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NASA NTRS1974

Contact and crack problems for an elastic wedge

The contact and the crack problems for an elastic wedge of arbitrary angle are considered. The problem is reduced to a singular integral equation which, in the general case, may have a generalized Cauchy kernel. The singularities under the stamp as well as at the wedge apex were studied, and the relevant stress intensity factors are defined. The problem was solved for various wedge geometries and loading conditions. The results may be applicable to certain foundation problems and to crack problems in symmetrically loaded wedges in which cracks initiate from the apex.

Erdogan, F.↗

NASA NTRS1983

Stability of streams and lakes on Mars

The freezing of small Martian streams is modelled for a variety of climatic conditions, on the supposition that the Martian atmosphere may have been considerably thicker in the past, at the time of the formation of the valley networks. This model examines the energy balance at the upper and lower surfaces of ice on streams, in order to determine the rate at which ice thickens with time. Results indicate that freezing rates are not strongly dependent on atmospheric pressure, and, under windy conditions, dependence on atmospheric pressure is even weaker. It is noted that the main problem in valley formation is in initiating the flow. Groundwater seepage alone is inadequate, due to the difficulty of groundwater system replenishment.

Carr, M. H.↗

NASA NTRS1997

Modelling the Centers of Galaxies

The key to studying central regions by means of nobody numerical experiments is to concentrate on the central few parsecs of a galaxy, replacing the remainder of the galaxy by a suitable boundary condition, rather after the manner in which stellar interiors can be studied without a detailed stellar atmosphere by replacing the atmosphere with a boundary condition. Replacements must be carefully designed because the long range gravitational force means that the core region is sensitive to mass outside that region and because particles can exchange between the outer galaxy and the core region. We use periodic boundary conditions, coupled with an iterative procedure to generate initial particle loads in isothermal equilibrium. Angular momentum conservation is ensured for problems including systematic rotation by a circular reflecting boundary and by integrating in a frame that rotates with the mean flow. Mass beyond the boundary contributes to the gravitational potential, but does not participate in the dynamics. A symplectic integration scheme has been developed for rotating coordinate systems. This combination works well, leading to robust configurations. Some preliminary results with this combination show that: (1) Rotating systems are extremely sensitive to non-axisymmetric external potentials, and (2) that a second core, orbiting near the main core (like the M31 second core system), shows extremely rapid orbital decay. The experimental setups will be discussed, along with preliminary results.

Smith, B. F.↗

NASA NTRS2005

Investigations of Spacecraft Orbits Around the L(2) Sun-Earth Libration Point

Space agencies are planning missions to the vicinity of the Sun-Earth L sub 2 point, some involving a distributed system of telescope spacecraft, configured in a plane about a hub. An improved understanding is developed of their relative motion. First, the telescope equations of motion are written relative to L sub 2 in the context of the classical circular restricted three-body problem, and expanded in terms of the distance from L sub 2. A basic examination is presented of the spacecraft configuration requirements and effects of small orbit insertion errors. Next, the telescope equations of motion relative to the hub are written and further expanded in terms of the hub-L sub 2 and hub-telescope distances. An analytical solution is developed and a halo telescope orbit investigated, with appropriate initial conditions. Then, the force model is extended to include perturbations to an accuracy of 10 to 20 m, east as additive contributions to the circular restricted problem. Perturbations include Earth's orbital eccentricity, lunar motion, solar radiation pressure, and small thrusting forces. Simulations are presented, along with solution sensitivity to errors in hub position knowledge.

RELATIVE MOTION↗

NASA NTRS1977

Minimum required capture radius in a coplanar model of the aerial combat problem

Coplanar aerial combat is modeled with constant speeds and specified turn rates. The minimum capture radius which will always permit capture, regardless of the initial conditions, is calculated. This 'critical' capture radius is also the maximum range which the evader can guarantee indefinitely if the initial range, for example, is large. A composite barrier is constructed which gives the boundary, at any heading, of relative positions for which the capture radius is less than critical.

Breakwell, J. V.↗

NASA NTRS1987

A global time-dependent model of thunderstorm electricity. I - Mathematical properties of the physical and numerical models

A time-dependent model is introduced that can be used to simulate the interaction of a thunderstorm with its global electrical environment. The model solves the continuity equation of the Maxwell current, which is assumed to be composed of the conduction, displacement, and source currents. Boundary conditions which can be used in conjunction with the continuity equation to form a well-posed initial-boundary value problem are determined. Properties of various components of solutions of the initial-boundary value problem are analytically determined. The results indicate that the problem has two time scales, one determined by the background electrical conductivity and the other by the time variation of the source function. A numerical method for obtaining quantitative results is introduced, and its properties are studied. Some simulation results on the evolution of the displacement and conduction currents during the electrification of a storm are presented.

Browning, G. L.↗

NASA NTRS1990

The analysis and simulation of compressible turbulence

Compressible turbulent flows at low turbulent Mach numbers are considered. Contrary to the general belief that such flows are almost incompressible, (i.e., the divergence of the velocity field remains small for all times), it is shown that even if the divergence of the initial velocity field is negligibly small, it can grow rapidly on a non-dimensional time scale which is the inverse of the fluctuating Mach number. An asymptotic theory which enables one to obtain a description of the flow in terms of its divergence-free and vorticity-free components has been developed to solve the initial-value problem. As a result, the various types of low Mach number turbulent regimes have been classified with respect to the initial conditions. Formulae are derived that accurately predict the level of compressibility after the initial transients have disappeared. These results are verified by extensive direct numerical simulations of isotropic turbulence.

Erlebacher, Gordon↗

NASA NTRS1990

The analysis and simulation of compressible turbulence

Compressible turbulent flows at low turbulent Mach numbers are considered. Contrary to the general belief that such flows are almost incompressible (i.e., the divergence of the velocity field remains small for all times), it is shown that even if the divergence of the initial velocity field is negligibly small, it can grow rapidly on a nondimensional time scale which is the inverse of the fluctuating Mach number. An asymptotic theory which enables one to obtain a description of the flow in terms of its divergence-free and vorticity-free components has been developed to solve the initial-value problem. As a result, the various types of low Mach number turbulent regimes have been classified with respect to the initial conditions. Formulae are derived that accurately predict the level of compressibility after the initial transients have disappeared. These results are verified by extensive direct numerical simulations of isotropic turbulence.

Erlebacher, Gordon↗

NASA NTRS1993

Boundary layer flow of air over water on a flat plate

A non-similar boundary layer theory for air blowing over a water layer on a flat plate is formulated and studied as a two-fluid problem in which the position of the interface is unknown. The problem is considered at large Reynolds number (based on x), away from the leading edge. A simple non-similar analytic solution of the problem is derived for which the interface height is proportional to x(sub 1/4) and the water and air flow satisfy the Blasius boundary layer equations, with a linear profile in the water and a Blasius profile in the air. Numerical studies of the initial value problem suggests that this asymptotic, non-similar air-water boundary layer solution is a global attractor for all initial conditions.

Nelson, John↗

NASA NTRS1973

Numerical experiments in climate stability.

Two semiempirical climate models are formulated as time-dependent problems in order to study the stability of their asymptotic steady state equilibrium solutions to perturbations in internal (in this case, initial) conditions. For fixed external conditions the time-dependent versions of these models exhibit fully transitive behavior to positive perturbations in initial conditions, slight intransivity to negative perturbations up to -18 K, and an ice-covered earth regime is obtained for extremely large negative perturbations in initial temperatures (below -18 K). The parametrization found to be most critical in these models is the albedo-temperature coupling, especially in tropical regions. The temperature albedo parametrization from an intransitive climate model used in a time-dependent form of one of the preceding models is found to be highly sensitive to negative perturbations in both internal and external conditions. Numerical experimentation with these semiempirical models shows the important role of the tropics in maintaining the equilibrium climate and suggests that the radiation balance in equatorial latitudes might have a controlling influence on the equatorward extent of the polar ice cap.

Schneider, S. H.↗

NASA NTRS1973

Relative motion of orbiting particles under the influence of perturbing forces. Volume 3: Construction of relative motion traces

Geometric traces can be simply constructed to illustrate the relative motions experienced by particles in a number of problem situations. These diagrams describe the displacements and hodographs which arise as a consequence of initial value inputs and selected disturbance (force) conditions. Due to the linearization which is imposed on the mathematical formulation there is a separation of the in-plane and out-of-plane coordinate solutions. The construction of in-plane traces is easier to represent and to visualize. The out-of-plane geometries are the more complicated cases and generally need some added specializations in order to acquire figures which have some degree of symmetry and simplicity.

Eades, J. B., Jr.↗

NASA NTRS1985

Vibration response of geometrically nonlinear elastic beams subjected to pulse and impact loading

An understanding of the response of composite laminates to foreign object impact is of fundamental importance to the design and the accurate evaluation of aircraft and aerospace structures. A characterization of the basic deformation response of the system is needed for the accurate and consistent modeling of the damage caused by the impact. The response to low velocity impact of foreign objects (less than 10 m/sec at 10 J of impact energy or less) is of particular interest because it corresponds to tool drop problems. An investigation of the vibration response of a simple square beam was conducted, taking into account both sharp initial velocity pulse conditions and simulated impact conditions. Attention is given to nonlinear beam theory, boundary conditions and materials modeling, initial conditions and impact modeling, and a solution of the obtained system of partial differential equations by the finite difference approach.

Moyer, E. T., Jr.↗

NASA NTRS1993

Summation by parts, projections, and stability

We have derived stability results for high-order finite difference approximations of mixed hyperbolic-parabolic initial-boundary value problems (IBVP). The results are obtained using summation by parts and a new way of representing general linear boundary conditions as an orthogonal projection. By slightly rearranging the analytic equations, we can prove strict stability for hyperbolic-parabolic IBVP. Furthermore, we generalize our technique so as to yield strict stability on curvilinear non-smooth domains in two space dimensions. Finally, we show how to incorporate inhomogeneous boundary data while retaining strict stability. Using the same procedure one can prove strict stability in higher dimensions as well.

Olsson, Pelle↗

NASA NTRS2017

Integrated Demand Management: Minimizing Unanticipated Excessive Departure Delay while Ensuring Fairness from a Traffic Management Initiative

This paper introduces NASA's Integrated Demand Management (IDM) concept and presents the results from an early proof-of-concept evaluation and an exploratory experiment. An initial development of the concept was focused on integrating two systems - i.e. the FAA's newly deployed Traffic Flow Management System (TFMS) tool called the Collaborative Trajectory Options Program (CTOP) and the Time-Based Flow Management (TBFM) system with Extended Metering (XM) capabilities to manage projected heavy traffic demand into a capacity-constrained airport. A human-in-the-loop (HITL) simulation experiment was conducted to demonstrate the feasibility of the initial development of the concept by adapting it to an arrival traffic problem at Newark Liberty International Airport (EWR) during clear weather conditions. In this study, the CTOP was utilized to strategically plan the arrival traffic demand by controlling take-off times of both short- and long-haul flights (long-hauls specify aircraft outside TBFM regions and short-hauls specify aircraft within TBFM regions) in a way that results in equitable delays among the groups. Such strategic planning allows less airborne delay to occur within TBFM by feeding manageable long-haul traffic demand while reserving sufficient slots in the overhead streams for the short-haul departures. The manageable traffic demand indicates the TBFM scheduler assigns no more airborne delay than its assigned airspace is capable of absorbing. TBFM then uses its time-based metering capabilities to deliver the desirable throughput by tactically rescheduling the TBFM entered long-haul flights and short-haul departures. Additional research was also performed to explore use of Required Time of Arrival (RTA) capabilities as a potential control mechanism for the airborne flights to improve arrival traffic delivery accuracy of scheduled long-haul traffic demand. The study results show that both short- and long-haul flights received similar ground delays. In addition, there was a noticeable reduction in the total amount of excessive unanticipated last-minute ground delays, i.e. delays that are frequently imposed on the short-haul flight in current day operations due to saturation in the overhead stream, commonly referred to as 'double penalty'. Furthermore, the concept achieved the target throughput while minimizing the expected cost associated with overall delays in arrival traffic. Assessment of the RTA capabilities showed that there was indeed improvement of the scheduled entry times into TBFM regions by using RTA capabilities. However, with respect to reduction in delays incurred within TBFM, there was no observable benefit of improving the precision of long-haul flights entry times.

Collaborative Trajectory Options Program (CTOP)↗

NASA NTRS1977

Calculations of the effects of angular momentum on the early evolution of Jupiter

Early phases in the evolutionary development of a rotating cloud of gas and dust are investigated to determine whether physically reasonable initial conditions are likely to produce a planet-satellite system with the values of specific orbital angular momentum currently observed for Jupiter and its inner satellites. Spherically symmetric computations of Jupiter's evolutionary history are reviewed, and transport of angular momentum is considered along with evolutionary sequences of individual differentially rotating models with decreasing values of equatorial radius. The problem of accounting for the present specific orbital angular momenta of Jupiter and its regular satellites is examined qualitatively, and a range of possible angular-momentum distributions in an initially tidally unstable cloud is assessed. The results show that particular physically plausible angular-momentum distributions, which could either be established as initial conditions or be caused by angular-momentum transport, will lead to favorable situations for the formation of a central planet and a surrounding rotating cloud with specific angular momentum comparable to that of the regular satellites.

Bodenheimer, P.↗

NASA NTRS1986

A practical assessment of spectral accuracy for hyperbolic problems with discontinuities

Physical and transform space filtering has been applied to the Fourier spectral collocation solution of the constant coefficient scalar wave equation with a discontinuous initial condition. High order accuracy can be extracted from the unfiltered solution. Smooth, high order Fourier space filtering gives expected polynomial order solutions away from the discontinuity. Spectral accuracy is observed with the physical space filter of Gottlieb and Tadmor.

Kopriva, David A.↗

NASA NTRS1987

A practical assessment of spectral accuracy for hyperbolic problems with discontinuities

Kopriva, David A.↗

NASA NTRS1984

STARS: A general-purpose finite element computer program for analysis of engineering structures

STARS (Structural Analysis Routines) is primarily an interactive, graphics-oriented, finite-element computer program for analyzing the static, stability, free vibration, and dynamic responses of damped and undamped structures, including rotating systems. The element library consists of one-dimensional (1-D) line elements, two-dimensional (2-D) triangular and quadrilateral shell elements, and three-dimensional (3-D) tetrahedral and hexahedral solid elements. These elements enable the solution of structural problems that include truss, beam, space frame, plane, plate, shell, and solid structures, or any combination thereof. Zero, finite, and interdependent deflection boundary conditions can be implemented by the program. The associated dynamic response analysis capability provides for initial deformation and velocity inputs, whereas the transient excitation may be either forces or accelerations. An effective in-core or out-of-core solution strategy is automatically employed by the program, depending on the size of the problem. Data input may be at random within a data set, and the program offers certain automatic data-generation features. Input data are formatted as an optimal combination of free and fixed formats. Interactive graphics capabilities enable convenient display of nodal deformations, mode shapes, and element stresses.

Gupta, K. K.↗