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

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

This proposal involves three projects. The first concerns modeling and efficient simulation of heteroepitaxial growth using kinetic Monte Carlo and will build from prior NSF support which resulted in the development of a Fourier multigrid method for the fast solution of discrete elastic equations for complex geometries. This work will be extended to develop methods for obtaining inexpensive upper bonds on rates, the use of local computations for elastic equations, and the inclusion of intermixing of multiple species. The second project involves the simulation of grain boundary motion in two and three dimensions using a recently developed multiphase variational level set framework which allows one to systematically deduce level set equations for a network of grains moving under curvature flow. We plan to extend this formulation to allow the simulation of thousands of seeds by using only a few level set functions. The efficient computation of high frequency wave propagation and the semi-classical limit of the Schrodinger equation is the third project. 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 this domain using a fast local convolution. It is planned to update the solutions by the computation of the matrix exponential using a Krylov subspace approach. 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 effects on both nanoscales and mesoscales. It is technologically relevant since quantum dot materials are made in this way. Our proposed techniques will greatly 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 s cience. 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.