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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 NTRS1992

Estimation of unknown variable parameters in moving boundary problems

The problem of estimating unknown variable parameters appearing in moving boundary problems is considered; these are specifically nonlinear diffusion equations defined on a moving spatial domain. A spline-based approximation method that results in a sequence of computationally tractable approximate parameter estimation problems has been developed. A convergence result is proved for a certain class of these moving boundary problems. The paper is concluded with a set of representative numerical examples.

Murphy, K. A.↗

NASA NTRS1992

Stable localized patterns in thin liquid films

A two-dimensional nonlinear evolution equation is studied which describes the three-dimensional spatiotemporal behavior of the air-liquid interface of a thin liquid film lying on the underside of a cooled horizontal plate. It is shown that the equation has a Liapunov functional, and this fact is exploited to demonstrate that the Marangoni effect can stabilize the destabilizing effect of gravity (the Rayleigh-Taylor instability), allowing for the existence of stable localized axisymmetric solutions for a wide range of parameter values. Various properties of these structures are discussed.

Deissler, Robert J.↗

NASA NTRS1990

Rotordynamics analysis of a Jeffcott model with deadband

A method is developed for determining the stability margins of a simple Jeffcott model with deadband via analysis of the discrete Fourier transform of the system response. The model in question is of a uniform, unbalanced, flexible shaft that is supported by a bearing as it rotates about its x axis. This model is represented by a system of coupled nonlinear differential equations.

Zalik, R. A.↗

NASA NTRS1992

On options for interdisciplinary analysis and design optimization

The interdisciplinary optimization of engineering systems is discussed from the standpoint of the computational alternatives available to the designer. The analysis of such systems typically requires the solution of coupled systems of nonlinear algebraic equations. The solution procedure is necessarily iterative in nature. It is shown that the system can be solved by fixed point iteration, by Newton's method, or by a combination of the two. However, the need for sensitivity analysis may affect the choice of analysis solution method. Similarly, the optimization of the system can be formulated in several ways that are discussed in the paper. It is shown that the effect of the topology of the interaction between disciplines is a key factor in the choice of analysis, sensitivity and optimization methods. Several examples are presented to illustrate the discussion.

Haftka, R. T.↗

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 NTRS1992

A hybrid approach to near-optimal launch vehicle guidance

This paper evaluates a proposed hybrid analytical/numerical approach to launch-vehicle guidance for ascent to orbit injection. The feedback-guidance approach is based on a piecewise nearly analytic zero-order solution evaluated using a collocation method. The zero-order solution is then improved through a regular perturbation analysis, wherein the neglected dynamics are corrected in the first-order term. For real-time implementation, the guidance approach requires solving a set of small dimension nonlinear algebraic equations and performing quadrature. Assessment of performance and reliability are carried out through closed-loop simulation for a vertically launched 2-stage heavy-lift capacity vehicle to a low earth orbit. The solutions are compared with optimal solutions generated from a multiple shooting code. In the example the guidance approach delivers over 99.9 percent of optimal performance and terminal constraint accuracy.

Leung, Martin S. K.↗

NASA NTRS1993

Development of an hp-version finite element method for computational optimal control

The purpose of this research effort was to begin the study of the application of hp-version finite elements to the numerical solution of optimal control problems. Under NAG-939, the hybrid MACSYMA/FORTRAN code GENCODE was developed which utilized h-version finite elements to successfully approximate solutions to a wide class of optimal control problems. In that code the means for improvement of the solution was the refinement of the time-discretization mesh. With the extension to hp-version finite elements, the degrees of freedom include both nodal values and extra interior values associated with the unknown states, co-states, and controls, the number of which depends on the order of the shape functions in each element. One possible drawback is the increased computational effort within each element required in implementing hp-version finite elements. We are trying to determine whether this computational effort is sufficiently offset by the reduction in the number of time elements used and improved Newton-Raphson convergence so as to be useful in solving optimal control problems in real time. Because certain of the element interior unknowns can be eliminated at the element level by solving a small set of nonlinear algebraic equations in which the nodal values are taken as given, the scheme may turn out to be especially powerful in a parallel computing environment. A different processor could be assigned to each element. The number of processors, strictly speaking, is not required to be any larger than the number of sub-regions which are free of discontinuities of any kind.

Hodges, Dewey H.↗

NASA NTRS1992

A multiple scales approach to sound generation by vibrating bodies

The problem of determining the acoustic field in an inviscid, isentropic fluid generated by a solid body whose surface executes prescribed vibrations is formulated and solved as a multiple scales perturbation problem, using the Mach number M based on the maximum surface velocity as the perturbation parameter. Following the idea of multiple scales, new 'slow' spacial scales are introduced, which are defined as the usual physical spacial scale multiplied by powers of M. The governing nonlinear differential equations lead to a sequence of linear problems for the perturbation coefficient functions. However, it is shown that the higher order perturbation functions obtained in this manner will dominate the lower order solutions unless their dependence on the slow spacial scales is chosen in a certain manner. In particular, it is shown that the perturbation functions must satisfy an equation similar to Burgers' equation, with a slow spacial scale playing the role of the time-like variable. The method is illustrated by a simple one-dimenstional example, as well as by three different cases of a vibrating sphere. The results are compared with solutions obtained by purely numerical methods and some insights provided by the perturbation approach are discussed.

Geer, James F.↗

NASA NTRS1992

Two-dimensional viscous accretion disk models. I - On meridional circulations in radiative regions

It has been suggested that thin Keplerian disks, like rotating stars, cannot attain hydrostatic and thermal equilibrium simultaneously if the angular velocity is a function of radius only. To address the problem of vertical structure and stability of thin accretion disks, we performed 2D axisymmetric calculations by solving the set of fully nonlinear hydrodynamic equations, including radiation transport. The results indicate that the variation of the rotational velocity with height is too small to sustain hydrostatic equilibrium. Instead, the variation of the radial (inflow) velocity with height suffices to establish a stationary state. In particular, we find that for low values of the viscosity parameter alpha, there is mass outflow in the central parts of the disk close to the equatorial plane, and inflow only near the surface. Only for modestly high values of alpha, the flow throughout the disk is directed inward. Thus, the flow within the disk is the result of viscous radial inflow plus a circulatory flow directed outward in the midplane and inward near the surface.

Kley, W.↗

NASA NTRS1992

Characteristics of obliquely propagating, nonlinear Alfven waves

Consideration is given to properties of obliquely propagating (with respect to the large-scale magnetic field) nonlinear Alfven and fast magnetosonic waves as described by the derivative nonlinear Schroedinger equation. The focus is on those waves which in the parallel-propagating limit are circularly polarized, exact solutions to the MHD equations. Results are presented of a study of the changes in the structure of these waves as the propagation angle is increased. Attention is also given to the evolution of initial conditions which do not correspond to stationary wave propagations. It is found that, as the propagation angle with respect to the mean field increases, the functional forms of the wave magnetic field components in the two directions orthogonal to the propagation direction become increasingly dissimilar. The angle about the mean field within which stationary waves resemble parallel-propagating, circularly polarized waves becomes smaller as the wavelength increases. If the 'initial condition' wave train is close to parallel-propagating, then there is little subsequent evolution in its form.

Spangler, Steven R.↗

NASA NTRS1993

The evolution of strongly modulated, low-frequency, moderate-amplitude wave packets in a dispersive plasma

The evolution of strongly modulated wave packets in a dispersive plasma that propagate parallel to the magnetic field is studied. Modulation effects are shown to reduce significantly (about 30 percent) the rate of spreading from that due to dispersion alone. For fluidlike behavior, nonlinearity has its greatest impact on evolution when the linear sound speed and initial wave packet speeds are well matched, resulting in a strong coupling between the wave magnetic and sonic components. Ion kinetic processes reduce the impact of nonlinearity and cause the rate of spreading to approach that expected from dispersion alone as the ratio of ion and electron temperatures, Ti/Te approaches 4. For Beta equal to or greater than 1 and Ti/Te of about 1, the coupled waveforms correspond qualitatively to kinetic treatments of the derivative nonlinear Schroedinger equation.

Vasquez, Bernard J.↗

NASA NTRS1993

Helicopter trim analysis by shooting and finite element methods with optimally damped Newton iterations

Helicopter trim settings of periodic initial state and control inputs are investigated for convergence of Newton iteration in computing the settings sequentially and in parallel. The trim analysis uses a shooting method and a weak version of two temporal finite element methods with displacement formulation and with mixed formulation of displacements and momenta. These three methods broadly represent two main approaches of trim analysis: adaptation of initial-value and finite element boundary-value codes to periodic boundary conditions, particularly for unstable and marginally stable systems. In each method, both the sequential and in-parallel schemes are used, and the resulting nonlinear algebraic equations are solved by damped Newton iteration with an optimally selected damping parameter. The impact of damped Newton iteration, including earlier-observed divergence problems in trim analysis, is demonstrated by the maximum condition number of the Jacobian matrices of the iterative scheme and by virtual elimination of divergence. The advantages of the in-parallel scheme over the conventional sequential scheme are also demonstrated.

Achar, N. S.↗

NASA NTRS1992

The evolution of large-amplitude MHD waves near quasi-parallel shocks in the solar wind

We discuss attempts to explain the steepening of magnetohydrodynamic waves upstream of the Earth's bow shock and the formation of short-wavelength 'shocklets' in terms of simple analytic models. The effort involves use of the Derivative Nonlinear Schroedinger Equation (DNLS) to describe the wave evolution. We review shortcomings of previous attempts to model foreshock wave phenomena in terms of wave packet evolution according to the DNLS. It is pointed out that: (1) the oblique propagation of the waves and (2) their growth by unstable particle distribution are conducive to the processes of wave steepening and shocklet formation. Numerical solutions of the DNLS with initial conditions corresponding to envelope-modulated, obliquely propagating wave packets yield evolved wave packets with many properties similar to the observed waves.

Spangler, S. R.↗

NASA NTRS1993

Nonlinear vibrations of thin arbitrarily laminated composite plates subjected to harmonic excitations using DKT elements

A finite element formulation is presented for determining the large-amplitude free and steady-state forced vibration response of arbitrarily laminated anisotropic composite thin plates using the Discrete Kirchhoff Theory (DKT) triangular elements. The nonlinear stiffness and harmonic force matrices of an arbitrarily laminated composite triangular plate element are developed for nonlinear free and forced vibration analyses. The linearized updated-mode method with nonlinear time function approximation is employed for the solution of the system nonlinear eigenvalue equations. The amplitude-frequency relations for convergence with gridwork refinement, triangular plates, different boundary conditions, lamination angles, number of plies, and uniform versus concentrated loads are presented.

Chiang, C. K.↗

NASA NTRS1993

Two-dimensional viscous accretion disk models. II - On viscous overstability

Accretion disks have been shown to be susceptible to pulsational overstability when the viscosity coefficient increases upon compression. In this study, the stability of an axisymmetric accretion disk with vertical structure is considered. 2D axisymmetric calculations were performed by solving the set of fully nonlinear hydrodynamic equations, including radiation transport. The time evolution of a representative section (ring) of a thin radiative accretion disk is studied. The existence of the overstability in 2D is confirmed. It is found that the presence of periodic oscillations near the outer boundary of a real disk might account for some of the low-amplitude flickering observed in dwarf novae.

Kley, W.↗

NASA NTRS1992

Thermosolutal convection during dendritic solidification of a binary alloy

The FEM model of Heinrich (1988) is presently extended to the case of dendritic vertical solidification under steady-state conditions. An analysis is conducted of the nonlinear governing equations for linear stability, and FEM calculations are performed in cases where instability has been predicted for Pb-Sn melt systems; the results obtained show that the model is more stable than the plane front liquid-solid model, and shows similar, finger-like covection when unstable. It is found that convection in the 'mush' zone is confined to the top of that region; convective motion is driven by convection in the bulk fluid, in the absence of natural convection.

Heinrich, Juan C.↗

NASA NTRS1993

Dynamics of aging magnetic clouds

The dynamics of radially expanding magnetic clouds is rigorously analyzed within the framework of ideal MHD. The cloud is modelled as a cylindrically symmetric magnetic flux rope. In the force balance we include the gas pressure gradient and the Lorentz force. Interaction with the ambient solar wind due to expansion of the magnetic cloud is represented by a drag force proportional to the bulk velocity. We consider the self-similar expansion of a polytrope, and reduce the problem to an ordinary nonlinear differential equation for the evolution function. Analyzing the asymptotic behavior of the evolution function, we formulate theoretical expectations for the long-term behavior of cloud parameters. We focus on the temporal evolution of (1) the magnetic field strength; (2) the twist of the field lines; (3) the asymmetry of the total field profile; and (4) the bulk flow speed. We present data from two magnetic clouds observed at 1 AU and 2 AU, respectively, and find good agreement with theoretical expectations. For a peak magnetic field strength at 1 AU of 25 nT and a polytropic index of 0.5, we find that a magnetic cloud can be distinguished from the background interplanetary field up to a distance of about 5 AU. Taking larger magnetic fields and bigger polytropic indices this distance can double.

Osherovich, V. A.↗