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All sourcesNASA NTRSDOE OSTI
NASA NTRS1973

Round-off errors in cutting plane algorithms based on the revised simplex procedure

This report statistically analyzes computational round-off errors associated with the cutting plane approach to solving linear integer programming problems. Cutting plane methods require that the inverse of a sequence of matrices be computed. The problem basically reduces to one of minimizing round-off errors in the sequence of inverses. Two procedures for minimizing this problem are presented, and their influence on error accumulation is statistically analyzed. One procedure employs a very small tolerance factor to round computed values to zero. The other procedure is a numerical analysis technique for reinverting or improving the approximate inverse of a matrix. The results indicated that round-off accumulation can be effectively minimized by employing a tolerance factor which reflects the number of significant digits carried for each calculation and by applying the reinversion procedure once to each computed inverse. If 18 significant digits plus an exponent are carried for each variable during computations, then a tolerance value of 0.1 x 10 to the minus 12th power is reasonable.

Moore, J. E.↗
NASA NTRS1979

A structure for digital notch filters

Narrow-band digital notch filters have their poles near the unit circle. As the sampling rate is increased, the poles move towards Z = +1. Implementing such filters requires long registers to overcome the sensitivity and roundoff errors. A filter structure based on digital incremental computers is proposed which has low sensitivity and round-off errors, and simple hardware implementation. The filter structure can be directly used on differentially pulse-code modulated signals. Hardware multipliers are not required as the poles approach z = +1, and excellent results can be obtained using multipliers with very short word lengths, or with small size read-only memories.

Abu-El-haija, A. I.↗
NASA NTRS1990

Optimal controllers for finite wordlength implementation

There exist realizations of a given controller transfer function yielding arbitrarily large effects from computational errors. Since, in general, there is no upper bound, it is important to have a systematic way of reducing these effects. Optimum controller designs are developed which take into account the digital round-off errors in the controller implementation and in the A/D and D/A converters. These results provide a natural extension to the LQG (linear quadratic Gaussian) theory since they reduce to the standard LQG controller when infinite-precision computation is used, but for finite precision the separation principle does not hold.

Liu, K.↗
NASA NTRS1972

Interplanetary Trajectories, Encke Method (ITEM)

Modified program has been developed using improved variation of Encke method which avoids accumulation of round-off errors and avoids numerical ambiguities arising from near-circular orbits of low inclination. Variety of interplanetary trajectory problems can be computed with maximum accuracy and efficiency.

Whitlock, F. H.↗
NASA NTRS1974

The modified equation approach to the stability and accuracy analysis of finite-difference methods

The stability and accuracy of finite-difference approximations to simple linear partial differential equations are analyzed by studying the modified partial differential equation. Aside from round-off error, the modified equation represents the actual partial differential equation solved when a numerical solution is computed using a finite-difference equation. The modified equation is derived by first expanding each term of a difference scheme in a Taylor series and then eliminating time derivatives higher than first order by certain algebraic manipulations. The connection between 'heuristic' stability theory based on the modified equation approach and the von Neumann (Fourier) method is established. In addition to the determination of necessary and sufficient conditions for computational stability, a truncated version of the modified equation can be used to gain insight into the nature of both dissipative and dispersive errors.

Warming, R. F.↗
NASA NTRS1979

A wave-envelope of sound propagation in nonuniform circular ducts with compressible mean flows

An acoustic theory is developed to determine the sound transmission and attenuation through an infinite, hard-walled or lined circular duct carrying compressible, sheared, mean flows and having a variable cross section. The theory is applicable to large as well as small axial variations, as long as the mean flow does not separate. The technique is based on solving for the envelopes of the quasi-parallel acoustic modes that exist in the duct instead of solving for the actual wave, thereby reducing the computation time and the round-off error encountered in purely numerical techniques. The solution recovers the solution based on the method of multiple scales for slowly varying duct geometry. A computer program was developed based on the wave-envelope analysis for general mean flows. Results are presented for the reflection and transmission coefficients as well as the acoustic pressure distributions for a number of conditions: both straight and variable area ducts with and without liners and mean flows from very low to high subsonic speeds are considered.

Nayfeh, A. H.↗
NASA NTRS1980

Transmission of sound through nonuniform circular ducts with compressible mean flows

An acoustic theory is developed to determine the sound transmission and attenuation through an infinite hard-walled or lined circular duct carrying compressible, sheared mean flows and having a variable cross section. The theory is applicable to large as well as small axial variations, as long as the mean flow does not separate. The technique is based on solving for the envelopes of the quasi-parallel acoustic modes that exist in the duct instead of solving for the actual wave, thereby reducing the computation time and the round-off error encountered in purely numerical techniques. A number of test cases that demonstrate the flexibility of the program are included. Convergence of the transmission coefficients and the acoustic pressure profiles with an increasing number of modes is illustrated.

Nayfeh, A. H.↗
NASA NTRS1985

Intersection of three-dimensional geometric surfaces

Calculating the line of intersection between two three-dimensional objects and using the information to generate a third object is a key element in a geometry development system. Techniques are presented for the generation of three-dimensional objects, the calculation of a line of intersection between two objects, and the construction of a resultant third object. The objects are closed surfaces consisting of adjacent bicubic parametric patches using Bezier basis functions. The intersection determination involves subdividing the patches that make up the objects until they are approximately planar and then calculating the intersection between planes. The resulting straight-line segments are connected to form the curve of intersection. The polygons in the neighborhood of the intersection are reconstructed and put back into the Bezier representation. A third object can be generated using various combinations of the original two. Several examples are presented. Special cases and problems were encountered, and the method for handling them is discussed. The special cases and problems included intersection of patch edges, gaps between adjacent patches because of unequal subdivision, holes, or islands within patches, and computer round-off error.

Crisp, V. K.↗
NASA NTRS1986

Desingularization of periodic vortex sheet roll-up

An analytical approach is used in an attempt to model the evolution of a vortex sheet past the critical time by means of a desingularization method. Evolution of the sheet, which is embedded in a two-dimensional flow, is described by approximating the total circulation between a fixed material point and an imaginary point on the curve of the flow. A linear dispersion relation is defined which shows that the short wavelength modes of the sheet are not unstable and therefore do adversely effect the computations as artifacts of round-off errors. The desingularization approach is demonstrated to converge beyond the critical time for the vortex sheet. Application of the technique for the study of the vortex sheet shed from an elliptically loaded wing is indicated.

Krasny, R.↗
NASA NTRS1987

A numerical simulation to verify the stress-free growth of silicon crystal ribbon

Thermal stresses developed during the growth of silicon crystal ribbon have been shown to be negligible, thus eliminating residual stresses and dislocations, if the temperature profile satisfies a second-order partial differential equation inside the ribbon. This has been numerically verified through a finite element model, an outline of which is presented here. This model shows that, for homogeneous isotropic material with temperature independent thermal expansion coefficients, thermal stresses will vanish if the temperature profile satisfies the Laplacian. A comparison of stresses due to uniform and nonuniform temperature gradients in the plane of the ribbon is also presented. The strategies employed to control the round-off error and to validate the computer model are discussed.

Ray, Sujit K.↗
NASA NTRS1987

Three dimensional numerical scattering from ocean-like surfaces

A new method of calculating the electric field scattered from two-dimensional conducting rough surfaces is presented. This method uses the operator conjugate gradients technique and the extended boundary condition equation. This avoids the build-up of round-off errors encountered in the inversion of large matrices. It also avoids the singularities of the integral kernels usually encountered in the method of moments. Techniques for improving the rate of convergence by applying physical constraints are discussed. This method is presently being used to calculate the polarization signatures from simulated ocean-like surfaces.

Rodriguez, Ernesto↗
NASA NTRS1987

Computation of vortex sheet roll-up in the Trefftz plane

Two vortex-sheet evolution problems arising in aerodynamics are studied numerically. The approach is based on desingularizing the Cauchy principal value integral which defines the sheet's velocity. Numerical evidence is presented which indicates that the approach converges with respect to refinement in the mesh-size and the smoothing parameter. For elliptic loading, the computed roll-up is in good agreement with Kaden's asymptoic spiral at early times. Some aspects of the solution's instability to short-wavelength perturbations, for a small value of the smoothing parameter, are inferred by comparing calculations performed with different levels of computer round-off error. The tip vortices' deformation, due to their mutual interaction, is shown in a long-time calculation. Computations for a simulated fuselage-flap configuration show a complicated process of roll-up, deformation and interaction involving the tip vortex and the inboard neighboring vortices.

Krasny, Robert↗
NASA NTRS1989

On the equivalence of Gaussian elimination and Gauss-Jordan reduction in solving linear equations

A novel general approach to round-off error analysis using the error complexity concepts is described. This is applied to the analysis of the Gaussian Elimination and Gauss-Jordan scheme for solving linear equations. The results show that the two algorithms are equivalent in terms of our error complexity measures. Thus the inherently parallel Gauss-Jordan scheme can be implemented with confidence if parallel computers are available.

Tsao, Nai-Kuan↗
NASA NTRS1990

Simulation Of Unsteady, Viscous, Incompressible Flow

Method for numerical solution of Navier-Stokes equations of viscous, incompressible flow developed based on use of fractional-step procedure. Accurate to second order in both space and time. Attempt made to minimize Poisson-equation difficulties by choosing pressures at centers and volume fluxes across faces of computational cells as dependent variables instead of familiar Cartesian components of velocity. Choice ensures satisfaction of discrete equation of conservation of mass to within round-off errors in any coordinate system and has favorable effects on convergence properties.

Rosenfeld, Moshe↗
NASA NTRS1988

An alternative to Guyan reduction of finite-element models

Structural modeling is a key part of structural system identification for large space structures. Finite-element structural models are commonly used in practice because of their general applicability and availability. The initial models generated by using a standard computer program such as NASTRAN, ANSYS, SUPERB, STARDYNE, STRUDL, etc., generally contain tens of thousands of degrees of freedom. The models must be reduced for purposes of identification. Not only does the magnitude of the identification effort grow exponentially as a function of the number of degrees of freedom, but numerical procedures may also break down because of accumulated round-off errors. Guyan reduction is usually applied after a static condensation. Misapplication of Guyan reduction can lead to serious modeling errors. It is quite unfortunate and disappointing, since the accuracy of the original detailed finite-element model one tries very hard to achieve is lost by the reduction. First, why and how Guyan reduction always causes loss of accuracy is examined. An alternative approach is then introduced. The alternative can be thought of as an improvement of Guyan reduction, the Rayleigh-Ritz method, and in particular the recent algorithm of Wilson, Yuan, and Dickens. Unlike Guyan reduction, the use of the alternative does not need any special insight, experience, or skill for partitioning the structural degrees of freedom. In addition to model condensation, this alternative approach can also be used for predicting analytically, quickly, and economically, what are those structural modes that are excitable by a force actuator at a given trial location. That is, in the excitation of the structural modes for identification, it can be used for guiding the placement of the force actuators.

Lin, Jiguan Gene↗