Large Scale Computational Dynamics in Engineered Granular Crystals

Large Scale Computational Dynamics in Engineered Granular Crystals

Matthew O. Williams, Dmitry Pozharskiy, Christopher Chong,
Panayotis G. Kevrekidis, Chiara Daraio, and Yannis G. Kevrekidis

May 13, 2013

Engineered granular crystals (EGCs), assemblies of particles in a closely packaged configuration contained inside of a matrix, are a relatively new computational and experimental testbed for nonlinear dynamics, with a large number of applications ranging from materials science [1] to biomedicine [2]. In particular, EGCs have been proposed for use in mechanical systems with tunable acoustic properties, fabricating devices with controllable acoustic bandgaps within the audible range targeting noise mitigation and vibration absorbing layers, and designing new acoustic lenses [3]. The flexibility of EGCs is, in large part, due to the number of degrees of freedom that exist both experimentally and computationally. The building blocks of an EGC are macroscopic particles, referred to here as beads, that are either spherical, toroidal, elliptical, or cylindrical in shape; by altering the shape, arrangement, and material composition of these building blocks, the transfer of energy in the system, and therefore the dynamics of the system, can be modified. With different choices of the parameter values, the EGC can be made to respond to external forcing in a periodic, quasi-periodic, or chaotic fashion [4].

To obtain a better understanding of the physics of the granular crystals, we conduct a bifurcation study of moderate- to large-size crystals in the presence of sinusoidal forcing. A key component in this work is efficiently and accurately computing time periodic solutions; for problems of this size, shooting methods are typically favored over the orthogonal collocation methods favored for smaller problems. To do so, we use large scale, parallelizable algorithms to compute limit cycles based on techniques from iterative linear algebra (e.g., quasi-Newton [5], Newton-Picard [6], or Newton-GMRES [7]) and to solve the extended systems associated with two parameter branches of bifurcations such as the limit point of a cycle or a Neimark-Sacker bifurcation [8]. Ultimately, we implement this procedure to compute one parameter branches of periodic solutions as well as two parameter branches of the bifurcations that occur.

These methods can be used in conjunction with matrix-free algorithms such as the Implicitly Restarted Arnoldi Method for solving high dimensional eigenproblems to characterize the stability of the resulting periodic orbits. This also enables us to locate the start of synchronization regions (Arnold tongues) on the two parameter “frontier” where quasi-periodic solutions first begin to appear. Although only the stable periodic orbits can be observed experimentally, the saddle periodic solutions in this region are also important because the interactions of their unstable manifolds may provide insight into the source of the chaotic dynamics observed in these types of systems. To complete the picture, we also investigate methods for computing invariant circles for the stroboscopic map (invariant tori for the flow) in the regions where synchronization does not occur. The larger purpose of this study is to reveal the impact that the construction of an EGC has on how it distributes/transfers energy upon impact; this will allow us to design methods that “sculpt” the linearized spectrum of an EGC by manipulating the composition and detailed geometry of the crystal. References

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