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

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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.