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DOE OSTI2023

Learning subgrid-scale models with neural ordinary differential equations

We propose a new approach to learning the subgrid-scale model when simulating partial differential equations (PDEs) solved by the method of lines and their representation in chaotic ordinary differential equations, based on neural ordinary differential equations (NODEs). Solving systems with fine temporal and spatial grid scales is an ongoing computational challenge, and closure models are generally difficult to tune. Machine learning approaches have increased the accuracy and efficiency of computational fluid dynamics solvers. In this approach neural networks are used to learn the coarse- to fine-grid map, which can be viewed as subgrid-scale parameterization. We propose a strategy that uses the NODE and partial knowledge to learn the source dynamics at a continuous level. Our method inherits the advantages of NODEs and can be used to parameterize subgrid scales, approximate coupling operators, and improve the efficiency of low-order solvers. Numerical results with the two-scale Lorenz 96 ODE, the convection-diffusion PDE, and the viscous Burgers' PDE are used to illustrate this approach.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Passive Islanding Detection of Inverter-Based Resources in a Noisy Environment

Islanding occurs when a load is energized solely by local generators and can result in frequency and voltage instability, changes in current, and poor power quality. Poor power quality can interrupt industrial operations, damage sensitive electrical equipment, and induce outages upon the resynchronization of the island with the grid. This study proposes an islanding detection method employing a Duffing oscillator to analyze voltage fluctuations at the point of common coupling (PCC) under a high-noise environment. Unlike existing methods, which overlook the noise effect, this paper mitigates noise impact on islanding detection. Power system noise in PCC measurements arises from switching transients, harmonics, grounding issues, voltage sags and swells, electromagnetic interference, and power quality issues that affect islanding detection. Transient events like lightning-induced traveling waves to the PCC can also introduce noise levels exceeding the voltage amplitude by more than seven times, thus disturbing conventional detection techniques. The noise interferes with measurements and increases the nondetection zone (NDZ), causing failed or delayed islanding detection. The Duffing oscillator nonlinear dynamics enable detection capabilities at a high noise level. The proposed method is designed to detect the PCC voltage fluctuations based on the IEEE standard 1547 through the Duffing oscillator. For the voltages beyond the threshold, the Duffing oscillator phase trajectory changes from periodic to chaotic mode and sends an islanded operation command to the inverter. The proposed islanding detection method distinguishes switching transients and faults from an islanded operation. Experimental validation of the method is conducted using a 3.6 kW PV setup.

Energy & Fuels↗

NASA NTRS1989

An introduction to chaotic and random time series analysis

The origin of chaotic behavior and the relation of chaos to randomness are explained. Two mathematical results are described: (1) a representation theorem guarantees the existence of a specific time-domain model for chaos and addresses the relation between chaotic, random, and strictly deterministic processes; (2) a theorem assures that information on the behavior of a physical system in its complete state space can be extracted from time-series data on a single observable. Focus is placed on an important connection between the dynamical state space and an observable time series. These two results lead to a practical deconvolution technique combining standard random process modeling methods with new embedded techniques.

Scargle, Jeffrey D.↗

NASA NTRS1994

Thermonuclear runaways in nova outbursts. 2: Effect of strong, instantaneous, local fluctuations

In an attempt to understand the manner in which nova outbursts are initiated on the surface of a white dwarf, we investigate the effects fluctuations have on the evolution of a thermonuclear runaway. Fluctuations in temperature density, or the composition of material in the burning shell may arise due to the chaotic flow field generated by convection when it occurs, or by the accretion process itself. With the aid of two-dimensional reactive flow calculations, we consider cases where a strong fluctutation in temperature arises during the early, quiescent accretion phase or during the later, more dynamic, explosion phase. In all cases we find that an instantaneous, local temperature fluctuation causes the affected material to become Rayleigh-Taylor unstable. The rapid rise and subsequent expansion of matter immediately cools the hot blob, which prevents the lateral propagation of burning. This suggests that local temperature fluctuations do not play a significant role in directly initiating the runaway, especially during the early stages. However, they may provide an efficient mechanism of mixing core material into the envelope (thereby pre-enriching the fuel for subsequent episodes of explosive hydrogen burning) and of mixing substantial amounts of the radioactive nucleus N-13 into the surface layers, making novae potential gamma-ray sources. This suggests that it is the global not the local, evolution of the core-envelope interface to high temperatures which dominates the development of the runaway. We also present a possible new scenario for the initiation of nova outbursts based on our results.

Shankar, Anurag↗

NASA NTRS1990

Non-linear dynamics of a spur gear pair

The backlash nonlinearity excited primarily by transmission error between spur gear pairs is studied for both external and internal excitations. The digital simulation technique and the method of harmonic balance are used to develop steady state solutions for the internal sinuosidal excitations. The analytic predictions agreed well with available experimental data. Digital simulation is used to observe that at the chaotic and subharmonic resonances may exist in a gear pair depending on the mean or design load, mean to alternating force ratio, damping, and backlash.

Kahraman, A.↗

NASA NTRS1989

On the Nature of Navier-stokes Turbulence

Several turbulent and nonturbulent solutions of the Navier-Stokes equations are obtained. The unaveraged equations are used numerically in conjunction with tools and concepts from nonlinear dynamics, including time series, phase portraits, Poincare sections, largest Liapunov exponents, power spectra, and strange attractors. Initially neighboring solutions for a low-Reynolds-number fully developed turbulence are compared. The solutions, separate exponentially with time, having a positive Liapunov exponent. Thus the turbulence is characterized as chaotic. In a search for solutions which contrast with the turbulent ones, the Reynolds number is reduced. Several qualitatively different flows are noted. These are, fully chaotic, complex period, weakly chaotic, simple periodic, and fixed-point. Of these, only the fully chaotic flows are classified as turbulent. Those flows have both a positive Liapunov exponent and Poincare sections without pattern. By contrast, the weakly chaotic flows have some pattern in their Poincare sections. The fixed-point and periodic flows are nonturbulent, since turbulence, is both time-dependent and aperiodic. Turbulent solutions are obtained in which energy cascades from large to small-scale motions. In general, the spectral energy transfer takes place between wavenumber bands that are considerably separated. The special transfer can occur either as a result of nonlinear turbulence self-interaction or by interaction of turbulence with mean gradients. Turbulent systems are compared with those studied in kinetic theory. The two types of systems are fundamentally different (continuous and dissipative as opposed to discrete and conservative), but there are similarities. For instance, both are nonlinear and show sensitive dependence on initial conditions. Also, the turbulent and molecular stress tensors are identical if the macroscopic velocities for the turbulent stress are replaced by molecular velocities.

Deissler, Robert G.↗

NASA NTRS2000

Controlling Chaos Via Knowledge of Initial Condition for a Curved Structure

Nonlinear response of a flexible curved panel exhibiting bifurcation to fully developed chaos is demonstrated along with the sensitivity to small perturbation from the initial conditions. The response is determined from the measured time series at two fixed points. The panel is forced by an external nonharmonic multifrequency and monofrequency sound field. Using a low power time-continuous feedback control, carefully tuned at each initial condition, produces large long-term effects on the dynamics toward taming chaos. Without the knowledge of the initial conditions, control may be achieved by destructive interference. In this case, the control power is proportional to the loading power. Calculation of the correlation dimension and the estimation of positive Lyapunov exponents, in practice, are the proof of chaotic response.

Maestrello, L.↗

NASA NTRS1989

Secular resonances

Theories and numerical experiments regarding secular resonances are reviewed. The basic dynamics and the positions of secular resonances are discussed, and secular perturbation theories for the nu16 resonance case, the nu6 resonance, and the nu5 resonance are addressed. What numerical experiments have revealed about asteroids located in secular resonances, the stability of secular resonances, variations of eccentricities and inclinations, and chaotic orbits is considered. Resonant transport of meteorites is discussed.

Scholl, H.↗

NASA NTRS1993

Transonic panel flutter

FEM is here used to ascertain the stability and aeroelastic response of thin, 2D panels subjected to Mach 0.8-2.5 flows. In the absence of shocks, it is found that the Euler equations used to represent the unsteady flowfield dynamics predict response behaviors resembling those obtained via potential flow methods. Where shocks do play a significant role in the overall motion of the panel, divergence and limit cycle flutter are observed. In the Mach 1.4-1.5 range, flutter involved the higher modes of the panel, tending toward possible chaotic motion.

Davis, Gary A.↗

NASA NTRS1996

Local and Global Bifurcations of Flow Fields During Physical Vapor Transport: Application to a Microgravity Experiment

The local bifurcation of the flow field, during physical vapor transport for a parametric range of experimental interest, shows that its dynamical state ranges from steady to aperiodic. Comparison of computationally predicted velocity profiles with laser doppler velocimetry measurements shows reasonable agreement in both magnitude and planform. Correlation of experimentally measured crystal quality with the predicted dynamical state of the flow field shows a degradation of quality with an increase in Rayleigh number. The global bifurcation of the flow field corresponding to low crystal quality indicates the presence of a traveling wave for Ra = 1.09 x 10(exp 5). For this Rayleigh number threshold a chaotic transport state occurs. However, a microgravity environment for this case effectively stabilizes the flow to diffusive-advective and provides the setting to grow crystals with optimal quality.

Duval, W. M. B.↗

NASA NTRS1992

Energetic ions as remote probes of X type neutral lines in the geomagnetic tail

The study presents a summary of particle orbit signatures of a neutral line in the geomagnetic tail current sheet (CS) as a function of spatial position along the CS boundary layer, as well as within the CS. Six types of CS particle orbits are reviewed: transient B sub z, transient X line, quasi-trapped, trapped, noncrossing X line, and limiting (singular) cases. It is found that for single particle dynamics in a current sheet with neutral line, the phase space ridge is predicted to be found throughout the current sheet if it is not destroyed by collective behavior. An initially tailward flowing distribution causes asymmetries in distributions earthward vs tailward of the neutral line. It is suggested that the chaotic pitch angle scattering may be a dominant mechanism to produce nightside proton isotropy in the auroral zones.

Speiser, T. W.↗

DOE OSTI2023

Rectified Rotational Dynamics of Mobile Inclusions in Two-Dimensional Active Nematics

We investigate the dynamics of mobile inclusions embedded in 2D active nematics. The interplay between the inclusion shape, boundary-induced nematic order, and autonomous flows powers the inclusion motion. Disks and achiral gears exhibit unbiased rotational motion, but with distinct dynamics. In comparison, chiral gear-shaped inclusions exhibit long-term rectified rotation, which is correlated with dynamics and polarization of nearby +1/2 topological defects. Furthermore, the chirality of defect polarities and the active nematic texture around the inclusion correlate with the inclusion’s instantaneous rotation rate. Inclusions provide a promising tool for probing the rheological properties of active nematics and extracting ordered motion from their inherently chaotic motion.

36 MATERIALS SCIENCE↗

NASA NTRS1987

Lyapunov exponents for infinite dimensional dynamical systems

Classically it was held that solutions to deterministic partial differential equations (i.e., ones with smooth coefficients and boundary data) could become random only through one mechanism, namely by the activation of more and more of the infinite number of degrees of freedom that are available to such a system. It is only recently that researchers have come to suspect that many infinite dimensional nonlinear systems may in fact possess finite dimensional chaotic attractors. Lyapunov exponents provide a tool for probing the nature of these attractors. This paper examines how these exponents might be measured for infinite dimensional systems.

Mhuiris, Nessan Mac Giolla↗

DOE OSTI2022

Trotter Errors from Dynamical Structural Instabilities of Floquet Maps in Quantum Simulation

We study the behavior of errors in the quantum simulation of spin systems with long-range multibody interactions resulting from the Trotter-Suzuki decomposition of the time-evolution operator. We identify a regime where the Floquet operator underlying the Trotter decomposition undergoes sharp changes even for small variations in the simulation step size. This results in a time evolution operator that is very different from the dynamics generated by the targeted Hamiltonian, which leads to a proliferation of errors in the quantum simulation. These regions of sharp change in the Floquet operator, referred to as structural instability regions, appear typically at intermediate Trotter step sizes and in the weakly interacting regime, and are thus complementary to recently revealed quantum chaotic regimes of the Trotterized evolution [L. M. Sieberer et al. npj Quantum Inf. 5, 78 (2019); M. Heyl, P. Hauke, and P. Zoller, Sci. Adv. 5, eaau8342 (2019)]. We characterize these structural instability regimes in p-spin models, transverse-field Ising models with all-to-all p-body interactions, and analytically predict their occurrence based on unitary perturbation theory. We further show that the effective Hamiltonian associated with the Trotter decomposition of the unitary time-evolution operator, when the Trotter step size is chosen to be in the structural instability region, is very different from the target Hamiltonian, which explains the large errors that can occur in the simulation in the regions of instability. These results have implications for the reliability of near-term gate-based quantum simulators, and reveal an important interplay between errors and the physical properties of the system being simulated.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS2002

Phase Synchronization and Desynchronization of Structural Response Induced by Turbulent and External Sound

Acoustic and turbulent boundary layer flow loadings over a flexible structure are used to study the spatial-temporal dynamics of the response of the structure. The stability of the spatial synchronization and desynchronization by an active external force is investigated with an array of coupled transducers on the structure. In the synchronous state, the structural phase is locked, which leads to the formation of spatial patterns while the amplitude peaks exhibit chaotic behaviors. Large amplitude, spatially symmetric loading is superimposed on broadband, but in the desynchronized state, the spectrum broadens and the phase space is lost. The resulting pattern bears a striking resemblance to phase turbulence. The transition is achieved by using a low power external actuator to trigger broadband behaviors from the knowledge of the external acoustic load inducing synchronization. The changes are made favorably and efficiently to alter the frequency distribution of power, not the total power level. Before synchronization effects are seen, the panel response to the turbulent boundary layer loading is discontinuously spatio-temporally correlated. The stability develops from different competing wavelengths; the spatial scale is significantly shorter than when forced with the superimposed external sound. When the external sound level decreases and the synchronized phases are lost, changes in the character of the spectra can be linked to the occurrence of spatial phase transition. These changes can develop broadband response. Synchronized responses of fuselage structure panels have been observed in subsonic and supersonic aircraft; results from two flights tests are discussed.

Maestrello, Lucio↗

NASA NTRS1989

Evolution of comets into asteroids

This paper presents observational evidence, together with recent theoretical developments, supporting the hypothesis that at least some asteroids might be extinct or dormant cometary nuclei. The observations include the discovery of a number of apparent asteroids in chaotic Jupiter-crossing orbits; the IRAS discovery of 1983 TB, an asteroid in the same orbit as the Geminid meteor shower; the apparent low activity levels determined for several short-period comet nuclei including Comet Halley; and observations of possible cometary activity in some earth-crossing asteroids. Theoretical developments include explorations of dynamical mechanisms capable of delivering main-belt asteroids into earth-crossing orbits, and an understanding of possible processes which may affect comets during their long residence in the Oort cloud and lead to the formation of nonvolatile crusts before and after they enter the planetary system.

Weissman, Paul R.↗

NASA NTRS1990

Symbolic description of periodic orbits for the quadratic Zeeman effect

The organization of the periodic orbits of a hydrogen atom in a uniform magnetic field is studied. It is shown that the dynamics is qualitatively different from that for weak fields and negative energies. The orbits can be mapped one-to-one onto a ternary symbolic tree, an organization which turns out to be applicable for all values of energy and field strength. The results indicate that the field-free hydrogen atom has a purely chaotic limit for positive energies.

Eckhardt, Bruno↗

NASA NTRS1993

Nonlinear vortex trail dynamics. II - Analytic solutions

Spatially periodic large amplitude solutions of the von Karman model are obtained in the neighborhood of singularities. These singularities correspond to vortex clusters in the physical plane. The quasi-periodic and unbounded solutions found analytically confirm earlier numerical work and show qualitative agreement with experimental observations of large-scale phenomena of vortex trails. Separatrices or heteroclinic orbits were explicitly found for an integrable approximate equation, which indicate that the von Karman model itself supports chaotic solutions.

Lim, Chjan C.↗