MATH 273A: Scientific Computation
Fall quarter, 2005
TOPICS IN APPLIED MATHEMATICS AND COMPUTATIONAL SCIENCE
Instructor: Bo Li
A Tentative List of Topics
(to be partially covered)
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1. Multiscale Methods
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The quasicontinuum method for defects in crystalline solids.
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Examples of the heterogeneous multiscale method:
stiff systems of ordinary differential equations for chemical reactions;
homogenization for composites and flow in a porous medium;
combined molecular dynamics and continuum simulation of solids.
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Coupling molecular dynamics and continuum simulations, boundary conditions.
Application to the contact line problem in fluids,
from no-slip to Navier to generalized Navier boundary conditions,
negative kinetic constants.
- Coupling the quantum level first-principle to molecular dynamics to continuum mechanics
simulations: the need? any hope? strategy? coupling? interface
conditions? implementation? and the mathematics?
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2. Interface Dynamics
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Examples: geometric motions;
solidification; epitaxial growth of thin films; microstructural evolution;
two-phase flow; and motion of biomolecules.
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Sharp interface models. The front-tracking method, application to the
surface evolution of solid films with dislocations.
The level-set method: a simple example.
A finite-element level-set method for the stress-driven interface motion.
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Phase-field models and numerical methods, time stepping, stability
beyond the Gronwall inequality, threshold dynamics.
3. Energy Minimization
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Concept of free energy, basics of thermodynamics, the second law.
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Examples: nonlinear elasticity; the Ginzburg-Landau functional for superconductivity;
the Cahn-Hilliard functional for phase separation;
the Poisson-Boltzmann model; the Helfrich membrane energy;
and the density-functional theory.
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Weak convergence methods, Gamma-limits as effective energies,
energies of martensitic thin films.
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Variational methods for coarsening in gradient systems,
application to the coarsening in epitaxial growth of thin films with or without slope selection.
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The Poisson-Boltzmann model and its improvement, boundary-value problems.
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The basics of the density-functional theory, the Kohn-Sham equations, real-space calculations
using parallel adaptive finite-element methods.
Last updated by Bo Li on October 17, 2005.