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A variety of numerical methods for fitting a function to data given at a set of points scattered throughout a domain in the plane are surveyed. Four classes of methods are discussed: (1) global interpolation; (2) local interpolation; (3) global approximation; and (4) local approximation. Also, two-stage methods and contouring are discussed. The surfaces constructed include polynomials, spline functions, and rational functions, among others.
The physical and geometric interpretation of a surface fitting technique, multiquadric equations (MQ), is discussed. The relation of this method and its reciprocal form to biharmonic and harmonic techniques respectively, is pointed out. In both forms the solutions may be viewed as being numerical approximations of an integral equation in which an unknown density function is the physical source for disturbing potential (reciprocal-MQ) or elastic displacement (MQ).
The interaction of a shock wave with a hot spot, a single vortex and a vortex street is studied within the framework of the two dimensional compressible Euler equations. The numerical results obtained by the pseudospectral method and the finite difference MacCormack method are compared. In both the methods the shock wave is fitted as a boundary of the computational domain.
The formulation of generalized models tracing the geometry and intensity of the synchrotron emission from precessing, twin, relativistic jets as projected on the plane of the sky is presented. It is shown that neither the shape of the image nor its relative intensities are altered by including the effects of a cosmological redshift and a relative velocity between the source and observer. The models are fitted to the available data for several quasars and radio galaxies and demonstrate the plausibility of the phenomenon. Probable selection effects are considered and diagnostics given for recognizing objects showing this behavior. In the radio galaxies considered, velocities up to about 0.2c and precession periods of 1,000,000 yr are deduced. In the QSOs investigated, velocities of 0.7c and greater are found and periods of order 10,000 yr. In some cases precession cone angles increase with time. Consequences in terms of lifetimes of QSO behavior and binary supermassive objects are discussed.
A comparison is presented of a computational investigation of the flowfield created by supersonic flow over an internal corner with rounding and experimental results at a freestream Mach number of 4.1. The computational study uses a shock-fitting finite difference scheme with explicit formulation of the transformation to computational coordinates. The solutions calculated agree with previous ones for the sharp corner case and with the experimental investigation results for the rounded corner case. Parametric effects were computed for variations in corner rounding, freestream Mach number, symmetric wedge angle and unsymmetric wedge angle. A region is described over which successful executions were obtained.
Conventional algorithms for the numerical integration of ordinary differential equations (ODEs) are based on the use of polynomial functions as interpolants. However, the exact solutions of stiff ODEs behave like decaying exponential functions, which are poorly approximated by polynomials. An obvious choice of interpolant are the exponential functions themselves, or their low-order diagonal Pade (rational function) approximants. A number of explicit, A-stable, integration algorithms were derived from the use of a three-parameter exponential function as interpolant, and their relationship to low-order, polynomial-based and rational-function-based implicit and explicit methods were shown by examining their low-order diagonal Pade approximants. A robust implicit formula was derived by exponential fitting the trapezoidal rule. Application of these algorithms to integration of the ODEs governing homogenous, gas-phase chemical kinetics was demonstrated in a developmental code CREK1D, which compares favorably with the Gear-Hindmarsh code LSODE in spite of the use of a primitive stepsize control strategy.
For the period considered, December 1977 through February 1978, bivariate Gaussian discriminant function cloud identification revealed that more than 93 percent of the 8-km resolution GOES infrared pixels were cloud contaminated. Cloud-free in-situ calibration points were distributed in nonrandom groups; this resulted in systematic errors when using least squares techniques. Surfaces and regression lines were least squares fitted between satellite and in-situ data; use was also made of differences and ratios. The best results were achieved with a regression in the form of the infrared radiative transfer equation; but this was no better than + or - 0.9 K. Because of extensive cloudiness, the linear regressions were seldom useful, and temperature ratios with + or - 1.3 K experimental errors best represent the applicability of GEOS data to sea surface temperatures.
Klein et al. (1977) have questioned the concept of endurance training as an appropriate means of preparing for prolonged space flights. Their opinion was mainly based on reports of endurance athletes who had a decreased tolerance to orthostatic or gravitational stress induced by lower body negative pressure (LBNP), upright tilt, or whole body water immersion. The present investigation had the objective to determine if the hemodynamic response to LBNP is different between a high and average fit group of subjects. In addition, the discrete aspect of cardiovascular function which had been altered by chronic training was to be identified. On the basis of the results of experiments conducted with 14 young male volunteers, it is concluded that the reflex response to central hypovolemia is altered by endurance exercise training.
The interaction of a planar shock wave with one or more vortexes is computed using a pseudospectral method and a finite difference method. The development of the spectral method is emphasized. In both methods the shock wave is fitted as a boundary of the computational domain. The results show good agreement between both computational methods. The spectral method is, however, restricted to smaller time steps and requires use of filtering techniques. Previously announced in STAR as N82-28061
A NASTRAN model of the Bearing Tester was developed to determine the fit of its components while in use. The model was executed for three loading conditions involving appropriate thermal and pressure loading throughout: rotation (30,000 rpm), rotation with 5000 lb axial shaft loading, and rotation with 2000 lb lateral shaft loading. A 30 deg slice of the Bearing Tester was modeled with solid isoparametric elements (CIHEX). Five more slices were generated to form a 180 deg symmetrical half. Each component was modeled separately for two reasons: (1) in the heat transfer version each component required covering with appropriate heat transfer coefficient elements, and (2) for the displacement version each component required independent radial expansion with respect to adjacent components. The components are connected at fastener and bearing locations axially and/or tangentially. Differences in radial displacements of adjacent cylindrical surfaces were calculated and added to those differences calculated for 180 deg opposite locations to determine the diametric change.
The RF loss properties and thermal capability of brazed helix circuits and interference fitted circuits were evaluated. The objective was to produce design circuits with minimum RF loss and maximum heat transfer. These circuits were to be designed to operate at 10 kV and at 20 GHz using a gamma a approximately equal to 1.0. This represents a circuit diameter of only 0.75 millimeters. The fabrication of this size circuit and the 0.48 millimeter high support rods required considerable refinements in the assembly techniques and fixtures used on lower frequency circuits. The transition from the helices to the waveguide was designed and the circuits were matched from 20 to 40 GHz since the helix design is a broad band circuit and at a gamma a of 1.0 will operate over this band. The loss measurement was a transmission measurement and therefore had two such transitions. This resulting double-ended match required tuning elements to achieve the broad band match and external E-H tuners at each end to optimize the match for each frequency where the loss measurement was made. The test method used was a substitution method where the test fixture was replaced by a calibrated attenuator.
In the present, simple analysis method for bolt hole clearance stresses in mechanically fastened joints, an inverse formulation is used with a finite element analysis. Conditions along the bolt hole contact arc are specified by displacement constraint equations. The method was applied to a single fastener clearance-fit joint with typical clearance values, for the cases of a rigid, frictionless bolt and a quasi-isotropic graphite/epoxy laminate. Results indicate that the contact arc, as well as the peak stresses around the hole (and their locations), were strongly influenced by the clearance. After a minor initial nonlinearity, peak stresses varied linearly with applied load. Clearance levels are found to have only a minor influence on overall joint stiffness.
Double-ring seal avoids expense of remachining or redesigning valve parts. Mating fittings on valve sealed by pair of rings - one O-ring and backup ring. Backup ring fills relatively large gap between parts. Prevents softer O-ring from being pushed into and through gap.
A shock fitting method applied to the transonic small disturbance (TSD) potential equation is described. This method is then applied to a simple, two dimensional (2-D) rotating disturbance which is analogous to a shock radiating from the tip of a rotor blade in high speed hover. A comparison is made between the results of this method and the more standard shock capturing method. This comparison makes it clear that the effect of the results on the acoustic signature of the 2-D model is significant, and similar results can be expected when the method is extended to the three dimensional (3-D) case.
An effort is currently under way by NASA Marshall Space Flight Center to upgrade the space shuttle main engine through the use of improved materials and processes. Under consideration is the use of the Langley Research Center explosive seam welding process. The objective is to demonstrate the feasibility of joining space shuttle main engine tube to fitting components in an oxygen heat exchanger, using the NASA LaRC explosive seam welding process. It was concluded that LaRC explosive joining is viable for this application; that there is no incompatability of materials; that ultrasonic inspection is the best nondestructive testing method; and that the .500 DIA joint experiences interface problems.
In this report, a numerical method for solving the equations of motion of three-dimensional incompressible flows in nonorthogonal body-fitted coordinate (BFC) systems has been developed. The equations of motion are transformed to a generalized curvilinear coordinate system from which the transformed equations are discretized using finite difference approximations in the transformed domain. The hybrid scheme is used to approximate the convection terms in the governing equations. Solutions of the finite difference equations are obtained iteratively by using a pressure-velocity correction algorithm (SIMPLE-C). Numerical examples of two- and three-dimensional, laminar and turbulent flow problems are employed to evaluate the accuracy and efficiency of the present computer code. The user's guide and computer program listing of the present code are also included.
A technique which measures load-contact variation in a clearance-fit, pin-loaded hole is presented in detail. A steel instrumented pin, which activates a make-or-break electrical circuit in the pin-hole contact region, was inserted into one aluminum and one polycarbonate specimen. The resulting load-contact variations are indicated schematically. The ability to accurately determine the arc of contact at any load was crucial to this measurement. It is noted that this simple experimental technique is applicable to both conducting and nonconducting materials.
In the present paper, a numerical method for solving the equations of motion of three-dimensional incompressible flows in nonorthogonal body-fitted coordinate (BEC) systems has been developed and evaluated. The equations of motion are transformed to a generalized curvilinear coordinate system from which the transformed equations are discretized using finite difference approximations in the transformed domain. The hybrid scheme and a central differencing plus artificial dissipation scheme are used to approximate the convection terms in the governing equations. Effects of these two schemes on the accuracy of numerical predictions are studied. Solutions of the finite difference equations are obtained iteratively by using a pressure-velocity correction algorithm, SIMPLE-C. Numerical examples of two- and three-dimensional, laminar and turbulent flow problems are employed to evaluate the accuracy and efficiency of the present numerical method.