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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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NASA NTRS2002

Variable Order and Distributed Order Fractional Operators

Many physical processes appear to exhibit fractional order behavior that may vary with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. This paper develops the concept of variable and distributed order fractional operators. Definitions based on the Riemann-Liouville definitions are introduced and behavior of the operators is studied. Several time domain definitions that assign different arguments to the order q in the Riemann-Liouville definition are introduced. For each of these definitions various characteristics are determined. These include: time invariance of the operator, operator initialization, physical realization, linearity, operational transforms. and memory characteristics of the defining kernels. A measure (m2) for memory retentiveness of the order history is introduced. A generalized linear argument for the order q allows the concept of "tailored" variable order fractional operators whose a, memory may be chosen for a particular application. Memory retentiveness (m2) and order dynamic behavior are investigated and applications are shown. The concept of distributed order operators where the order of the time based operator depends on an additional independent (spatial) variable is also forwarded. Several definitions and their Laplace transforms are developed, analysis methods with these operators are demonstrated, and examples shown. Finally operators of multivariable and distributed order are defined in their various applications are outlined.

Lorenzo, Carl F.↗
NASA NTRS1981

Free-Vibration Analysis of Rotating Beams by a Variable-Order Finite-Element Method

The free vibration of rotating beams is analyzed by means of a finite-element method of variable order. This method entails displacement functions that are a complete power series of a variable number of terms. The terms are arranged so that the generalized coordinates are composed of displacements and slopes at the element extremities and, additionally, displacements at certain points within the element. The displacement is assumed to be analytic within an element and thus can be approximated to any degree of accuracy desired by a complete power series. Numerical results are presented for uniform beams with zero and nonzero hub radii, tapered beams, and a nonuniform beam with discontinuities. Since the present method reduces to a conventional beam finite-element method for a cubic displacement function, the results are compared and found to be superior to the conventional results in terms of accuracy for a given number of degrees of freedom. Indeed, essentially exact eigenvalues and eigenvectors are obtained with this technique, which is far more rapidly convergent than other approaches in the literature.

Hodges, Dewey H.↗
NASA NTRS1993

A variable-order laminated plate theory based on the variational-asymptotical method

The variational-asymptotical method is a mathematical technique by which the three-dimensional analysis of laminated plate deformation can be split into a linear, one-dimensional, through-the-thickness analysis and a nonlinear, two-dimensional, plate analysis. The elastic constants used in the plate analysis are obtained from the through-the-thickness analysis, along with approximate, closed-form three-dimensional distributions of displacement, strain, and stress. In this paper, a theory based on this technique is developed which is capable of approximating three-dimensional elasticity to any accuracy desired. The asymptotical method allows for the approximation of the through-the-thickness behavior in terms of the eigenfunctions of a certain Sturm-Liouville problem associated with the thickness coordinate. These eigenfunctions contain all the necessary information about the nonhomogeneities along the thickness coordinate of the plate and thus possess the appropriate discontinuities in the derivatives of displacement. The theory is presented in this paper along with numerical results for the eigenfunctions of various laminated plates.

Lee, Bok W.↗
NASA NTRS1982

Comment on 'Time-optimal orbit transfer trajectory for solar sail spacecraft'

An analytic proof is presented to show that the orbital transfer times of an earth-to-Mars solar-sail propelled spacecraft trajectory as calculated by Jayaraman (1980) are incorrect. In particular, different boundary conditions are defined, which indicate that a minimization of the Hamiltonian, which Jayaraman used, can yield the wrong stationary solution. Transfer times are calculated using a neighboring extremal algorithm based on numerical differentiation in conjunction with Krogh's variable order, variable step size integrator, resulting in a transfer time of 322 days at 2 mm/sec-sq, with endpoint restraints satisfied to within 1/1 billion. Finally, it is concluded that minimization of flight time is secondary in importance to maximization of delivered payload and minimization of overall mission cost and risk.

Wood, L. J.↗
NASA NTRS2011

Rapid Calculation of Spacecraft Trajectories Using Efficient Taylor Series Integration

A variable-order, variable-step Taylor series integration algorithm was implemented in NASA Glenn's SNAP (Spacecraft N-body Analysis Program) code. SNAP is a high-fidelity trajectory propagation program that can propagate the trajectory of a spacecraft about virtually any body in the solar system. The Taylor series algorithm's very high order accuracy and excellent stability properties lead to large reductions in computer time relative to the code's existing 8th order Runge-Kutta scheme. Head-to-head comparison on near-Earth, lunar, Mars, and Europa missions showed that Taylor series integration is 15.8 times faster than Runge- Kutta on average, and is more accurate. These speedups were obtained for calculations involving central body, other body, thrust, and drag forces. Similar speedups have been obtained for calculations that include J2 spherical harmonic for central body gravitation. The algorithm includes a step size selection method that directly calculates the step size and never requires a repeat step. High-order Taylor series integration algorithms have been shown to provide major reductions in computer time over conventional integration methods in numerous scientific applications. The objective here was to directly implement Taylor series integration in an existing trajectory analysis code and demonstrate that large reductions in computer time (order of magnitude) could be achieved while simultaneously maintaining high accuracy. This software greatly accelerates the calculation of spacecraft trajectories. At each time level, the spacecraft position, velocity, and mass are expanded in a high-order Taylor series whose coefficients are obtained through efficient differentiation arithmetic. This makes it possible to take very large time steps at minimal cost, resulting in large savings in computer time. The Taylor series algorithm is implemented primarily through three subroutines: (1) a driver routine that automatically introduces auxiliary variables and sets up initial conditions and integrates; (2) a routine that calculates system reduced derivatives using recurrence relations for quotients and products; and (3) a routine that determines the step size and sums the series. The order of accuracy used in a trajectory calculation is arbitrary and can be set by the user. The algorithm directly calculates the motion of other planetary bodies and does not require ephemeris files (except to start the calculation). The code also runs with Taylor series and Runge-Kutta used interchangeably for different phases of a mission.

Scott, James R.↗
NASA NTRS1973

Numerical integration routines for near-earth operations

Two general purpose numerical integration schemes were built into the NASA-JSC computer system. The state-of-the-art of numerical integration, the particular integrators built into the JSC computer system, and the use of the new integration packages are described. Background information about numerical integration and the variable-order, variable-stepsize Adams numerical integration technique is discussed. Results concerning the PEACE parameter optimization program are given along with recommendations and conclusions.

Powers, W. F.↗
NASA NTRS1989

Focus of attention in an activity-based scheduler

Earlier research in job shop scheduling has demonstrated the advantages of opportunistically combining order-based and resource-based scheduling techniques. An even more flexible approach is investigated where each activity is considered a decision point by itself. Heuristics to opportunistically select the next decision point on which to focus attention (i.e., variable ordering heuristics) and the next decision to be tried at this point (i.e., value ordering heuristics) are described that probabilistically account for both activity precedence and resource requirement interactions. Preliminary experimental results indicate that the variable ordering heuristic greatly increases search efficiency. While least constraining value ordering heuristics have been advocated in the literature, the experimental results suggest that other value ordering heuristics combined with our variable-ordering heuristic can produce much better schedules without significantly increasing search.

Sadeh, Norman↗
NASA NTRS1992

JIGSAW: Preference-directed, co-operative scheduling

Techniques that enable humans and machines to cooperate in the solution of complex scheduling problems have evolved out of work on the daily allocation and scheduling of Tactical Air Force resources. A generalized, formal model of these applied techniques is being developed. It is called JIGSAW by analogy with the multi-agent, constructive process used when solving jigsaw puzzles. JIGSAW begins from this analogy and extends it by propagating local preferences into global statistics that dynamically influence the value and variable ordering decisions. The statistical projections also apply to abstract resources and time periods--allowing more opportunities to find a successful variable ordering by reserving abstract resources and deferring the choice of a specific resource or time period.

Linden, Theodore A.↗
NASA NTRS1995

KRESKA: A compression system for small and very large images

An effective lossless compression system for grayscale images is presented using finite context variable order Markov models. A new method to accurately estimate the probability of the escape symbol is proposed. The choice of the best model order and rules for selecting context pixels are discussed. Two context precision and two symbol precision techniques to handle noisy image data with Markov models are introduced. Results indicate that finite context variable order Markov models lead to effective lossless compression systems for small and very large images. The system achieves higher compression ratios than some of the better known image compression techniques such as lossless JPEG, JBIG, or FELICS.

Ohnesorge, Krystyna W.↗
NASA NTRS1987

Solving Ordinary Differential Equations

Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Krogh, F. T.↗
NASA NTRS2007

Variable High Order Multiblock Overlapping Grid Methods for Mixed Steady and Unsteady Multiscale Viscous Flows

Flows containing steady or nearly steady strong shocks in parts of the flow field, and unsteady turbulence with shocklets on other parts of the flow field are difficult to capture accurately and efficiently employing the same numerical scheme even under the multiblock grid or adaptive grid refinement framework. On one hand, sixth-order or higher shock-capturing methods are appropriate for unsteady turbulence with shocklets. On the other hand, lower order shock-capturing methods are more effective for strong steady shocks in terms of convergence. In order to minimize the shortcomings of low order and high order shock-capturing schemes for the subject flows,a multi- block overlapping grid with different orders of accuracy on different blocks is proposed. Test cases to illustrate the performance of the new solver are included.

Multiscale viscous flows↗
NASA NTRS1974

Comparison of numerical integration techniques for orbital applications

The present work gives a brief comparison of the performance of programs for integrating differential equations for orbital applications. The evaluation criteria and the method of testing are described, and the results of the test problem set are included. Integration methods that were chosen for comparison include high-order Runge-Kutta methods; a rational extrapolation method (Bulirsch and Stoer, 1966); a variable step, variable order, multistep method (Krogh, 1969); classical multistep methods of Adams and Cowell; and modified multistep methods. The high-order Runge-Kutta methods used in the comparison include RKF 7(8) and RKF 8(9) (Fehlberg, 1968), and RKS 8-10 (Shanks, 1966).

Moore, H.↗
NASA NTRS1993

Decision theory for computing variable and value ordering decisions for scheduling problems

Heuristics that guide search are critical when solving large planning and scheduling problems, but most variable and value ordering heuristics are sensitive to only one feature of the search state. One wants to combine evidence from all features of the search state into a subjective probability that a value choice is best, but there has been no solid semantics for merging evidence when it is conceived in these terms. Instead, variable and value ordering decisions should be viewed as problems in decision theory. This led to two key insights: (1) The fundamental concept that allows heuristic evidence to be merged is the net incremental utility that will be achieved by assigning a value to a variable. Probability distributions about net incremental utility can merge evidence from the utility function, binary constraints, resource constraints, and other problem features. The subjective probability that a value is the best choice is then derived from probability distributions about net incremental utility. (2) The methods used for rumor control in Bayesian Networks are the primary way to prevent cycling in the computation of probable net incremental utility. These insights lead to semantically justifiable ways to compute heuristic variable and value ordering decisions that merge evidence from all available features of the search state.

Linden, Theodore A.↗
NASA NTRS1982

Second order closure for variable density free shear layer

A full second order closure turbulence model for the prediction of free shear flows with variable density is developed. The importance of density fluctuations due to inhomogeneities in fluid composition is emphasized. Some simplicity is achieved by using density weighted averaging to account implicitly for density fluctuation effects. Transport equations for various second order correlations - such as Reynolds stress, turbulent mass fluxes or density velocity correlations - are derived, modelled and solved, together with the mean equations for momentum and species conservation. Solutions of these equations are compared with the two different sets of experimental data for low speed free mixing layers (helium into nitrogen and freon 12 into air).

Vandromme, D.↗