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At least 1,081 records · Page 60
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NASA NTRS1975

Interpolation in numerical optimization

The present work discusses the generation of the cubic-spline interpolator in numerical optimization methods which use a variable-step integrator with step size control based on local relative truncation error. An algorithm for generating the cubic spline with successive over-relaxation is presented which represents an improvement over that given by Ralston and Wilf (1967). Rewriting the code reduces the number of N-vectors from eight to one. The algorithm is formulated in such a way that the solution of the linear system set up yields the first derivatives at the nodal points. This method is as accurate as other schemes but requires the minimum amount of storage.

Hall, K. R.↗
NASA NTRS1975

Finite element computation of a viscous compressible free shear flow governed by the time dependent Navier-Stokes equations

A finite element algorithm for solution of fluid flow problems characterized by the two-dimensional compressible Navier-Stokes equations was developed. The program is intended for viscous compressible high speed flow; hence, primitive variables are utilized. The physical solution was approximated by trial functions which at a fixed time are piecewise cubic on triangular elements. The Galerkin technique was employed to determine the finite-element model equations. A leapfrog time integration is used for marching asymptotically from initial to steady state, with iterated integrals evaluated by numerical quadratures. The nonsymmetric linear systems of equations governing time transition from step-to-step are solved using a rather economical block iterative triangular decomposition scheme. The concept was applied to the numerical computation of a free shear flow. Numerical results of the finite-element method are in excellent agreement with those obtained from a finite difference solution of the same problem.

Cooke, C. H.↗
NASA NTRS1975

Fully unsteady subsonic and supersonic potential aerodynamics for complex aircraft configurations with applications to flutter

A general formulation is presented for the analysis of steady and unsteady, subsonic and supersonic aerodynamics for complex aircraft configurations. The theoretical formulation, the numerical procedure, the description of the program SOUSSA (steady, oscillatory and unsteady, subsonic and supersonic aerodynamics) and numerical results are included. In particular, generalized forces for fully unsteady (complex frequency) aerodynamics for a wing-body configuration, AGARD wing-tail interference in both subsonic and supersonic flows as well as flutter analysis results are included. The theoretical formulation is based upon an integral equation, which includes completely arbitrary motion. Steady and oscillatory aerodynamic flows are considered. Here small-amplitude, fully transient response in the time domain is considered. This yields the aerodynamic transfer function (Laplace transform of the fully unsteady operator) for frequency domain analysis. This is particularly convenient for the linear systems analysis of the whole aircraft.

Tseng, K.↗
NASA NTRS1975

A discussion of dynamic stability measurement techniques

Techniques for the measurement of the dynamic stability of linear systems are discussed. Particular attention is given to an analysis of the errors in the procedures, and to methods for calculating the system damping from the data. The techniques discussed include: transient decay, moving block analysis, spectral analysis, random decrement signatures, transfer function analysis, and parameter identification methods. The special problems of rotorcraft dynamic stability testing are discussed.

Johnson, W.↗
NASA NTRS1976

Recursive partitioned inversion of large (1500 x 1500) symmetric matrices

A recursive algorithm was designed to invert large, dense, symmetric, positive definite matrices using small amounts of computer core, i.e., a small fraction of the core needed to store the complete matrix. The described algorithm is a generalized Gaussian elimination technique. Other algorithms are also discussed for the Cholesky decomposition and step inversion techniques. The purpose of the inversion algorithm is to solve large linear systems of normal equations generated by working geodetic problems. The algorithm was incorporated into a computer program called SOLVE. In the past the SOLVE program has been used in obtaining solutions published as the Goddard earth models.

Putney, B. H.↗
NASA NTRS1976

Optimization of multi-constrained structures based on optimality criteria

A weight-reduction algorithm is developed for the optimal design of structures subject to several multibehavioral inequality constraints. The structural weight is considered to depend linearly on the design variables. The algorithm incorporates a simple recursion formula derived from the Kuhn-Tucker necessary conditions for optimality, associated with a procedure to delete nonactive constraints based on the Gauss-Seidel iterative method for linear systems. A number of example problems is studied, including typical truss structures and simplified wings subject to static loads and with constraints imposed on stresses and displacements. For one of the latter structures, constraints on the fundamental natural frequency and flutter speed are also imposed. The results obtained show that the method is fast, efficient, and general when compared to other competing techniques. Extensions to the generality of the method to include equality constraints and nonlinear merit functions is discussed.

Rizzi, P.↗
NASA NTRS1976

Numerical solution of the state dependent noise problem

A numerical technique is given for solving the matrix quadratic equation that arises in the optimal stationary control of linear systems with state (and/or control) dependent noise. The technique exploits fully existing, efficient algorithms for the matrix Lyapunov and Ricatti equations. The computational requirements are discussed, with an associated example.

Kleinman, D. L.↗
NASA NTRS1976

Optimal command generation for tracking a class of discontinuous trajectories

Commands are found to drive a linear system to optimally track a class of prescribed trajectories, each of which contains a point of discontinuity. The paper focuses on the guidance problem of command generation, to be implemented in a feedforward fashion; the necessary additional control or feedback regulation structure is not studied in detail, but assumed to be provided in applications as a conventional error-feedback regulator. Solutions are found for the infinite-time problem which are optimal with respect to a quadratic performance criterion; suboptimal controls which satisfy a continuity condition are also found. The controls have applications to certain problems in aircraft guidance where command trajectories are piecewise continuous. Several examples are worked out in detail, with comparisons with conventional, nonfeedforward solutions to the problem, and a brief discussion of a simpler, suboptimal solution.

Weissenberger, S.↗
NASA NTRS1976

A variant of nested dissection for solving n by n grid problems

Nested dissection orderings are known to be very effective for solving the sparse positive definite linear systems which arise from n by n grid problems. In this paper nested dissection is shown to be the final step of incomplete nested dissection, an ordering which corresponds to the premature termination of dissection. Analyses of the arithmetic and storage requirements for incomplete nested dissection are given, and the ordering is shown to be competitive with nested dissection under certain conditions.

George, A.↗
NASA NTRS1976

Consideration of computer limitations in implementing on-line controls

A formal statement of the optimal control problem which includes the interval of dicretization as an optimization parameter, and extend this to include selection of a control algorithm as part of the optimization procedure, is formulated. The performance of the scalar linear system depends on the discretization interval. Discrete-time versions of the output feedback regulator and an optimal compensator, and the use of these results in presenting an example of a system for which fast partial-state-feedback control better minimizes a quadratic cost than either a full-state feedback control or a compensator, are developed.

Roberts, G. K.↗
NASA NTRS1977

A new algorithm for the integration of exponential and logarithmic functions

An algorithm for symbolic integration of functions built up from the rational functions by repeatedly applying either the exponential or logarithm functions is discussed. This algorithm does not require polynomial factorization nor partial fraction decomposition and requires solutions of linear systems with only a small number of unknowns. It is proven that if this algorithm is applied to rational functions over the integers, a computing time bound for the algorithm can be obtained which is a polynomial in a bound on the integer length of the coefficients, and in the degrees of the numerator and denominator of the rational function involved.

Rothstein, M.↗
NASA NTRS1976

On the design of optimal controllers with certain structural constraints

This paper considers the problem of designing certain structurally constrained optimal regulators for linear systems subjected to additive white process noise and measurement noise. Three types of controller structures are considered, using direct output feedback, prespecified time constant filters, and optimal dynamic compensators. Necessary conditions are obtained for minimizing quadratic performance criteria. The techniques are demonstrated by application to a helicopter/slung load system, and a flexible space station.

Joshi, S. M.↗
NASA NTRS1977

Routh's algorithm - A centennial survey

One hundred years have passed since the publication of Routh's fundamental work on determining the stability of constant linear systems. The paper presents an outline of the algorithm and considers such aspects of it as the distribution of zeros and applications of it that relate to the greatest common divisor, the abscissa of stability, continued fractions, canonical forms, the nonnegativity of polynomials and polynomial matrices, the absolute stability, optimality and passivity of dynamic systems, and the stability of two-dimensional circuits.

Barnett, S.↗
NASA NTRS1977

Effect of chordwise forces and deformations and deformations due to steady lift on wing flutter

This investigation explores the effects of chordwise forces and deformations and steady-state deformation due to lift on the static and dynamic aeroelastic stability of a uniform cantilever wing. Results of this analysis are believed to have practical applications for high-performance sailplanes and certain RPV's. The airfoil cross section is assumed to be symmetric and camber bending is neglected. Motions in vertical bending, fore-and-aft bending, and torsion are considered. A differential equation model is developed, which included the nonlinear elastic bending-torsion coupling that accompanies even moderate deflections. A linearized expansion in small time-dependent deflections is made about a steady flight condition. The stability determinant of the linearized system then contains coefficients that depend on steady displacements. Loads derived from two-dimensional incompressible aerodynamic theory are used to obtain the majority of the results, but cases using three-dimensional subsonic compressible theory are also studied. The stability analysis is carried out in terms of the dynamically uncoupled natural modes of vibration of the uniform cantilever.

Boyd, W. N.↗
NASA NTRS1978

On decentralized estimation

A multilevel scheme is proposed to construct decentralized estimators for large linear systems. The scheme is numerically attractive since only observability tests of low-order subsystems are required. Equally important is the fact that the constructed estimators are reliable under structural perturbations and can tolerate a wide range of nonlinearities in coupling among the subsystems.

Siljak, D. D.↗
NASA NTRS1978

A double shooting scheme for certain unstable and singular boundary value problems

A scheme is presented to obtain the unique bounded solution for an exponentially unstable linear system. The scheme consists of choosing random data at large initial values and integrating forwards and backwards until accurate regular boundary values are obtained. Proofs of convergence are given for the case that the homogeneous equation has an exponential dichotomy. Applications to other types of problems are discussed and numerical results are presented.

Bayliss, A.↗
NASA NTRS1978

On the optimization of discrete structures with aeroelastic constraints

The paper deals with the problem of dynamic structural optimization where constraints relating to flutter of a wing (or other dynamic aeroelastic performance) are imposed along with conditions of a more conventional nature such as those relating to stress under load, deflection, minimum dimensions of structural elements, etc. The discussion is limited to a flutter problem for a linear system with a finite number of degrees of freedom and a single constraint involving aeroelastic stability, and the structure motion is assumed to be a simple harmonic time function. Three search schemes are applied to the minimum-weight redesign of a particular wing: the first scheme relies on the method of feasible directions, while the other two are derived from necessary conditions for a local optimum so that they can be referred to as optimality-criteria schemes. The results suggest that a heuristic redesign algorithm involving an optimality criterion may be best suited for treating multiple constraints with large numbers of design variables.

Mcintosh, S. C., Jr.↗
NASA NTRS1978

Incomplete nested dissection for solving n by n grid problems

Nested dissection orderings are known to be very effective for solving sparse positive definite linear systems which arise from n by n grid problems. In this paper we consider incomplete nested dissection, an ordering which corresponds to the premature termination of nested dissection. Analyses of the arithmetic and storage requirements for incomplete nested dissection are given and the ordering is shown to be competitive with nested dissection with regard to arithmetic operations and superior to that ordering in storage requirements.

George, A.↗