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NASA NTRS1976

Construction of analytically solvable models for interacting species

The basic form of a model representation for systems of n interacting biological species is a set of essentially nonlinear autonomous ordinary differential equations. A generic canonical expression for the rate functions in the equations is reported which permits the analytical general solution to be obtained by elementary computation. It is shown that a general analytical solution is directly obtainable for models where the rate functions are prescribed by the generic canonical expression from the outset. Some illustrative examples are given which demonstrate that the generic canonical expression can be used to construct analytically solvable models for two interacting species with limit-cycle dynamics as well as for a three-species interdependence.

Rosen, G.↗
NASA NTRS1977

Approximate method for calculating free vibrations of a large-wind-turbine tower structure

A set of ordinary differential equations were derived for a simplified structural dynamic lumped-mass model of a typical large-wind-turbine tower structure. Dunkerley's equation was used to arrive at a solution for the fundamental natural frequencies of the tower in bending and torsion. The ERDA-NASA 100-kW wind turbine tower structure was modeled, and the fundamental frequencies were determined by the simplified method described. The approximate fundamental natural frequencies for the tower agree within 18 percent with test data and predictions analyzed.

Das, S. C.↗
NASA NTRS1978

Error behaviour of multistep methods applied to unstable differential systems

The problem of modelling a dynamic system described by a system of ordinary differential equations which has unstable components for limited periods of time is discussed. It is shown that the global error in a multistep numerical method is the solution to a difference equation initial value problem, and the approximate solution is given for several popular multistep integration formulae. Inspection of the solution leads to the formulation of four criteria for integrators appropriate to unstable problems. A sample problem is solved numerically using three popular formulae and two different stepsizes to illustrate the appropriateness of the criteria.

Brown, R. L.↗
NASA NTRS1978

Investigation of ODE integrators using interactive graphics

Two FORTRAN programs using an interactive graphic terminal to generate accuracy and stability plots for given multistep ordinary differential equation (ODE) integrators are described. The first treats the fixed stepsize linear case with complex variable solutions, and generates plots to show accuracy and error response to step driving function of a numerical solution, as well as the linear stability region. The second generates an analog to the stability region for classes of non-linear ODE's as well as accuracy plots. Both systems can compute method coefficients from a simple specification of the method. Example plots are given.

Brown, R. L.↗
NASA NTRS1979

Stability of sequences generated by nonlinear differential systems

A local stability analysis is presented for both the analytic and numerical solutions of the initial value problem for a system of ordinary differential equations. It is shown that, using a proper choice of Liapunov function, a connected region of stable initial values of both the analytic solution and the one-leg k-step numerical solution can be approximated. Attention is given to the example of the two-dimensional problem involving the stability of the longitudinal equations of motion of a gliding jet aircraft.

Brown, R. L.↗
NASA NTRS1978

Investigation of ODE integrators using interactive graphics

In the present paper, two FORTRAN programs are described which use an interactive graphic terminal to generate accuracy and stability plots for a given multistep integrator of ordinary differential equations (ODE). One program treats the fixed-stepsize linear case with complex variable solutions, and generates plots to show accuracy and error response to step driving function of a numerical solution. It also generates the linear stability region. The other program generates an analog with an accuracy plot. Both systems are capable of computing method coefficients from a simple specification of the method. Example plots are given.

Brown, R. L.↗
NASA NTRS1980

Analysis and testing of numerical formulas for the initial value problem

Three computer programs for evaluating and testing numerical integration formulas used with fixed stepsize programs to solve initial value systems of ordinary differential equations are described. A program written in PASCAL SERIES, takes as input the differential equations and produces a FORTRAN subroutine for the derivatives of the system and for computing the actual solution through recursive power series techniques. Both of these are used by STAN, a FORTRAN program that interactively displays a discrete analog of the Liapunov stability region of any two dimensional subspace of the system. The derivatives may be used by CLMP, a FORTRAN program, to test the fixed stepsize formula against a good numerical result and interactively display the solutions.

Brown, R. L.↗
NASA NTRS1979

Free oscillations in a climate model with ice-sheet dynamics

A study of stable periodic solutions to a simple nonlinear model of the ocean-atmosphere-ice system is presented. The model has two dependent variables: ocean-atmosphere temperature and latitudinal extent of the ice cover. No explicit dependence on latitude is considered in the model. Hence all variables depend only on time and the model consists of a coupled set of nonlinear ordinary differential equations. The globally averaged ocean-atmosphere temperature in the model is governed by the radiation balance. The reflectivity to incoming solar radiation, i.e., the planetary albedo, includes separate contributions from sea ice and from continental ice sheets. The major physical mechanisms active in the model are (1) albedo-temperature feedback, (2) continental ice-sheet dynamics and (3) precipitation-rate variations. The model has three-equilibrium solutions, two of which are linearly unstable, while one is linearly stable. For some choices of parameters, the stability picture changes and sustained, finite-amplitude oscillations obtain around the previously stable equilibrium solution. The physical interpretation of these oscillations points to the possibility of internal mechanisms playing a role in glaciation cycles.

Kallen, E.↗
NASA NTRS1981

Multistep integration formulas for the numerical integration of the satellite problem

The use of two Class 2/fixed mesh/fixed order/multistep integration packages of the PECE type for the numerical integration of the second order, nonlinear, ordinary differential equation of the satellite orbit problem. These two methods are referred to as the general and the second sum formulations. The derivation of the basic equations which characterize each formulation and the role of the basic equations in the PECE algorithm are discussed. Possible starting procedures are examined which may be used to supply the initial set of values required by the fixed mesh/multistep integrators. The results of the general and second sum integrators are compared to the results of various fixed step and variable step integrators.

Lundberg, J. B.↗
NASA NTRS1981

Scaled Runge-Kutta algorithms for handling dense output

Low order Runge-Kutta algorithms are developed which determine the solution of a system of ordinary differential equations at any point within a given integration step, as well as at the end of each step. The scaled Runge-Kutta methods are designed to be used with existing Runge-Kutta formulas, using the derivative evaluations of these defining algorithms as the core of the system. For a slight increase in computing time, the solution may be generated within the integration step, improving the efficiency of the Runge-Kutta algorithms, since the step length need no longer be severely reduced to coincide with the desired output point. Scaled Runge-Kutta algorithms are presented for orders 3 through 5, along with accuracy comparisons between the defining algorithms and their scaled versions for a test problem.

Horn, M. K.↗
NASA NTRS1981

A new solution-adaptive grid generation method for transonic airfoil flow calculations

The clustering algorithm is controlled by a second-order, ordinary differential equation which uses the airfoil surface density gradient as a forcing function. The solution to this differential equation produces a surface grid distribution which is automatically clustered in regions with large gradients. The interior grid points are established from this surface distribution by using an interpolation scheme which is fast and retains the desirable properties of the original grid generated from the standard elliptic equation approach.

Nakamura, S.↗
NASA NTRS1981

Parameter estimation of large flexible aerospace structures with application to the control of the Maypole Deployable Reflector

Systems such as the Maypole deployable reflector have a distributed parameter nature. The flexible column and hoop structure and the circular antenna of 30-100 meter diameter which it supports are described by partial, rather than ordinary, differential equations. Progress completed in reduced order modelling andd controller design and digital parameter estimation and control is summarized. Topics covered include depolyment and on-orbit operation; quasi-static (steady state) operation; dynamic distributed parameter system; autoregressive moving average identification; frequency domain procedures; direct or implicit active control; adaptive observers; parameter estimation using a linear reinforcement learning factor; feedback control; and reduced order modeling for nonlinear systems.

Balas, M. J.↗
NASA NTRS1982

Solidification of a binary mixture

The time dependent concentration and temperature profiles of a finite layer of a binary mixture are investigated during solidification. The coupled time dependent Stefan problem is solved numerically using an implicit finite differencing algorithm with the method of lines. Specifically, the temporal operator is approximated via an implicit finite difference operator resulting in a coupled set of ordinary differential equations for the spatial distribution of the temperature and concentration for each time. Since the resulting differential equations set form a boundary value problem with matching conditions at an unknown spatial point, the method of invariant imbedding is used for its solution.

Antar, B. N.↗
NASA NTRS1982

Algorithmic aspects of transient heat transfer problems in structures

It is noted that the application of finite element or finite difference techniques to the solution of transient heat transfer problems in structures often results in a stiff system of ordinary differential equations. Such systems are usually handled most efficiently by implicit integration techniques which require the solution of large and sparse systems of algebraic equations. The assembly and solution of these systems using the incomplete Cholesky conjugate gradient algorithm is examined. Several examples are used to demonstrate the advantage of the algorithm over other techniques.

Haftka, R. T.↗
NASA NTRS1981

The three-dimensional boundary layer on a rotating helical blade

The laminar boundary layer on a twisted helical blade is considered. The blade geometry is the same as that proposed by Horlock and Wordsworth (1965). However, the blade is twisted about the leading edge in the manner described by Miyake and Fujita (1974). The flow may be considered to be the analog, in a rotating reference frame, of the flat-plate boundary layer in a stationary frame. It is shown that a coordinate system which is orthogonal in the blade surface may be developed. With the appropriate scaling of the dependent variables a solution for the boundary layer flow is readily obtained. The systems of ordinary differential equations for the stream function of the primary flow and the cross-flow are solved numerically.

Morris, P. J.↗
NASA NTRS1982

High frequency sound emission from moving point multipole sources embedded in arbitrary transversely sheared mean flows

Formulas are derived for the high frequency sound emission from moving point multipole sources embedded in an arbitrary unidirectional transversely sheared mean flow. The results are used to study the sound generated by non-axisymmetric turbulent jets. The effect of the asymmetry in both the mean flow and the source distribution is accounted for by a 'circumferential directivity factor', which is easily calculated from the solution of a second order ordinary differential equation in the general case and from an explicit formula when the mean flow is symmetric but the source location is not. This factor is used to assess the potential of employing asymmetric velocity profiles that redirect the sound upward to reduce the noise radiation below the flight path of a jet aircraft.

Goldstein, M. E.↗
NASA NTRS1982

Bifurcation and stability of low-order steady flows in horizontally and vertically forced convection

A nonlinear spectral model of two-dimensional, shallow Boussinesq convection which responds to heating in both the horizontal and vertical directions is examined. The governing partial differential system is converted to an infinite set of ordinary differential equations and truncated to a small set to permit detailed study of the number and types of transitions from one flow configuration to another. The Hadley number and the Rayleigh number are defined as the horizontal and vertical thermal forcing mechanisms, respectively, for inclusion in the nonlinear spectral model, which is composed of three equations. The model is then used to describe steady states, linearly stable solutions, and balancing factors in unstable stratification. The number and the distribution of the steady states are found to be qualitatively independent of the aspect ratio and the Prandtl number.

Yost, D. A.↗
NASA NTRS1982

Effects of internal heat transfer on the structure of self-similar blast waves

An analysis of the problem of self-similar, nonadiabatic blast waves, where both conduction and radiation are allowed to take place, show the problem to be reducible to the integration of a system of six coupled nonlinear ordinary differential equations. Consideration of these equations shows that although radiation tends to produce uniform fields through temperature gradient attenuation, all the energy carried by radiation is deposited on the front and the bounding shock becomes increasingly overdriven. When conduction is taken into account, the distribution of gasdynamic parameters in blast waves in the case of Rosseland diffusion radiation is more uniform than in the case of the Planck emission radiation.

Ghoniem, A. F.↗