The immersed interface method is a quite general methodology for solving interface problems governed by differential equations. In this talk, I will first give an overview of this method. I will then present a boundary condition capturing immersed interface method, which can enforce the prescribed or free motions of rigid objects in a fluid wtih desirable numerical stability, accuracy and efficiency. Last, I will derive some principal jump conditions across a two-fluid interface, and present some thoughts on how to use them in the immersed interface method for simulation of two-fluid flows.

We discuss an advection-diffusion equation that has been proposed by Keith Moffatt as a model for the Geodynamo. Even though the drift velocity can be strongly singular, we prove that the critically diffusive PDE is globally well-posed. We examine the nonlinear instability of a particular steady state and use continued fractions to construct a lower bound on the growth rate of a solution. This lower bound grows as the inverse of the diffusivity coefficient. In the Earth's fluid core this coefficient is expected to be very small. Thus the model does indeed produce very strong Geodynamo action.

This work is joint with Vlad Vicol.

Piezoelectric materials exhibit hysteresis in the field-strain relation at
essentially all drive levels. Furthermore, this nonlinear relation is
dependent upon both prestresses and dynamic stresses generated during
employment of the materials. The accurate characterization of this nonlinear
and hysteretic material behavior is critical for material characterization,
device design, and model-based control design. In this talk, we will
discuss the characterization of hysteresis using the homogenized energy
model (HEM) framework. At the mesoscale, energy relations characterizing
field and stress-dependent 90 and 180 degree switching are used to develop
fundamental kernels or hysterons. Material and field nonhomogeneities are
subsequently incorporated by assuming that certain parameters are
manifestations of underlying densities. This yields a macroscopic model
that accurately characterizes the fundamental material behavior yet is
sufficiently efficient for optimization and control implementation.
Attributes of the model will be illustrated through comparison to
experimental data.