(651a) Large Scale Computational Dynamics in Engineered Granular Crystals

Williams, M. - Presenter, Princeton University
Kevrekidis, Y. G., Princeton University
Kevrekidis, P., University of Massachusetts, Amherst
Pozharskiy, D., Princeton University

Large Scale Computational Dynamics in Engineered Granular
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

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

[1]   F. Melo, S. Job, F. Santibanez, and F. Tapia, “Experimental evidence of shock mitigation in a Hertzian tapered chain,” Physical Review E, vol. 73, p. 041305, Apr. 2006.

[2]   A. Spadoni and C. Daraio, “Generation and control of sound bullets with a nonlinear acoustic lens,” Proceedings of the National Academy of Sciences, vol. 107, pp. 7230–7234, Apr. 2010.

[3]   C. Hoogeboom, Y. Man, N. Boechler, G. Theocharis, P. G. Kevrekidis, I. G. Kevrekidis, and C. Daraio, “Hysteresis Loops and Multi-stability: From Periodic Orbits to Chaotic Dynamics (and Back) in Diatomic Granular Crystals,” arXiv.org, pp. 1–7, Nov. 2012.

[4]   G. Theocharis, N. Boechler, and C. Daraio, “Nonlinear Periodic Phononic Structures and Granular Crystals,” pp. 217–251, Berlin, Heidelberg: Springer Berlin Heidelberg, Oct. 2012.

[5]   M. O. Williams, J. Wilkening, E. Shlizerman, and J. N. Kutz, “Continuation of periodic solutions in the waveguide array mode-locked laser,” Physica D, vol. 240, no. 22, pp. 1791–1804, 2011.

[6]   K. Lust and D. Roose, “An Adaptive Newton–Picard Algorithm with Subspace Iteration for Computing Periodic Solutions,” SIAM Journal on Scientific Computing, vol. 19, pp. 1188–1209, July 1998.

[7]   C. T. Kelley, Solving Nonlinear Equations with Newton’s Method. Siam, 2003.

[8]   E. Doedel, H. B. Keller, and J. P. Kernevez, “Numerical analysis and control of bifurcation problems (I): Bifurcation in finite dimensions,” International Journal of Bifurcation and Chaos, vol. 1, no. 03, pp. 493–520, 1991.


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