Numerical Spectral Study of Elliptic Operators

Claremont Mckenna College, Claremont CA

Investigators

Abstract

Since Lord Rayleigh conjectured more than a century ago that the disk should minimize the first Laplace-Dirichlet eigenvalue among all shapes of equal area, spectral study of elliptic operators has been an active research topic with applications including mechanical vibration, optical resonators, photonic crystals, and population dynamics. In mechanical vibration, for example, it is interesting to explore what shapes or what density distributions can generate minimal fundamental frequency; in photonic crystals, one seeks to design semiconductor structures with a periodic variation of refractive index to maximize photonic bandgap in which the propagation of light is forbidden. This project aims to advance numerical approaches to these kinds of questions for classes of eigenvalue problems that arise in design of containers to minimize fluid sloshing and in vibration control. Researchers have turned their attention to these questions with a renewed interest due to surprising recent discoveries, which include symmetry structure found in the optimizer of Steklov eigenvalue problems, optimal density arrangements in thin plates, and localization of vibration induced by interior clamped points. The project will explore a range of related open questions and aims to develop numerical approaches to solve them. The project also provides opportunities for mentoring students and engaging interested scientists, including those from underrepresented groups. Results are expected to have important potential application in systems that reduce liquid sloshing in missiles and other vessels, in noise and vibration control, and in medical and geophysical imaging. This project aims to develop numerical approaches to solve Steklov and biharmonic eigenvalue problems and study their related shape and topology optimization problems. The forward solvers for this project are based on boundary integral methods, finite element methods, and spectral methods. The optimization solvers are based on shape/topology derivatives and rearrangement methods. The investigator will study a wide range of problems arising from applications in liquid sloshing and plate vibrations, including (1) computation of the k-th Steklov eigenvalue problem and its optimization among star-shaped domains in three dimensions, (2) computation of principal eigenvalue of mixed Steklov eigenvalue problems and its related shape optimization, (3) Steklov eigenvalue problems on general manifolds, (4) spectral study of buckled plate eigenvalue problems, (5) localization of eigenfunctions induced by clamped points, and (6) multiphase shape optimization problems involving biharmonic operators. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.