Generalized relaxation methods applied to problems in transonic flow
Generalized relaxation methods application to transonic flow problems, combining with numerical integration theory for ordinary differential equations
Lomax, H.↗
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Algorithm for Liapunov stability analysis
Development of algorithm provides automatic computation of quadratic estimate of domain of stability for stable equilibrium states of nonlinear systems of ordinary differential equations.
Geiss, G. R.↗
Dissipative systems
The abstract theory presented shows how the theory of dissipative systems of ordinary differential equations can be extended to include a wide class of functional and partial differential equations. Since the basic hypotheses are all in terms of boundedness, finding sufficient conditions in terms of Liapunov functions was not difficult. Work is being undertaken to solve some nontrivial examples to illustrate how the theory can applied.
Lasalle, J. P.↗
An indirect optimization method with improved convergence characteristics
A method is presented for obtaining numerical solutions to a certain class of two-point boundary value problems which arise in optimal control theory. The problems are characterized by systems of nonlinear ordinary differential equations with nonlinear boundary conditions.
Doiron, H. H.↗
Nonlinear axisymmetric flexural vibration of spherical shells
Axisymmetric responses are presented of a nonshallow thin-walled spherical shell on the basis of nonlinear bending theory. An ordinary differential equation with nonlinearity of quadratic as well as cubic terms associated with variable time is derived. The derivation is based on the assumption that the deflection mode is the sum of four Legendre polynomials, and the Galerkin procedure is applied. The equation is solved by asymptotic expansion, and a first approximate solution is adopted. Unstable regions of this solution are discussed.
Kunieda, H.↗
Applications of the theory of optimal control of distributed-parameter systems to structural optimization
An extension of classical methods of optimal control theory for systems described by ordinary differential equations to distributed-parameter systems described by partial differential equations is presented. An application is given involving the minimum-mass design of a simply-supported shear plate with a fixed fundamental frequency of vibration. An optimal plate thickness distribution in analytical form is found. The case of a minimum-mass design of an elastic sandwich plate whose fundamental frequency of free vibration is fixed. Under the most general conditions, the optimization problem reduces to the solution of two simultaneous partial differential equations involving the optimal thickness distribution and the modal displacement. One equation is the uniform energy distribution expression which was found by Ashley and McIntosh for the optimal design of one-dimensional structures with frequency constraints, and by Prager and Taylor for various design criteria in one and two dimensions. The second equation requires dynamic equilibrium at the preassigned vibration frequency.
Armand, J. P.↗
IBM system/360 assembly language interval arithmetic software
Computer software designed to perform interval arithmetic is described. An interval is defined as the set of all real numbers between two given numbers including or excluding one or both endpoints. Interval arithmetic consists of the various elementary arithmetic operations defined on the set of all intervals, such as interval addition, subtraction, union, etc. One of the main applications of interval arithmetic is in the area of error analysis of computer calculations. For example, it has been used sucessfully to compute bounds on sounding errors in the solution of linear algebraic systems, error bounds in numerical solutions of ordinary differential equations, as well as integral equations and boundary value problems. The described software enables users to implement algorithms of the type described in references efficiently on the IBM 360 system.
Phillips, E. J.↗
A study of nozzle and ejector flow problems by the method of integral relations
The application of the method of integral relations to nozzle and ejector flow problems was examined. For nozzle flow problems, the general formulation is that the approaching flow may be rotational. Particular attention was given to the phenomenon of choking under nonuniform flow conditions. Numerical integration of the governing ordinary differential equations was also investigated. This scheme of analysis was also applied to study the interacting flow field within an ejector system.
Howlett, L. D.↗
Dissipative systems.
A theory is presented that shows how the concept of dissipative systems of ordinary differential equations can be extended to include a broad class of functional and partial differential equations, such as retarded functional differential equations and parabolic partial differential equations. Since the basic hypotheses are all in terms of boundedness, finding sufficient conditions in terms of Liapunov functions would not be too difficult.
Lasalle, J. P.↗
Control theory for random systems
A survey is presented of the current knowledge available for designing and predicting the effectiveness of controllers for dynamic systems which can be modeled by ordinary differential equations. A short discussion of feedback control is followed by a description of deterministic controller design and the concept of system state. The need for more realistic disturbance models led to the use of stochastic process concepts, in particular the Gauss-Markov process. A compensator controlled system, with random forcing functions, random errors in the measurements, and random initial conditions, is treated as constituting a Gauss-Markov random process; hence the mean-square behavior of the controlled system is readily predicted. As an example, a compensator is designed for a helicopter to maintain it in hover in a gusty wind over a point on the ground.
Bryson, A. E., Jr.↗
Analysis and optimization of cyclic methods in orbit computation
The mathematical analysis and computation of the K=3, order 4; K=4, order 6; and K=5, order 7 cyclic methods and the K=5, order 6 Cowell method and some results of optimizing the 3 backpoint cyclic multistep methods for solving ordinary differential equations are presented. Cyclic methods have the advantage over traditional methods of having higher order for a given number of backpoints while at the same time having more free parameters. After considering several error sources the primary source for the cyclic methods has been isolated. The free parameters for three backpoint methods were used to minimize the effects of some of these error sources. They now yield more accuracy with the same computing time as Cowell's method on selected problems. This work is being extended to the five backpoint methods. The analysis and optimization are more difficult here since the matrices are larger and the dimension of the optimizing space is larger. Indications are that the primary error source can be reduced. This will still leave several parameters free to minimize other sources.
Pierce, S.↗
A Runge-Kutta Nystrom algorithm.
A Runge-Kutta algorithm of order five is presented for the solution of the initial value problem where the system of ordinary differential equations is of second order and does not contain the first derivative. The algorithm includes the Fehlberg step control procedure.
Bettis, D. G.↗
A highly stable explicit integration technique for computational continuum mechanics
A user-oriented subroutine package is built around a highly stable explicit integration algorithm for solution of large order systems of ordinary differential equations, as result from discretization of initial-boundary value problems in continuum mechanics. Fast and accurate solutions, for problems in laminar and turbulent, two and three-dimensional viscous flow fields and multi-dimensional transient heat transfer, are presented using this algorithm, as embodied within a general purpose finite element computer program.
Baker, A. J.↗
Transmission of sound through annular ducts of varying cross sections and sheared mean flow
An asymptotic solution is presented for the transmission and attenuation of acoustic waves in an annular duct of slowly varying cross section which carries a sheared mean flow. The analysis also takes into account the growth of the boundary layer as well as any slow variations in the acoustic liner properties. The problem is reduced to the solution of a first-order ordinary differential equation for the amplitude. The analysis is used to estimate the effects of the variations of the duct cross section and boundary layer growth on the different acoustic modes.
Nayfeh, A. H.↗
Effect of the acoustic boundary layer on the wave propagation in ducts
An analysis is presented for the wave propagation in two-dimensional and circular lined ducts taking into account the effects of viscosity in both the mean and the acoustic problems. The method of composite expansions is used to express each acoustic flow quantity as the sum of an inviscid part and a boundary layer part insignificant outside a thin layer next to the wall. The problem is reduced to solving a second-order ordinary differential equation for the pressure perturbation as in the inviscid acoustic case but with a modified specific wall admittance. An analytic expression is presented for the variation of the modified admittance with the wall and flow parameters, such as the acoustic boundary layer thickness, the mean velocity and temperature gradients at the wall, the frequency of oscillation, and the wavelength.
Nayfeh, A. H.↗
Use of Green's functions in the numerical solution of two-point boundary value problems
This study investigates the use of Green's functions in the numerical solution of the two-point boundary value problem. The first part deals with the role of the Green's function in solving both linear and nonlinear second order ordinary differential equations with boundary conditions and systems of such equations. The second part describes procedures for numerical construction of Green's functions and considers briefly the conditions for their existence. Finally, there is a description of some numerical experiments using nonlinear problems for which the known existence, uniqueness or convergence theorems do not apply. Examples here include some problems in finding rendezvous orbits of the restricted three body system.
Gallaher, L. J.↗
Optimal control problems with mixed control-phase variable equality and inequality constraints
In this paper, necessary conditions are obtained for optimal control problems containing equality constraints defined in terms of functions of the control and phase variables. The control system is assumed to be characterized by an ordinary differential equation, and more conventional constraints, including phase inequality constraints, are also assumed to be present. Because the first-mentioned equality constraint must be satisfied for all t (the independent variable of the differential equation) belonging to an arbitrary (prescribed) measurable set, this problem gives rise to infinite-dimensional equality constraints. To obtain the necessary conditions, which are in the form of a maximum principle, an implicit-function-type theorem in Banach spaces is derived.
Makowski, K.↗
Mathematical model of one-man air revitalization system
A mathematical model was developed for simulating the steady state performance in electrochemical CO2 concentrators which utilize (NMe4)2 CO3 (aq.) electrolyte. This electrolyte, which accommodates a wide range of air relative humidity, is most suitable for one-man air revitalization systems. The model is based on the solution of coupled nonlinear ordinary differential equations derived from mass transport and rate equations for the processes which take place in the cell. The boundary conditions are obtained by solving the mass and energy transport equations. A shooting method is used to solve the differential equations.
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