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NASA NTRS1973

Wind-driven currents in a shallow lake or sea.

A technique is given for obtaining exact analytical solutions for wind-driven currents in a shallow lake or sea with a large class of bottom topographies which may or may not contain an island. Welander's second-order partial differential equation for the lake's surface displacement is used for determining the lake circulation. The technique, together with some results based on conformal mapping, is used to reduce the problem of finding solutions to Welander's equation for a closed body of water to the problem of solving an ordinary differential equation. The method is applied to an elliptically shaped lake and a circular lake containing an eccentrically located circular island.

Goldstein, M.↗

NASA NTRS1997

Mathematical Analysis and Optimization of Infiltration Processes

A variety of infiltration techniques can be used to fabricate solid materials, particularly composites. In general these processes can be described with at least one time dependent partial differential equation describing the evolution of the solid phase, coupled to one or more partial differential equations describing mass transport through a porous structure. This paper presents a detailed mathematical analysis of a relatively simple set of equations which is used to describe chemical vapor infiltration. The results demonstrate that the process is controlled by only two parameters, alpha and beta. The optimization problem associated with minimizing the infiltration time is also considered. Allowing alpha and beta to vary with time leads to significant reductions in the infiltration time, compared with the conventional case where alpha and beta are treated as constants.

Chang, H.-C.↗

NASA NTRS1991

From differential to difference equations for first order ODEs

When constructing an algorithm for the numerical integration of a differential equation, one should first convert the known ordinary differential equation (ODE) into an ordinary difference equation. Given this difference equation, one can develop an appropriate numerical algorithm. This technical note describes the derivation of two such ordinary difference equations applicable to a first order ODE. The implicit ordinary difference equation has the same asymptotic expansion as the ODE itself, whereas the explicit ordinary difference equation has an asymptotic that is similar in structure but different in value when compared with that of the ODE.

Freed, Alan D.↗

NASA NTRS1956

Application of the Green's function method for 2- and 3-dimensional steady transonic flows

A Time-Domain Green's function method for the nonlinear time-dependent three-dimensional aerodynamic potential equation is presented. The Green's theorem is being used to transform the partial differential equation into an integro-differential-delay equation. Finite-element and finite-difference methods are employed for the spatial and time discretizations to approximate the integral equation by a system of differential-delay equations. Solution may be obtained by solving for this nonlinear simultaneous system of equations in time. This paper discusses the application of the method to the Transonic Small Disturbance Equation and numerical results for lifting and nonlifting airfoils and wings in steady flows are presented.

Tseng, K.↗

NASA NTRS1967

Nonlinear combustion instability in liquid-propellant rocket engines. The transverse mode. I. The hydrodynamic differential and difference equations and computer algorithm

Hydrodynamic differential and difference equations and computer algorithm for nonlinear combustion instability in liquid propellant rocket engines, and Fortran IV listing

ALGORITHM↗

NASA NTRS1973

Singular perturbation equations for flexible satellites

The dynamic model of a flexible satellite with n generalized structural position coordinates requires the solution of a set of N coupled nonlinear ordinary or partial differential equations. For single composite body satellites, N is equal to (n + 3). For dual-spin systems, N is equal to (n + 9). These equations usually involve time derivatives up to the second order. For large values of n, linearization of the system has so far been the only practicable way of solution. The present study shows a method of avoiding this linearization by reducing the N equations to three or nine nonlinear, coupled, first-order ordinary differential equations involving only the angular velocities of the composite bodies. The solutions for these angular velocities lead to linear equations in the n generalized structural position coordinates, which can then be solved by known methods.

Huang, T. C.↗

NASA NTRS1984

Variational objective analyses for cyclone studies

The basic analysis equations, i.e., the two horizontal momentum equations, the hydrostatic equation, and the integrated continuity equation were derived for the nonlinear vertical coordinate, nondimensionalized, and expressed in finite differences on a staggered grid. Special care was taken to transform the hydrostatic equation and the pressure gradient terms of the horizontal momentum equations to nearly eliminate truncation error over steeply sloping terrain. This formulation also eliminated explicit reference to orographically induced variations in the thermodynamic variables so that the variational adjustments are on the scale of the meteorological perturbations. The analysis equations were subjected to the Euler-Lagrange operations as expressed for finite differences and an additional set of five partial differential equations was derived, bringing to nine the number of equations in Model I. Higher order terms, terms containing observed quantities, and terms containing none of the variables to be adjusted were grouped into forcing functions and the equations were solved for the zero order terms. Zero order variables were eliminated between these equations and there resulted two diagnostic equations which take the form of general linear second order partial differential equations with nonconstant coefficients.

Achtemeier, G. L.↗

NASA NTRS1971

On the integrability cases of the equation of motion for a satellite in an axially symmetric gravitational field.

Investigation of two cases of integrability of a second-order differential equation describing the projection of an axisymmetric satellite orbit on to a plane perpendicular to the rotation axis. It is demonstrated that for these two cases the integration can be carried out either by quadratures or reduced to a first-order differential equation. Analytical and physical properties are expressed, and it is shown that the equation can be derived from the classical plane eikonal equation of geometric optics.

Ghaffari, A.↗

NASA NTRS1990

Quadrature imposition of compatibility conditions in Chebyshev methods

This paper handles a problem encountered in the solution by way of Chebyshev spectral collocation discretization of the Poisson equation with homogeneous Neumann boundary conditions. When pseudospectral Chebyshev methodology is utilized to discretize the partial differential equation these integrals have to be approximated by an appropriate quadrature formula. The correct quadrature formula is developed for these problems, taking into account the degree of the polynomials involved. It is shown that this formula leads to a well-conditioned Chebyshev approximation to the differential equations and that the compatibility condition is automatically satisfied.

Gottlieb, D.↗

NASA NTRS1994

Modeling of outgassing and matrix decomposition in carbon-phenolic composites

Work done in the period Jan. - June 1994 is summarized. Two threads of research have been followed. First, the thermodynamics approach was used to model the chemical and mechanical responses of composites exposed to high temperatures. The thermodynamics approach lends itself easily to the usage of variational principles. This thermodynamic-variational approach has been applied to the transpiration cooling problem. The second thread is the development of a better algorithm to solve the governing equations resulting from the modeling. Explicit finite difference method is explored for solving the governing nonlinear, partial differential equations. The method allows detailed material models to be included and solution on massively parallel supercomputers. To demonstrate the feasibility of the explicit scheme in solving nonlinear partial differential equations, a transpiration cooling problem was solved. Some interesting transient behaviors were captured such as stress waves and small spatial oscillations of transient pressure distribution.

Mcmanus, Hugh L.↗

NASA NTRS1972

Simulation of a steady-state integrated human thermal system.

The mathematical model of an integrated human thermal system is formulated. The system consists of an external thermal regulation device on the human body. The purpose of the device (a network of cooling tubes held in contact with the surface of the skin) is to maintain the human body in a state of thermoneutrality. The device is controlled by varying the inlet coolant temperature and coolant mass flow rate. The differential equations of the model are approximated by a set of algebraic equations which result from the application of the explicit forward finite difference method to the differential equations. The integrated human thermal system is simulated for a variety of combinations of the inlet coolant temperature, coolant mass flow rate, and metabolic rates. Two specific cases are considered: (1) the external thermal regulation device is placed only on the head and (2) the devices are placed on the head and the torso. The results of the simulation indicate that when the human body is exposed to hot environment, thermoneutrality can be attained by localized cooling if the operating variables of the external regulation device(s) are properly controlled.

Hsu, F. T.↗

NASA NTRS1972

Simulation of a steady-state integrated human thermal system.

Hsu, F. T.↗

NASA NTRS1983

Finite element or Galerkin type semidiscrete schemes

A finite element of Galerkin type semidiscrete method is proposed for numerical solution of a linear hyperbolic partial differential equation. The question of stability is reduced to the stability of a system of ordinary differential equations for which Dahlquist theory applied. Results of separating the part of numerical solution which causes the spurious oscillation near shock-like response of semidiscrete scheme to a step function initial condition are presented. In general all methods produce such oscillatory overshoots on either side of shocks. This overshoot pathology, which displays a behavior similar to Gibb's phenomena of Fourier series, is explained on the basis of dispersion of separated Fourier components which relies on linearized theory to be satisfactory. Expository results represented.

Durgun, K.↗

NASA NTRS1993

A new approach for designing self-organizing systems and application to adaptive control

There is tremendous interest in the design of intelligent machines capable of autonomous learning and skillful performance under complex environments. A major task in designing such systems is to make the system plastic and adaptive when presented with new and useful information and stable in response to irrelevant events. A great body of knowledge, based on neuro-physiological concepts, has evolved as a possible solution to this problem. Adaptive resonance theory (ART) is a classical example under this category. The system dynamics of an ART network is described by a set of differential equations with nonlinear functions. An approach for designing self-organizing networks characterized by nonlinear differential equations is proposed.

Ramamoorthy, P. A.↗

NASA NTRS1993

Implementing abstract multigrid or multilevel methods

Multigrid methods can be formulated as an algorithm for an abstract problem that is independent of the partial differential equation, domain, and discretization method. In such an abstract setting, problems not arising from partial differential equations can be treated. A general theory exists for linear problems. The general theory was motivated by a series of abstract solvers (Madpack). The latest version was motivated by the theory. Madpack now allows for a wide variety of iterative and direct solvers, preconditioners, and interpolation and projection schemes, including user callback ones. It allows for sparse, dense, and stencil matrices. Mildly nonlinear problems can be handled. Also, there is a fast, multigrid Poisson solver (two and three dimensions). The type of solvers and design decisions (including language, data structures, external library support, and callbacks) are discussed. Based on the author's experiences with two versions of Madpack, a better approach is proposed. This is based on a mixed language formulation (C and FORTRAN + preprocessor). Reasons for not using FORTRAN, C, or C++ (individually) are given. Implementing the proposed strategy is not difficult.

Douglas, Craig C.↗

NASA NTRS1995

Numerical Stability In Hyperbolic Boundary-Value Problems

Technical memorandum discusses stability of numerical solutions involving semidiscrete approximations to hyperbolic partial differential equations in initial-and-boundary-value problems. Topic of practical significance because hyperbolic partial differential equations arise in mathematical modeling of waves and blasts. Solutions often needed over restricted regions of unbounded spaces. Outer boundaries artificial, introduced only to limit domains of numerical solutions. Conditions at such artificial boundaries cause numerical instabilities that degrade computed solutions.

Warming, Robert F.↗

NASA NTRS1949

On the Theory of the Laval Nozzle

In the present paper, the motion of a gas in a plane-parallel Laval nozzle in the neighborhood of the transition from subsonic to supersonic velocities is studied. In a recently published paper, F. I. Frankl, applying the holograph method of Chaplygin, undertook a detailed investigation of the character of the flow near the line of transition from subsonic to supersonic velocities. From the results of Tricomi's investigation on the theory of differential equations of the mixed elliptic-hyperbolic type, Frankl introduced as one of the independent variables in place of the modulus of the velocity, a certain specially chosen function of this modulus. He thereby succeeded in explaining the character of the flow at the point of intersection of the transition line and the axis of symmetry (center of the nozzle) and in studying the behavior of the stream function in the neighborhood of this point by separating out the principal term having, together with its derivatives, the maximum value as compared with the corresponding corrections. This principal term is represented in Frankl's paper in the form of a linear combination of two hypergeometric functions. In order to find this linear combination, it is necessary to solve a number of boundary problems, which results in a complex analysis. In the investigation of the flow with which this paper is concerned, a second method is applied. This method is based on the transformation of the equations of motion to a form that may be called canonical for the system of differential equations of the mixed elliptic-hyperbolic type to which the system of equations of the motion of an ideal compressible fluid refers. By studying the behavior of the integrals of this system in the neighborhood of the parabolic line, the principal term of the solution is easily separated out in the form of a polynomial of the third degree. As a result, the computation of the transitional part of the nozzle is considerably simplified.

Falkovich, S. V.↗