Mathematical
modeling in materials science and in crystal growth
Numerical methods for problems with surfaces and interfaces
Pattern formation on surfaces
Thin films
My work is in the area of modeling in
materials science and crystal growth.
Me and my co-authors developed PDE-based models of
micro(nano)-scale crystal growth on patterned substrates and of
grain-boundary grooving in polycrystalline
thin films. The work I do is by definition interdisciplinary
and requires insights into real-world physical phenomena. Applied
mathematics steps in (i) when components and ideas
from multiple fields are united in a single model, and (b) when reduced
models amenable for analytical treatment could be derived.
Given the complex and non-standard nature of the problems I usually
deal with, the latter is often impossible and/or
does not represent the phenomena in question, and this is when the
computation steps in.
Below is samples from completed and partially completed
projects. Most often, the surface or bulk diffusion
is the major component of the process; mathematically, such
model is formulated as IBVP for either a single parabolic
PDE, or a system of parabolic PDEs.
These PDEs are heavily nonlinear, often very high order (up to sixth)
and unstable.