Two Studies of Complex Nonlinear Systems: Engineered Granular Crystals and Coarse-Graining Optimization Problems

Abstract

In recent years a nonlinear, acoustic metamaterial, named granular crystals, has

gained prominence due to its high accessibility, both experimentally and compu-

tationally. The observation of a wide range of dynamical phenomena in the system,

due to its inherent nonlinearities, has suggested its importance in many engineering

applications related to wave propagation.

In the first part of this dissertation, we explore the nonlinear dynamics of damped-

driven granular crystals. In one case, we consider a highly nonlinear setting, also

known as a sonic vacuum, and derive a nonlinear analogue of a linear spectrum,

corresponding to resonant periodic propagation and antiresonances. Experimental

studies confirm the computational findings and the assimilation of experimental data

into a numerical model is demonstrated. In the second case, global bifurcations in a

precompressed granular crystal are examined, and their involvement in the appear-

ance of chaotic dynamics is demonstrated. Both results highlight the importance of

exploring the nonlinear dynamics, to gain insight into how a granular crystal responds

to different external excitations.

In the second part, we borrow established ideas from coarse-graining of dynamical

systems, and extend them to optimization problems. We combine manifold learning

algorithms, such as Diffusion Maps, with stochastic optimization methods, such as

Simulated Annealing, and show that we can retrieve an ensemble, of few, important

parameters that should be explored in detail. This framework can lead to acceleration

of convergence when dealing with complex, high-dimensional optimization, and could

potentially be applied to design engineered granular crystals.