AMSC 698V Course Description - B. Li
AMSC 698V: Advanced Topics in Applied Mathematics, Fall, 2003
MATHEMATICAL AND COMPUTATIONAL PROBLEMS IN MATERIALS SCIENCE
MWF, 11:00-11:50, Math 0405
Instructor: Bo Li (Office: Math 4304,
The past decade has witnessed the rapid development of mathematical
and computational researches in materials science. Applied mathematics
has made much contribution to the understanding of some of the
fundamental principles and basic mechanisms of complex material
processes that are often characterized by singularities, fluctuation,
and multi-scale. The advance in materials science and nanotechnology
today, however, has posed new challenges and opportunities to applied
and computational mathematics.
This course will review some of the mathematical and computational
aspects of the materials science. It will cover a variety of interesting
topics, some of which have been much studied and some are still
relatively new. They include:
Most of the course references are recently published research articles.
But the following is a brief list of articles, lecture notes, and
monographs that contain the subjects listed above.
Weak convergence methods for variational problems modeling
Nonlinear evolutionary differential equations modeling the
dynamics of internal layers, phase separation, and coarsening;
Continuum models for epitaxial growth of thin films;
The level set method for the simulation of interface motion;
The heterogeneous multi-scale methods for material simulations.
A.-L. Barabási and H. E. Stanley, Fractal concepts in surface
growth, Cambridge Univ. Press, 1995.
K. Bhattacharya, Theory of martensitic microstructure and the
shape-memory effect, 2003.
R. E. Caflisch, Mathematical problems in materials science,
Unpublished lecture notes, UCLA, 1999.
W. E and B. Engquist, The heterogeneous multi-scale methods,
Comm. Math. Sci., 1:87-133, 2003.
P. Fife, Dynamics of internal layers and diffusive interfaces,
B. Li, Weak convergence methods for variational problems modeling
crystalline microstructure, Unpublished lecture notes, UCLA, 1998.
S. Müller, Variational models for microstructure and phase
transitions, Lecture notes, Max-Planck Inst. for Math. in Sci.,
S. Osher and R. Fedkiw, Level set methods and dynamic implicit
surfaces, Springer, 2003.
A. Pimpinelli and J. Villain, Physics of crystal growth, Cambridge
Univ. Press, 1998.
J. A. Sethian, Level set methods and fast marching methods, 1999.
This course is designed for graduate students who are interested in
applied partial differential equations, nonlinear dynamics, scientific
computing, and numerical analysis. The undergraduate course on partial
differential equations or equivalent will be a prerequisite. But no
knowledge on materials science is assumed. The material presented in
the course will be essentially self-closed. Along the course, reference
material and lecture notes will be distributed, and possible research
topics will be discussed.
For more information, please contact the instructor.