Georgia Institute of Technology College of Sciences

Abstract: In the field of structural dynamics, engineers heavily rely on high-fidelity models of the structure at hand to predict its dynamic response and identify potential threats to its integrity.

The structure under investigation, be it an aircraft wing or a MEMS device, is typically discretised with finite elements, giving rise to a very large system of nonlinear ODEs. Due to the high dimensionality, the solution of such systems is very expensive in terms of computational time. For this reason, a large amount of literature in this field is devoted to the development of reduced order models of much lower dimensionality, able to accurately reproduce the original system’s dynamics. Not only the lower dimensionality increases the computational speed, but also provides engineers with interpretable and manageable models of complex systems, which can be easily coupled with data and uncertainty quantification, and whose parameter space can be easily explored. Slow invariant manifolds prove to be the perfect candidate for dimensionality reduction, however their computation for large scale systems has only been proposed in recent years (see Gonzalez et al. (2019), Haller et al. (2020), AV et al. (2019)).

In this talk, the Direct Parametrisation of Invariant Manifolds method (DPIM) will be presented. The theoretical basis of the method is provided by the results of Cabré, Fontich and de la Llave and its algorithmic implementation relies on the parametrisation method for invariant manifolds proposed by Haro et al.. The idea is to parametrise the invariant manifold around a fixed point through a power series expansion which can be solved recursively for each monomial in the reduced coordinates. The main limitation of the original algorithm is the necessity to operate in diagonal representation, which is unfeasible for large finite element systems as it would require the computation of the whole eigenspectrum. The main novelty of the proposed method lies in the expression of the normal homological equation directly in physical coordinates, which is the key aspect that permits its application to large scale systems.

The talk will focus on problems in structural dynamics in both autonomous and nonautonomous settings. The accuracy of the reduction will be shown on several examples, covering phenomena like internal resonances and parametric resonances. Finally, the current limitations and future developments of the method will be discussed.

Given a divergence-free vector field on the torus, we consider the mixing properties of the associated flow. There is a rich body of work studying the dependence of the mixing scale on various norms of the vector field. We will discuss some interesting examples of vector fields that mix at the optimal rate, and an improved bound on the mixing scale under the extra assumption that the vector field is a shear at each time.

Symmetries are  fundamental concepts in modern physics and biology. Spontaneous symmetry breaking often leads to fascinating  dynamical patterns such as  chimera states in which structurally and dynamically identical oscillators  split into coherent and incoherent clusters.  Solitary states in which one oscillator separates from the coherent cluster and oscillates with a different frequency represent  “weak” chimeras. While a rigorous stability analysis of a “strong” chimera with a multi-oscillator incoherent cluster  is typically out of reach for finite-size networks, solitary states offer a unique test bed for the development of stability approaches to large chimeras. In this talk, we will present such an approach and study the stability of solitary states in Kuramoto networks of identical 2D phase oscillators with inertia and a phase-lagged coupling.   We will derive asymptotic stability conditions for such solitary states as a function of inertia, network size, and phase lag that may yield either attractive or repulsive coupling. Counterintuitively, our analysis demonstrates that (i) increasing the size of the coherent cluster can promote the stability of the solitary state in the attractive coupling case and (ii) the solitary state can be stable in small-size networks with all repulsive coupling. We also discuss the implications of our analysis for the emergence of rotatory chimeras and splay states. This is a joint work with V. Munyaev, M. Bolotov, L. Smirnov, and G. Osipov.

In this talk, we study the existence of quasi periodic solutions to the generalized Surface Quasi-Geostropic (gSQG) equations. Despite its similar structure with the 2D Euler equation, the global existence/finite time singularity formation of gSQG equations have been open for a long time. Exploiting its Hamiltonian structure, we are able to construct a quasi periodic solutions with the initial date that are sufficiently close to its steady states. This is a joint work with Javier Gomez-Serrano and Alex Ionescu.