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

该提案涉及三个项目。第一个是利用动力学蒙特卡罗对异质外延生长进行建模和有效模拟，并将建立在先前NSF的支持基础上，该支持导致了傅里叶多重网格方法的发展，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速率的方法，使用弹性方程的局部计算，以及包含多物种的混合。第二个项目涉及使用最近开发的多相变分水平集框架在二维和三维空间模拟晶界运动，该框架使人们能够系统地推导出在曲率流下运动的颗粒网络的水平集方程。我们计划扩展这个公式，只使用几个水平集函数就可以模拟数千种种子。高频波传播的有效计算和薛定谔方程的半经典极限是第三个项目。所提出的算法是基于这样的观察，即大多数时候，在这些极限情况下，解非常局限于波数域。这可以通过使用快速局部卷积在该域中求解方程来利用。计划用克雷洛夫子空间方法计算矩阵指数来更新解。每一个拟议的项目都有可能对具有根本性和技术重要性的问题产生重大影响。由于异质外延生长在纳米尺度和中尺度上都有影响，因此在科学上很有意义。它在技术上是相关的，因为量子点材料是用这种方式制造的。我们提出的技术将大大提高仿真速度，从而促进模型的开发。利用曲率流动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。由于没有在三维空间中对大量颗粒进行可靠的模拟，因此建议的项目应该具有重大影响。高频波传播的有效计算具有从天线设计到地震传感等各个方面的重要意义。另一方面，对薛定谔方程的半经典极限的快速模拟可以为化学反应动力学、分子表面散射和光解等提供更深入的了解。

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.