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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 469 records · Page 26
All sourcesNASA NTRSDOE OSTI
NASA NTRS1999

Nonlinear Response of Composite Panels Under Combined Acoustic Excitation and Aerodynamic Pressure

A finite element formulation is presented for the analysis of large deflection response of composite panels subjected to aerodynamic pressure- at supersonic flow and high acoustic excitation. The first-order shear deformation theory is considered for laminated composite plates, and the von Karman nonlinear strain-displacement relations are employed for the analysis of large deflection panel response. The first-order piston theory aerodynamics and the simulated Gaussian white noise are employed for the aerodynamic and acoustic loads, respectively. The nonlinear equations of motion for an arbitrarily laminated composite panel subjected to a combined aerodynamic and acoustic pressures are formulated first in structure node degrees-of-freedom. The system equations are then transformed and reduced to a set of coupled nonlinear equations in modal coordinates. Modal participation is defined and the in-vacuo modes to be retained in the analysis are based on the modal participation values. Numerical results include root mean square values of maximum deflections, deflection and strain response time histories, probability distributions, and power spectrum densities. Results showed that combined acoustic and aerodynamic loads have to be considered for panel analysis and design at high dynamic pressure values.

Abdel-Motagaly, K.↗
NASA NTRS2001

Multi-Tethered Space-Based Interferometers: Particle System Model

Dynamics models are presented for a class of space-based interferometers comprised of multiple component bodies, interconnected in various arrangements, by low-mass flexible tethers of variable length. The tethered constellations are to perform coordinated rotational scanning accompanied by baseline dimensional changes, as well as spin axis realignments and spin-up/spin-down maneuvers. The mechanical idealization is a system of N point masses interconnected by massless tethers of variable length. Both extensible and inextensible tethers are considered. Expressions for system angular and linear momenta are developed. The unrestricted nonlinear motion equations are derived via Lagranges equations. Rheonomic constraints are introduced to allow prescribed motion of any degrees of freedom, and the associated physical forces are determined. The linearized equations of motion are obtained for the steady rotation of a system with extensible tethers of constant unstrained length.

Gates, Stephen S.↗
NASA NTRS1985

Performance of direct and iterative algorithms on an optical systolic processor

The frequency-multiplexed optical linear algebra processor (OLAP) is treated in detail with attention to its performance in the solution of systems of linear algebraic equations (LAEs). General guidelines suitable for most OLAPs, including digital-optical processors, are advanced concerning system and component error source models, guidelines for appropriate use of direct and iterative algorithms, the dominant error sources, and the effect of multiple simultaneous error sources. Specific results are advanced on the quantitative performance of both direct and iterative algorithms in the solution of systems of LAEs and in the solution of nonlinear matrix equations. Acoustic attenuation is found to dominate iterative algorithms and detector noise to dominate direct algorithms. The effect of multiple spatial errors is found to be additive. A theoretical expression for the amount of acoustic attenuation allowed is advanced and verified. Simulations and experimental data are included.

Ghosh, A. K.↗
NASA NTRS1994

A parametric sensitivity study for single-stage-to-orbit hypersonic vehicles using trajectory optimization

The class of hypersonic vehicle configurations with single stage-to-orbit (SSTO) capability reflect highly integrated airframe and propulsion systems. These designs are also known to exhibit a large degree of interaction between the airframe and engine dynamics. Consequently, even simplified hypersonic models are characterized by tightly coupled nonlinear equations of motion. In addition, hypersonic SSTO vehicles present a major system design challenge; the vehicle's overall mission performance is a function of its subsystem efficiencies including structural, aerodynamic, propulsive, and operational. Further, all subsystem efficiencies are interrelated, hence, independent optimization of the subsystems is not likely to lead to an optimum design. Thus, it is desired to know the effect of various subsystem efficiencies on overall mission performance. For the purposes of this analysis, mission performance will be measured in terms of the payload weight inserted into orbit. In this report, a trajectory optimization problem is formulated for a generic hypersonic lifting body for a specified orbit-injection mission. A solution method is outlined, and results are detailed for the generic vehicle, referred to as the baseline model. After evaluating the performance of the baseline model, a sensitivity study is presented to determine the effect of various subsystem efficiencies on mission performance. This consists of performing a parametric analysis of the basic design parameters, generating a matrix of configurations, and determining the mission performance of each configuration. Also, the performance loss due to constraining the total head load experienced by the vehicle is evaluated. The key results from this analysis include the formulation of the sizing problem for this vehicle class using trajectory optimization, characteristics of the optimal trajectories, and the subsystem design sensitivities.

Lovell, T. Alan↗
DOE OSTI2026

Toward real-time optimization through model reduction and model discrepancy sensitivities

Optimization problems arise in a range of scenarios, from optimal control to model parameter estimation. In many applications, such as the development of digital twins, it is essential to solve these optimization problems within wall-clock-time limitations. However, this is often unattainable for complex systems, such as those modeled by nonlinear partial differential equations. One strategy for mitigating this issue is to construct a reduced-order model (ROM) that enables more rapid optimization. In particular, the use of nonintrusive ROMs—those that do not require access to the full-order model at evaluation time—is popular because they facilitate the computation of optimization solutions within the wall-clock time requirements. However, the optimization solution will be unreliable if the iterates move outside the ROM training data. This article proposes the use of hyper-differential sensitivity analysis with respect to model discrepancy (HDSA-MD) as a computationally efficient tool to augment ROM-constrained optimization and improve its reliability. The proposed approach consists of two phases: (i) an offline phase where several full-order model evaluations are computed to train the ROM, and (ii) an online phase where a ROM-constrained optimization problem is solved, a limited number of full-order model evaluations are computed, and HDSA-MD is used to enhance the optimization solution. Numerical results are demonstrated for two examples, atmospheric contaminant control and wildfire ignition location estimation, in which a ROM is trained offline using inaccurate atmospheric data. In conclusion, the HDSA-MD update yields a significant improvement in the ROM-constrained optimization solution using only one full-order model evaluation online with corrected atmospheric data.

PDE-constrained optimization↗
DOE OSTI2021

Sensitive parameter identification and uncertainty quantification for the stability of pipeline conveying fluid

In this study, several uncertainty quantification and sensitivity analysis methods are used to determine the most sensitive geometric and material input parameters of a cantilevered pipeline conveying fluid when uncertainty is introduced to the system at the onset of instability. The full nonlinear equations of motion are modeled using the extended Hamilton’s principle and then discretized using Galerkin’s method. A parametric study is first performed, and the Morris elementary effects are calculated to obtain a preliminary understanding of how the onset speed changes when each parameter is introduced to a ± 5% uncertainty. Then, four different input uncertainty distributions, mainly, uniform and Gaussian distribution, are chosen to investigate how input distributions affect uncertainty in the output. A convergence analysis is used to determine the number of samples needed to maintain simulation accuracy while saving the most computational time. Then, Monte Carlo simulations are run, and the output distributions for each input distribution at ± 1%, ± 3% and ± 5% input uncertainty range are found and discussed. Additionally, the Pearson correlation coefficients are evaluated for different uncertainty ranges. A final Monte Carlo study is performed in which single parameters are held constant while all others still have uncertainty. Overall, the flow speed at the onset of instability is the most sensitive to changes in the outer diameter of the pipe.

36 MATERIALS SCIENCE↗
DOE OSTI2026

An efficient explicit implementation of a near-optimal quantum algorithm for simulating linear dissipative differential equations

We propose an efficient block-encoding technique for the implementation of the Linear Combination of Hamiltonian Simulations (LCHS) for simulating dissipative initial-value problems. This algorithm approximates a target nonunitary operator as a weighted sum of Hamiltonian evolutions, thereby emulating a dissipative problem by mixing various time scales. We introduce an efficient encoding of the LCHS into a quantum circuit based on a simple coordinate transformation that turns the dependence on the summation index into a trigonometric function. Classically, this method is equivalent to the use of a highly accurate Fejér-Clenshaw-Curtis quadrature formula. Quantumly, this significantly simplifies block-encoding of a dissipative problem and allows one to perform an exponential number of Hamiltonian simulations by a single Quantum Signal Processing (QSP) circuit. The resulting LCHS circuit has high success probability and the selector scales logarithmically with the number of terms in the LCHS sum and linearly with time. Careful analysis of error convergence proves that this method is more efficient than other LCHS circuits that have recently appeared in the literature. We verify the quantum circuit and its scaling by simulating it on a digital emulator of fault-tolerant quantum computers and, as a test problem, solve the advection-diffusion equation. The proposed algorithm can be used for simulating a wide class of nonunitary initial-value problems including the Liouville equation with added dissipation and linear embeddings of nonlinear systems, such as the Koopman-von Neumann and Carleman embeddings.

Novikau, I [Lawrence Livermore National Laboratory↗
DOE OSTI2020

A forced Korteweg–de Vries model for nonlinear mixing of oscillations in a dusty plasma

Nonlinear mixing of oscillations in a dusty plasma due to the harmonic time varying modulation of a nonlinear compressional oscillation is analyzed using a simple mathematical model consisting of a forced Korteweg-de Vries equation. An exact analytic solution of this equation is found to exhibit nonlinear mixing in the system. Here, the model solution can be usefully employed to predict the existence of nonlinear mixing of oscillations in a two-dimensional dusty plasma system of a particular experimental configuration.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS1972

An initial value method for the Ambarzumian integral equation.

It is shown that a general class of nonlinear integral equations may be transformed into a Cauchy system. That this leads to an effective numerical scheme is demonstrated by solving the Ambarzumian integral equation. The new method does not involve successive approximations or series expansions.

Buell, J.↗
NASA NTRS1977

A multiscale numerical study into the cascade of kinetic energy leading to severe local storms

The cascade of kinetic energy from macro- through mesoscales is studied on the basis of a nested grid system used to solve a set of nonlinear differential equations. The kinetic energy cascade and the concentration of vorticity through the hydrodynamic spectrum provide a means for predicting the location and intensity of severe weather from large-scale data sets. A mechanism described by the surface pressure tendency equation proves to be important in explaining how initial middle-tropospheric mass-momentum imbalances alter the low-level pressure field.

Paine, D. A.↗
NASA NTRS1983

Computational methods for ideal compressible flow

Conservative dissipative difference schemes for computing one dimensional flow are introduced, and the recognition and representation of flow discontinuities are discussed. Multidimensional methods are outlined. Second order finite volume schemes are introduced. Conversion of difference schemes for a single linear convection equation into schemes for the hyperbolic system of the nonlinear conservation laws of ideal compressible flow is explained. Approximate Riemann solvers are presented. Monotone initial value interpolation; and limiters, switches, and artificial dissipation are considered.

Vanleer, B.↗
NASA NTRS1988

A general-purpose approach to computer-aided dynamic analysis of a flexible helicopter

A general purpose mathematical formulation is described for dynamic analysis of a helicopter consisting of flexible and/or rigid bodies that undergo large translations and rotations. Rigid body and elastic sets of generalized coordinates are used. The rigid body coordinates define the location and the orientation of a body coordinate frame (global frame) with respect to an inertial frame. The elastic coordinates are introduced using a finite element approach in order to model flexible components. The compatibility conditions between two adjacent elements in a flexible body are imposed using a Boolean matrix, whereas the compatibility conditions between two adjacent bodies are imposed using the Lagrange multiplier approach. Since the form of the constraint equations depends upon the type of kinematic joint and involves only the generalized coordinates of the two participating elements, then a library of constraint elements can be developed to impose the kinematic constraint in an automated fashion. For the body constraints, the Lagrange multipliers yield the reaction forces and torques of the bodies at the joints. The virtual work approach is used to derive the equations of motion, which are a system of differential and algebraic equations that are highly nonlinear. The formulation presented is general and is compared with hard-wired formulations commonly used in helicopter analysis.

Agrawal, Om P.↗
NASA NTRS1991

Exponential integration algorithms applied to viscoplasticity

Four, linear, exponential, integration algorithms (two implicit, one explicit, and one predictor/corrector) are applied to a viscoplastic model to assess their capabilities. Viscoplasticity comprises a system of coupled, nonlinear, stiff, first order, ordinary differential equations which are a challenge to integrate by any means. Two of the algorithms (the predictor/corrector and one of the implicits) give outstanding results, even for very large time steps.

Freed, Alan D.↗
NASA NTRS1990

A computational algorithm for spacecraft control and momentum management

Developments in the area of nonlinear control theory have shown how coordinate changes in the state and input spaces of a dynamical system can be used to transform certain nonlinear differential equations into equivalent linear equations. These techniques are applied to the control of a spacecraft equipped with momentum exchange devices. An optimal control problem is formulated that incorporates a nonlinear spacecraft model. An algorithm is developed for solving the optimization problem using feedback linearization to transform to an equivalent problem involving a linear dynamical constraint and a functional approximation technique to solve for the linear dynamics in terms of the control. The original problem is transformed into an unconstrained nonlinear quadratic program that yields an approximate solution to the original problem. Two examples are presented to illustrate the results.

Dzielski, John↗
NASA NTRS1991

Coupled rotor-flexible fuselage vibration reduction using open loop higher harmonic control

A fundamental study of vibration prediction and vibration reduction in helicopters using active controls was performed. The nonlinear equations of motion for a coupled rotor/flexible fuselage system have been derived using computer algebra on a special purpose symbolic computer facility. The trim state and vibratory response of the helicopter are obtained in a single pass by applying the harmonic balance technique and simultaneously satisfying the trim and the vibratory response of the helicopter for all rotor and fuselage degrees of freedom. The influence of the fuselage flexibility on the vibratory response is studied. It is shown that the conventional single frequency higher harmonic control is capable of reducing either the hub loads or only the fuselage vibrations but not both simultaneously. It is demonstrated that for simultaneous reduction of hub shears and fuselae vibrations a new scheme called multiple higher harmonic control is required.

Papavassiliou, I.↗
NASA NTRS1990

A study of coupled rotor-fuselage vibration with higher harmonic control using a symbolic computing facility

A fundamental study of vibration prediction and vibration reduction in helicopters using active controls was performed. The nonlinear equations of motion for a coupled rotor/flexible fuselage system have been derived using computer algebra on a special purpose symbolic computing facility. The details of the derivation using the MACSYMA program are described. The trim state and vibratory response of the helicopter are obtained in a single pass by applying the harmonic balance technique and simultaneously satisfying the trim and the vibratory response of the helicopter for all rotor and fuselage degrees of freedom. The influence of the fuselage flexibility on the vibratory response is studied. It is shown that the conventional single frequency higher harmonic control (HHC) capable of reducing either the hub loads or only the fuselage vibrations but not both simultaneously. It is demonstrated that for simultaneous reduction of hub shears and fuselage vibrations a new scheme called multiple higher harmonic control (MHHC) is required. The fundamental aspects of this scheme and its uniqueness are described in detail, providing new insight on vibration reduction in helicopters using HHC.

Papavassiliou, I.↗