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, Email: bli@math.umd.edu)

Course Description

References

• Major Course References

• References on Variational Models and Weak Convergence Methods for Crysalline Solids

• References on Epitaxial Growth

• References on Surface Diffusion

• References on Stress-Induced Morphological Instabilities

Homework Assignments

• Homework Assignment 1 (PDF file)

• Homework Assignment 2 (PDF file)

• Homework Assignment 3 (PDF file)

Lecture Notes (partially distributed to the class)

• Weak convergence methods for variational problems modeling crystalline solids
  • The theory of quasi-convexity: sequential weak lower semi-continuity; existence; partial regularity; various notion of convexity and null-Lagrangians.
  • The theory of compensated compactness: the Div-Curl lemma and its generalizations; necessary conditions; sufficient conditions; Young measures and gradient Young measures.
  • Mechanics of crystalline solids: Bravais lattices; martensitic phase transformation; the Cauchy-Born rule; twinning; Hadamard compatibility condition; classification of interfaces.
  • Mathematical theory of microstructure: simply laminated microstructure; the two-well problem; reduction of multi-well problems; kinematics of microstructure; restrictions on microstructure.
  • Additional topics: surface energy and scaling; martensitic thin films; dynamics.

• Epitaxial growth of thin films
  • Island dynamics models
  • Continuum models: coarsening and dynamic scaling
  • Mathematical and numerical aspects of surface diffusion

• The level-set method for interface motion