Young-Ju Lee
UCLA
In this talk, we shall discuss the construction of discrete
analogues of continuous theories and their vital roles in two
research areas, fluid dynamics and material sciences.
For the simulation of rate-type non-Newtonian fluid flows, we
present new numerical discretization techniques obtained from the
observation that the constitutive equations can be recast into the
well-known symmetric matrix Riccati differential equations. Our
discretization schemes are then shown to preserve some important
physical characteristics that have been believed to be crucial for
the numerical stability of any discretization scheme. The
confirmation of such a belief shall be clearly demonstrated by
providing the discrete analogue of the energy estimate for the
continuous models.
A discrete strain model is essential for understanding the strain
effects in the thin film heteroepitaxial growth (one atomic
species grows on a substrate of another material). New techniques
are necessary to impose certain artificial boundary condition for
the solution of the model, since the thickness of the substrate is
considered to be infinite compared to the film thickness. We
develop the discrete analogue of the conditions that ensure the
existence of the exact artificial boundary condition, whose
validation leads to the reduction of the computational domain
siginificantly with no loss of accuracy.
Some illustrative but nontrivial applications of our methodologies
shall be provided.