## Research - Applied Mathematics

Applied mathematics is a common thread that runs through almost all of my research activities. Within the broad field of applied mathematics, I work primarily on the following areas:

• Numerical analysis and methods,
• Level set and phase field methods,
• Asymptotic analysis, and
• Partial differential equations.

My interests in these subfields of applied mathematics are a natural outgrowth of my focus on problems in science and engineering:

While most of my research has a strong science/engineering component, the following projects are of a distinctly mathematical flavor.

## Shape Optimization via The Level Set Method

(with Salvatore Torquato and Youngjean Jung)

Schoen G triply periodic, minimal surface generated by minimizing the total surface area under a constraint in the voulme fractions of the two regions separated by the surface. The optimization procedure used to generate this surface is based on a variational level set method approach. The simulation that generated this figure was implemented using LSMLIB. (Image generated by Y. Jung)

The level set method is naturally suited to shape optimization problems where the objective function and constraints are representable (possibly implicitly) as a functionals of the level set function. In these situations, it is possible to use variational calculus to devise a constrained, steepest descent optimization procedure. In addition to yielding a computational procedure, the variational level set method approach also provides us with a means to theoretically characterize optimal surfaces.

Using a variational, level set-based shape optimization approach, we investigated the properties of triply periodic microstructures. Specifically, we explored the structure of triply periodic surfaces that minimize the total surface area while satisfying a constraint on the volume fractions of the two phases it separates. The main outcomes of this work were

• a proof that surfaces that optimize the total surface area under a volume fraction constraint are precisely those possessing a constant mean curvature;
• a computational exploration of the properties of optimal structures as a function of the the volume fractions of the two phases separated by surface.