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Programmable calculator runs simple finite-series approximation for Laplace transform inversions. Utilizing family of orthonormal functions, approximation is used for wide range of transforms, including those encountered in feedback control problems. Method works well as long as F(t) decays to zero as it approaches infinity and so is appliable to most physical systems.
A method of inverting Laplace transforms by using a set of orthonormal functions is reported. As a byproduct of the inversion, approximation of complicated Laplace transforms by a transform with a series of simple poles along the left half plane real axis is shown. The inversion and approximation process is simple enough to be put on a programmable hand calculator.
The implementation of the approximate factorization algorithm and its ability to efficiently and accurately describe transonic flow about an NACA 64A010 airfoil section is examined. The approximate factorization algorithm is developed from the nondimensional, conservative, vectorized Navier-Stokes equations expressed in curvilinear coordinates. Equations of state and transport coefficient relations appropriate to atmospheric air are appended to close the system of partial differential equations. An algebraic turbulence model is also incorporated into the equation set. This algorithm was verified by investigating the flow about an NACA 64A010 airfoil at 0, 2, and 3.5 deg angle of attack for free-stream conditions of 2,000,000 Reynolds number and 0.8 Mach number. Overall results were in good qualitative agreement with wind tunnel data sets. However, while nondimensional times of six were attained, numerical difficulties prevented any case from reaching a true steady state.
The paper analyzes features of particle behavior in magnetospheric convection using simple analytic forms for the electrostatic potential. It is shown that analytic approximations for the shape and position of surfaces delimiting closed and open particle orbits can be derived. These approximations are good for any pitch angle and at most energies. They break down only for the medium-energy protons that are predicted by such models to penetrate far inside the plasmapause. The derived expressions allow rapid tests of a variety of electric field models when analyzing particle data, and this is illustrated by intercomparing field models with several observations.
Constructive proofs and several generalizations of approximation results of J. H. Bramble and S. R. Hilbert are presented. Using an averaged Taylor series, we represent a function as a polynomial plus a remainder. The remainder can be manipulated in many ways to give different types of bounds. Approximation of functions in fractional order Sobolev spaces is treated as well as the usual integer order spaces and several nonstandard Sobolev-like spaces.
A user's guide is presented for ACCESS-3, a research oriented program which combines dual methods and a collection of approximation concepts to achieve excellent efficiency in structural synthesis. The finite element method is used for structural analysis and dual algorithms of mathematical programming are applied in the design optimization procedure. This program retains all of the ACCESS-2 capabilities and the data preparation formats are fully compatible. Four distinct optimizer options were added: interior point penalty function method (NEWSUMT); second order primal projection method (PRIMAL2); second order Newton-type dual method (DUAL2); and first order gradient projection-type dual method (DUAL1). A pure discrete and mixed continuous-discrete design variable capability, and zero order approximation of the stress constraints are also included.
The thin-layer approximation is extended to an axial corner that is formed by the intersection of two perpendicular plates, one of which has an inclination angle with respect to the free stream. A computer code developed by Hung and MacCormack (1978) is modified for the thin-layer approximation, and a case with Mach 5.9 and a wedge angle of 6 deg is computed. In addition, it is shown that it is not necessary to solve the complete Navier-Stokes equations for a three-dimensional high-Reynolds-number corner flow.
Vertical-electronic static-exchange photoexcitation and ionization cross sections are reported which provide a first approximation to the complete dipole spectrum of CO2. Separated-channel static-exchange calculations of vertical-electronic transition energies and oscillator strengths, and Stieltjes-Chebyshev moment methods were used in the development. Detailed comparisons were made of the static-exchange excitation and ionization spectra with photoabsorption, electron-impact excitation, and quantum-defect estimates of discrete transition energies and intensities, and with partial-channel photoionization cross sections obtained from fluorescence measurements and from tunable-source and (e, 2e) photoelectron spectroscopy. Results show that the separate-channel static-exchange approximation is generally satisfactory in CO2.
The approximation of Penndorf (1962) and Shifrin-Punina (1968) to the Mie solution at forward scattering angles are extended to small size parameters. The proposed semiempirical approximation accurately represents the Mie results down to x = 0.5-1 for refractive index m = 1.33, and to x = 2.0 for larger index values. The implications of the result for the inversion of particle size distribution from single scattering data in the forward direction are discussed.
An engineering method has been developed for computing the windward-symmetry plane convective heat-transfer rates on Shuttle-like vehicles at large angles of attack. The engineering code includes an approximate inviscid flowfield technique, laminar and turbulent heating-rate expressions, an approximation to account for the variable-entropy effects on the surface heating and the concept of an equivalent axisymmetric body to model the windward-ray flowfields of Shuttle-like vehicles at angles of attack from 25 to 45 degrees. The engineering method is validated by comparing computed heating results with corresponding experimental data measured on Shuttle and advanced transportation models over a wide range of flow conditions and angles of attack from 25 to 40 degrees and also with results of existing prediction techniques. The comparisons are in good agreement.
Modifications and corrections are presented to relations obtained in an investigation conducted by Szebehely (1978), who has discussed the problem of Hill's (1878) stability of satellites in the restricted problem of three bodies. Attention is given to an approximation of the Jacobian constant for the satellite, the critical value of the Jacobian constant, and approximate solutions.
An empirical expression is derived to account for the molecular multiple scattering contribution to the almucantar radiance field. Formulas for the correction factors which incorporate the effects of multiple scattering and nonzero ground albedo are also given. The use and accuracy of the multiple-scattering approximation in direct problems of radiative transfer associated with almucantar radiance are discussed and illustrated by examples. It is shown that in almost all instances, inclusion of the molecular multiple-scattering contribution reduces the errors obtained with the single-scattering approximation by a factor of at least 2.
A method is presented for the minimum mass design of three dimensional space frames constructed of thin walled rectangular cross-section members. Constraints on nodal displacements and rotations, material stress, local buckling, and cross sectional dimensions are included. A high quality separable approximate problem is formed in terms of the reciprocals of the four section properties of the frame element cross section, replacing all implicit functions with simplified explicit relations. The cross sectional dimensions are efficiently calculated without using multilevel techniques. Several test problems are solved, demonstrating that a series of approximate problem solutions converge rapidly to an optimal design.
An approximate method is developed for predicting laminar and turbulent heating rates on the windward side of the Space Shuttle Orbiter for both the wind-tunnel and flight environments. The method is based on a 'local infinite swept cylinder' analysis and includes both equilibrium-air chemistry and variable boundary-layer-edge entropy. The method is validated by comparing with data from wind-tunnel experiments and from the first and second Space Shuttle flights both along the windward-symmetry plane and in a lateral direction off the symmetry plane. Agreement with the flight data is good from approximately 67 km (peak heating) downward.
An approximate technique for solving the radiative transfer problem for fluxes reflected, transmitted, and absorbed by a homogeneous scattering layer is presented. The technique is a straightforward delta-function modification of a solution to the transfer equation in closed form via the discrete-ordinates method with four streams. By the use of four streams, significant improvement in accuracy is obtained for anisotropic phase functions over that obtained by similarly employed two-stream approximations with delta-functions. However, the computational effort is increased only slightly. Equations are developed for the general case of a layer underlaid by a reflecting Lambert surface. Graphical comparisons are given of fractional error resulting from the use of this method with that resulting from the use of typical four-stream and delta-two-stream techniques.
Stahara et al. (1978) have considered the use of an approximation technique which employs two or more nonlinear base solutions determined by the full computational method to predict entire families of related nonlinear solutions. The present investigation provides results for several applications of that method which demonstrate both its accuracy and its utility for engineering applications. Attention is given to the perturbation concept and methods, aspects of coordinate straining, aspects of analytical formulation, and an application to surface properties. In a discussion of the results, single and multiple parameter perturbations are considered along with a combination of the approximation method with optimization procedures. The results show that it is possible to combine in certain cases large savings in computational cost with improved optimization.
A procedure for analyzing low-energy (less than approximately 100 eV) ion data from the plasma composition experiment on ISEE 1 is set forth. The method is based on a derived analytic expression for particle flux to a limited aperture retarding potential analyzer (RPA) in the thin sheath approximation, which makes allowance for some effects of a charged spacecraft on plasma particle trajectories. Calculations using simulated data are employed in testing the efficacy and accuracy of the technique. On the basis of an analysis of these calculation results and the mathematical model, the method is seen as being able to provide accurate ion temperatures from all good plasmaspheric RPA data. It is noted that corresponding densities and spacecraft potentials should be accurate when spacecraft potentials are negative but that they are subject to error for positive spacecraft potentials, particularly when ion Mach numbers are much less than 1. An analysis of data from a representative ISEE 1 pass produces a plasmasphere temperature profile that is consistent in overall structure with previous observations.
One-sided confidence regions for continuous cumulative distribution functions are constructed using empirical cumulative distribution functions and the generalized Kolmogorov-Smirnov distance. The band width of such regions becomes narrower in the right or left tail of the distribution. To avoid tedious computation of confidence levels and critical values, an approximation based on the Poisson process is introduced. This aproximation provides a conservative confidence region; moreover, the approximation error decreases monotonically to 0 as sample size increases. Critical values necessary for implementation are given. Applications are made to the areas of risk analysis, investment modeling, reliability assessment, and analysis of fault tolerant systems.