A numerical method for the evaluation of complex integrals
Numerical methods, FORTRAN program and table form for complex integrals evaluation along unit circle in positive direction
Agniel, R. G.↗
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Results for “NUMERICAL INTEGRATION”
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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A study of accuracy in selected numerical-analysis integration techniques
Report discusses several methods of performing numerical integration with computer. When data can be expressed as state vector that is dependent variable in a differential equation, self-starting integrators can be used to predict future data.
Lear, W. M.↗
Numerical Digital Computer Method for Determining the Transient Responses of Nonlinear Automatic Systems Based on Calculation of the Convolution Integral
Numerical digital computer method for determining transient responses of nonlinear automatic systems based on calculation of convolution integral
CONVOLUTION THEORY↗
Computer program ETC improves computation of elastic transfer matrices of Legendre polynomials P/0/ and P/1/
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.
Gibson, G.↗
Solution of multi-center molecular integrals of Slater-type orbitals
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.
Tai, H.↗
An approximation to midcourse correction direction errors.
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.
Kibler, J. F.↗
Analytic Method for Computing Instrument Pointing Jitter
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.
Bayard, David↗
Predicting Spacecraft Trajectories by the WeavEncke Method
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.
Weaver, Jonathan K.↗
Design tradeoff study for reflector antenna systems for the shuttle imaging microwave system
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.
Hansen, R. C.↗
The chaotic obliquity of Mars
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.
Touma, Jihad↗
Impacts of Bulk Microphysics Scheme Structural Choices on Simulations of Rain Initiation Through Drop Coalescence
This study examines how different structural choices in bulk microphysics schemes impact the simulation of warm rain initiation. A single liquid category (SLC) approach prognosing up to four moments of a single drop size distribution (DSD) is compared to the traditional two-category, two-moment approach with separate DSDs for cloud and rain (four total prognostic variables). Different methods for calculating tendencies of the prognostic variables from drop collision-coalescence are also tested: a discretized numerical-integration approach, machine learning via neural networks, lookup tables, and traditional power law fits. Relative to simulations using a bin microphysics model, SLC gives smaller error overall than the two-category approach when numerical integration is used to calculate the collision-coalescence tendencies for both. Replacing the numerical integration with a pre-computed lookup table reduces computational cost with little loss of accuracy. However, using fitted power laws with SLC to represent the collision-coalescence tendencies substantially reduces accuracy and leads to an order of magnitude increase in error. It is also demonstrated that with SLC, reasonably accurate solutions are obtained using only three prognostic moments, while a two-moment SLC scheme leads to substantial error. Overall, both the choice of prognostic moments (e.g., SLC vs. two-category) and method to calculate the collision-coalescence tendencies are important to consider for minimizing errors in bulk schemes. SLC with a sufficiently detailed calculation of the collision-coalescence tendencies provides accurate solutions for a reasonable computational cost, providing a viable alternative to the traditional two-category, two-moment approach for bulk microphysics.
320 (cloud physics and chemistry)↗
Numerical calculations of velocity and pressure distribution around oscillating airfoils
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.
Bratanow, T.↗
Collision Integrals for the Exponential Attractive Potential
Numerical integration calculation of collision integrals for exponential attractive potential of gases
GAS TRANSPORT↗
A general time element for orbit integration in Cartesian coordinates
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.
Janin, G.↗
Satellite attitude motion models for capture and retrieval investigations
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.
Cochran, John E., Jr.↗
Numerical simulation of laminar-turbulent transition in a spatially-developing flat plate wake
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.
Dratler, D. I.↗
Cross-Section Parameterizations for Pion and Nucleon Production From Negative Pion-Proton Collisions
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.
Norbury, John W.↗
Romberg integration
Theoretical method of numerical integration of definite integral f/t/ dt
Omalley, M. J.↗