| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | June 25, 2008 |
| Latest Amendment Date: | April 16, 2010 |
| Award Number: | 0810113 |
| Award Instrument: | Continuing Grant |
| Program Manager: |
Junping Wang
jwang@nsf.gov (703)292-4488 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | July 1, 2008 |
| End Date: | June 30, 2013 (Estimated) |
| Total Intended Award Amount: | $256,429.00 |
| Total Awarded Amount to Date: | $256,429.00 |
| Funds Obligated to Date: |
FY 2009 = $92,163.00 FY 2010 = $94,869.00 |
| History of Investigator: |
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| Recipient Sponsored Research Office: |
1109 GEDDES AVE STE 3300 ANN ARBOR MI US 48109-1015 (734)763-6438 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
1109 GEDDES AVE STE 3300 ANN ARBOR MI US 48109-1015 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Parent UEI: |
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| NSF Program(s): | COMPUTATIONAL MATHEMATICS |
| Primary Program Source: |
01000910DB NSF RESEARCH & RELATED ACTIVIT 01001011DB NSF RESEARCH & RELATED ACTIVIT |
| Program Reference Code(s): |
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| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
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.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
This grant supported the development of efficient computational methods for three important applications thereby allowing simulations on length and time scales not previously attained in prior work. The first project was concerned with the simulation of strained epitaxial growth using kinetic Monte Carlo (KMC). Epitaxial growth is a process by which electronic devices can be manufactured and elastic strain can result if different materials are used. KMC is a modeling approach that allows one to formulate atomistic scale models that have correct thermodynamic behavior. However, KMC simulation for strained epitaxial growth is slow because it requires one to compute the elastic field billions of time during the course of a simulation. This project focused on both speeding up the computation of the elastic field and reducing the number of times it needed to be updated.
The second project involved the simulation of grain networks which are ubiquitous in many metals such as steel. The properties of these grains can determine the strength of the metal. In this project, we developed a computational method capable of the simulation of the hundreds of the thousands of grains. Our approach uses an implicit representation which is similar to a level set method and thereby naturally handles splitting and merging of grains. However the approach we use is based on the diffusion of distance functions and provides an easy way to incorporate the Herring condition when three grains meet.
The third project was concerned with the computation of the semiclassical limit of the Schroedinger equation. This is important for some problems in chemistry, for example photo dissociation. Mathematically, this is a challenging problem since the solutions are highly oscillatory in both space and time. We developed a transformation based on the Gaussian wavepacket solution that allows one to transform the Schroedinger equation into a new equation whose solutions are not highly oscillatory. This has allowed us to compute accurate solutions of the Schroedinger equation in the semiclassical limit for up to four dimensions.
These projects also involved the training of graduate students in areas of material science and computational methods for both partial differential equations and stochastic processes.
Last Modified: 11/25/2013
Modified by: Peter Smereka
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