Project I: Capillary Instability of Concentric Cylindrical Shells

Keywords: Computational Fluid Dynamics, Level-Set Methods, Capillary Instability, Linear Stability Analysis

Computational Tools: C, MATLAB, MEX, OpenMP, FFTW, Lapack.

A classic example of capillary instability (Plateau-Rayleigh instability) is the breakup of a fluid thread into droplets due to surface tension force. Motivated by complex multi-fluid geometries currently being explored in fibre-device manufacturing, we theoretically and computationally studied capillary instabilities in concentric cylindrical flows of N fluids with arbitrary viscosities, thicknesses, densities, and surface tensions in both the Stokes regime and for the full Navier-Stokes problem.

The mathematical model we built can quickly predict the breakup length-scale and time-scale of concentric cylindrical fluids, and provides useful guidance for material selections in fiber-drawing experiments. For example, in our experimental collaborators' recent work published in Nature, they developed a technique to produce uniform sized, structured core-shell particles by capillary instability, and our model captured the main physics in this experiment and predicted correct length-scale of these spherical particles.

In addition to the theoretically linear stability analysis, we also implemented a full 3D Stokes simulation by spectral and level-set methods to understand the physics from a computational perspective. One interesting example is to predict which interface (inner or outer) breaks up first in a three-fluid system. For various surface tensions of the outer interface γ₂, the following simulation movies confirmed the prediction from the mathematical analysis that the inner interface breaks up first when γ₂=6 (figure a), the outer interface breaks up first for γ₂=25 (figure c), and near-simultaneous breakup occurs for γ₂=15 (figure b).

Project II: Large-Scale Microcavity Topology Optimization

Keywords: Computational Electromagnetism, Parallel Computing, Nonlinear PDE-Constrained Optimization

Computational Tools: C, PETSc, MPI, NLOPT, PaStiX, Mumps.

Applications such as lasers and nonlinear devices require optical microcavities with long lifetimes Q and small modal volumes V. We formulate and solve a full 3d optimization scheme, over all possible 2d-lithography patterns in a thin dielectric film. The key to our formulation is a frequency-averaged local density of states (LDOS), where the frequency averaging corresponds to the desired bandwidth, evaluated by a novel technique: solving a single scattering problem at a complex frequency. To compute the objective LDOS, we implement a full 3D Maxwell's equation frequency domain parallel solver in C (PETSc). For the optimization, we use gradient-based algorithms from the optimization library NLOPT. The optimized structure we obtained for 3D slab is four times better (same order of Q, but much smaller V) than previous hand designs in the literature.

The optimization of a 2D microcavity for TM polarization from photonic crystal initial guess (blue for air and red for silicon):