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

这项建议涉及三个项目。第一个问题是使用动力学蒙特卡罗对异质外延生长进行建模和高效模拟，并将建立在先前NSF的支持基础上，该支持导致了傅立叶多重网格方法的发展，用于复杂几何形状的离散弹性方程的快速求解。 这项工作将被扩展到开发方法，以获得廉价的上债券onrates，使用本地计算的弹性方程，并列入intermixing的多个物种。 第二个项目涉及在二维和三维晶界运动的模拟usinga最近开发的多相变分水平集框架whichallows一个系统地推导水平集方程的网络下移动的晶粒曲率流。我们计划扩展这一提法，允许模拟成千上万的种子，只使用几个水平setfunctions。 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，在这些限制制度，解决方案是非常本地化的波数域。这可以通过使用快速局部卷积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础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是令人感兴趣的，因为它对纳米尺度和介观尺度都有影响。 这是技术相关的，因为量子点材料是以这种方式制造的。 我们提出的技术将大大提高模拟速度，从而方便模型的开发。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。由于没有强大的模拟大量晶粒在三维空间中，拟议的项目应该有显着的影响。高频波传播的有效计算具有从天线设计到地震感测的重要方面。 另一方面，对薛定谔方程的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的更深入了解。

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.