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Results for “Linear and nonlinear operators”

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 145 records · Page 8
All sourcesNASA NTRSDOE OSTI
NASA NTRS1975

Upwind second-order difference schemes and applications in unsteady aerodynamic flows

Explicit second-order upwind difference schemes in combination with spatially symmetric schemes can produce larger stability bounds and better numerical resolution than symmetric schemes alone. However, if conservation form is essential, a special operator is required for transition between schemes. An operational approach has been devised for deriving transition operators so that strict conservation and local consistency are maintained. Various aspects of hybrid schemes are studied numerically for model linear and nonlinear equations. To demonstrate the utility of combining two different algorithms, MacCormack's explicit, noncentered, second-order method is combined with a completely upwind version, and numerical solutions of the Euler equations are obtained for two-dimensional, transonic flows with embedded supersonic regions and shock waves. The general utility of the operational approach for combining schemes is emphasized by deriving a second-order conservative scheme for the steady transonic small-disturbance potential equation.

Warming, R. F.↗
DOE OSTI2021

Tuning successive linear programming to solve AC optimal power flow problem for large networks

Successive linear programming (SLP) is a practical approach for solving large-scale nonlinear optimization problems. Alternating current optimal power flow (ACOPF) is no exception, particularly the large size of real-world networks. However, in order to achieve tractability, it is essential to tune the SLP algorithm presented in the literature. This paper presents a modified SLP algorithm to solve the ACOPF problem, specified by the U.S. Department of Energy’s (DOE) Grid Optimization (GO) Competition Challenge 1, within strict time limits. The algorithm first finds a near-optimal solution for the relaxed problem (i.e., Stage 1). Then, it finds a feasible solution in the proximity of the near-optimal solution (i.e., Stage 2 and Stage 3). The numerical experiments on test cases ranging from 500-bus to 30,000-bus systems show that the algorithm is tractable. Here the results show that our proposed algorithm is tractable and can solve more than 80% of test cases faster than the well-known Interior Point Method while significantly reduce the number of iterations required to solve ACOPF. The number of iterations is considered an important factor in the examination of tractability which can drastically reduce the computational time required within each iteration.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2023

Deep Koopman learning of nonlinear time-varying systems

Here this paper presents a data-driven approach to approximate the dynamics of a nonlinear time-varying system (NTVS) by a linear time-varying system (LTVS), which results from the Koopman operator and deep neural networks. Analysis of the approximation error between states of the NTVS and the resulting LTVS is presented. Simulations on a representative NTVS show that the proposed method achieves small approximation errors, even when the system changes rapidly. Furthermore, simulations in an example of quadcopters demonstrate the computational efficiency of the proposed approach.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Data-driven learning of Mori–Zwanzig operators for isotropic turbulence

Developing reduced-order models for turbulent flows, which contain dynamics over a wide range of scales, is an extremely challenging problem. In statistical mechanics, the Mori–Zwanzig (MZ) formalism provides a mathematically exact procedure for constructing reduced-order representations of high-dimensional dynamical systems, where the effects due to the unresolved dynamics are captured in the memory kernel and orthogonal dynamics. Turbulence models based on MZ formalism have been scarce due to the limited knowledge of the MZ operators, which originates from the difficulty in deriving MZ kernels for complex nonlinear dynamical systems. In this work, we apply a recently developed data-driven learning algorithm, which is based on Koopman's description of dynamical systems and Mori's linear projection operator, on a set of fully resolved isotropic turbulence datasets to extract the Mori–Zwanzig operators. With data augmentation using known turbulence symmetries, the extracted Markov term, memory kernel, and orthogonal dynamics are statistically converged and the generalized fluctuation–dissipation relation can be verified. The properties of the memory kernel and orthogonal dynamics, and their dependence on the choices of observables are investigated to address the modeling assumptions that are commonly used in MZ-based models. A series of numerical experiments are then constructed using the extracted kernels to evaluate the memory effects on prediction. The results show that the prediction errors are strongly affected by the choice of observables and can be further reduced by including the past history of the observables in the memory kernel.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Analysis of the linear and nonlinear stability of Alfven eigenmodes and fish-bones in JET DT discharges: mode identification and shear flows generation

The plasma in future nuclear fusion reactors will be heated by neutral beam injectors (NBIs) and high frequency electromagnetic waves as well as fusion born alpha particles. Energetic particles (EPs), with energies up to two orders of magnitude larger than the thermal plasma, can trigger EP driven modes and induce harmful EP losses, reducing the plasma heating efficiency and the economical viability of the reactor. The present study is dedicated to analyze the Alfven Eigenmode (AE) activity in JET D–T discharges, the closest experiment to reactor-like operation performed until now. There, EP driven modes are induced by the combined effect of tangential NBIs and ion cyclotron resonance heating (ICRH) driven EP. Linear and nonlinear simulations are performed with the gyro-fluid FAR3d code to analyze the AE activity observed in the discharge 99896. The linear simulations reproduce the unstable n = 3 to 5 toroidal AEs (TAE) at the inner plasma region observed in the experiment, triggered by highly energetic passing deuterium populations injected by the tangential NBIs, further accelerated by the effect of the ICRH up to 1 MeV. In addition, fish-bones triggered by energetic trapped hydrogen induced by the ICRH are also reproduced. On the other hand, the alpha particles density is too small to destabilize AEs in the experiment. Nonetheless, increasing artificially the alpha density by one order of magnitude, an n = 1 beta induced AE can be destabilized in the inner plasma region. Nonlinear simulations indicate the generation of zonal structures during the AE/fish-bone saturation phase. TAE and fish-bones causes a rather weak increase of the passing D and trapped H EP (around 2%), respectively. Shear flows and zonal currents are generated during the saturation of TAE and fish-bones. Nonlinear simulations performed for D–T and pure deuterium thermal plasma indicate AE/fish-bone activity is weaker and shear flows are less intense in the pure deuterium case, trends consistent with the experimental observations that also indicates a deterioration of the thermal plasma confinement. Therefore, both numerical studies and experimental evidence indicate the generation of shear flows by AE/fish-bones could be connected with an improvement of the thermal plasma confinement.

AE↗
NASA NTRS1991

A Lyapunov based nonlinear control scheme for stabilizing a basic compression system using a close-coupled control valve

The use of a closed-loop control to allow surge-free operation of a compression system beyond its uncontrolled surge line is addressed. In contrast to previous analyses which used a linearized model, the approach described directly addresses the nonlinear nature of the compressor characteristic using a Liapunov-based control law design formulation. The proposed approach is fairly generic and should be of interest for gas turbine engines as well as other applications.

Simon, J. S.↗
DOE OSTI2023

Multi-level optimization with the koopman operator for data-driven, domain-aware, and dynamic system security

Cyber-Physical Systems (CPSs) like the power grid are critically important but also increasingly vulnerable; ensuring reliable system operation in the face of disruptions is becoming more and more challenging. Multi-Level Optimization (MLO) is a powerful way to model adversarial interactions, which naturally makes it applicable to studying CPS security. However, MLO typically does not address underlying system dynamics, and incorporating nonlinear dynamics is generally infeasible. In this paper, we show how to combine MLO with the Koopman Operator (KO) to remedy this. The KO maps nonlinear dynamics to a lifted space in which those dynamics are linear, thus making it ideal for use with MLO. Moreover, the structure of the KO also provides convenient ways to incorporate domain knowledge into the data-driven process of learning the KO representation of a given system. Here we then demonstrate the use of MLO-KO on a small example problem taken from the power grid domain, discuss the scalability and computational cost of MLO-KO, and identify future research directions for this work.

42 ENGINEERING↗
DOE OSTI2020

Multi-species collisions for delta-f gyrokinetic simulations: Implementation and verification with GENE

we report a multi-species linearized collision operator based on the model developed by Sugama et al. has been implemented in the nonlinear gyrokinetic code, GENE. Such a model conserves particles, momentum, and energy to machine precision, and is shown to have negative definite free energy dissipation characteristics, satisfying Boltzmann’s H-theorem, including for realistic mass ratio. Finite Larmor Radius (FLR) effects have also been implemented into the local version of the code. For the global version of the code, the collision operator has been developed to allow for block-structured velocity space grids, allowing for computationally tractable collisional global simulations. The validity of the collision operator has been demonstrated by relaxation and conservation tests, as well as appropriate benchmarks. The newly implemented operator shall be used in future simulations to study magnetically confined fusion plasma turbulence and transport in more extreme regions with higher collisionality.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Multi-species collisions for delta-f gyrokinetic simulations: Implementation and verification with GENE

A multi-species linearized collision operator based on the model developed by Sugama et al. has been implemented in the nonlinear gyrokinetic code, GENE. Such a model conserves particles, momentum, and energy to machine precision, and is shown to have negative definite free energy dissipation characteristics, satisfying Boltzmann’s H-theorem, including for realistic mass ratio. Finite Larmor Radius (FLR) effects have also been implemented into the local version of the code. For the global version of the code, the collision operator has been developed to allow for block-structured velocity space grids, allowing for computationally tractable collisional global simulations. The validity of the collision operator has been demonstrated by relaxation and conservation tests, as well as appropriate benchmarks. The newly implemented operator shall be used in future simulations to study magnetically confined fusion plasma turbulence and transport in more extreme regions with higher collisionality.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS1975

Implicit approximate-factorization schemes for the low-frequency transonic equation

Two- and three-level implicit finite-difference algorithms for the low-frequency transonic small disturbance-equation are constructed using approximate factorization techniques. The schemes are unconditionally stable for the model linear problem. For nonlinear mixed flows, the schemes maintain stability by the use of conservatively switched difference operators for which stability is maintained only if shock propagation is restricted to be less than one spatial grid point per time step. The shock-capturing properties of the schemes were studied for various shock motions that might be encountered in problems of engineering interest. Computed results for a model airfoil problem that produces a flow field similar to that about a helicopter rotor in forward flight show the development of a shock wave and its subsequent propagation upstream off the front of the airfoil.

Ballhaus, W. F.↗
NASA NTRS1975

The human controller in car following

The man machine interface between the human driver and his automobile during car following tasks was examined. A direct connection between models of the human operator and the theory of optimal control was established for car following situations involving both linear and nonlinear car dynamics. Results obtained from several types of mathematical models are discussed. Conventional parameter identification techniques were used to obtain the parameters of these models by comparing their performance to data obtained by aerial photogrammetry. The results indicate that there are two general classes of model structures: models assumed to have preview information and models without preview.

Burnham, G. O.↗
NASA NTRS1988

A flight-test methodology for identification of an aerodynamic model for a V/STOL aircraft

Described is a flight test methodology for developing a data base to be used to identify an aerodynamic model of a vertical and short takeoff and landing (V/STOL) fighter aircraft. The aircraft serves as a test bed at Ames for ongoing research in advanced V/STOL control and display concepts. The flight envelope to be modeled includes hover, transition to conventional flight, and back to hover, STOL operation, and normaL cruise. Although the aerodynamic model is highly nonlinear, it has been formulated to be linear in the parameters to be identified. Motivation for the flight test methodology advocated in this paper is based on the choice of a linear least-squares method for model identification. The paper covers elements of the methodology from maneuver design to the completed data base. Major emphasis is placed on the use of state estimation with tracking data to ensure consistency among maneuver variables prior to their entry into the data base. The design and processing of a typical maneuver is illustrated.

Bach, Ralph E., Jr.↗
NASA NTRS1989

A flight-test methodology for identification of an aerodynamic model for a V/STOL aircraft

This paper describes a flight-test methodology for developing a data base to be used to identify an aerodynamic model of a V/STOL fighter aircraft. The aircraft serves as a test bed at NASA Ames for ongoing research in advanced V/STOL control and display concepts. The flight envelope to be modeled includes hover, transition to conventional flight and back to hover, STOL operation, and normal cruise. Although the aerodynamic model is highly nonlinear, it has been formulated to be linear in the parameters to be identified. Motivation for the flight-test methodology advocated in this paper is based on the choice of a linear least-squares method for model identification. The paper covers elements of the methodology from maneuver design to the completed data base. Major emphasis is placed on the use of state estimation with tracking data to ensure consistency among maneuver variables prior to their entry into the data base. The design and processing of a typical maneuver are illustrated.

Bach, R. E., Jr.↗
NASA NTRS1984

Optical systolic solutions of linear algebraic equations

The philosophy and data encoding possible in systolic array optical processor (SAOP) were reviewed. The multitude of linear algebraic operations achievable on this architecture is examined. These operations include such linear algebraic algorithms as: matrix-decomposition, direct and indirect solutions, implicit and explicit methods for partial differential equations, eigenvalue and eigenvector calculations, and singular value decomposition. This architecture can be utilized to realize general techniques for solving matrix linear and nonlinear algebraic equations, least mean square error solutions, FIR filters, and nested-loop algorithms for control engineering applications. The data flow and pipelining of operations, design of parallel algorithms and flexible architectures, application of these architectures to computationally intensive physical problems, error source modeling of optical processors, and matching of the computational needs of practical engineering problems to the capabilities of optical processors are emphasized.

Neuman, C. P.↗
NASA NTRS1993

Robust, nonlinear, high angle-of-attack control design for a supermaneuverable vehicle

High angle-of-attack flight control laws are developed for a supermaneuverable fighter aircraft. The methods of dynamic inversion and structured singular value synthesis are combined into an approach which addresses both the nonlinearity and robustness problems of flight at extreme operating conditions. The primary purpose of the dynamic inversion control elements is to linearize the vehicle response across the flight envelope. Structured singular value synthesis is used to design a dynamic controller which provides robust tracking to pilot commands. The resulting control system achieves desired flying qualities and guarantees a large margin of robustness to uncertainties for high angle-of-attack flight conditions. The results of linear simulation and structured singular value stability analysis are presented to demonstrate satisfaction of the design criteria. High fidelity nonlinear simulation results show that the combined dynamics inversion/structured singular value synthesis control law achieves a high level of performance in a realistic environment.

Adams, Richard J.↗
DOE OSTI2026

Multibody for Everybody (M4E) - A Linearization Approach to Enable Frequency Domain Analysis, Time Integration and Control Co-Design

1.1 Background/Objectives: Marine energy represents a promising yet underexploited source of power. To increase the harvested power, significant efforts have been made to improve wave energy converter (WEC) modeling capabilities and optimize power take-off (PTO) performance; however, these efforts have often treated WEC dynamics, PTO design, and controller development sequentially. In contrast, control co-design (CCD) is emerging as a promising strategy to address these issues directly, creating a growing need for fast analysis tools suitable for repeated simulation and parametric studies [1]. To support this need, this work presents the Multibody for Everybody (M4E) [2] linearization module, which employs a symbolic toolbox to provide deeper insight of WEC design parameters. The objective is to demonstrate that a minimal-coordinate linearization of articulated WEC dynamics can provide accurate wave response predictions and substantial computational savings relative to nonlinear time-domain simulation, while preserving compatibility with broader wave-energy analysis workflows, enabling CCD. 1.2 Approach/Activities: The proposed approach linearizes the equations of motion, generated by M4E, in minimal coordinates about a selected operating point and combines the resulting system with frequencydomain hydrodynamic terms to incorporate the reduced mass, damping, stiffness, and forcing operators. The linearized model is used for both impedance-based response amplitude operator (RAO) prediction and rapid regular-wave time integration. The methodology is demonstrated on a single-flap device and a FOSWEC configuration, with linearized M4E responses compared against the corresponding nonlinear M4E simulations and WEC-Sim results. Regular-wave time histories, RAO trends, and runtime differences are assessed. The framework is also compatible with broader wave-energy workflows, including coupling to WecOptTool, although that capability is not the focus of this work [3]. 1.3 Results/Lessons: The linearized M4E model reproduces key regularwave response characteristics such as integration and Response Amplitude over multiple frequencies. This module matches nonlinear M4E and WEC-Sim results while substantially reducing integration cost. Thus, the proposed framework can serve as a rapid analysis layer for articulated WEC design, parameter studies, and controls-oriented workflows. The analysis is most appropriate in the near-equilibrium regime, about the linearization point.

16 TIDAL AND WAVE POWER↗
NASA NTRS1985

Bounding solutions of geometrically nonlinear viscoelastic problems

Integral transform techniques, such as the Laplace transform, provide simple and direct methods for solving viscoelastic problems formulated within a context of linear material response and using linear measures for deformation. Application of the transform operator reduces the governing linear integro-differential equations to a set of algebraic relations between the transforms of the unknown functions, the viscoelastic operators, and the initial and boundary conditions. Inversion either directly or through the use of the appropriate convolution theorem, provides the time domain response once the unknown functions have been expressed in terms of sums, products or ratios of known transforms. When exact inversion is not possible approximate techniques may provide accurate results. The overall problem becomes substantially more complex when nonlinear effects must be included. Situations where a linear material constitutive law can still be productively employed but where the magnitude of the resulting time dependent deformations warrants the use of a nonlinear kinematic analysis are considered. The governing equations will be nonlinear integro-differential equations for this class of problems. Thus traditional as well as approximate techniques, such as cited above, cannot be employed since the transform of a nonlinear function is not explicitly expressible.

Stubstad, J. M.↗
NASA NTRS1986

Bounding solutions of geometrically nonlinear viscoelastic problems

Integral transform techniques, such as the Laplace transform, provide simple and direct methods for solving viscoelastic problems formulated within a context of linear material response and using linear measures for deformation. Application of the transform operator reduces the governing linear integro-differential equations to a set of algebraic relations between the transforms of the unknown functions, the viscoelastic operators, and the initial and boundary conditions. Inversion either directly or through the use of the appropriate convolution theorem, provides the time domain response once the unknown functions have been expressed in terms of sums, products or ratios of known transforms. When exact inversion is not possible approximate techniques may provide accurate results. The overall problem becomes substantially more complex when nonlinear effects must be included. Situations where a linear material constitutive law can still be productively employed but where the magnitude of the resulting time dependent deformations warrants the use of a nonlinear kinematic analysis are considered. The governing equations will be nonlinear integro-differential equations for this class of problems. Thus traditional as well as approximate techniques, such as cited above, cannot be employed since the transform of a nonlinear function is not explicitly expressible.

Stubstad, J. M.↗