要的问题产生重大影响。异质外延生长在科学上很有趣，因为它对纳米尺度和介观尺度都有影响。这在技术上是相关的，因为量子点材料是以这种方式制成的。利用曲率流研究晶界运动是应用数学和计算数学中的经典问题，在材料科学中具有重要意义。由于没有对三维的大量颗粒进行稳健的模拟，因此拟议的项目应该会产生重大影响。从天线设计到地震传感，高频波传播的有效计算具有重要的意义。另一方面，对薛定谔方程半经典极限的快速模拟可以提供对化学反应动力学、分子表面散射和光解离等更深入的了解。

This proposal involves three projects. The first concerns modeling andefficient simulation of heteroepitaxial growth using kinetic Monte Carlo and will build from prior NSF support which resulted in thedevelopment of a Fourier multigrid method for the fast solution ofdiscrete elastic equations for complex geometries. This work will be extended to develop methods for obtaining inexpensive upper bonds onrates, the use of local computations for elastic equations, and the inclusion ofintermixing of multiple species. The second project involves the simulation of grain boundary motion in two and three dimensions usinga recently developed multiphase variational level set framework whichallows one to systematically deduce level set equations for a network ofgrains moving under curvature flow. We plan to extend this formulation to allow the simulation of thousands of seeds by using only a few level setfunctions. The efficient computation of high frequency wave propagation and the semi-classical limit of the Schrodinger equation is the thirdproject. The proposed algorithm is based on the observation that most of the time, in these limiting regimes, the solutions are very localized in the wavenumber domain. This can be exploited by solving the equations in thisdomain using a fast local convolution. It is planned to update the solutionsby the computation of the matrix exponential using a Krylov subspaceapproach.Each of the proposed projects has the potential to have a significant impact on problems that are both fundamental and technologically important. Heteroepitaxial growth is scientifically interesting since it has effectson both nanoscales and mesoscales. It is technologically relevant sincequantum dot materials are made in this way. Our proposed techniques willgreatly increase the simulation speed thereby facilitating model development.The study of grain boundary motion using curvature flow is a classic problem in applied and computational mathematics which has importance in material science. Since there are no robust simulations of a large number grains in three dimensions the proposed project should have significant impact. The efficient computation of high frequency wave propagation has important facets ranging from antenna design to seismic sensing. On the other hand,fast simulation of the semi-classical limit of the Schrodinger equation could provide deeper insight into chemical reaction dynamics, molecular-surface scattering, and photodissociation, for example.