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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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All sourcesNASA NTRSDOE OSTI
NASA NTRS1981

Vibrations of twisted rotating blades

The literature dealing with vibrations of turbomachinery blades is voluminous, but the vast majority of it treats the blades as beams. In a previous paper a two-dimensional analytical procedure was developed and demonstrated on simple models of blades having camber. The procedure utilizes shallow shell theory along with the classical Ritz method for solving the vibration problem. Displacement functions are taken as algebraic polynomials. In the present paper the method is demonstrated on blade models having camber. Comparisons are first made with results in the literature for nonrotating twisted plates and various disagreements between results are pointed out. A method for depicting mode shape information is demonstrated, permitting one to examine all three components of displacement. Finally, the analytical procedure is demonstrated on rotating twisted blade modes, both without and with camber.

Leissa, A. W.↗
NASA NTRS1983

Acceleration of convergence of vector sequences

A general approach to the construction of convergence acceleration methods for vector sequence is proposed. Using this approach, one can generate some known methods, such as the minimal polynomial extrapolation, the reduced rank extrapolation, and the topological epsilon algorithm, and also some new ones. Some of the new methods are easier to implement than the known methods and are observed to have similar numerical properties. The convergence analysis of these new methods is carried out, and it is shown that they are especially suitable for accelerating the convergence of vector sequences that are obtained when one solves linear systems of equations iteratively. A stability analysis is also given, and numerical examples are provided. The convergence and stability properties of the topological epsilon algorithm are likewise given.

Sidi, A.↗
NASA NTRS1984

Fourier-Legendre spectral methods for incompressible channel flow

An iterative collocation technique is described for modeling implicit viscosity in three-dimensional incompressible wall bounded shear flow. The viscosity can vary temporally and in the vertical direction. Channel flow is modeled with a Fourier-Legendre approximation and the mean streamwise advection is treated implicitly. Explicit terms are handled with an Adams-Bashforth method to increase the allowable time-step for calculation of the implicit terms. The algorithm is applied to low amplitude unstable waves in a plane Poiseuille flow at an Re of 7500. Comparisons are made between results using the Legendre method and with Chebyshev polynomials. Comparable accuracy is obtained for the perturbation kinetic energy predicted using both discretizations.

Zang, T. A.↗
NASA NTRS1986

Acceleration of convergence of vector sequences

A general approach to the construction of convergence acceleration methods for vector sequence is proposed. Using this approach, one can generate some known methods, such as the minimal polynomial extrapolation, the reduced rank extrapolation, and the topological epsilon algorithm, and also some new ones. Some of the new methods are easier to implement than the known methods and are observed to have similar numerical properties. The convergence analysis of these new methods is carried out, and it is shown that they are especially suitable for accelerating the convergence of vector sequences that are obtained when one solves linear systems of equations iterative. A stability analysis is also given, and numerical examples are provided. The convergence and stability properties of the topological epsilon algorithm are likewise given.

Sidi, A.↗
NASA NTRS1989

On the parallel solution of parabolic equations

Parallel algorithms for the solution of linear parabolic problems are proposed. The first of these methods is based on using polynomial approximation to the exponential. It does not require solving any linear systems and is highly parallelizable. The two other methods proposed are based on Pade and Chebyshev approximations to the matrix exponential. The parallelization of these methods is achieved by using partial fraction decomposition techniques to solve the resulting systems and thus offers the potential for increased time parallelism in time dependent problems. Experimental results from the Alliant FX/8 and the Cray Y-MP/832 vector multiprocessors are also presented.

Gallopoulos, E.↗
NASA NTRS1991

A spectral collocation method for compressible, non-similar boundary layers

An efficient and highly accurate algorithm based on a spectral collocation method is developed for numerical solution of the compressible, two-dimensional and axisymmetric boundary layer equations. The numerical method incorporates a fifth-order, fully implicit marching scheme in the streamwise (timelike) dimension and a spectral collocation method based on Chebyshev polynomial expansions in the wall-normal (spacelike) dimension. The spectral collocation algorithm is used to derive the nonsimilar mean velocity and temperature profiles in the boundary layer of a 'fuselage' (cylinder) in a high-speed (Mach 5) flow parallel to its axis. The stability of the flow is shown to be sensitive to the gradual streamwise evolution of the mean flow and it is concluded that the effects of transverse curvature on stability should not be ignored routinely.

Pruett, C. D.↗
NASA NTRS1993

An approach for finding long period elliptical orbits for precursor SEI missions

Precursors for Solar System Exploration Initiative (SEI) missions may require long period elliptical orbits about a planet. These orbits will typically have periods on the order of tens to hundreds of days. Some potential uses for these orbits may include the following: studying the effects of galactic cosmic radiation, parking orbits for engineering and operational test of systems, and ferrying orbits between libration points and low altitude orbits. This report presents an approach that can be used to find these orbits. The approach consists of three major steps. First, it uses a restricted three-body targeting algorithm to determine the initial conditions which satisfy certain desired final conditions in a system of two massive primaries. Then the initial conditions are transformed to an inertial coordinate system for use by a special perturbation method. Finally, using the special perturbation method, other perturbations (e.g., sun third body and solar radiation pressure) can be easily incorporated to determine their effects on the nominal trajectory. An algorithm potentially suitable for on-board guidance will also be discussed. This algorithm uses an analytic method relying on Chebyshev polynomials to compute the desired position and velocity of the satellite as a function of time. Together with navigation updates, this algorithm can be implemented to predict the size and timing for AV corrections.

Fraietta, Michael F.↗
NASA NTRS1990

On well-partial-order theory and its application to combinatorial problems of VLSI design

We nonconstructively prove the existence of decision algorithms with low-degree polynomial running times for a number of well-studied graph layout, placement, and routing problems. Some were not previously known to be in p at all; others were only known to be in p by way of brute force or dynamic programming formulations with unboundedly high-degree polynomial running times. Our methods include the application of the recent Robertson-Seymour theorems on the well-partial-ordering of graphs under both the minor and immersion orders. We also briefly address the complexity of search versions of these problems.

Fellows, M.↗
NASA NTRS1992

Reformulating Non-Monotonic Theories for Inference and Updating

We aim to help build programs that do large-scale, expressive non-monotonic reasoning (NMR): especially, 'learning agents' that store, and revise, a body of conclusions while continually acquiring new, possibly defeasible, premise beliefs. Currently available procedures for forward inference and belief revision are exhaustive, and thus impractical: they compute the entire non-monotonic theory, then re-compute from scratch upon updating with new axioms. These methods are thus badly intractable. In most theories of interest, even backward reasoning is combinatoric (at least NP-hard). Here, we give theoretical results for prioritized circumscription that show how to reformulate default theories so as to make forward inference be selective, as well as concurrent; and to restrict belief revision to a part of the theory. We elaborate a detailed divide-and-conquer strategy. We develop concepts of structure in NM theories, by showing how to reformulate them in a particular fashion: to be conjunctively decomposed into a collection of smaller 'part' theories. We identify two well-behaved special cases that are easily recognized in terms of syntactic properties: disjoint appearances of predicates, and disjoint appearances of individuals (terms). As part of this, we also definitionally reformulate the global axioms, one by one, in addition to applying decomposition. We identify a broad class of prioritized default theories, generalizing default inheritance, for which our results especially bear fruit. For this asocially monadic class, decomposition permits reasoning to be localized to individuals (ground terms), and reduced to propositional. Our reformulation methods are implementable in polynomial time, and apply to several other NM formalisms beyond circumscription.

Grosof, Benjamin N.↗
NASA NTRS2004

Higher-Order Finite Elements for Computing Thermal Radiation

Two variants of the finite-element method have been developed for use in computational simulations of radiative transfers of heat among diffuse gray surfaces. Both variants involve the use of higher-order finite elements, across which temperatures and radiative quantities are assumed to vary according to certain approximations. In this and other applications, higher-order finite elements are used to increase (relative to classical finite elements, which are assumed to be isothermal) the accuracies of final numerical results without having to refine computational meshes excessively and thereby incur excessive computation times. One of the variants is termed the radiation sub-element (RSE) method, which, itself, is subject to a number of variations. This is the simplest and most straightforward approach to representation of spatially variable surface radiation. Any computer code that, heretofore, could model surface-to-surface radiation can incorporate the RSE method without major modifications. In the basic form of the RSE method, each finite element selected for use in computing radiative heat transfer is considered to be a parent element and is divided into sub-elements for the purpose of solving the surface-to-surface radiation-exchange problem. The sub-elements are then treated as classical finite elements; that is, they are assumed to be isothermal, and their view factors and absorbed heat fluxes are calculated accordingly. The heat fluxes absorbed by the sub-elements are then transferred back to the parent element to obtain a radiative heat flux that varies spatially across the parent element. Variants of the RSE method involve the use of polynomials to interpolate and/or extrapolate to approximate spatial variations of physical quantities. The other variant of the finite-element method is termed the integration method (IM). Unlike in the RSE methods, the parent finite elements are not subdivided into smaller elements, and neither isothermality nor other unrealistic physical conditions are assumed. Instead, the equations of radiative heat transfer are integrated numerically over the parent finite elements by use of a computationally efficient Gaussian integration scheme.

Gould, Dana C.↗
NASA NTRS2013

Entropy Stable Spectral Collocation Schemes for the Navier-Stokes Equations: Discontinuous Interfaces

Nonlinear entropy stability and a summation-by-parts framework are used to derive provably stable, polynomial-based spectral collocation methods of arbitrary order. The new methods are closely related to discontinuous Galerkin spectral collocation methods commonly known as DGFEM, but exhibit a more general entropy stability property. Although the new schemes are applicable to a broad class of linear and nonlinear conservation laws, emphasis herein is placed on the entropy stability of the compressible Navier-Stokes equations.

Carpenter, Mark H.↗
NASA NTRS2015

Kodiak: An Implementation Framework for Branch and Bound Algorithms

Recursive branch and bound algorithms are often used to refine and isolate solutions to several classes of global optimization problems. A rigorous computation framework for the solution of systems of equations and inequalities involving nonlinear real arithmetic over hyper-rectangular variable and parameter domains is presented. It is derived from a generic branch and bound algorithm that has been formally verified, and utilizes self-validating enclosure methods, namely interval arithmetic and, for polynomials and rational functions, Bernstein expansion. Since bounds computed by these enclosure methods are sound, this approach may be used reliably in software verification tools. Advantage is taken of the partial derivatives of the constraint functions involved in the system, firstly to reduce the branching factor by the use of bisection heuristics and secondly to permit the computation of bifurcation sets for systems of ordinary differential equations. The associated software development, Kodiak, is presented, along with examples of three different branch and bound problem types it implements.

Smith, Andrew P.↗
NASA NTRS2024

Quantification of Uncertainty and Risk Sensitivity for Safety of Emerging Operations

The growing need to develop and deploy small unmanned aerial vehicles (sUAVs) for various applications in the airspace necessitates reliable tools to accurately predict the flight trajectories of the sUAVs. The knowledge of the predicted trajectories help decision makers anticipate potential conflict, assess the risk, and take appropriate risk mitigation actions. In addition, uncertainties in vehicle models, weather, and controller action further highlights the need for reliable prediction tools. In this project, the application of mixed sparse grid-based quadrature and generalized polynomial chaos(gPC) expansion method for uncertainty quantification and collision assessment in air traffic consisting of fixed-wing small unmanned aerial vehicles (sUAV) was studied. From the results obtained, it can be concluded that this provides a reliable framework to carry out quantitative conflict assessment in an unmanned air traffic, which when employed, can improve the functionalities of the unmanned traffic management system. It was observed that the results from the gPC expansion framework developed in the project can be utilized to conduct rapid probabilistic collision assessment for near real-time unmanned traffic management in the airspace. From the vehicle models, position updates, and wind-field data, a priori gPC based 3-σcon-fidence ellipses can provide estimates of potential conflict at some future instants. The computational costs scaled linearly when the uncertain inputs were fewer. Further, the largest allowable distribution of para-metric uncertainties that leads to the smallest risk of collision in traffic of small unmanned aerial vehicles could be calculated. The time of closest approach between two sUAVs can be established paving way for development of proactive mitigation strategies. The separation between the sUAVs was found to be most significantly affected by uncertainties in the maximum available thrusts, zero-lift drag coefficients, and wing planform areas of the sUAVs. The study of uncertain wind-fields indicated that a heterogeneous traffic mix resulted in an increased probability of conflict. Increased measurement update rate reduced the uncertain-ties in the trajectories of the vehicles, further reducing the probability of conflict but rapid updates of all vehicles in the airspace poses a stringent communication limitation. The gPC framework also provided the means to analyze vehicle impact (crash region) due to loss of control resulting from actuator failure in sUAS traffic, essentially to predict impact and crash zones for representative vehicles. The predicted regions when compared with non-participant density, provides a means to develop an early mitigation strategy, should the sUAV detect an imminent actuator failure.

Rajnish Bhusal↗
NASA NTRS1985

Comparison of Two Algebraic Methods for Curve/curve Intersection

Most geometric modeling systems use either polynomial or rational functions to represent geometry. In such systems most computational problems can be formulated as systems of polynomials in one or more variables. Classical elimination theory can be used to solve such systems. Here Cayley's method of elimination is summarized and it is shown how it can best be used to solve the curve/curve intersection problem. Cayley's method was found to be a more straightforward approach. Furthermore, it is computationally simpler, since the elements of the Cayley matrix are one variable instead of two variable polynomials. Researchers implemented and tested both methods and found Cayley's to be more efficient. Six pairs of curves, representing mixtures of lines, circles, and cubic arcs were used. Several examples had multiple intersection points. For all six cases Cayley's required less CPU time than the other method. The average time ratio of method 1 to method 2 was 3.13:1, the least difference was 2.33:1, and the most dramatic was 6.25:1. Both of the above methods can be extended to solve the surface/surface intersection problem.

Demontaudouin, Y.↗
NASA NTRS1984

ADS: A FORTRAN program for automated design synthesis, version 1.00

A new general-purpose optimization program for engineering design is described. ADS-1 (Automated Design Synthesis - Version 1) is a FORTRAN program for solution of nonlinear constrained optimization problems. The program is segmented into three levels, being strategy, optimizer, and one-dimensional search. At each level, several options are available so that a total of over 100 possible combinations can be created. Examples of available strategies are sequential unconstrained minimization, the Augmented Lagrange Multiplier method, and Sequential Linear Programming. Available optimizers include variable metric methods and the Method of Feasible Directions as examples and one-dimensional search options include polynomial interpolation and the Golden Section method as examples. Emphasis is placed on ease of use of the program. All information is transferred via a single parameter list. Default values are provided for all internal program parameters such as convergence criteria, and the user is given a simple means to over-ride these, if desired. The program is demonstrated with a simple structural design example.

Vanderplaats, G. N.↗
NASA NTRS1984

A pseudospectral Legendre method for hyperbolic equations with an improved stability condition

A new pseudospectral method is introduced for solving hyperbolic partial differential equations. This method uses different grid points than previously used pseudospectral methods: in fact the grid are related to the zeroes of the Legendre polynomials. The main advantage of this method is that the allowable time step is proportional to the inverse of the number of grid points 1/N rather than to 1/n(2) (as in the case of other pseudospectral methods applied to mixed initial boundary value problems). A highly accurate time discretization suitable for these spectral methods is discussed.

Tal-Ezer, H.↗
NASA NTRS1985

ADS: A FORTRAN program for automated design synthesis: Version 1.10

A new general-purpose optimization program for engineering design is described. ADS (Automated Design Synthesis - Version 1.10) is a FORTRAN program for solution of nonlinear constrained optimization problems. The program is segmented into three levels: strategy, optimizer, and one-dimensional search. At each level, several options are available so that a total of over 100 possible combinations can be created. Examples of available strategies are sequential unconstrained minimization, the Augmented Lagrange Multiplier method, and Sequential Linear Programming. Available optimizers include variable metric methods and the Method of Feasible Directions as examples, and one-dimensional search options include polynomial interpolation and the Golden Section method as examples. Emphasis is placed on ease of use of the program. All information is transferred via a single parameter list. Default values are provided for all internal program parameters such as convergence criteria, and the user is given a simple means to over-ride these, if desired.

Vanderplaats, G. N.↗
DOE OSTI2025

Optimal Polynomial Smoothers and One‐Sided V‐Cycles for Poisson Problems

The solution to the Poisson equation arising from the spectral element discretization of the incompressible Navier‐Stokes equations needs robust preconditioning strategies. One such strategy is multigrid. To realize the potential of multigrid methods, effective smoothing strategies are needed. Chebyshev polynomial smoothers, in conjunction with pointwise Jacobi or additive Schwarz methods (ASMs), prove to be an effective smoother. Other polynomial smoothers, however, may provide superior convergence to the multigrid preconditioner. The authors compare the standard Chebyshev polynomial smoothers to both the novel fourth‐kind Chebyshev polynomial smoothers proposed by Lottes as well as smoothers based on the polynomial of best uniform approximation to as proposed by Kraus, Vassilevski, and Zikatanov. At the cost of symmetry, further improvements may be made. For example, a order polynomial smoother on both sides of the V‐cycle may be substituted with an order polynomial smoother on one side at no additional cost. The choice of omitting the postsmoother in favor of higher‐order polynomial presmoothing is advantageous in cases where the multigrid approximation property constant is large. The authors consider a 2D model problem based on finite differences to motivate the choice of polynomial smoother, order, and whether to apply postsmoothing for the target application of high‐order ‐geometric multigrid methods for GPU architectures. Results from both domains demonstrate the substantial improvement of these approaches over the standard Chebyshev polynomial smoother with a symmetric V‐cycle.

97 MATHEMATICS AND COMPUTING↗