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All sourcesNASA NTRSDOE OSTI
NASA NTRS1977

Stability of numerical integration techniques for transient rotor dynamics

A finite element model of a rotor bearing system was analyzed to determine the stability limits of the forward, backward, and centered Euler; Runge-Kutta; Milne; and Adams numerical integration techniques. The analysis concludes that the highest frequency mode determines the maximum time step for a stable solution. Thus, the number of mass elements should be minimized. Increasing the damping can sometimes cause numerical instability. For a uniform shaft, with 10 mass elements, operating at approximately the first critical speed, the maximum time step for the Runge-Kutta, Milne, and Adams methods is that which corresponds to approximately 1 degree of shaft movement. This is independent of rotor dimensions.

Kascak, A. F.↗
NASA NTRS1976

Conservation properties of numerical integration methods for systems of ordinary differential equations

If a system of ordinary differential equations represents a property conserving system that can be expressed linearly (e.g., conservation of mass), it is then desirable that the numerical integration method used conserve the same quantity. It is shown that both linear multistep methods and Runge-Kutta methods are 'conservative' and that Newton-type methods used to solve the implicit equations preserve the inherent conservation of the numerical method. It is further shown that a method used by several authors is not conservative.

Rosenbaum, J. S.↗
NASA NTRS1974

Inelastic, nonlinear analysis of stiffened shells of revolution by numerical integration

This paper describes the latest addition to the STARS system of computer programs, STARS-2P, for the plastic, large deflection analysis of axisymmetrically loaded shells of revolution. The STARS system uses a numerical integration scheme to solve the governing differential equations. Several unique features for shell of revolution programs that are included in the STARS-2P program are described. These include orthotropic nonlinear kinematic hardening theory, a variety of shell wall cross sections and discrete ring stiffeners, cyclic and nonproportional mechanical and thermal loading capability, the coupled axisymmetric large deflection elasto-plastic torsion problem, an extensive restart option, arbitrary branching capability, and the provision for the inelastic treatment of smeared stiffeners, isogrid, and waffle wall constructions. To affirm the validity of the results, comparisons with available theoretical and experimental data are presented.

Levine, H. S.↗
NASA NTRS1973

Feasibility study of the numerical integration of shell equations using the field method

The field method is developed for arbitrary open branch domains subjected to general linear boundary conditions. Although closed branches are within the scope of the method, they are not treated here. The numerical feasibility of the method has been demonstrated by implementing it in a computer program for the linear static analysis of open branch shells of revolution under asymmetric loads. For such problems the field method eliminates the well-known numerical problem of long subintervals associated with the rapid growth of extraneous solutions. Also, the method appears to execute significantly faster than other numerical integration methods.

Cohen, G. A.↗
NASA NTRS1975

New methods for the numerical integration of ordinary differential equations and their application to the equations of motion of spacecraft

A number of hybrid methods for solving Cauchy problems are described on the basis of an evaluation of advantages of single and multiple-point numerical integration methods. The selection criterion is the principle of minimizing computer time. The methods discussed include the Nordsieck method, the Bulirsch-Stoer extrapolation method, and the method of recursive Taylor-Steffensen power series.

Banyukevich, A.↗
NASA NTRS1993

Comparison of symbolic and numerical integration methods for an assumed-stress hybrid shell element

Hybrid shell elements have long been regarded with reserve by the commercial finite element developers despite the high degree of reliability and accuracy associated with such formulations. The fundamental reason is the inherent higher computational cost of the hybrid approach as compared to the displacement-based formulations. However, a noteworthy factor in favor of hybrid elements is that numerical integration to generate element matrices can be entirely avoided by the use of symbolic integration. In this paper, the use of the symbolic computational approach is presented for an assumed-stress hybrid shell element with drilling degrees of freedom and the significant time savings achieved is demonstrated through an example.

Rengarajan, Govind↗
NASA NTRS1983

DE 102 - A numerically integrated ephemeris of the moon and planets spanning forty-four centuries

It is pointed out that the 1960's were the turning point for the generation of lunar and planetary ephemerides. All previous measurements of the positions of solar system bodies were optical angular measurements. New technological improvements leading to immense changes in observational accuracy are related to developments concerning radar, Viking landers on Mars, and laser ranges to lunar corner cube retroreflectors. Suitable numerical integration techniques and more comprehensive physical models were developed to match the accuracy of the modern data types. The present investigation is concerned with the first integrated ephemeris, DE 102, which covers the entire span of the historical astronomical observations of usable accuracy which are known. The fit is made to modern data. The integration spans the time period from 1411 BC to 3002 AD.

Newhall, X. X.↗
NASA NTRS1975

The use of back corrections in multistep methods of numerical integration

Generalized multistep methods for the numerical solution of nonlinear systems of ordinary differential equations are introduced which allow the correction of previously computed values of the solution at internal points of the grid. These methods are shown to possess enhanced numerical stability. Preliminary numerical results indicate that for some satellite orbit problems these methods also possess greater overall efficiency. A uniformly converging theory of error propagation is presented which is valid for nonasymptotic values of the step size. Experimental results are seen to conform with theory.

Beaudet, P.↗
NASA NTRS1973

Numerical integration of shell equations using the field method.

The 'field method' for the numerical solution of even-order linear boundary-value problems in ordinary differential equations is formulated. This method converts the boundary-value problem into two successive initial-value problems, which may be solved by standard forward integration techniques. The method has been implemented in a computer program to calculate the static response of ring-stiffened branched shells of revolution to asymmetric loads. For such problems the field method eliminates the well-known numerical problem of 'long subintervals' and also executes operations significantly faster than other numerical integration methods.

Cohen, G. A.↗
NASA NTRS1976

Multistep methods of numerical integration using back-corrections

A class of linear multistep methods is proposed for the solution of the equations of motion of certain dynamical systems encountered in celestial mechanics and astrodynamics. These methods are distinguished from the classical predictor-corrector methods in that they permit 'back-corrections' of the solution to be made. As the integration advances in time, the numerical solution is corrected or improved at certain points in the past. The enhanced numerical stability of these methods allows the meaningful application of high-order algorithms. Consequently, step sizes larger than those attainable with the classical methods may be adopted, and greater overall efficiency may be realized. These methods are applied to the problem of determining the orbit of an artificial satellite, and the results are compared with those obtained using classical methods.

Feagin, T.↗
NASA NTRS1988

An improvement in the numerical integration procedure used in the NASA Marshall engineering thermosphere model

A proposed replacement scheme for the integration of the barometric and diffusion equations in the NASA Marshall Engineering Thermosphere (MET) model is presented. This proposed integration scheme is based on Gaussian Quadrature. Extensive numerical testing reveals it to be faster, more accurate and more reliable than the present integration scheme (a modified form of Simpson's Rule) used in the MET model. Numerous graphical examples are provided, along with a listing of a modified form of the MET model in which subroutine INTEGRATE (using Simpson's Rule) is replaced by subroutine GAUSS (which uses Gaussian Quadrature). It is recommended that the Gaussian Quadrature integration scheme, as used here, be used in the MET model.

Hickey, Michael Philip↗
NASA NTRS1972

A method for direct numerical integration of the Boltzmann equation

The principal difficulties in numerical solution of the Boltzmann equation are considered. The study is aimed at formulating a numerical solution in such a manner that it contains a minimum amount of excess information at the distribution function level. It is pointed out that the accurate calculation of the distribution function at each point in phase space requires a tremendous number of operations, due to the necessity of solving five-fold quadratures in the collision integral. This results in the operational memory of the digital computer being insufficient to store all the data on the distribution functions at the necessary points in phase space. An algorithm is constructed involving successive iterations of the Boltzmann equation which does not require storage of each step of the new distribution function.

Cheremisin, F. G.↗
NASA NTRS1966

Numerical integration of periodic orbits in the elliptic restricted three-body problem

The problem of finding periodic orbits in the circular restricted three-body problem has been very extensively studied in celestial mechanics. It is well known that continuous families of periodic orbits exist, for which the period varies in a continuous way. However, all the applications which are found in the solar system correspond to cases with non-zero eccentricities, and the elliptic restricted three-body problem is thus a better approximation than the circular one. For instance, for the motion of a satellite in the Earth-Moon system, as a first approximation, we may assume that the Moon moves around the Earth in circular motion; but as a much better approximation, we can also assume that the Moon moves in an elliptic orbit around the Earth.

NUMERICAL INTEGRATION↗
NASA NTRS1965

Numerical integration of kinetic equations.

Relaxation to Maxwellian of Balescu-Lenard equation solved numerically as initial value problem for isotropic distributions of electrons in background of positive charge

MAXWELL DISTRIBUTION↗
NASA NTRS1975

A technique for accelerating iterative convergence in numerical integration, with application in transonic aerodynamics

A technique is described for the efficient numerical solution of nonlinear partial differential equations by rapid iteration. In particular, a special approach is described for applying the Aitken acceleration formula (a simple Pade approximant) for accelerating the iterative convergence. The method finds the most appropriate successive approximations, which are in a most nearly geometric sequence, for use in the Aitken formula. Simple examples are given to illustrate the use of the method. The method is then applied to the mixed elliptic-hyperbolic problem of steady, inviscid, transonic flow over an airfoil in a subsonic free stream.

Martin, E. D.↗
NASA NTRS1976

A technique for accelerating iterative convergence in numerical integration, with application in transonic aerodynamics

A technique is described for the efficient numerical solution of nonlinear partial differential equations by rapid iteration. In particular, a special approach is described for applying the Aitken acceleration formula (a simple Pade approximant) for accelerating the iterative convergence. The method finds the most appropriate successive approximations, which are in a most nearly geometric sequence, for use in the Aitken formula. Simple examples are given to illustrate the use of the method. The method is then applied to the mixed elliptic-hyperbolic problem of steady, inviscid, transonic flow over an airfoil in a subsonic free stream.

Martin, E. D.↗