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NASA NTRS1992

Approximate Solutions Of Equations Of Steady Diffusion

Rigorous analysis yields reliable criteria for "best-fit" functions. Improved "curve-fitting" method yields approximate solutions to differential equations of steady-state diffusion. Method applies to problems in which rates of diffusion depend linearly or nonlinearly on concentrations of diffusants, approximate solutions analytic or numerical, and boundary conditions of Dirichlet type, of Neumann type, or mixture of both types. Applied to equations for diffusion of charge carriers in semiconductors in which mobilities and lifetimes of charge carriers depend on concentrations.

Edmonds, Larry D.↗

NASA NTRS1992

Lightning-Sensor Data Help In Understanding Thunderstorms

NASA technical memorandum discusses research on use of data from network of ground-based magnetic direction-finding ground-strike lightning sensors to diagnose and predict occurrence and evolution of thunderstorms. Purposes of study to explore applicability and limitations of extrapolation techniques used to generate forecasts from data; to examine physically-based, nonlinear mathematical models for applicability to lightning-forecast problem; and to determine valid extrapolation ranges of such models for various weather scenarios.

Goodman, Steven J.↗

NASA NTRS1993

Advanced launch system trajectory optimization using suboptimal control

The maximum-final mass trajectory of a proposed configuration of the Advanced Launch System is presented. A model for the two-stage rocket is given; the optimal control problem is formulated as a parameter optimization problem; and the optimal trajectory is computed using a nonlinear programming code called VF02AD. Numerical results are presented for the controls (angle of attack and velocity roll angle) and the states. After the initial rotation, the angle of attack goes to a positive value to keep the trajectory as high as possible, returns to near zero to pass through the transonic regime and satisfy the dynamic pressure constraint, returns to a positive value to keep the trajectory high and to take advantage of minimum drag at positive angle of attack due to aerodynamic shading of the booster, and then rolls off to negative values to satisfy the constraints. Because the engines cannot be throttled, the maximum dynamic pressure occurs at a single point; there is no maximum dynamic pressure subarc. To test approximations for obtaining analytical solutions for guidance, two additional optimal trajectories are computed: one using untrimmed aerodynamics and one using no atmospheric effects except for the dynamic pressure constraint. It is concluded that untrimmed aerodynamics has a negligible effect on the optimal trajectory and that approximate optimal controls should be able to be obtained by treating atmospheric effects as perturbations.

Shaver, Douglas A.↗

NASA NTRS1995

Abort performance for a winged-body single-stage to orbit vehicle

Optimal control theory is employed to determine the performance of abort to orbit (ATO) and return to launch site (RTLS) maneuvers for a single-stage to orbit vehicle. The vehicle configuration examined is a seven engine, winged-body vehicle, that lifts-off vertically and lands horizontally. The abort maneuvers occur as the vehicle ascends to orbit and are initiated when the vehicle suffers an engine failure. The optimal control problems are numerically solved in discretized form via a nonlinear programming (NLP) algorithm. A description highlighting the attributes of this NLP method is provided. ATO maneuver results show that the vehicle is capable of ascending to orbit with a single engine failure at lift-off. Two engine out ATO maneuvers are not possible from the launch pad, but are possible after launch when the thrust to weight ratio becomes sufficiently large. Results show that single engine out RTLS maneuvers can be made for up to 180 seconds after lift-off and that there are scenarios for which RTLS maneuvers should be performed instead of ATP maneuvers.

Lyon, Jeffery A.↗

NASA NTRS1994

Spectral Methods for Computational Fluid Dynamics

As a tool for large-scale computations in fluid dynamics, spectral methods were prophesized in 1944, born in 1954, virtually buried in the mid-1960's, resurrected in 1969, evangalized in the 1970's, and catholicized in the 1980's. The use of spectral methods for meteorological problems was proposed by Blinova in 1944 and the first numerical computations were conducted by Silberman (1954). By the early 1960's computers had achieved sufficient power to permit calculations with hundreds of degrees of freedom. For problems of this size the traditional way of computing the nonlinear terms in spectral methods was expensive compared with finite-difference methods. Consequently, spectral methods fell out of favor. The expense of computing nonlinear terms remained a severe drawback until Orszag (1969) and Eliasen, Machenauer, and Rasmussen (1970) developed the transform methods that still form the backbone of many large-scale spectral computations. The original proselytes of spectral methods were meteorologists involved in global weather modeling and fluid dynamicists investigating isotropic turbulence. The converts who were inspired by the successes of these pioneers remained, for the most part, confined to these and closely related fields throughout the 1970's. During that decade spectral methods appeared to be well-suited only for problems governed by ordinary diSerential eqllations or by partial differential equations with periodic boundary conditions. And, of course, the solution itself needed to be smooth. Some of the obstacles to wider application of spectral methods were: (1) poor resolution of discontinuous solutions; (2) inefficient implementation of implicit methods; and (3) drastic geometric constraints. All of these barriers have undergone some erosion during the 1980's, particularly the latter two. As a result, the applicability and appeal of spectral methods for computational fluid dynamics has broadened considerably. The motivation for the use of spectral methods in numerical calculations stems from the attractive approximation properties of orthogonal polynomial expansions.

Zang, T. A.↗

NASA NTRS1997

A Comparison of PETSC Library and HPF Implementations of an Archetypal PDE Computation

Two paradigms for distributed-memory parallel computation that free the application programmer from the details of message passing are compared for an archetypal structured scientific computation a nonlinear, structured-grid partial differential equation boundary value problem using the same algorithm on the same hardware. Both paradigms, parallel libraries represented by Argonne's PETSC, and parallel languages represented by the Portland Group's HPF, are found to be easy to use for this problem class, and both are reasonably effective in exploiting concurrency after a short learning curve. The level of involvement required by the application programmer under either paradigm includes specification of the data partitioning (corresponding to a geometrically simple decomposition of the domain of the PDE). Programming in SPAM style for the PETSC library requires writing the routines that discretize the PDE and its Jacobian, managing subdomain-to-processor mappings (affine global- to-local index mappings), and interfacing to library solver routines. Programming for HPF requires a complete sequential implementation of the same algorithm, introducing concurrency through subdomain blocking (an effort similar to the index mapping), and modest experimentation with rewriting loops to elucidate to the compiler the latent concurrency. Correctness and scalability are cross-validated on up to 32 nodes of an IBM SP2.

Hayder, M. Ehtesham↗

NASA NTRS2000

Motion Cueing Algorithm Development: Initial Investigation and Redesign of the Algorithms

In this project four motion cueing algorithms were initially investigated. The classical algorithm generated results with large distortion and delay and low magnitude. The NASA adaptive algorithm proved to be well tuned with satisfactory performance, while the UTIAS adaptive algorithm produced less desirable results. Modifications were made to the adaptive algorithms to reduce the magnitude of undesirable spikes. The optimal algorithm was found to have the potential for improved performance with further redesign. The center of simulator rotation was redefined. More terms were added to the cost function to enable more tuning flexibility. A new design approach using a Fortran/Matlab/Simulink setup was employed. A new semicircular canals model was incorporated in the algorithm. With these changes results show the optimal algorithm has some advantages over the NASA adaptive algorithm. Two general problems observed in the initial investigation required solutions. A nonlinear gain algorithm was developed that scales the aircraft inputs by a third-order polynomial, maximizing the motion cues while remaining within the operational limits of the motion system. A braking algorithm was developed to bring the simulator to a full stop at its motion limit and later release the brake to follow the cueing algorithm output.

Telban, Robert J.↗

NASA NTRS2002

Computational AeroAcoustics for Fan Noise Prediction

An overview of the current state-of-the-art in computational aeroacoustics as applied to fan noise prediction at NASA Glenn is presented. Results from recent modeling efforts using three dimensional inviscid formulations in both frequency and time domains are summarized. In particular, the application of a frequency domain method, called LINFLUX, to the computation of rotor-stator interaction tone noise is reviewed and the influence of the background inviscid flow on the acoustic results is analyzed. It has been shown that the noise levels are very sensitive to the gradients of the mean flow near the surface and that the correct computation of these gradients for highly loaded airfoils is especially problematic using an inviscid formulation. The ongoing development of a finite difference time marching code that is based on a sixth order compact scheme is also reviewed. Preliminary results from the nonlinear computation of a gust-airfoil interaction model problem demonstrate the fidelity and accuracy of this approach. Spatial and temporal features of the code as well as its multi-block nature are discussed. Finally, latest results from an ongoing effort in the area of arbitrarily high order methods are reviewed and technical challenges associated with implementing correct high order boundary conditions are discussed and possible strategies for addressing these challenges ore outlined.

Envia, Ed↗

NASA NTRS2006

Dynamic Theory of Relativistic Electrons Stochastic Heating by Whistler Mode Waves with Application to the Earth Magnetosphere

In the Hamiltonian approach an electron motion in a coherent packet of the whistler mode waves propagating along the direction of an ambient magnetic field is studied. The physical processes by which these particles are accelerated to high energy are established. Equations governing a particle motion by group symmetries of the problem were transformed in to a closed pair of nonlinear difference equations. The solutions of these equations have shown there exists the energetic threshold below that the electron motion is regular, and when the initial energy is above the threshold an electron moves stochastically. It is proved that the upper boundary of particle stochastic heating is conditioned by intrinsic property of the particle chaotic motion. Particle energy spectra and pitch angle electron scattering are described by the Fokker-Planck-Kolmogorov equations. It is shown that significant pitch angle diffusion occurs for the Earth radiation belt electrons with energies from a few keV up to a few MeV.

Khazanov, G. V.↗

NASA NTRS2006

A robust model predictive control algorithm for uncertain nonlinear systems that guarantees resolvability

A robustly stabilizing MPC (model predictive control) algorithm for uncertain nonlinear systems is developed that guarantees resolvability. With resolvability, initial feasibility of the finite-horizon optimal control problem implies future feasibility in a receding-horizon framework. The control consists of two components; (i) feed-forward, and (ii) feedback part. Feed-forward control is obtained by online solution of a finite-horizon optimal control problem for the nominal system dynamics. The feedback control policy is designed off-line based on a bound on the uncertainty in the system model. The entire controller is shown to be robustly stabilizing with a region of attraction composed of initial states for which the finite-horizon optimal control problem is feasible. The controller design for this algorithm is demonstrated on a class of systems with uncertain nonlinear terms that have norm-bounded derivatives and derivatives in polytopes. An illustrative numerical example is also provided.

feed - forward↗

NASA NTRS2014

Spacecraft Stabilization and Control for Capture of Non-Cooperative Space Objects

This paper addresses stabilization and control issues in autonomous capture and manipulation of non-cooperative space objects such as asteroids, space debris, and orbital spacecraft in need of servicing. Such objects are characterized by unknown mass-inertia properties, unknown rotational motion, and irregular shapes, which makes it a challenging control problem. The problem is further compounded by the presence of inherent nonlinearities, signi cant elastic modes with low damping, and parameter uncertainties in the spacecraft. Robust dissipativity-based control laws are presented and are shown to provide global asymptotic stability in spite of model uncertainties and nonlinearities. It is shown that robust stabilization can be accomplished via model-independent dissipativity-based controllers using thrusters alone, while stabilization with attitude and position control can be accomplished using thrusters and torque actuators.

Joshi, Suresh↗

NASA NTRS2017

Improving Magnetosphere In Situ Observations Using Solar Sails

Past and current magnetosphere missions employ conventional spacecraft formations for in situ observations of the geomagnetic tail. Conventional spacecraft flying in inertially fixed Keplerian orbits are only aligned with the geomagnetic tail once per year, since the geomagnetic tail is always aligned with the Earth-Sun line, and therefore, rotates annually. Solar sails are able to artificially create sunsynchronous orbits such that the orbit apse line remains aligned with the geomagnetic tail line throughout the entire year. This continuous presence in the geomagnetic tail can significantly increase the science phase for magnetosphere missions. In this paper, the problem of solar sail formation design is explored using nonlinear programming to design optimal two-craft, triangle, and tetrahedron solar sail formations, in terms of formation quality and formation stability. The designed formations are directly compared to the formations used in NASA's Magnetospheric Multi-Scale mission.

Sails↗

NASA NTRSDate not supplied

Near Rectilinear Halo Orbit Determination with Simulated DSN Observations

This paper presents the results of a high-fidelity simulation of spacecraft orbit determination in a near rectilinear halo orbit (NRHO). Others in the literature have examined this problem with linear covariance analysis, but the highly-nonlinear dynamics of this orbit challenge the assumptions underlying such analyses. The present work builds on similar analysis performed by other authors to contribute a fuller understanding of the operational requirements for NRHO navigation. The present work serves as a check to the assumptions of previous studies and an independent verification of those results. The results from the literature are extended by quantifying the space of orbital states from which a spacecraft with given control authority can safely return to the nominal path. Spacecraft state uncertainty estimates are evaluated as a function of time. Simulated range and range-rate measurements with the Deep Space Network (DSN) ground stations are used to model orbit determination accuracy. Orbit maintenance maneuvers are performed using both short-horizon and long-horizon stationkeeping targeting. Monte Carlo analysis of orbit determination and stationkeeping is performed. This paper quantifies the achievable state uncertainty with deep space network (DSN)-only range and range-rate observations. This paper also addresses requirements on the frequency of DSN observation periods and correlates ground contact frequency with navigation accuracy. The results of several related studies are presented and discussed: the effect of missing ground station passes, the effect of missing stationkeeping maneuvers, the sensitivity of the spacecraft state estimate to realistic error sources, and stationkeeping propellant budget.

Nathan L Parrish↗

NASA NTRSDate not supplied

Linearized Frequency-Domain Gust Analysis and Adjoint-Based Sensitivities

Gust analysis is added to a linearized frequency-domain method in FUN3D, a NASA computational fluid dynamics solver. The method linearizes about a nonlinear static equilibrium condition and is therefore appropriate for problems with small perturbations such as transonic stochastic gust analysis. In addition to the gust analysis, adjoint-based sensitivities of stochastic gust constraints are implemented for multidisciplinary design optimization. The linearized frequency-domain gust model and adjoint-based sensitivities are described and verified. The method is applied to an optimization for mass minimization of the AGARD 445.6 wing subject to a stochastic gust constraint limiting the displacement of the wing tip.

Aeroelasticity↗

NASA NTRS2005

Mathematical Metaphors: Problem Reformulation and Analysis Strategies

This paper addresses the critical need for the development of intelligent or assisting software tools for the scientist who is working in the initial problem formulation and mathematical model representation stage of research. In particular, examples of that representation in fluid dynamics and instability theory are discussed. The creation of a mathematical model that is ready for application of certain solution strategies requires extensive symbolic manipulation of the original mathematical model. These manipulations can be as simple as term reordering or as complicated as discovery of various symmetry groups embodied in the equations, whereby Backlund-type transformations create new determining equations and integrability conditions or create differential Grobner bases that are then solved in place of the original nonlinear PDEs. Several examples are presented of the kinds of problem formulations and transforms that can be frequently encountered in model representation for fluids problems. The capability of intelligently automating these types of transforms, available prior to actual mathematical solution, is advocated. Physical meaning and assumption-understanding can then be propagated through the mathematical transformations, allowing for explicit strategy development.

Thompson, David E.↗

DOE OSTI2020

On differentiable local bounds preserving stabilization for Euler equations

This work presents the design of nonlinear stabilization techniques for the finite element discretization of Euler equations in both steady and transient form. Implicit time integration is used in the case of the transient form. A differentiable local bounds preserving method has been developed, which combines a Rusanov artificial diffusion operator and a differentiable shock detector. Nonlinear stabilization schemes are usually stiff and highly nonlinear. This issue is mitigated by the differentiability properties of the proposed method. Moreover, in order to further improve the nonlinear convergence, we also propose a continuation method for a subset of the stabilization parameters. The resulting method has been successfully applied to steady and transient problems with complex shock patterns. Numerical experiments show that it is able to provide sharp and well resolved shocks. Furthermore, the importance of the differentiability is assessed by comparing the new scheme with its non-differentiable counterpart. Numerical experiments suggest that, for up to moderate nonlinear tolerances, the method exhibits improved robustness and nonlinear convergence behavior for steady problems. Additionally, in the case of transient problem, we also observe a reduction in the computational cost.

42 ENGINEERING↗

NASA NTRS1974

Theoretical investigation of nonlinear three-dimensional instabilities in liquid rockets with real nozzles

An approximate analytical technique has been developed for the solution of nonlinear three-dimensional, transverse and axial combustion-instability problems that are frequently observed in liquid-propellant rocket motors. This theory is an extension and generalization of previous analyses, which could analyze either transverse or axial instabilities in liquid combustors with quasi-steady nozzles, to the practical situations of three-dimensional instabilities in combustors with conventional DeLaval nozzles. Unlike the quasi-steady nozzle, the presence of a conventional nozzle imposes restrictions upon the behavior of both the amplitudes and phases of the oscillations at the nozzle entrance plane. The applicability of the solution technique, which is based on the Galerkin method, is demonstrated by analyzing the nonlinear stability of a cylindrical liquid-rocket combustor with uniform injection of propellants at one end and a conventional nozzle at the other end. Crocco's pressure-sensitive time-lag model is used to describe the unsteady combustion process.

Powell, E. A.↗

NASA NTRS1987

Substructure analysis using NICE/SPAR and applications of force to linear and nonlinear structures

Parallel computing studies are presented for a variety of structural analysis problems. Included are the substructure planar analysis of rectangular panels with and without a hole, the static analysis of space mast, using NICE/SPAR and FORCE, and substructure analysis of plane rigid-jointed frames using FORCE. The computations are carried out on the Flex/32 MultiComputer using one to eighteen processors. The NICE/SPAR runstream samples are documented for the panel problem. For the substructure analysis of plane frames, a computer program is developed to demonstrate the effectiveness of a substructuring technique when FORCE is enforced. Ongoing research activities for an elasto-plastic stability analysis problem using FORCE, and stability analysis of the focus problem using NICE/SPAR are briefly summarized. Speedup curves for the panel, the mast, and the frame problems provide a basic understanding of the effectiveness of parallel computing procedures utilized or developed, within the domain of the parameters considered. Although the speedup curves obtained exhibit various levels of computational efficiency, they clearly demonstrate the excellent promise which parallel computing holds for the structural analysis problem. Source code is given for the elasto-plastic stability problem and the FORCE program.

Razzaq, Zia↗