Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 109 records · Page 6
All sourcesNASA NTRSDOE OSTI
NASA NTRS1990

Kinetic effects on Alfven wave nonlinearity. II - The modified nonlinear wave equation

A previously developed Vlasov theory is used here to study the role of resonant particle and other kinetic effects on Alfven wave nonlinearity. A hybrid fluid-Vlasov equation approach is used to obtain a modified version of the derivative nonlinear Schroedinger equation. The differences between a scalar model for the plasma pressure and a tensor model are discussed. The susceptibilty of the modified nonlinear wave equation to modulational instability is studied. The modulational instability normally associated with the derivative nonlinear Schroedinger equation will, under most circumstances, be restricted to left circularly polarized waves. The nonlocal term in the modified nonlinear wave equation engenders a new modulational instability that is independent of beta and the sense of circular polarization. This new instability may explain the occurrence of wave packet steepening for all values of the plasma beta in the vicinity of the earth's bow shock.

Spangler, Steven R.↗
NASA NTRS1989

Assessment of numerical methods for the solution of fluid dynamics equations for nonlinear resonance systems

The capability of accurate nonlinear flow analysis of resonance systems is essential in many problems, including combustion instability. Classical numerical schemes are either too diffusive or too dispersive especially for transient problems. In the last few years, significant progress has been made in the numerical methods for flows with shocks. The objective was to assess advanced shock capturing schemes on transient flows. Several numerical schemes were tested including TVD, MUSCL, ENO, FCT, and Riemann Solver Godunov type schemes. A systematic assessment was performed on scalar transport, Burgers' and gas dynamic problems. Several shock capturing schemes are compared on fast transient resonant pipe flow problems. A system of 1-D nonlinear hyperbolic gas dynamics equations is solved to predict propagation of finite amplitude waves, the wave steepening, formation, propagation, and reflection of shocks for several hundred wave cycles. It is shown that high accuracy schemes can be used for direct, exact nonlinear analysis of combustion instability problems, preserving high harmonic energy content for long periods of time.

Przekwas, A. J.↗
NASA NTRS1989

An iterative method for systems of nonlinear hyperbolic equations

An iterative algorithm for the efficient solution of systems of nonlinear hyperbolic equations is presented. Parallelism is evident at several levels. In the formation of the iteration, the equations are decoupled, thereby providing large grain parallelism. Parallelism may also be exploited within the solves for each equation. Convergence of the interation is established via a bounding function argument. Experimental results in two-dimensions are presented.

Scroggs, Jeffrey S.↗
NASA NTRS1976

An improved method for the prediction of completely three-dimensional aerodynamic load distributions of configurations with leading edge vortex separation

The application of a higher-order subsonic potential flow panel method to the solution of three-dimensional flow about wing and wing-body combinations with leading-edge vortex separation is presented. The governing equations are the linear flow differential equation and nonlinear boundary conditions which require that the flow be parallel to the wing and body surfaces and that the free vortex sheet, springing from the leading and trailing edges, be aligned with the local flow and support no pressure jump. The vortex core is modeled as a simple line vortex which receives vorticity from the free sheet through a connecting sheet. The Kutta condition is imposed on all appropriate edges of the wing. This set of nonlinear equations is solved by an iterative procedure. The Goethert rule accounts for compressibility. The method has been programmed for the CDC 6600. Delta wings, gothic wings, arrow wings, cambered wings, and wing with body have been analyzed. Initial studies involving variations of panel density, vortex sheet sizing, Jacobian update, and initial geometry demonstrate that the present method generally exhibits good convergence characteristics.

Rubbert, p. E.↗
DOE OSTI2024

Collocation methods for nonlinear differential equations on low-rank manifolds

We introduce new methods for integrating nonlinear differential equations on low-rank manifolds. These methods rely on interpolatory projections onto the tangent space, enabling low-rank time integration of vector fields that can be evaluated entry-wise. A key advantage of our approach is that it does not require the vector field to exhibit low-rank structure, thereby overcoming significant limitations of traditional dynamical low-rank methods based on orthogonal projection. To construct the interpolatory projectors, we develop a sparse tensor sampling algorithm based on the discrete empirical interpolation method (DEIM) that parameterizes tensor train manifolds and their tangent spaces with cross interpolation. Using these projectors, we propose two time integration schemes on low-rank tensor train manifolds. The first scheme integrates the solution at selected interpolation indices and constructs the solution with cross interpolation. The second scheme generalizes the well-known orthogonal projector-splitting integrator to interpolatory projectors. We demonstrate the proposed methods with applications to several tensor differential equations arising from the discretization of partial differential equations.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1980

Study of solution procedures for nonlinear structural equations

A method for the redution of the cost of solution of large nonlinear structural equations was developed. Verification was made using the MARC-STRUC structure finite element program with test cases involving single and multiple degrees of freedom for static geometric nonlinearities. The method developed was designed to exist within the envelope of accuracy and convergence characteristic of the particular finite element methodology used.

Young, C. T., II↗
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 NTRS1978

Numerical method for solution of systems of non-stationary spatially one-dimensional nonlinear differential equations

A computational scheme and a standard program is proposed for solving systems of nonstationary spatially one-dimensional nonlinear differential equations using Newton's method. The proposed scheme is universal in its applicability and its reduces to a minimum the work of programming. The program is written in the FORTRAN language and can be used without change on electronic computers of type YeS and BESM-6. The standard program described permits the identification of nonstationary (or stationary) solutions to systems of spatially one-dimensional nonlinear (or linear) partial differential equations. The proposed method may be used to solve a series of geophysical problems which take chemical reactions, diffusion, and heat conductivity into account, to evaluate nonstationary thermal fields in two-dimensional structures when in one of the geometrical directions it can take a small number of discrete levels, and to solve problems in nonstationary gas dynamics.

Morozov, S. K.↗
NASA NTRS2012

High Energy Laser Beam Propagation in the Atmosphere: The Integral Invariants of the Nonlinear Parabolic Equation and the Method of Moments

The method of moments is used to define and derive expressions for laser beam deflection and beam radius broadening for high-energy propagation through the Earth s atmosphere. These expressions are augmented with the integral invariants of the corresponding nonlinear parabolic equation that describes the electric field of high-energy laser beam to propagation to yield universal equations for the aforementioned quantities; the beam deflection is a linear function of the propagation distance whereas the beam broadening is a quadratic function of distance. The coefficients of these expressions are then derived from a thin screen approximation solution of the nonlinear parabolic equation to give corresponding analytical expressions for a target located outside the Earth s atmospheric layer. These equations, which are graphically presented for a host of propagation scenarios, as well as the thin screen model, are easily amenable to the phase expansions of the wave front for the specification and design of adaptive optics algorithms to correct for the inherent phase aberrations. This work finds application in, for example, the analysis of beamed energy propulsion for space-based vehicles.

Manning, Robert M.↗
NASA NTRS1990

A parallel iterative solution method for systems of nonlinear hyperbolic equations

An iterative algorithm suitable for the solution of a system of nonlinear hyperbolic partial differentiation equations in multiple dimensions is discussed. Current numerical methods for systems of nonlinear PDEs have limited parallelism due to strong coupling between the equations. This method decouples the PDEs by linearizing the convention coefficient for a space-time domain. This provides large grain parallelism. The linearization also allows the treatment of some terms in the equations as source terms, providing more freedom to choose from a wider variety of numerical methods. Smaller grain parallelism may be exploited within the solves for each equation. Thus, the method has potential for parallelism at several levels.

Scroggs, Jeffrey S.↗
NASA NTRS1989

A method for exponential propagation of large systems of stiff nonlinear differential equations

A new time integrator for large, stiff systems of linear and nonlinear coupled differential equations is described. For linear systems, the method consists of forming a small (5-15-term) Krylov space using the Jacobian of the system and carrying out exact exponential propagation within this space. Nonlinear corrections are incorporated via a convolution integral formalism; the integral is evaluated via approximate Krylov methods as well. Gains in efficiency ranging from factors of 2 to 30 are demonstrated for several test problems as compared to a forward Euler scheme and to the integration package LSODE.

Friesner, Richard A.↗
NASA NTRS1989

The Jeffcott equations in nonlinear rotordynamics

The solutions of the Jeffcott equations describing the behavior of a rotating shaft are investigated analytically, with a focus on the case where deadband is taken into account. Bounds on the solutions are obtained from those for the linearized equations, and the onset of destructive vibrations is predicted by analyzing the Fourier transforms of the solutions; good agreement with numerical solutions and power-spectrum density plots is demonstrated. It is suggested that the present analytical approach could be applied to determine cryogenic-pump stability margins in flight and hot-fire ground testing of launch vehicles such as the Space Shuttle.

Zalik, R. A.↗
NASA NTRS1977

Nonlinear dynamic response of wind turbine rotors

The nonlinear equations of motion for a rigid rotor restrained by three flexible springs representing the flapping, lagging and feathering motions are derived using Lagrange's equations for arbitrary angular rotations. These are reduced to a consistent set of nonlinear equations using nonlinear terms up to third order.

Chopra, I.↗