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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 217 records · Page 12
All sourcesNASA NTRSDOE OSTI
NASA NTRS1981

Construction of composite three dimensional grids from subregion grids generated by elliptic systems

A technique is presented for constructing a boundary-conforming grid throughout a general three-dimensional flow region as a composite of subregion grids. Each subregion grid is generated numerically by solving a nonlinear system of elliptic equations. The boundary values represent nodal points in a quasi-two-dimensional grid that covers the curved surface bounding the subregion, and are generated numerically by a modified elliptic system. A primary feature of the technique is that the composite three-dimensional grid remains both continuous and smooth across the surface of juncture between any two adjoining subregions. Sample results are given for a wing-body combination.

Thomas, P. D.↗
NASA NTRS1987

Construction of explicit and implicit symmetric TVD schemes and their applications

A one-parameter family of explicit and implicit total variation diminishing (TVD) schemes is developed which permits incorporation of an expanded group of slope and flux limiters. The numerical technique is intended for use in calculations which include a time-differencing scheme and an optional Lax-Wendroff scheme. Methods of extending the TVD models to nonlinear scalar equations and systems of hyperbolic conservation equations are described. Sample results are presented from calculations of shocked flows around NACA 0012 and NACA 0018 airfoils.

Yee, H. C.↗
NASA NTRS1987

Optimal time free nodal transfers between elliptical orbits

The paper presents the necessary conditions for the minimum fuel, time-free transfer between two noncoplanar elliptical orbits. It is shown that the solution is obtained by solving a system of three nonlinear equations for three unknowns. The case where the impulses are applied along the line of nodes is discussed. In general, this nodal transfer is nonoptimal, but the characteristic velocity for the best nodal transfer, called the minimizing nodal transfer, is reasonably close to the one for the optimal transfer for it to be useful as a substitute for a practical transfer. Furthermore, when the relative position of the two terminal orbits is varied, the two characteristic velocities, for the minimizing nodal transfer and the optimal transfer, exhibit the same trend in the sense that they pass through their maxima and minima at nearly the same relative position. This makes the set of explicit formulas for computing the minimizing nodal transfer, as presented in this paper, a useful tool for designing a minimum fuel transfer between several orbits.

Vinh, Nguyen X.↗
NASA NTRS1989

Inclusion of filter modulation in synthetic-discriminant-function construction

A technique in which the filter modulation is included in the synthesis of a synthetic-discriminant-function (SDF) matched spatial filter is presented. In the filter synthesis, a system of simultaneous nonlinear equations is solved with an iteration procedure. A computer simulation of the new method using thresholded images of the Space Shuttle over a range of aspect angles was performed for phase-only filters (POFs) and binary-phase-only filters (BPOFs). The filters constructed are capable of obtaining the specified peak-correlation response to within 1 percent with a high signal-to-clutter-ratio for the one-class problem, the two-class problem, and the multilevel problem. In contrast, conventional projection SDF POFs and BPOFs are unable to produce the desired peak-correlation response.

Jared, David A.↗
NASA NTRS1990

Linear iterative solvers for implicit ODE methods

The numerical solution of stiff initial value problems, which lead to the problem of solving large systems of mildly nonlinear equations are considered. For many problems derived from engineering and science, a solution is possible only with methods derived from iterative linear equation solvers. A common approach to solving the nonlinear equations is to employ an approximate solution obtained from an explicit method. The error is examined to determine how it is distributed among the stiff and non-stiff components, which bears on the choice of an iterative method. The conclusion is that error is (roughly) uniformly distributed, a fact that suggests the Chebyshev method (and the accompanying Manteuffel adaptive parameter algorithm). This method is described, also commenting on Richardson's method and its advantages for large problems. Richardson's method and the Chebyshev method with the Mantueffel algorithm are applied to the solution of the nonlinear equations by Newton's method.

Saylor, Paul E.↗
NASA NTRS1990

Time-optical spinup maneuvers of flexible spacecraft

Attitude controllers for spacecraft have been based on the assumption that the bodies being controlled are rigid. Future spacecraft, however, may be quite flexible. Many applications require spinning up/down these vehicles. In this work the minimum time control of these maneuvers is considered. The time-optimal control is shown to possess an important symmetry property. Taking advantage of this property, the necessary and sufficient conditions for optimality are transformed into a system of nonlinear algebraic equations in the control switching times during one half of the maneuver, the maneuver time, and the costates at the mid-maneuver time. These equations can be solved using a homotopy approach. Control spillover measures are introduced and upper bounds on these measures are obtained. For a special case these upper bounds can be expressed in closed form for an infinite dimensional evaluation model. Rotational stiffening effects are ignored in the optimal control analysis. Based on a heuristic argument a simple condition is given which justifies the omission of these nonlinear effects. This condition is validated by numerical simulation.

Singh, G.↗
NASA NTRS1989

Kinetic theory model for the flow of a simple gas from a two-dimensional nozzle

A system of nonlinear integral equations equivalent to the Krook kinetic equation for the steady state is the mathematical basis used to develop a computer code to model the flowfields for low-thrust two-dimensional nozzles. The method of characteristics was used to solve numerically by an iteration process the approximated Boltzmann equation for the number density, temperature, and velocity profiles of a simple gas as it exhausts into a vacuum. Results predict backscatter and show the effect of the inside wall boundary layer on the flowfields external to the nozzle.

Riley, B. R.↗
NASA NTRS1992

A staggered mesh finite difference scheme for the computation of compressible flows

A simple high resolution finite difference technique is presented to approximate weak solutions to hyperbolic systems of conservation laws. The method does not rely on Riemann problem solvers and is therefore easy to extend to a wide variety of problems. The overall performance (resolution and CPU requirements) is competitive, with other state-of-the-art techniques offering sharp nonoscillatory shocks and contacts. Theoretical results confirm the reliability of the approach for linear systems and nonlinear scalar equations.

Sanders, Richard↗
NASA NTRS1991

Kinetic theory model for the flow of a simple gas from a three-dimensional axisymmetric nozzle

A system of nonlinear integral equations equivalent to the Krook kinetic equations for the steady state is the mathematical basis used to develop a computer code to model the flowfields for low-thrust three-dimensional axisymmetric nozzles. The method of characteristics is used to solve numerically by an iteration process the approximated Boltzmann equation for the number density, temperature, and velocity profiles of a simple gas as it expands into a vacuum. Results predict backscatter and show the effect of the nozzle wall boundary layer on the external flowfields.

Riley, B. R.↗
NASA NTRS1958

Investigation of a Nonlinear Control System

A discontinuous variation of coefficients of the differential equation describing the linear control system before nonlinear elements are added is studied in detail. The nonlinear feedback is applied to a second-order system. Simulation techniques are used to study performance of the nonlinear control system and to compare it with the linear system for a wide variety of inputs. A detailed quantitative study of the influence of relay delays and of a transport delay is presented.

Flugge-Lotz, I↗
NASA NTRS1993

Kinetic theory model predictions compared with low-thrust axisymmetric nozzle plume data

A system of nonlinear integral equations equivalent to the steady-state Krook kinetic equation was used to model the flow from a low-thrust axisymmetric nozzle. The mathematical model was used to numerically calculate the number density, temperature, and velocity of a simple gas as it expands into a near vacuum. With these quantities the gas pressure and flow directions of the gas near the exit plane were calculated and compared with experimental values for a low-thrust nozzle of the same geometry and mass flow rate.

Riley, B. R.↗
NASA NTRS1996

Reliability, Risk and Cost Trade-Offs for Composite Designs

Risk and cost trade-offs have been simulated using a probabilistic method. The probabilistic method accounts for all naturally-occurring uncertainties including those in constituent material properties, fabrication variables, structure geometry and loading conditions. The probability density function of first buckling load for a set of uncertain variables is computed. The probabilistic sensitivity factors of uncertain variables to the first buckling load is calculated. The reliability-based cost for a composite fuselage panel is defined and minimized with respect to requisite design parameters. The optimization is achieved by solving a system of nonlinear algebraic equations whose coefficients are functions of probabilistic sensitivity factors. With optimum design parameters such as the mean and coefficient of variation (representing range of scatter) of uncertain variables, the most efficient and economical manufacturing procedure can be selected. In this paper, optimum values of the requisite design parameters for a predetermined cost due to failure occurrence are computationally determined. The results for the fuselage panel analysis show that the higher the cost due to failure occurrence, the smaller the optimum coefficient of variation of fiber modulus (design parameter) in longitudinal direction.

Shiao, Michael C.↗
NASA NTRS1999

Continuum Modeling of the Dynamics of Externally Injection-locked Coupled Oscillator Arrays

Mutually injection-locked arrays of electronic oscillators provide a novel means of controlling the aperture phase of a phased-array antenna, thus achieving the advantages of spatial power combining while retaining the ability to steer the radiated beam. In a number of design concepts, one or more of the oscillators are injection locked to a signal from an external master-oscillator. The behavior of such a system has been analyzed by numerical solution of a system of nonlinear differential equations which, due to its complexity, yields limited insight into the relationship between the injection signals and the aperture phase. In this paper, we develop a continuum model, which results in a single partial differential equation for the aperture phase as a function of time. Solution of the equation is effected by means of the Laplace transformation and yields detailed information concerning the dynamics of the array under the influence of the external injection signals.

injection locked↗
NASA NTRS2014

Static Aeroelastic Analysis with an Inviscid Cartesian Method

An embedded-boundary, Cartesian-mesh flow solver is coupled with a three degree-of-freedom structural model to perform static, aeroelastic analysis of complex aircraft geometries. The approach solves a nonlinear, aerostructural system of equations using a loosely-coupled strategy. An open-source, 3-D discrete-geometry engine is utilized to deform a triangulated surface geometry according to the shape predicted by the structural model under the computed aerodynamic loads. The deformation scheme is capable of modeling large deflections and is applicable to the design of modern, very-flexible transport wings. The coupling interface is modular so that aerodynamic or structural analysis methods can be easily swapped or enhanced. After verifying the structural model with comparisons to Euler beam theory, two applications of the analysis method are presented as validation. The first is a relatively stiff, transport wing model which was a subject of a recent workshop on aeroelasticity. The second is a very flexible model recently tested in a low speed wind tunnel. Both cases show that the aeroelastic analysis method produces results in excellent agreement with experimental data.

Aeroelastic↗
NASA NTRS1982

Improved Numerical Differencing Analyzer

SINDA, Systems Improved Numerical Differencing Analyzer, solves differential and algebric equations representing physical systems. SINDA solves numerically almost any set of ordinary differential equations that represent transient behavior of a lumped-parameter system or any set of nonlinear algebraic equations that represents the steady state conditions of a physical system.

Skladany, J. T.↗
NASA NTRS1991

Integrated analysis and design for controlled nonlinear multibody systems

A hybrid approach for integrated analysis and design of nonlinear multibody systems is proposed. Based on the approach, a general-purpose design tool is developed for nonlinear dynamic systems subjected to nonlinear design constraints. For analysis purposes, second-order nonlinear equations of motion of the system are automatically generated using general-purpose multibody formulations. Once the equations are solved, they are written in first-order form to take advantage of the first-order formulations of design sensitivity analysis for dynamic systems. A nonlinear programming technique is then used to optimize the nonlinear systems, such that a nonlinear cost function is minimized and performance constraints are satisfied. The approach proved to be very general and useful for design and analysis of controlled multibody systems. The approach is applied to the design of passive dynamic controllers for slewing control of multibody systems.

Wu, Shih-Chin↗
DOE OSTI2025

Bayesian learning with Gaussian processes for low-dimensional representations of time-dependent nonlinear systems

This work presents a data-driven method for learning low-dimensional time-dependent physics-based surrogate models whose predictions are endowed with uncertainty estimates. We use the operator inference approach to model reduction that poses the problem of learning low-dimensional model terms as a regression of state space data and corresponding time derivatives by minimizing the residual of reduced system equations. Standard operator inference models perform well with accurate training data that are dense in time, but producing stable and accurate models when the state data are noisy and/or sparse in time remains a challenge. Another challenge is the lack of uncertainty estimation for the predictions from the operator inference models. Our approach addresses these challenges by incorporating Gaussian process surrogates into the operator inference framework to (1) probabilistically describe uncertainties in the state predictions and (2) procure analytical time derivative estimates with quantified uncertainties. The formulation leads to a generalized least-squares regression and, ultimately, reduced-order models that are described probabilistically with a closed-form expression for the posterior distribution of the operators. The resulting probabilistic surrogate model propagates uncertainties from the observed state data to reduced-order predictions. Furthermore, we demonstrate the method is effective for constructing low-dimensional models of two nonlinear partial differential equations representing a compressible flow and a nonlinear diffusion–reaction process, as well as for estimating the parameters of a low-dimensional system of nonlinear ordinary differential equations representing compartmental models in epidemiology.

Data-driven model reduction↗
DOE OSTI2020

A model for discrete fracture-clay rock interaction incorporating electrostatic effects on transport

Abstract A model based on the code CrunchClay is presented for a fracture-clay matrix system that takes electrostatic effects on transport into account. The electrostatic effects on transport include those associated with the development of a diffusion potential as captured by the Nernst-Planck equation, and the formation of a diffuse layer bordering negatively charged clay particles within which partial anion exclusion occurs. The model is based on a dual continuum formulation that accounts for diffuse layer and bulk water pore space, providing a more flexible framework than is found in the classical mean electrostatic potential models. The diffuse layer model is obtained by volume averaging ion concentrations in the Poisson-Boltzmann equation, but also includes the treatment of longitudinal transport within this continuum. The calculation of transport within the bulk and diffuse layer porosity is based on a new formulation for the Nernst-Planck equation that considers averaging of diffusion coefficients and accumulation factors at grid cell interfaces. Equations for function residuals and the associated Jacobian matrix are presented such that the system of nonlinear differential-algebraic equations can be solved with Newton’s method. As an example, we consider a 2D system with a single discrete fracture within which flow and advective transport occurs that is coupled to diffusion in the clay-rich matrix. The simulation results demonstrate the lack of retardation for anions (e.g., 36 Cl − ) of the contaminant plume within the fracture flow system because they are largely excluded from the charged clay rock, while the migration of cations (e.g., 90 Sr ++ ) is more strongly attenuated. The diffusive loss of divalent cations in particular from the fracture is accentuated by their accumulation in the diffuse layer within the clay-rich matrix.

58 GEOSCIENCES↗