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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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Computer program ETC improves computation of elastic transfer matrices of Legendre polynomials P/0/ and P/1/. Rather than carrying out a double integration numerically, one of the integrations is accomplished analytically and the numerical integration need only be carried out over one variable.
The troublesome multi-center molecular integrals of Slater-type orbitals (STO) in molecular physics calculations can be evaluated by using the Fourier transform and proper coupling of the two center exchange integrals. A numerical integration procedure is then readily rendered to the final expression in which the integrand consists of well known special functions of arguments containing the geometrical arrangement of the nuclear centers and the exponents of the atomic orbitals. A practical procedure was devised for the calculation of a general multi-center molecular integrals coupling arbitrary Slater-type orbitals. Symmetry relations and asymptotic conditions are discussed. Explicit expressions of three-center one-electron nuclear-attraction integrals and four-center two-electron repulsion integrals for STO of principal quantum number n=2 are listed. A few numerical results are given for the purpose of comparison.
A new approximation to the components of midcourse correction direction errors is described that is more accurate than a previously used approximation. The calculation effort involved is much less than that for numerical integration. A comparison of numerical integration results with those of the new approximation is presented in a diagram.
A new method of calculating the root-mean-square (rms) pointing jitter of a scientific instrument (e.g., a camera, radar antenna, or telescope) is introduced based on a state-space concept. In comparison with the prior method of calculating the rms pointing jitter, the present method involves significantly less computation. The rms pointing jitter of an instrument (the square root of the jitter variance shown in the figure) is an important physical quantity which impacts the design of the instrument, its actuators, controls, sensory components, and sensor- output-sampling circuitry. Using the Sirlin, San Martin, and Lucke definition of pointing jitter, the prior method of computing the rms pointing jitter involves a frequency-domain integral of a rational polynomial multiplied by a transcendental weighting function, necessitating the use of numerical-integration techniques. In practice, numerical integration complicates the problem of calculating the rms pointing error. In contrast, the state-space method provides exact analytic expressions that can be evaluated without numerical integration.
A combination of methods is proposed of predicting spacecraft trajectories that possibly include multiple maneuvers and/or perturbing accelerations, with greater speed, accuracy, and repeatability than were heretofore achievable. The combination is denoted the WeavEncke method because it is based on unpublished studies by Jonathan Weaver of the orbit-prediction formulation of the noted astronomer Johann Franz Encke. Weaver evaluated a number of alternatives that arise within that formulation, arriving at an orbit-predicting algorithm optimized for complex trajectory operations. In the WeavEncke method, Encke's method of prediction of perturbed orbits is enhanced by application of modern numerical methods. Among these methods are efficient Kepler s-equation time-of-flight solutions and self-starting numerical integration with time as the independent variable. Self-starting numerical integration satisfies the requirements for accuracy, reproducibility, and efficiency (and, hence, speed). Self-starting numerical integration also supports fully analytic regulation of integration step sizes, thereby further increasing speed while maintaining accuracy.
A general tradeoff is made of the symmetric Cassegrain antenna with regard to the possibility of meeting a 90% beam efficiency. The effects of aperture taper and blockage are calculated using an adjustable sidelobe circular distribution. Numerical integration is used. For the feed spillover calculation, a low sidelobe symmetric feed pattern is used with the equivalent parabola and numerical integration. Reflector cross polarization is calculated using double numerical integration. Reflector back lobes are estimated from radiation pattern envelopes of commercial common carrier dish antennas. The curves allow a range of f/D to be determined for a specified edge taper and blockage diameter ratio, and with a table of Cassegrain parameters, a range of possible designs that meet the 90% beam efficiency is obtained. It is shown that the feed and reflector design and implementation must be carefully done.
The discovery (by Laskar, 1989, 1990) that the evolution of the solar system is chaotic, made in a numerical integration of the averaged secular approximation of the equations of motions for the planets, was confirmed by Sussman and Wisdom (1992) by direct numerical integration of the whole solar system. This paper presents results of direct integrations of the rotation of Mars in the chaotically evolved planetary system, made using the same model as that used by Sussman and Wisdom. The numerical integration shows that the obliquity of Mars undergoes large chaotic variations, which occur as the system evolves in the chaotic zone associated with a secular spin-orbit resonance.
An analytical procedure based on the Navier-Stokes equations was developed for analyzing and representing properties of unsteady viscous flow around oscillating obstacles. A variational formulation of the vorticity transport equation was discretized in finite element form and integrated numerically. At each time step of the numerical integration, the velocity field around the obstacle was determined for the instantaneous vorticity distribution from the finite element solution of Poisson's equation. The time-dependent boundary conditions around the oscillating obstacle were introduced as external constraints, using the Lagrangian Multiplier Technique, at each time step of the numerical integration. The procedure was then applied for determining pressures around obstacles oscillating in unsteady flow. The obtained results for a cylinder and an airfoil were illustrated in the form of streamlines and vorticity and pressure distributions.
Numerical integration calculation of collision integrals for exponential attractive potential of gases
Two techniques are discussed for increasing the accuracy of the numerical integration of eccentric orbits in Cartesian coordinates. One involves the use of an independent variable different from time; this increases the efficiency of the numerical integration. The other uses a time element, which reduces the in-track error. A general expression is given of a time element valid for an arbitrary independent variable. It is pointed out that this time element makes it possible to switch the independent variable merely by applying a scaling factor; there is no need to change the differential equations of the motion. Eccentric, true, and elliptic anomalies are used as independent variables in the case of a transfer orbit for a geosynchronous orbit. The elliptic anomaly is shown to perform much better than the other classical anomalies.
The primary purpose of this research is to provide mathematical models which may be used in the investigation of various aspects of the remote capture and retrieval of uncontrolled satellites. Emphasis has been placed on analytical models; however, to verify analytical solutions, numerical integration must be used. Also, for satellites of certain types, numerical integration may be the only practical or perhaps the only possible method of solution. First, to provide a basis for analytical and numerical work, uncontrolled satellites were categorized using criteria based on: (1) orbital motions, (2) external angular momenta, (3) internal angular momenta, (4) physical characteristics, and (5) the stability of their equilibrium states. Several analytical solutions for the attitude motions of satellite models were compiled, checked, corrected in some minor respects and their short-term prediction capabilities were investigated. Single-rigid-body, dual-spin and multi-rotor configurations are treated. To verify the analytical models and to see how the true motion of a satellite which is acted upon by environmental torques differs from its corresponding torque-free motion, a numerical simulation code was developed. This code contains a relatively general satellite model and models for gravity-gradient and aerodynamic torques. The spacecraft physical model for the code and the equations of motion are given. The two environmental torque models are described.
Laminar-turbulent transition of an incompressible flat-plate wake is investigated by direct numerical integration of the Navier-Stokes equations. For the numerical integration, a combination of finite-difference and spectral methods along with an ADI/Crank-Nicolson/Adams-Bashforth time integration scheme is employed. Subject to 2D forcing, the wake exhibited a rapidly-growing fundamental disturbance that quickly saturated. This saturation was due partly to the stabilizing effect of the mean flow distortion. Downstream of the saturation point, disturbance energy was concentrated in the fundamental disturbance, the second harmonic, and the mean flow distortion component. At large amplitude levels, a Karman vortex street formed. Variations in the 2D forcing level did not alter the qualitative behavior of the disturbances. Simulations of 3D breakdown indicates that the presence of large-amplitude, 2D disturbances tends to initially suppress small-amplitude 3D disturbance growth. Following this initial suppression, a resumption of 3D growth is observed that may have been due to a secondary instability mechanism. For high levels of 3D disturbance energy, lambda-shaped vortical structures formed between adjacent Karman vortices.
Ranft has provided parameterizations of Lorentz invariant differential cross sections for pion and nucleon production in pion-proton collisions that are compared to some recent data. The Ranft parameterizations are then numerically integrated to form spectral and total cross sections. These numerical integrations are further parameterized to provide formula for spectral and total cross sections suitable for use in radiation transport codes. The reactions analyzed are for charged pions in the initial state and both charged and neutral pions in the final state.
Theoretical method of numerical integration of definite integral f/t/ dt
Report discusses spurious numerical solutions sometimes produced by explicit Runge-Kutta numerical integrations. Includes steady-state solutions, periodic orbits, and chaotic phenomena.
Limited word size in contemporary microprocessors causes numerical problems in autonomous satellite navigation applications. Numerical error introduced in navigation computations performed on small wordlength machines can cause divergence of sequential estimation algorithms. To insure filter reliability, square root algorithms have been adopted in many applications. The optimal navigation algorithm requires a careful match of the estimation algorithm, dynamic model, and numerical integrator. In this investigation, the relationship of several square root filters and numerical integration methods is evaluated to determine their relative performance for satellite navigation applications. The numerical simulations are conducted using the Phase I GPS constellation to determine the orbit of a LANDSAT-D type satellite. The primary comparison is based on computation time and relative estimation accuracy.
The analysis of lunar laser ranging data requires very accurate calculations of the lunar physical librations. Libration terms are given which arise from the additive and planetary terms in the lunar theory. The large size of the recently discovered terms due to third degree gravitational harmonics will allow some of these harmonics to be measured by laser ranging to the moon. Numerical integration promises to be an effective method of calculating librations. Comparison of numerical integrations with analytic series indicates that the calculation of the series due to third and fourth degree harmonics is not yet as accurate as the more extensively developed second degree terms.-
We describe how orbital tunnels could be used to transport payloads through the Earth. If you use a brachistochrone for the tunnel, the body forces in the tunnel become overwhelmingly large for small angular distances traveled. Projectiles move along an orbital tunnel faster than they would along abrachistochrone connecting the same points but the body force components cancel. We describe how parabolic Keplerian orbits outside the object merge onto quasi-Keplerian orbits inside the object. We use models of the interior of the Earth with three values of the polytropic index (n) to calculate interior or bits that travel between surface points. The n¼3 results are also scaled to the Sun. Numerical integrations of the equations describing polytropes were used to generate the initial models. Numerical integration of the equations of motion are then used to calculate the angular distance you can travel along the surface and the traversal time as a function of the parabolic periaps is distance for each model. Trajectories through objects of low central condensation show a focusing effect that decreases as the central condensation increases. Analytic solutions for the trajectories in a homogeneous sphere are derived and compared to the numeric results. The results can be scaled to other planets, stars, or even globular clusters.