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DOE OSTI2023

Bound-preserving finite element approximations of the Keller–Segel equations

We report this paper aims to develop numerical approximations of the Keller–Segel equations that mimic at the discrete level the lower bounds and the energy law of the continuous problem. We solve these equations for two unknowns: the organism (or cell) density, which is a positive variable, and the chemoattractant density, which is a non-negative variable. We propose two algorithms, which combine a stabilized finite element method and a semi-implicit time integration. The stabilization consists of a nonlinear artificial diffusion that employs a graph-Laplacian operator and a shock detector that localizes local extrema. As a result, both algorithms turn out to be nonlinear and can generate cell and chemoattractant numerical densities fulfilling lower bounds. However, the first algorithm requires a suitable constraint between the space and time discrete parameters, whereas the second one does not. We design the latter to attain a discrete energy law on acute meshes. We report some numerical experiments to validate the theoretical results on blowup and nonblowup phenomena. In the blowup setting, we identify a locking phenomenon that relates the L ∞ (Ω)-norm to the L 1 (Ω)-norm limiting the growth of the singularity when supported on a macroelement.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Dynamics of Intra-Cell Thermal Front Propagation in Lithium-Ion Battery Safety Issues

Thermal runaway (TR), a critical failure mode in lithium-ion batteries (LIBs), poses significant safety risks and hinders wider application of LIBs. TR typically begins at a localized heat source and spreads across the cell. Understanding thermal front propagation (TFP) characteristics, such as front and velocity, is crucial for assessing energy release and temperature distribution for battery hazardous estimation. Recent studies assume that TR within cells propagates at a near-constant velocity, based on the reaction kinetics and thermal properties. Here, in this study, an intra-battery TR model is further proposed and it indicates that TFP velocity stabilizes when the front is distanced from the heat source. Theoretical estimates for propagation velocity and front are developed and validated through numerical simulations and experimental tests from the NREL Battery Failure Databank. The energy release rate during TFP and the impact of preheating based on a point heat source are explored. This work clarifies the long-standing clouds of the thermal font propagation behaviors within the single cell, highlights the power and beauty of mathematics modeling to describe the complicated thermal behaviors, and provides important guidelines for thermal hazardous understanding for next-generation batteries.

25 ENERGY STORAGE↗

DOE OSTI2020

Reversible structural transformations in supercooled liquid water from 135 to 245 K

Water has many anomalous properties compared to “simple” liquids, and these anomalies are typically enhanced in supercooled water.1-3 While numerous models have been proposed, including the liquid-liquid critical point,4,5 the singularity-free scenario,6 and the stability limit conjecture,1 a molecular-level understanding remains elusive. The main difficulty in determining which, if any, of these models is correct is the limited amount of data in the relevant temperature and pressure ranges. For water at ambient pressures, which is the focus of this work, data is largely missing from 160 – 232 K (“No Man’s Land”) due to rapid crystallization.2,3 Whether rapid crystallization is just an experimental obstacle, or a fundamental problem signaling the inability of water to thermally equilibrate prior to crystallization is also a major unanswered question.5,7 Here, we investigate the structural transformations of transiently-heated, supercooled water with nanosecond time resolution using infrared vibrational spectroscopy. The experiments demonstrate three key results. First, water’s structure relaxes from its initial configuration to a “steady-state” configuration prior to the onset of crystallization over a wide temperature range. Second, water’s steady-state structure can be reproduced by a linear combination of two, temperature-independent structures that correspond to a “high-temperature liquid” and a “low-temperature liquid.” Third, the observed structural changes are reversible over the full temperature range. Taken together, these results show that supercooled water can equilibrate prior to crystallization for temperatures from the homogeneous nucleation temperature, TH ~232 K,3 down to the glass transition temperature (Tg ~ 136 K). Second, the results provide support for the hypothesis that supercooled water can be described as a mixture of two, structurally-distinct, interconvertible liquids from 135 K to 245 K.5,8-18

Kringle, Loni M.↗

DOE OSTI2022

Cyber-Physical Secure Observer-Based Corrective Control under Compromised Sensor Measurements

In this paper we introduce the objectives and design principles of corrective control under cyber-physical attacks. We propose two types of observer-based corrective control for both the open-loop stable and the open-loop unstable LTI systems. The basic idea of our corrective control design is to use the observer as the ground-truth during the attack, making the plant dynamics follow the observer behavior. This is the opposite to the no-attack-detected period in which the observer is designed to follow the plant dynamics. We show stability of the proposed control under compromised sensor measurements, and quantify the effects of the discrepancy between the observer and the plant. Numerical examples, with illustrations using microgrid energy dynamics, are presented to show benefits of the proposed corrective control.

Wu, Dan↗

DOE OSTI2020

Multi-frequency General Relativistic Radiation-hydrodynamics with M 1 Closure

We report on recent upgrades to our general relativistic radiation-magnetohydrodynamics code, Cosmos++ , which expands the two-moment, M 1 , radiation treatment from gray to multi-frequency transport, including Doppler and gravitational frequency shifts. The solver accommodates either photon (Bose–Einstein) or neutrino (Fermi–Dirac) statistical distribution functions with absorption, emission, and elastic scattering processes. An implicit scheme is implemented to simultaneously solve the primitive inversion problem together with the radiation–matter coupling source terms, providing stability over a broad range of opacities and optical depths where the interaction terms can be stiff. We discuss our formulations and numerical methods, and validate our methods against a wide variety of test problems spanning optically thin to thick regimes in flat, weakly curved, and strongly curved spacetimes.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1971

A study of the Nordsieck-type predictor-corrector methods

Predictor-corrector schemes are discussed for the numerical integration of initial value problems by computer. Descloux formulas, which are considered modified Nordsieck formulas are studied because of their desirable stability and truncation properties along with the standard Nordsieck formulas. The stability of Nordsieck and modified Nordsieck algorithms are computed, and the results are graphed. Test problems are included as examples for comparing the efficiency of the algorithms.

Eiserike, H. M.↗

NASA NTRS1978

Investigation of the burning configuration of a coaxial injector in a combustion chamber

An analytical investigation was made into the stability of the burning configuration of a single coaxial injector surrounded by similar injectors. The stability criteria was based on an average pressure difference along the boundaries of the adjacent stream tubes as calculated using Spaulding's numerical method. The results indicate qualitatively that there is a tendency for the injectors to have different burning configurations. It is believed that the configuration achieved is random, however once the burning configuration is established, it is believed to persist.

Ohara, J.↗

NASA NTRS1983

On the interaction of a sound pulse with the shear layer of an axisymmetric jet. II - Heated jets

The fluctuating field of a heated jet excited by a sound pulse is simulated numerically. The fluctuations in both the flow field and the far field are studied. The flow field results depend crucially on the stability properties of the heated jet. The altered stability properties due to the heating cause a reduction in the overall fluctuations and a shift of the instability waves into lower frequencies. These changes cause a similar downward shift in the far field spectrum and a reduction in the total far field noise. The results provide a partial explanation in terms of stability theory of experimentally observed properties of the noise of heated jets.

Bayliss, A.↗

NASA NTRS1984

Three-dimensional stability of an elliptical vortex in a straining field

The three-dimensional linear stability of a rectilinear vortex of elliptical cross-section existing as a steady state in an irrotational straining field is studied numerically in the case of finite strain. It is shown that the instability predicted analytically for weak strain persists for finite strain and that the weak-strain results continue to be quantitatively valid for finite strain. The dependence of the growth rates of the unstable modes on the strain and the axial-disturbance wavelength is discussed. It is also shown that a three-dimensional instability is always more unstable than a two-dimensional instability in the range of parameters of most interest.

Robinson, A. C.↗

NASA NTRS1987

Stability analysis of two coupled Lorenz lasers and the coupling-induced periodic to chaotic transition

A linear stability analysis of two Lorenz lasers coupled by their electric fields has been performed, and it is shown that the bad cavity condition becomes a function of coupling and that a good cavity instability may occur if the injected fields are inverted before injection. In addition, it is shown that the symmetrically coupled Lorenz system is isomorphic to the original Lorenz system with new parameters. The stability analysis also predicts a lowering of the second laser threshold with coupling for both the chaotic and self-pulsing regimes. Numerical integration of the equations is in agreement with these predictions and has revealed a coupling induced transition from self-pulsing to chaotic behavior. The classification of the behavior of the coupled system in the parameter space of the coupling constants has been investigated and shows that the results of symmetric coupling allow enough of a margin for an experimental test of the theory. This would allow experimentalists to observe the actual Lorenz instability at excitations as low as 4-5 times above threshold.

Lawandy, N. M.↗

NASA NTRS1994

Algorithms For Integrating Nonlinear Differential Equations

Improved algorithms developed for use in numerical integration of systems of nonhomogenous, nonlinear, first-order, ordinary differential equations. In comparison with integration algorithms, these algorithms offer greater stability and accuracy. Several asymptotically correct, thereby enabling retention of stability and accuracy when large increments of independent variable used. Accuracies attainable demonstrated by applying them to systems of nonlinear, first-order, differential equations that arise in study of viscoplastic behavior, spread of acquired immune-deficiency syndrome (AIDS) virus and predator/prey populations.

Freed, A. D.↗

NASA NTRS1998

Boundary and Interface Conditions for High Order Finite Difference Methods Applied to the Euler and Navier-Strokes Equations

Boundary and interface conditions for high order finite difference methods applied to the constant coefficient Euler and Navier-Stokes equations are derived. The boundary conditions lead to strict and strong stability. The interface conditions are stable and conservative even if the finite difference operators and mesh sizes vary from domain to domain. Numerical experiments show that the new conditions also lead to good results for the corresponding nonlinear problems.

Nordstrom, Jan↗

NASA NTRS2000

Wave Fluid Film Bearing Tests for an Aviation Gearbox

An oil-lubricated wave journal-thrust bearing assembly was successfully tested at conditions found in general aviation engine gearboxes. The bearing performed well at both steady state conditions and in start-stop tests. It ran stably under all loading conditions, including zero load, at all speeds up to 16 000 rpm. The bearing carried 25 percent more load than required for the gearbox application, supporting 8900 N (94 bars average pressure), and showed very good thermal stability. 450 start-stop cycles were also performed, including 350 cycles without oil supply during starting and stopping. Test results and numerical predictions were in good agreement.

Dimofte, Florin↗

NASA NTRS2016

Periodic Orbit-Attitude Solutions in the Planar Elliptic Restricted Three-Body Problem

The pitch motion of a spacecraft in the planar elliptic restricted three-body system is studied. Previous studies laid the foundation for spacecraft stability analysis with a small perturbation to the zero pitch motion. In this study, a cell mapping approach that combines analytical and numerical techniques is used to study the global behavior of the full nonlinear spacecraft attitude in which coupling between orbital dynamics and attitude occurs. The effect of gravity gradient torques, orbital eccentricity, and the spacecraft configuration at different Lagrangian points is analyzed. Multiple-period periodic solutions and invariant surfaces are presented for different cases. Reference trajectories around the Lagrangian points are also considered to study coupled dynamics.

Koh, Dayung↗

DOE OSTI2023

Chaotic and integrable magnetic fields in one-dimensional hybrid Vlasov–Maxwell equilibria

The construction of kinetic equilibrium states is important for studying stability and wave propagation in collisionless plasmas. Thus, many studies over the past decades have been focused on calculating Vlasov–Maxwell equilibria using analytical and numerical methods. However, the problem of kinetic equilibrium of hybrid models is less studied, and self-consistent treatments often adopt restrictive assumptions ruling out cases with irregular and chaotic behaviour, although such behaviour is observed in spacecraft observations of space plasmas. In this paper, we develop a one-dimensional (1-D), quasineutral, hybrid Vlasov–Maxwell equilibrium model with kinetic ions and massless fluid electrons and derive associated solutions. The model allows for an electrostatic potential that is expressed in terms of the vector potential components through the quasineutrality condition. The equilibrium states are calculated upon solving an inhomogeneous Beltrami equation that determines the magnetic field, where the inhomogeneous term is the current density of the kinetic ions and the homogeneous term represents the electron current density. We show that the corresponding 1-D system is Hamiltonian, with position playing the role of time, and its trajectories have a regular, periodic behaviour for ion distribution functions that are symmetric in the two conserved particle canonical momenta. For asymmetric distribution functions, the system is nonintegrable, resulting in irregular and chaotic behaviour of the fields. The electron current density can modify the magnetic field phase space structure, inducing orbit trapping and the organization of orbits into large islands of stability. Thus, the electron contribution can be responsible for the emergence of localized electric field structures that induce ion trapping. We also provide a paradigm for the analytical construction of hybrid equilibria using a rotating two-dimensional harmonic oscillator Hamiltonian, enabling the calculation of analytic magnetic fields and the construction of the corresponding distribution functions in terms of Hermite polynomials.

Kaltsas, Dimitrios A. (ORCID:0000000300769015)↗

NASA NTRS1983

Numerical studies of laminar and turbulent drag reduction, part 2

The flow over wave shaped surfaces is studied using a Navier Stokes solver. Detailed comparisons with theoretical results are presented, including the stability of a laminar flow over wavy surfaces. Drag characteristics of nonplanar surfaces are predicted using the Navier-Stokes solver. The secondary instabilities of wall bounded and free shear flows are also discussed.

Balasubramanian, R.↗

NASA NTRS1988

Approximation theory for LQG (Linear-Quadratic-Gaussian) optimal control of flexible structures

An approximation theory is presented for the LQG (Linear-Quadratic-Gaussian) optimal control problem for flexible structures whose distributed models have bounded input and output operators. The main purpose of the theory is to guide the design of finite dimensional compensators that approximate closely the optimal compensator. The optimal LQG problem separates into an optimal linear-quadratic regulator problem and an optimal state estimation problem. The solution of the former problem lies in the solution to an infinite dimensional Riccati operator equation. The approximation scheme approximates the infinite dimensional LQG problem with a sequence of finite dimensional LQG problems defined for a sequence of finite dimensional, usually finite element or modal, approximations of the distributed model of the structure. Two Riccati matrix equations determine the solution to each approximating problem. The finite dimensional equations for numerical approximation are developed, including formulas for converting matrix control and estimator gains to their functional representation to allow comparison of gains based on different orders of approximation. Convergence of the approximating control and estimator gains and of the corresponding finite dimensional compensators is studied. Also, convergence and stability of the closed-loop systems produced with the finite dimensional compensators are discussed. The convergence theory is based on the convergence of the solutions of the finite dimensional Riccati equations to the solutions of the infinite dimensional Riccati equations. A numerical example with a flexible beam, a rotating rigid body, and a lumped mass is given.

Gibson, J. S.↗

NASA NTRS1993

Shell stability analysis in a computer aided engineering (CAE) environment

The development of 'DISDECO', the Delft Interactive Shell DEsign COde is described. The purpose of this project is to make the accumulated theoretical, numerical and practical knowledge of the last 25 years or so readily accessible to users interested in the analysis of buckling sensitive structures. With this open ended, hierarchical, interactive computer code the user can access from his workstation successively programs of increasing complexity. The computational modules currently operational in DISDECO provide the prospective user with facilities to calculate the critical buckling loads of stiffened anisotropic shells under combined loading, to investigate the effects the various types of boundary conditions will have on the critical load, and to get a complete picture of the degrading effects the different shapes of possible initial imperfections might cause, all in one interactive session. Once a design is finalized, its collapse load can be verified by running a large refined model remotely from behind the workstation with one of the current generation 2-dimensional codes, with advanced capabilities to handle both geometric and material nonlinearities.

Arbocz, J.↗