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Results for “System of nonlinear equations”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 289 records · Page 16
All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

MFiX: Fractional-Step Method Implementation

A comprehensive, multiphase computational fluid dynamics (CFD) simulation solves several coupled transport equations including continuity, momentum, species, and energy. Chemical reactions further couple these equations through heats of reaction and rates of formation of products and rates of destruction of reactants. A fractional-step method separates changes attributed to chemical reactions from transport phenomena like convection and diffusion. When the governing equations are split into the transport and reacting components, efficient and independent methodologies can be exploited to solve the different systems. Specifically, discretization of field variable transport equations results in large, sparse matrices which are loosely coupled. These systems are solved in succession using iterative techniques that take advantage of the matrix structure. In contrast, chemical reactions tightly couple field variables locally within the domain (e.g., within a single computational cell) resulting in low dimensional but dense, nonlinear systems that are better solved using direct integration techniques.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Generalized optimization-based synthesis of membrane systems for multicomponent gas mixture separation

Synthesizing a membrane system to separate multicomponent gas mixture is challenging due to the combinatorial number of feasible configurations and the difficulties in describing the multicomponent permeators. Here we present a mixed-integer nonlinear programming (MINLP) model for synthesizing membrane systems for multicomponent gas mixture separation. The approach employs a richly connected superstructure to represent numerous potential system configurations, and different physics-based surrogate permeator models, such as countercurrent flow or crossflow, to be used in each stage. Moreover, to describe realistic systems, pressure drop equations can be included. We also present solution methods to accelerate the solution process. Through a case study of natural gas sweetening, we demonstrate that the proposed approach is able to obtain good solutions using an off-the-shelf global optimization solver. Finally, we expand the conventional membrane system synthesis problem by introducing feed variability in our model through a case study of an integrated reactor-separation system.

42 ENGINEERING↗
DOE OSTI2023

Evolution of coupled weakly driven waves in a dissipative plasma

The nonlinear collisional dynamics of coupled driven plasma waves in the presence of background dissipation is studied analytically within kinetic theory. Sufficiently near marginal stability, phase space correlations are poorly preserved and time delays become unimportant. The system is then shown to be governed by two first-order coupled autonomous differential equations of cubic order for the wave amplitudes and two complementary first-order equations for the evolution of their phases. That system of equations can be decoupled and further simplified to a single second-order differential equation of Liénard's type for each amplitude. Numerical solutions for this equation are obtained in the general case, while analytic solutions are obtained for special cases in terms of parameters related to the spacing of the resonances of the two waves in frequency space, e.g., wave lengths and oscillation frequencies. These parameters are further analyzed to find classes of quasi-steady saturation and pulsating scenarios. To classify equilibrium points, local stability analysis is applied, and bifurcation conditions are determined. When the two waves saturate at similar amplitude levels, their combined signal is shown to invariably exhibit amplitude beating and phase jumps of nearly π. In conclusion, the obtained analytical results can be used to benchmark simulations and to interpret eigenmode amplitude measurements in fusion experiments.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Grey-Box System Identification of Grid-Forming Inverters

This paper demonstrates the use of grey-box system identification methods for simplifying and understanding the nonlinear power dynamics of grid-forming inverters (GFMs). The power and frequency outputs of complex high-order GFM models are fed into system identification software in order to fit them to a predetermined LTI system and learn system parameters such as (synthetic) inertia and droop constants. The same process is then run for a high-order synchronous generator model, and the outputs are fit to the same set of LTI equations. Simulation of a network of GFM inverters with diverse control architecture is also performed for the same process. The intent is threefold: first, to demonstrate the appropriateness of unified LTI models for describing the power and frequency dynamics of individual resources and connected networks, in order to facilitate analysis of larger heterogeneous networked systems; second, to discover the relationship between internal control parameters of GFMs and their externally observed values; and third, to validate that grey-box data-driven system identification techniques can be a valuable tool to discover the values of important parameters in the absence of explicit vendor models.

analytical models↗
DOE OSTI2025

Central Compact Finite‐Difference Scheme With High Spectral Resolution for KdV Equation

This work presents a combination of cell‐node and cell‐centered compact finite difference scheme for the approximation of third derivatives involved in Korteweg–de Vries (KdV) equations. This approach employs a half‐shifted derivative construction at cell centers, avoiding the need for compact interpolation, thereby removing transfer errors; hence, it improves spectral resolution and maintains high‐order accuracy. Fourier analysis is performed to show the spectral properties of the proposed formulation, which provides higher spectral resolutions as compared to node‐based compact schemes. A filtering strategy is incorporated to suppress high‐frequency oscillations without compromising the accuracy of the numerical scheme, and the total variation diminishing Runge Kutta (TVDRK3) method is applied for time integration. Numerical experiments on linear, nonlinear, and coupled KdV systems are conducted, and a comparative analysis with cell‐node compact schemes confirms that the proposed scheme consistently reduces errors by up to an order of magnitude and achieves high spectral resolution properties.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Hybridizing Machine Learning and Physically-based Earth System Models to Improve Prediction of Multivariate Extreme Events (AI Exploration of Wildland Fire Prediction)

Focal Areas: This project responds to two focal areas identified in the DOE Call for AI4ESP White Papers: 1) Predictive modeling through the use of artificial intelligence (AI) techniques, and 2) insights gleaned from complex data using explainable AI and big data analytics. Science Challenge: Large wildland fires (hereafter wildfires) appearing as high-impact compound climate extreme events are closely related to hydroclimate and water cycle extremes that modulate surface fuel supply and combustibility. These compound events have multivariate climatic features (e.g., temperature, precipitation, relative humidity, wind, lightning) and societal drivers (e.g., forest management, land use change, human caused ignitions). Meanwhile, they induce strong feedbacks to the coupled atmosphere, biosphere, and hydrosphere by perturbing regional and global radiation budget as well as ecological, biogeochemical, and water cycles across multiple spatiotemporal scales. The nonlinear interactions between these natural and anthropogenic components of the Earth system are too complex to be completely and adequately represented in today’s Earth system models (ESMs). The inherent stochastic nature of fire activity at all scales further increases the difficulty of its prediction using ESMs that are usually developed from deterministic equations and parameterizations. Besides, concurrence of long-term (decadal to interdecadal) global climate change and fire regime shifts overlapping with short-term (intraseasonal to interannual) variations of regional fire weather and burning activity confound predictability of these compound extreme events. We propose to address the above scientific challenges by using machine learning (ML)-based data-driven modeling techniques to integrate observations and physically-based ESMs’ simulations in a computationally efficient hybrid prediction system. This prediction system is supposed to characterize the wildfire’s sensitivity to climate and exogenous drivers at high resolution (~ 0.25°) on subseasonal to seasonal (S2S) timescales providing improved predictability and explainability. We will use the system to help identify: (1) What are the computational elements of a hybrid system needed to predict compound climate extreme events such as global wildfires? (2) What are the key drivers (either natural or anthropogenic) that modulate short-term variations of multivariate fire weather and burning activity over different regions? How can one take advantage of those driver-response relationships to improve the predictability of large wildfires on S2S time scales? (3) What are the underlying physical mechanisms and sources of improved predictability? Which ML techniques are optimal in revealing and adapting these mechanisms?

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

On how fast ions enhance the regulation of drift wave turbulence by zonal flows

Abstract This paper presents a mechanism for enhanced regulation of drift wave turbulence by zonal flows in the presence of a fast ion population. It demonstrates that dilution effects due to the energetic particles (EPs) have a far-reaching impact on all aspects of the nonlinear dynamics. The modulational growth of zonal flow shear and the corresponding evolution of drift wave energy are calculated with dilution effects. The coupled zonal flow growth and drift wave energy equations are reduced to a predator–prey model. This is solved for the fixed points, which represents the various states of the system. Results display a strong dependence on dilution, which leads to greatly reduced levels of saturated turbulence and transport. Implications for the FIRE mode plasma of KSTAR are discussed in detail. This model is perhaps the simplest dynamical one which captures the beneficial effects of EPs on confinement.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Fast nonlinear iterative solver for an implicit, energy-conserving, asymptotic-preserving charged-particle orbit integrator

Here recently, an asymptotic-preserving (AP) particle orbit integrator has been proposed with remarkable properties including exact energy conservation, the ability to capture of all first-order drifts (including the ∇B-drift), the ability to capture trapped-passing boundaries with parallel velocity extremely close to the critical velocity, and the ability to transition from strongly to weakly magnetized spatial regions. The new AP orbit integrator is implicit, employing a Crank-Nicolson (CN) temporal discretization to ensure exact energy conservation. This, in turn, requires a local nonlinear iteration involving particle velocities and positions, and the local electromagnetic fields, to obtain the new-time solution. Ref. [1] did not attempt to provide an efficient solver for this system, and employed a brute-force GMRES-driven Jacobian-free Newton-Krylov (JFNK) solver to invert the particle orbit equations at every timestep for expediency. While JFNK is robust and reliable, it is also expensive and very intrusive for practical implementations of the method (it requires having the JFNK machinery available and solving a 6 x 6 Jacobian system iteratively once per iteration per particle).

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Dynamic gain and frequency comb formation in exceptional-point lasers

Abstract Exceptional points (EPs)—singularities in the parameter space of non-Hermitian systems where two nearby eigenmodes coalesce—feature unique properties with applications such as sensitivity enhancement and chiral emission. Existing realizations of EP lasers operate with static populations in the gain medium. By analyzing the full-wave Maxwell–Bloch equations, here we show that in a laser operating sufficiently close to an EP, the nonlinear gain will spontaneously induce a multi-spectral multi-modal instability above a pump threshold, which initiates an oscillating population inversion and generates a frequency comb. The efficiency of comb generation is enhanced by both the spectral degeneracy and the spatial coalescence of modes near an EP. Such an “EP comb” has a widely tunable repetition rate, self-starts without external modulators or a continuous-wave pump, and can be realized with an ultra-compact footprint. We develop an exact solution of the Maxwell–Bloch equations with an oscillating inversion, describing all spatiotemporal properties of the EP comb as a limit cycle. We numerically illustrate this phenomenon in a 5-μm-long gain-loss coupled AlGaAs cavity and adjust the EP comb repetition rate from 20 to 27 GHz. This work provides a rigorous spatiotemporal description of the rich laser behaviors that arise from the interplay between the non-Hermiticity, nonlinearity, and dynamics of a gain medium.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Subspace recursive Fermi-operator expansion strategies for large-scale DFT eigenvalue problems on HPC architectures

Quantum mechanical calculations for material modeling using Kohn–Sham density functional theory (DFT) involve the solution of a nonlinear eigenvalue problem for N smallest eigenvector-eigenvalue pairs, with N proportional to the number of electrons in the material system. Here, these calculations are computationally demanding and have asymptotic cubic scaling complexity with the number of electrons. Large-scale matrix eigenvalue problems arising from the discretization of the Kohn–Sham DFT equations employing a systematically convergent basis traditionally rely on iterative orthogonal projection methods, which are shown to be computationally efficient and scalable on massively parallel computing architectures. However, as the size of the material system increases, these methods are known to incur dominant computational costs through the Rayleigh–Ritz projection step of the discretized Kohn–Sham Hamiltonian matrix and the subsequent subspace diagonalization of the projected matrix. This work explores the potential of polynomial expansion approaches based on recursive Fermi-operator expansion as an alternative to the subspace diagonalization of the projected Hamiltonian matrix to reduce the computational cost. Subsequently, we perform a detailed comparison of various recursive polynomial expansion approaches to the traditional approach of explicit diagonalization on both multi-node central processing unit and graphics processing unit architectures and assess their relative performance in terms of accuracy, computational efficiency, scaling behavior, and energy efficiency.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Catastrophic Cooling Instability in Optically Thin Plasmas

The solar corona is the prototypical example of a low-density environment heated to high temperatures by external sources. The plasma cools radiatively, and because it is optically thin to this radiation, it becomes possible to model the density, velocity, and temperature structure of the system by modifying the MHD equations to include an energy source term that approximates the local heating and cooling rates. The solutions can be highly inhomogeneous and even multiphase because the well-known linear instability associated with this source term, thermal instability, leads to a catastrophic heating and cooling of the plasma in the nonlinear regime. Here we show that there is a separate, much simpler linear instability accompanying this source term that can rival thermal instability in dynamical importance. The stability criterion is the isochoric one identified by Parker (1953), and we demonstrate that cooling functions derived from collisional ionization equilibrium are highly prone to violating this criterion. If catastrophic cooling instability can act locally in global simulations, then it is an alternative mechanism for forming condensations, and due to its nonequilibrium character, it may be relevant to explaining a host of phenomena associated with the production of cooler gas in hot, low density plasmas.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2023

Mean Flow from Phase Averages in the 2D Boussinesq Equations

The atmosphere and ocean are described by highly oscillatory PDEs that challenge both our understanding of their dynamics and their numerical approximation. This paper presents a preliminary numerical study of one type of phase averaging applied to mean flows in the 2D Boussinesq equations that also has application to numerical methods. The phase averaging technique, well-known in dynamical systems theory, relies on a mapping using the exponential operator, and then an averaging over the phase. The exponential operator has connections to the Craya–Herring basis pioneered by Jack Herring to study the fluid dynamics of oscillatory, nonlinear fluid dynamics. In this paper, we perform numerical experiments to study the effect of this averaging technique on the time evolution of the solution. We explore its potential as a definition for mean flows. We also show that, as expected from theory, the phase-averaging method can reduce the magnitude of the time rate of change in the PDEs, making them potentially suitable for time stepping methods.

Wingate, Beth A.↗
DOE OSTI2020

Quantifying Information without Entropy: Identifying Intermittent Disturbances in Dynamical Systems

A system’s response to disturbances in an internal or external driving signal can be characterized as performing an implicit computation, where the dynamics of the system are a manifestation of its new state holding some memory about those disturbances. Identifying small disturbances in the response signal requires detailed information about the dynamics of the inputs, which can be challenging. This paper presents a new method called the Information Impulse Function (IIF) for detecting and time-localizing small disturbances in system response data. The novelty of IIF is its ability to measure relative information content without using Boltzmann’s equation by modeling signal transmission as a series of dissipative steps. Since a detailed expression of the informational structure in the signal is achieved with IIF, it is ideal for detecting disturbances in the response signal, i.e., the system dynamics. Those findings are based on numerical studies of the topological structure of the dynamics of a nonlinear system due to perturbated driving signals. The IIF is compared to both the Permutation entropy and Shannon entropy to demonstrate its entropy-like relationship with system state and its degree of sensitivity to perturbations in a driving signal.

42 ENGINEERING↗
DOE OSTI2020

Spontaneous Creation of Circularly Polarized Photons in Chiral Astrophysical Systems

This work establishes a relation between chiral anomalies in curved spacetimes and the radiative content of the gravitational field. In particular, we show that a flux of circularly polarized gravitational waves triggers the spontaneous creation of photons with net circular polarization from the quantum vacuum. Using waveform catalogs, we identify precessing binary black holes as astrophysical configurations that emit such gravitational radiation and then solve the fully nonlinear Einstein’s equations with numerical relativity to evaluate the net effect. The quantum amplitude for a merger is comparable to the Hawking emission rate of the final black hole and small to be directly observed. However, the implications for the inspiral of binary neutron stars could be more prominent, as argued on symmetry grounds.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Generalized fractional approach to solving partial differential equations with arbitrary dispersion relations

The dynamics of physical systems are typically modeled by partial differential equations (PDEs). Because of the complexity in physical systems, PDE models rely on approximations or limiting cases. To model the full complexity of physical systems, it is necessary to use multiscale approaches where appropriate models are used at each level. Alternatively, complexity can be directly tackled by fractional integrodifferential equations that must be derived for each system. Here, we argue that complexity can be restored in PDEs by describing them from a fractional calculus perspective. Rather than deriving a fractional integrodifferential equation, we reinterpret the dispersion relation of the system by use of the Riesz definition, which contains the required information relating the energy and momentum space of the system and thus fully describes their dynamics. The approach is demonstrated by two examples: the Landau–Lifshitz equation in a 1D ferromagnetic chain and a modified KdV equation supporting surface gravity waves or Euler dispersion. The presented approach is applicable to fluids, soft matter, and solid-state matter and can be readily generalized to higher dimensions and more complex systems. While numerical calculations are needed to determine the fractional operator, the approach is analytical and can be utilized to determine analytical solutions and investigate nonlinear problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Fast Computational Algorithms for Partial Differential Equations and Uncertainty Quantifications

This project concerned the construction, testing and analysis of computational algorithms for solving parameterized and stochastic partial differential equations. The study and understanding of equations of this type is of fundamental importance in numerous engineering and scientific applications. Examples include simulation of plasma dynamics in models of electric propulsion and nuclear fusion, simulation of multiphase flows, such as the flow of water, gas and oil in reservoirs, and structural analysis of the dependence of structures on materials. Parametrization is used in such settings when properties of the models such as viscosity of fluids or electric resistivity of materials are not precisely understood and instead are treated as random variables. The resulting solutions are themselves random, and having such solutions will enable engineers to use probabilistic methods to assess the likelihood of events, for example, whether a pollutant in a liquid will exceed a limit, and to use such analyses to develop ways to ensure positive outcomes. Construction of accurate (high resolution) computational solutions is expensive, requiring significant computer time and computational resources, and there is need to reduce computational cost to make simulation useful and effective. The aim of the project was to construct algorithms to efficiently compute surrogate solutions to parameterized problems to allow for efficient and accurate simulation. The technical approach used focused on two related strategies, based on rank-reduction methods and reduced-order models. These methods construct surrogate solutions of parameter-dependent models by projection or interpolation into low-dimensional approximation spaces. Cost savings are achieved if the low-dimensional spaces can be identified and constructed efficiently and if the resulting low-dimensional algebraic systems can be solved cheaply. Accomplishments include: Theoretical and empirical demonstration of the effectiveness of fast multigrid solution strategies for computing low-rank representations of parameter-dependent solutions to discrete partial differential equations, including the first proof establishing so-called textbook convergence properties for low-rank methods. Development of efficient solution algorithms for solving nonlinear parameter-dependent partial differential equations used in models of fluid dynamics. Developent of efficient algorithms for low-rank representation of solutions of time-dependent simulations of fluid dynamics using multi-dimensional tensor representations of solutions.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Identifying stochastic dynamics via finite expression methods

Modeling stochastic differential equations (SDEs) is crucial for understanding complex dynamical systems in various scientific fields. Recent methods often employ neural network-based models, which typically represent SDEs through a combination of deterministic and stochastic terms. However, these models usually lack interpretability and have difficulty in generalizing beyond their training domain. Here, this paper introduces the Finite Expression Method (FEX), a symbolic learning approach designed to derive interpretable mathematical representations of the deterministic component of SDEs. For the stochastic component, we integrate FEX with advanced generative modeling techniques to provide a comprehensive representation of SDEs. The numerical experiments on linear, nonlinear, and multidimensional SDEs demonstrate that FEX generalizes well beyond the training domain and delivers more accurate long-term predictions compared to neural network-based methods. The symbolic expressions identified by FEX not only improve prediction accuracy but also offer valuable scientific insights into the underlying dynamics of the systems.

Complex dynamical systems↗
DOE OSTI2022

Entropy–Preserving and Entropy–Stable Relaxation IMEX and Multirate Time–Stepping Methods

In this work, we propose entropy-preserving and entropy-stable partitioned Runge–Kutta (RK) methods. In particular, we extend the explicit relaxation Runge–Kutta methods to IMEX–RK methods and a class of explicit second-order multirate methods for stiff problems arising from scale-separable or grid-induced stiffness in a system. The proposed approaches not only mitigate system stiffness but also fully support entropy-preserving and entropy-stability properties at a discrete level. The key idea of the relaxation approach is to adjust the step completion with a relaxation parameter so that the time-adjusted solution satisfies the entropy condition at a discrete level. The relaxation parameter is computed by solving a scalar nonlinear equation at each timestep in general; however, as for a quadratic entropy function, we theoretically derive the explicit form of the relaxation parameter and numerically confirm that the relaxation parameter works the Burgers equation. Several numerical results for ordinary differential equations and the Burgers equation are presented to demonstrate the entropy-conserving/stable behavior of these methods. We also compare the relaxation approach and the incremental direction technique for the Burgers equation with and without a limiter in the presence of shocks.

97 MATHEMATICS AND COMPUTING↗