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)
1. Multiscale Methods
The quasicontinuum method for defects in crystalline solids.
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.
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?
2. Interface Dynamics
Examples: geometric motions;
solidification; epitaxial growth of thin films; microstructural evolution;
two-phase flow; and motion of biomolecules.
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.
Phase-field models and numerical methods, time stepping, stability
beyond the Gronwall inequality, threshold dynamics.
3. Energy Minimization
Concept of free energy, basics of thermodynamics, the second law.
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.
Weak convergence methods, Gamma-limits as effective energies,
energies of martensitic thin films.
Variational methods for coarsening in gradient systems,
application to the coarsening in epitaxial growth of thin films with or without slope selection.
The Poisson-Boltzmann model and its improvement, boundary-value problems.
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.