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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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At least 91 records · Page 5
All sourcesNASA NTRSDOE OSTI
NASA NTRS1977

Aeroelastic stability of complete rotors with application to a teetering rotor in forward flight

The derivation of a set of non-linear coupled flap-lag-torsion equations of motion for moderately large deflections of an elastic, two-bladed teetering helicopter rotor in forward flight is concisely outlined. The following degrees of freedom are included in the mathematical model: rigid body flapping, rigid body lead-lag, elastic bending in flap and lead-lag, blade root torsion, and shaft torsion. Quasi-steady aerodynamic loads are considered and the effects of reversed flow are included. The aeroelastic stability of the complete rotor is investigated by using a linearized system of equations of motion. The equilibrium position about which the equations are linearized is obtained by considering the trim state of the helicopter, in true or simulated forward flight conditions. The sensitivity of the aeroelastic stability boundaries to interblade structural and mechanical coupling is illustrated by comparing the complete rotor stability boundaries with those obtained from a single blade analysis for a number of hover and forward flight cases.

Shamie, J.↗
NASA NTRS1974

Parallel tridiagonal equation solvers

Three parallel algorithms were compared for the direct solution of tridiagonal linear systems of equations. The algorithms are suitable for computers such as ILLIAC 4 and CDC STAR. For array computers similar to ILLIAC 4, cyclic odd-even reduction has the least operation count for highly structured sets of equations, and recursive doubling has the least count for relatively unstructured sets of equations. Since the difference in operation counts for these two algorithms is not substantial, their relative running times may be more related to overhead operations, which are not measured in this paper. The third algorithm, based on Buneman's Poisson solver, has more arithmetic operations than the others, and appears to be the least favorable. For pipeline computers similar to CDC STAR, cyclic odd-even reduction appears to be the most preferable algorithm for all cases.

Stone, H. S.↗
NASA NTRS1989

An interpretation and solution of ill-conditioned linear equations

Data insufficiency, poorly conditioned matrices and singularities in equations occur regularly in complex optimization, correlation, and interdisciplinary model studies. This work concerns itself with two methods of obtaining certain physically realistic solutions to ill-conditioned or singular algebraic systems of linear equations arising from such studies. Two efficient computational solution procedures that generally lead to locally unique solutions are presented when there is insufficient data to completely define the model, or a least-squares error formulation of this system results in an ill-conditioned system of equations. If it is assumed that a reasonable estimate of the uncertain data is available in both cases cited above, then we shall show how to obtain realistic solutions efficiently, in spite of the insufficiency of independent data. The proposed methods of solution are more efficient than singular-value decomposition for dealing with such systems, since they do not require solutions for all the non-zero eigenvalues of the coefficient matrix.

Ojalvo, I. U.↗
NASA NTRS1981

On a fourth order accurate implicit finite difference scheme for hyperbolic conservation laws. II - Five-point schemes

This paper presents a family of two-level five-point implicit schemes for the solution of one-dimensional systems of hyperbolic conservation laws, which generalized the Crank-Nicholson scheme to fourth order accuracy (4-4) in both time and space. These 4-4 schemes are nondissipative and unconditionally stable. Special attention is given to the system of linear equations associated with these 4-4 implicit schemes. The regularity of this system is analyzed and efficiency of solution-algorithms is examined. A two-datum representation of these 4-4 implicit schemes brings about a compactification of the stencil to three mesh points at each time-level. This compact two-datum representation is particularly useful in deriving boundary treatments. Numerical results are presented to illustrate some properties of the proposed scheme.

Harten, A.↗
NASA NTRS2001

A Curved, Elastostatic Boundary Element for Plane Anisotropic Structures

The plane-stress equations of linear elasticity are used in conjunction with those of the boundary element method to develop a novel curved, quadratic boundary element applicable to structures composed of anisotropic materials in a state of plane stress or plane strain. The curved boundary element is developed to solve two-dimensional, elastostatic problems of arbitrary shape, connectivity, and material type. As a result of the anisotropy, complex variables are employed in the fundamental solution derivations for a concentrated unit-magnitude force in an infinite elastic anisotropic medium. Once known, the fundamental solutions are evaluated numerically by using the known displacement and traction boundary values in an integral formulation with Gaussian quadrature. All the integral equations of the boundary element method are evaluated using one of two methods: either regular Gaussian quadrature or a combination of regular and logarithmic Gaussian quadrature. The regular Gaussian quadrature is used to evaluate most of the integrals along the boundary, and the combined scheme is employed for integrals that are singular. Individual element contributions are assembled into the global matrices of the standard boundary element method, manipulated to form a system of linear equations, and the resulting system is solved. The interior displacements and stresses are found through a separate set of auxiliary equations that are derived using an Airy-type stress function in terms of complex variables. The capabilities and accuracy of this method are demonstrated for a laminated-composite plate with a central, elliptical cutout that is subjected to uniform tension along one of the straight edges of the plate. Comparison of the boundary element results for this problem with corresponding results from an analytical model show a difference of less than 1%.

Smeltzer, Stanley S.↗
NASA NTRS1985

Parallel, iterative solution of sparse linear systems - Models and architectures

Solving large, sparse, linear systems of equations is a fundamental problem in large scale scientific and engineering computation. A model of a general class of asynchronous, iterative solution methods for linear systems is developed. In the model, the system is solved by creating several cooperating tasks that each compute a portion of the solution vector. A data transfer model predicting both the probability that data must be transferred between two tasks and the amount of data to be transferred is presented. This model is used to derive an execution time model for predicting parallel execution time and an optimal number of tasks given the dimension and sparsity of the coefficient matrix and the costs of computation, synchronization, and communication. The suitability of different parallel architectures for solving randomly sparse linear systems is discussed. Based on the complexity of task scheduling, one parallel architecture, based on a broadcast bus, is presented and analyzed.

Reed, D. A.↗
NASA NTRS1992

Formally biorthogonal polynomials and a look-ahead Levinson algorithm for general Toeplitz systems

Systems of linear equations with Toeplitz coefficient matrices arise in many important applications. The classical Levinson algorithm computes solutions of Toeplitz systems with only O(n(sub 2)) arithmetic operations, as compared to O(n(sub 3)) operations that are needed for solving general linear systems. However, the Levinson algorithm in its original form requires that all leading principal submatrices are nonsingular. An extension of the Levinson algorithm to general Toeplitz systems is presented. The algorithm uses look-ahead to skip over exactly singular, as well as ill-conditioned leading submatrices, and, at the same time, it still fully exploits the Toeplitz structure. In our derivation of this algorithm, we make use of the intimate connection of Toeplitz matrices with formally biorthogonal polynomials.

Freund, Roland W.↗
NASA NTRS1976

The effect of adhesive layer on crack propagation in laminates

The effect of the adhesive layer on crack propagation in composite materials is investigated. The composite medium consists of parallel load carrying laminates and buffer strips arranged periodically and bonded with thin adhesive layers. The strips, assumed to be isotropic and linearly elastic, contain symmetric cracks of arbitrary lengths located normal to the interfaces. Two problems are considered: (1) thin adhesive layers are approximated by uncoupled tension and shear springs distributed along the interfaces of the strips for which only the case of internal cracks can be treated rigorously; (2) broken laminates and the true singular behavior in the presence of the adhesive layer are studied. The adhesive is then treated as an isotropic, linearly elastic continuum. General expressions for field quantities are obtained in terms of infinite Fourier integrals. These expressions give a system of singular integral equations in terms of the crack surface displacement derivatives. By using appropriate quadrature formulas, the integral equations reduce to a system of linear algebraic equations which are solved numerically.

Gecit, M. R.↗
NASA NTRS1984

A model of asynchronous iterative algorithms for solving large, sparse, linear systems

Solving large, sparse, linear systems of equations is one of the fundamental problems in large scale scientific and engineering computation. A model of a general class of asynchronous, iterative solution methods for linear systems is developed. In the model, the system is solved by creating several cooperating tasks that each compute a portion of the solution vector. This model is then analyzed to determine the expected intertask data transfer and task computational complexity as functions of the number of tasks. Based on the analysis, recommendations for task partitioning are made. These recommendations are a function of the sparseness of the linear system, its structure (i.e., randomly sparse or banded), and dimension.

Reed, D. A.↗
NASA NTRS1976

Finite difference procedure for boundary layers including effects of longitudinal and transverse curvatures

A second order viscous layer solution procedure has been developed that does in a consistent way include curvature effects and the corresponding normal pressure gradients. In the present system, the normal momentum equation is retained. The parabolic system of nonlinear partial differential equations is converted by linear finite differencing procedures to a system of linear algebraic equations and solved in primitive coordinates. The solutions have been shown to give smooth stable distributions for all the variables, most particularly the normal velocity which plays an important role in the interaction procedure. An algorithm for matching the viscous layer solution with a rotational characteristics outer solution has been developed.

Tassa, Y.↗
NASA NTRS1972

Interpolation using surface splines.

A surface spline is a mathematical tool for interpolating a function of two variables. It is based upon the small deflection equation of an infinite plate. The surface spline depends upon the solution of a system of linear equations, and thus, will ordinarily require the use of a digital computer. The closed form solution involves no functions more complicated than logarithms, and is easily coded. Several modifications which can be incorporated are discussed.

Harder, R. L.↗
NASA NTRS1989

Parallel solution of pentadiagonal systems using generalized odd-even elimination

A method for the solution of pentadiagonal systems of linear equations is presented. The method is a generalization of ordinary odd-even elimination used for tridiagonal systems. Using n processors, an n x n pentadiagonal system can be solved using the new method (generalized odd-even elimination) in time proportional to log(2) n.

Levit, Creon↗
NASA NTRS1989

A method for exponential propagation of large systems of stiff nonlinear differential equations

A new time integrator for large, stiff systems of linear and nonlinear coupled differential equations is described. For linear systems, the method consists of forming a small (5-15-term) Krylov space using the Jacobian of the system and carrying out exact exponential propagation within this space. Nonlinear corrections are incorporated via a convolution integral formalism; the integral is evaluated via approximate Krylov methods as well. Gains in efficiency ranging from factors of 2 to 30 are demonstrated for several test problems as compared to a forward Euler scheme and to the integration package LSODE.

Friesner, Richard A.↗
NASA NTRS1979

Lyapunov stability and its application to systems of ordinary differential equations

An outline and a brief introduction to some of the concepts and implications of Lyapunov stability theory are presented. Various aspects of the theory are illustrated by the inclusion of eight examples, including the Cartesian coordinate equations of the two-body problem, linear and nonlinear (Van der Pol's equation) oscillatory systems, and the linearized Kustaanheimo-Stiefel element equations for the unperturbed two-body problem.

Kennedy, E. W.↗
NASA NTRS1987

Automated problem scheduling and reduction of synchronization delay effects

It is anticipated that in order to make effective use of many future high performance architectures, programs will have to exhibit at least a medium grained parallelism. A framework is presented for partitioning very sparse triangular systems of linear equations that is designed to produce favorable preformance results in a wide variety of parallel architectures. Efficient methods for solving these systems are of interest because: (1) they provide a useful model problem for use in exploring heuristics for the aggregation, mapping and scheduling of relatively fine grained computations whose data dependencies are specified by directed acrylic graphs, and (2) because such efficient methods can find direct application in the development of parallel algorithms for scientific computation. Simple expressions are derived that describe how to schedule computational work with varying degrees of granularity. The Encore Multimax was used as a hardware simulator to investigate the performance effects of using the partitioning techniques presented in shared memory architectures with varying relative synchronization costs.

Saltz, Joel H.↗
NASA NTRS1984

Two-dimensional computer simulation of EMVJ and grating solar cells under AMO illumination

A computer program, SCAP2D (Solar Cell Analysis Program in 2-Dimensions), is used to evaluate the Etched Multiple Vertical Junction (EMVJ) and grating solar cells. The aim is to demonstrate how SCAP2D can be used to evaluate cell designs. The cell designs studied are by no means optimal designs. The SCAP2D program solves the three coupled, nonlinear partial differential equations, Poisson's Equation and the hole and electron continuity equations, simultaneously in two-dimensions using finite differences to discretize the equations and Newton's Method to linearize them. The variables solved for are the electrostatic potential and the hole and electron concentrations. Each linear system of equations is solved directly by Gaussian Elimination. Convergence of the Newton Iteration is assumed when the largest correction to the electrostatic potential or hole or electron quasi-potential is less than some predetermined error. A typical problem involves 2000 nodes with a Jacobi matrix of order 6000 and a bandwidth of 243.

Gray, J. L.↗
NASA NTRS1993

Preconditioned conjugate-gradient methods for low-speed flow calculations

An investigation is conducted into the viability of using a generalized Conjugate Gradient-like method as an iterative solver to obtain steady-state solutions of very low-speed fluid flow problems. Low-speed flow at Mach 0.1 over a backward-facing step is chosen as a representative test problem. The unsteady form of the two dimensional, compressible Navier-Stokes equations is integrated in time using discrete time-steps. The Navier-Stokes equations are cast in an implicit, upwind finite-volume, flux split formulation. The new iterative solver is used to solve a linear system of equations at each step of the time-integration. Preconditioning techniques are used with the new solver to enhance the stability and convergence rate of the solver and are found to be critical to the overall success of the solver. A study of various preconditioners reveals that a preconditioner based on the Lower-Upper Successive Symmetric Over-Relaxation iterative scheme is more efficient than a preconditioner based on Incomplete L-U factorizations of the iteration matrix. The performance of the new preconditioned solver is compared with a conventional Line Gauss-Seidel Relaxation (LGSR) solver. Overall speed-up factors of 28 (in terms of global time-steps required to converge to a steady-state solution) and 20 (in terms of total CPU time on one processor of a CRAY-YMP) are found in favor of the new preconditioned solver, when compared with the LGSR solver.

Ajmani, Kumud↗