arXiv subjects

George Haller

Publications and source records attributed to George Haller.

At least 73 records · Page 4Linked to original sources

arXiv2020

How do Conservative Backbone Curves Perturb into Forced Responses? A Melnikov Function Analysis

Weakly damped mechanical systems under small periodic forcing tend to exhibit periodic response in a close vicinity of certain periodic orbits of their conservative limit. Specifically, amplitude frequency plots for the conservative limit have often been noted, both numerically and experimentally, to serve as backbone curves for the near resonance peaks of the forced response. In other cases, such a relationship between the unforced and forced response was not observed. Here we provide a systematic mathematical analysis that predicts which members of conservative periodic orbit families will serve as backbone curves for the forced-damped response. We also obtain mathematical conditions under which approximate numerical and experimental approaches, such as energy balance and force appropriation, are justifiable. Finally, we derive analytic criteria for the birth of isolated response branches (isolas) whose identification is otherwise challenging from numerical continuation.

math.DS↗

arXiv2020

Explicit Unsteady Navier-Stokes Solutions and their Analysis via Local Vortex Criteria

We construct a class of spatially polynomial velocity fields that are exact solutions of the planar unsteady Navier-Stokes equation. These solutions can be used as simple benchmarks for testing numerical methods or verifying the feasibility of flow-feature identification principles. We use examples from the constructed solution family to illustrate deficiencies of streamlines-based feature detection and of the Okubo-Weiss criterion, which is the common two-dimensional version of the broadly used Q-, Delta-, Lambda-2- and Lambda-Ci-criteria for vortex-detection. Our planar polynomial solutions also extend directly to explicit, three-dimensional unsteady Navier-Stokes solutions with a symmetry.

physics.flu-dyn↗

arXiv2019

Exact Model Reduction and Fast Forced Response Calculation in High-Dimensional Nonlinear Mechanical Systems

We show how spectral submanifold (SSM) theory can be used to extract forced-response curves, including isolas, without any numerical simulation in high-degree-of-freedom, periodically forced mechanical systems. We use multivariate recurrence relations to construct the SSMs, achieving a major speed-up relative to earlier autonomous SSM algorithms. The increase in computational efficiency promises to close the current gap between studying lower-dimensional academic examples and analyzing larger systems obtained from finite-element modeling, as we illustrate on a discretization of a damped-forced beam model.

math.DS↗

arXiv2019

Vortex boundaries as barriers to diffusive vorticity transport in two-dimensional flows

We put forward the idea of defining vortex boundaries in planar flows as closed material barriers to the diffusive transport of vorticity. Such diffusive vortex boundaries minimize the leakage of vorticity from the fluid mass they enclose when compared to other nearby material curves. Building on recent results on passive diffusion barriers, we develop an algorithm for the automated identification of such structures from general, two-dimensional unsteady flow data. As examples, we identify vortex boundaries as vorticity diffusion barriers in two flows: an explicitly known laminar flow and a numerically generated turbulent Navier--Stokes flow.

physics.flu-dyn↗

arXiv2019

When does a Periodic Response Exist in a Periodically Forced Multi-Degree-of-Freedom Mechanical System?

While periodic responses of periodically forced dissipative nonlinear mechanical systems are commonly observed in experiments and numerics, their existence can rarely be concluded in rigorous mathematical terms. This lack of a priori existence criteria for mechanical systems hinders definitive conclusions about periodic orbits from approximate numerical methods, such as harmonic balance. In this work, we establish results guaranteeing the existence of a periodic response without restricting the amplitude of the forcing or the response. Our results provide a priori justification for the use of numerical methods for the detection of periodic responses. We illustrate on examples that each condition of the existence criterion we discuss is essential.

math.DS↗

arXiv2019

Machine-Learning Ocean Dynamics from Lagrangian Drifter Trajectories

Lagrangian ocean drifters provide highly accurate approximations of ocean surface currents but are sparsely located across the globe. As drifters passively follow ocean currents, there is minimal control on where they will be making measurements, providing limited temporal coverage for a given region. Complementary Eulerian velocity data is available with global coverage, but is itself limited by the spatial and temporal resolution possible with satellite altimetry measurements. Additionally, altimetry measurements approximate geostrophic components of ocean currents but neglect smaller sub-mesoscale motions and require smoothing and interpolation from raw satellite track measurements. In an effort to harness the rich dynamics available in ocean drifter datasets, we have trained a recurrent neural network on the time history of drifter motion to minimize the error in a reduced-order Maxey-Riley drifter model. This approach relies on a slow-manifold approximation to determine the most mathematically-relevant variables with which to train, subsequently improving the temporal and spatial resolution of the underlying velocity field. By adding this neural network component, we also correct drifter trajectories near sub-mesoscale features missed by deterministic models using only satellite and wind reanalysis data. The effect of varying similarity between training and testing trajectory datasets for the blended model was evaluated, as was the effect of seasonality in the Gulf of Mexico.

math.DS↗

arXiv2019

Search and rescue at sea aided by hidden flow structures

Every year hundreds of people die at sea because of vessel and airplane accidents. A key challenge in reducing the number of these fatalities is to make Search and Rescue (SAR) algorithms more efficient. Here we address this challenge by uncovering hidden TRansient Attracting Profiles (TRAPs) in ocean-surface velocity data. Computable from a single velocity-field snapshot, TRAPs act as short-term attractors for all floating objects. In three different ocean field experiments, we show that TRAPs computed from measured as well as modelled velocities attract deployed drifters and manikins emulating people fallen in the water. TRAPs, which remain hidden to prior flow diagnostics, thus provide critical information for hazard responses, such as SAR and oil spill containment, and hence have the potential to save lives and limit environmental disasters.

physics.ao-ph↗

arXiv2019

Explicit Third-Order Model Reduction Formulas for General Nonlinear Mechanical Systems

For general nonlinear mechanical systems, we derive closed-form, reduced-order models up to cubic order based on rigorous invariant manifold results. For conservative systems, the reduction is based on Lyapunov Subcenter Manifold (LSM) theory, whereas for damped-forced systems, we use Spectral Submanifold (SSM) theory. To evaluate our explicit formulas for the reduced model, no coordinate changes are required beyond an initial linear one. The reduced-order models we derive are simple and depend only on physical and modal parameters, allowing us to extract fundamental characteristics, such as backbone curves and forced-response curves, of multi-degree-of-freedom mechanical systems. To numerically verify the accuracy of the reduced models, we test the reduction formulas on several mechanical systems, including a higher-dimensional nonlinear Timoshenko beam.

math.DS↗

arXiv2019

Analytic Reconstruction of a Two-Dimensional Velocity Field from an Observed Diffusive Scalar

Inverting an evolving diffusive scalar field to reconstruct the underlying velocity field is an underdetermined problem. Here we show, however, that for two-dimensional incompressible flows, this inverse problem can still be uniquely solved if high-resolution tracer measurements, as well as velocity measurements along a curve transverse to the instantaneous scalar contours, are available. Such measurements enable solving a system of partial differential equations for the velocity components by the method of characteristics. If the value of the scalar diffusivity is known, then knowledge of just one velocity component along a transverse initial curve is sufficient. These conclusions extend to the shallow-water equations and to flows with spatially dependent diffusivity. We illustrate our results on velocity reconstruction from tracer fields for planar Navier- Stokes flows and for a barotropic ocean circulation model. We also discuss the use of the proposed velocity reconstruction in oceanographic applications to extend localised velocity measurements to larger spatial domains with the help of remotely sensed scalar fields.

physics.flu-dyn↗

arXiv2019

Material spike formation in high Reynolds number flow separation

We apply the recent frame-invariant theory of separation spike formation to complex unsteady flows including a turbulent separation bubble, an impinging jet, and flows around a freely moving cylinder and a freely rotating ellipse. We show how the theory captures the onset of material spike formation, without any assumption on the flow type (steady, periodic, unsteady) or separation type (on- or off-wall, fixed or moving boundaries). We uncover new phenomena, such as the transition from on-wall to off-wall separation, the merger of initially distinct spikes, and the presence of severe material spikes that remain hidden to previous approaches. These results unveil how an involved network of spikes arises, interacts, and merges dynamically, leading to the final ejection of particles from the wall in highly transient flow separation processes.

physics.flu-dyn↗

arXiv2019

Barriers to the Transport of Diffusive Scalars in Compressible Flows

Our recent work identifies material surfaces in incompressible flows that extremize the transport of an arbitrary, weakly diffusive scalar field relative to neighboring surfaces. Such barriers and enhancers of transport can be located directly from the deterministic component of the velocity field without diffusive or stochastic simulations. Here we extend these results to compressible flows and to diffusive concentration fields affected by sources or sinks, as well as by spontaneous decay. We construct diffusive transport extremizers with and without constraining them on a specific initial concentration distribution. For two-dimensional flows, we obtain explicit differential equations and a diagnostic scalar field that identify the most observable extremizers with pointwise uniform transport density. We illustrate our results by uncovering diffusion barriers and enhancers in analytic, numerical, and observational velocity fields.

physics.flu-dyn↗

arXiv2018

Analytic Prediction of Isolated Forced Response Curves from Spectral Submanifolds

We show how spectral submanifold theory can be used to provide analytic predictions for the response of periodically forced multi-degree-of-freedom mechanical systems. These predictions include an explicit criterion for the existence of isolated forced responses that will generally be missed by numerical continuation techniques. Our analytic predictions can be refined to arbitrary precision via an algorithm that does not require the numerical solutions of the mechanical system. We illustrate all these results on low- and high-dimensional nonlinear vibration problems. We find that our SSM-based forced-response predictions remain accurate in high-dimensional systems, in which numerical continuation of the periodic response is no longer feasible.

math.DS↗

arXiv2018

Fast Computation of Steady-State Response for Nonlinear Vibrations of High-Degree-of-Freedom Systems

We discuss an integral equation approach that enables fast computation of the response of nonlinear multi-degree-of-freedom mechanical systems under periodic and quasi-periodic external excitation. The kernel of this integral equation is a Green's function that we compute explicitly for general mechanical systems. We derive conditions under which the integral equation can be solved by a simple and fast Picard iteration even for non-smooth mechanical systems. The convergence of this iteration cannot be guaranteed for near-resonant forcing, for which we employ a Newton--Raphson iteration instead, obtaining robust convergence. We further show that this integral-equation approach can be appended with standard continuation schemes to achieve an additional, significant performance increase over common approaches to computing steady-state response.

math.DS↗

arXiv2018

Material Barriers to Diffusive and Stochastic Transport

We seek transport barriers and transport enhancers as material surfaces across which the transport of diffusive tracers is minimal or maximal in a general, unsteady flow. We find that such surfaces are extremizers of a universal, non-dimensional transport functional whose leading-order term in the diffusivity can be computed directly from the flow velocity. The most observable (uniform) transport extremizers are explicitly computable as null-surfaces of an objective transport tensor. Even in the limit of vanishing diffusivity, these surfaces differ from all previously identified coherent structures for purely advective fluid transport. Our results extend directly to stochastic velocity fields and hence enable transport barrier and enhancer detection under uncertainties.

physics.geo-ph↗

arXiv2018

Explicit Backbone Curves from Spectral Submanifolds of Forced-Damped Nonlinear Mechanical Systems

Spectral submanifolds (SSMs) have recently been shown to provide exact and unique reduced-order models for nonlinear unforced mechanical vibrations. Here we extend these results to periodically or quasiperiodically forced mechanical systems, obtaining analytic expressions for forced responses and backbone curves on modal (i.e. two-dimensional) time dependent SSMs. A judicious choice of the parameterization of these SSMs allows us to simplify the reduced dynamics considerably. We demonstrate our analytical formulae on three numerical examples and compare them to results obtained from available normal form methods.

math.DS↗

arXiv2018

Automated Computation of Autonomous Spectral Submanifolds for Nonlinear Modal Analysis

We discuss an automated computational methodology for computing two-dimensional spectral submanifolds (SSMs) in autonomous nonlinear mechanical systems of arbitrary degrees of freedom. In our algorithm, SSMs, the smoothest nonlinear continuations of modal subspaces of the linearized system, are constructed up to arbitrary orders of accuracy, using the parameterization method. An advantage of this approach is that the construction of the SSMs does not break down when the SSM folds over its underlying spectral subspace. A further advantage is an automated a posteriori error estimation feature that enables a systematic increase in the orders of the SSM computation until the required accuracy is reached. We find that the present algorithm provides a major speed-up, relative to numerical continuation methods, in the computation of backbone curves, especially in higher-dimensional problems. We illustrate the accuracy and speed of the automated SSM algorithm on lower- and higher-dimensional mechanical systems.

math.DS↗

arXiv2017

Transport by Lagrangian Vortices in the Eastern Pacific

Rotationally coherent Lagrangian vortices (RCLVs) are identified from satellite-derived surface geostrophic velocities in the Eastern Pacific (180$^\circ$-130$^\circ$ W) using the objective (frame-invariant) finite-time Lagrangian-coherent-structure detection method of Haller et al. (2016) based on the Lagrangian-averaged vorticity deviation. RCLVs are identified for 30, 90, and 270 day intervals over the entire satellite dataset, beginning in 1993. In contrast to structures identified using Eulerian eddy-tracking methods, the RCLVs maintain material coherence over the specified time intervals, making them suitable for material transport estimates. Statistics of RCLVs are compared to statistics of eddies identified from sea-surface height (SSH) by Chelton et al. 2011. RCLVs and SSH eddies are found to propagate westward at similar speeds at each latitude, consistent with the Rossby wave dispersion relation. However, RCLVs are uniformly smaller and shorter-lived than SSH eddies. A coherent eddy diffusivity is derived to quantify the contribution of RCLVs to meridional transport; it is found that RCLVs contribute less than 1% to net meridional dispersion and diffusion in this sector, implying that eddy transport of tracers is mostly due to incoherent motions, such as swirling and filamentation outside of the eddy cores, rather than coherent meridional translation of eddies themselves. These findings call into question prior estimates of coherent eddy transport based on Eulerian eddy identification methods.

physics.ao-ph↗

arXiv2017

Exact Model Reduction for Damped-Forced Nonlinear Beams: An Infinite-Dimensional Analysis

We use invariant manifold results on Banach spaces to conclude the existence of spectral submanifolds (SSMs) in a class of nonlinear, externally forced beam oscillations. SSMs are the smoothest nonlinear extensions of spectral subspaces of the linearized beam equation. Reduction of the governing PDE to SSMs provides an explicit low-dimensional model which captures the correct asymptotics of the full, infinite-dimensional dynamics. Our approach is general enough to admit extensions to other types of continuum vibrations. The model-reduction procedure we employ also gives guidelines for a mathematically self-consistent modeling of damping in PDEs describing structural vibrations.

math-ph↗