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NASA NTRS1987

Decoupled direct method for sensitivity analysis in combustion kinetics

An efficient, decoupled direct method for calculating the first order sensitivity coefficients of homogeneous, batch combustion kinetic rate equations is presented. In this method the ordinary differential equations for the sensitivity coefficients are solved separately from , but sequentially with, those describing the combustion chemistry. The ordinary differential equations for the thermochemical variables are solved using an efficient, implicit method (LSODE) that automatically selects the steplength and order for each solution step. The solution procedure for the sensitivity coefficients maintains accuracy and stability by using exactly the same steplengths and numerical approximations. The method computes sensitivity coefficients with respect to any combination of the initial values of the thermochemical variables and the three rate constant parameters for the chemical reactions. The method is illustrated by application to several simple problems and, where possible, comparisons are made with exact solutions and those obtained by other techniques.

Radhakrishnan, Krishnan↗

NASA NTRS1982

Compressibility effects in turbulence modeling

Numerical turbulence modeling is discussed with attention given to fluid property variations caused by compressibility in an adiabatic flow. The models are considered in terms of integral quantities expressed by ordinary differential equations and by those formulated as partial differential equations. Compressibility corrections for both integral and partial differential methods are reviewed. Eddy-viscosity models are explored for their capability to characterize the mass-weighted Reynolds stress, which can be accounted for with primitive and/or mass-weighted variables. Compressible flow simulations are currently constrained to low Re and zero mean dilation. The effects of compressibility are defined in wave number space by resolving the Fourier transforms of the velocity vectors into components which are perpendicular and parallel to the wave number vector. Statistical correlations then permit obtaining a value for each contribution.

Rubesin, M. W.↗

NASA NTRS1990

The method of lines in analyzing solids containing cracks

A semi-numerical method is reviewed for solving a set of coupled partial differential equations subject to mixed and possibly coupled boundary conditions. The line method of analysis is applied to the Navier-Cauchy equations of elastic and elastoplastic equilibrium to calculate the displacement distributions in various, simple geometry bodies containing cracks. The application of this method to the appropriate field equations leads to coupled sets of simultaneous ordinary differential equations whose solutions are obtained along sets of lines in a discretized region. When decoupling of the equations and their boundary conditions is not possible, the use of a successive approximation procedure permits the analytical solution of the resulting ordinary differential equations. The use of this method is illustrated by reviewing and presenting selected solutions of mixed boundary value problems in three dimensional fracture mechanics. These solutions are of great importance in fracture toughness testing, where accurate stress and displacement distributions are required for the calculation of certain fracture parameters. Computations obtained for typical flawed specimens include that for elastic as well as elastoplastic response. Problems in both Cartesian and cylindrical coordinate systems are included. Results are summarized for a finite geometry rectangular bar with a central through-the-thickness or rectangular surface crack under remote uniaxial tension. In addition, stress and displacement distributions are reviewed for finite circular bars with embedded penny-shaped cracks, and rods with external annular or ring cracks under opening mode tension. The results obtained show that the method of lines presents a systematic approach to the solution of some three-dimensional mechanics problems with arbitrary boundary conditions. The advantage of this method over other numerical solutions is that good results are obtained even from the use of a relatively coarse grid.

Gyekenyesi, John P.↗

NASA NTRS1972

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.↗

NASA NTRS1999

A Textbook for a First Course in Computational Fluid Dynamics

This paper describes and discusses the textbook, Fundamentals of Computational Fluid Dynamics by Lomax, Pulliam, and Zingg, which is intended for a graduate level first course in computational fluid dynamics. This textbook emphasizes fundamental concepts in developing, analyzing, and understanding numerical methods for the partial differential equations governing the physics of fluid flow. Its underlying philosophy is that the theory of linear algebra and the attendant eigenanalysis of linear systems provides a mathematical framework to describe and unify most numerical methods in common use in the field of fluid dynamics. Two linear model equations, the linear convection and diffusion equations, are used to illustrate concepts throughout. Emphasis is on the semi-discrete approach, in which the governing partial differential equations (PDE's) are reduced to systems of ordinary differential equations (ODE's) through a discretization of the spatial derivatives. The ordinary differential equations are then reduced to ordinary difference equations (O(Delta)E's) using a time-marching method. This methodology, using the progression from PDE through ODE's to O(Delta)E's, together with the use of the eigensystems of tridiagonal matrices and the theory of O(Delta)E's, gives the book its distinctiveness and provides a sound basis for a deep understanding of fundamental concepts in computational fluid dynamics.

Zingg, D. W.↗

NASA NTRS1985

Unsteady Heat Transfer Due to Time-dependent Free Stream Velocity

The present work was undertaken in order to study the unsteady combined convection from a horizontal circular cylinder to a transverse flow. A coordinate perturbation method is used to transform the governing set of partial differential equations into a system of ordinary differential equations. The free stream time-dependent velocity was assumed to be sinusoidal and the boundary layer response due to both low as well as high frequencies of oscillation will be studied. Currently numerical solutions are being obtained for the distribution of the unsteady Nusselt number and the friction factor.

Gorla, R. S. R.↗

NASA NTRS1978

On approximating hereditary dynamics by systems of ordinary differential equations

The paper deals with methods of obtaining approximate solutions to linear retarded functional differential equations (hereditary systems). The basic notion is to project the infinite dimensional space of initial functions for the hereditary system onto a finite dimensional subspace. Within this framework, two particular schemes are discussed. The first uses well-known piecewise constant approximations, while the second is a new method based on piecewise linear approximating functions. Numerical results are given.

Cliff, E. M.↗

NASA NTRS1991

Active control of panel vibrations induced by boundary-layer flow

Some problems in active control of panel vibration excited by a boundary layer flow over a flat plate are studied. In the first phase of the study, the optimal control problem of vibrating elastic panel induced by a fluid dynamical loading was studied. For a simply supported rectangular plate, the vibration control problem can be analyzed by a modal analysis. The control objective is to minimize the total cost functional, which is the sum of a vibrational energy and the control cost. By means of the modal expansion, the dynamical equation for the plate and the cost functional are reduced to a system of ordinary differential equations and the cost functions for the modes. For the linear elastic plate, the modes become uncoupled. The control of each modal amplitude reduces to the so-called linear regulator problem in control theory. Such problems can then be solved by the method of adjoint state. The optimality system of equations was solved numerically by a shooting method. The results are summarized.

Chow, Pao-Liu↗

NASA NTRS1976

Nonlinear Equations of Motion for Cantilever Rotor Blades in Hover with Pitch Link Flexibility, Twist, Precone, Droop, Sweep, Torque Offset, and Blade Root Offset

Nonlinear equations of motion for a cantilever rotor blade are derived for the hovering flight condition. The blade is assumed to have twist, precone, droop, sweep, torque offset and blade root offset, and the elastic axis and the axes of center of mass, tension, and aerodynamic center coincident at the quarter chord. The blade is cantilevered in bending, but has a torsional root spring to simulate pitch link flexibility. Aerodynamic forces acting on the blade are derived from strip theory based on quasi-steady two-dimensional airfoil theory. The equations are hybrid, consisting of one integro-differential equation for root torsion and three integro-partial differential equations for flatwise and chordwise bending and elastic torsion. The equations are specialized for a uniform blade and reduced to nonlinear ordinary differential equations by Galerkin's method. They are linearized for small perturbation motions about the equilibrium operating condition. Modal analysis leads to formulation of a standard eigenvalue problem where the elements of the stability matrix depend on the solution of the equilibrium equations. Two different forms of the root torsion equation are derived that yield virtually identical numerical results. This provides a reasonable check for the accuracy of the equations.

Hodges, D. H.↗

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 NTRS1989

Semidiscrete Galerkin modelling of compressible viscous flow past a circular cone at incidence

A numerical study of the laminar and compressible boundary layer, about a circular cone in a supersonic free stream, is presented. It is thought that if accurate and efficient numerical schemes can be produced to solve the boundary layer equations, they can be joined to numerical codes that solve the inviscid outer flow. The combination of these numerical codes is competitive with the accurate, but computationally expensive, Navier-Stokes schemes. The primary goal is to develop a finite element method for the calculation of 3-D compressible laminar boundary layer about a yawed cone. The proposed method can, in principle, be extended to apply to the 3-D boundary layer of pointed bodies of arbitrary cross section. The 3-D boundary layer equations governing supersonic free stream flow about a cone are examined. The 3-D partial differential equations are reduced to 2-D integral equations by applying the Howarth, Mangler, Crocco transformations, a linear relation between viscosity, and a Blasius-type of similarity variable. This is equivalent to a Dorodnitsyn-type formulation. The reduced equations are independent of density and curvature effects, and resemble the weak form of the 2-D incompressible boundary layer equations in Cartesian coordinates. In addition the coordinate normal to the wall has been stretched, which reduces the gradients across the layer and provides high resolution near the surface. Utilizing the parabolic nature of the boundary layer equations, a finite element method is applied to the Dorodnitsyn formulation. The formulation is presented in a Petrov-Galerkin finite element form and discretized across the layer using linear interpolation functions. The finite element discretization yields a system of ordinary differential equations in the circumferential direction. The circumferential derivatives are solved by an implicit and noniterative finite difference marching scheme. Solutions are presented for a 15 deg half angle cone at angles of attack of 5 and 10 deg. The numerical solutions assume a laminar boundary layer with free stream Mach number of 7. Results include circumferential distribution of skin friction and surface heat transfer, and cross flow velocity distributions across the layer.

Meade, Andrew James, Jr.↗

NASA NTRS1994

Spectral Methods for Computational Fluid Dynamics

As a tool for large-scale computations in fluid dynamics, spectral methods were prophesized in 1944, born in 1954, virtually buried in the mid-1960's, resurrected in 1969, evangalized in the 1970's, and catholicized in the 1980's. The use of spectral methods for meteorological problems was proposed by Blinova in 1944 and the first numerical computations were conducted by Silberman (1954). By the early 1960's computers had achieved sufficient power to permit calculations with hundreds of degrees of freedom. For problems of this size the traditional way of computing the nonlinear terms in spectral methods was expensive compared with finite-difference methods. Consequently, spectral methods fell out of favor. The expense of computing nonlinear terms remained a severe drawback until Orszag (1969) and Eliasen, Machenauer, and Rasmussen (1970) developed the transform methods that still form the backbone of many large-scale spectral computations. The original proselytes of spectral methods were meteorologists involved in global weather modeling and fluid dynamicists investigating isotropic turbulence. The converts who were inspired by the successes of these pioneers remained, for the most part, confined to these and closely related fields throughout the 1970's. During that decade spectral methods appeared to be well-suited only for problems governed by ordinary diSerential eqllations or by partial differential equations with periodic boundary conditions. And, of course, the solution itself needed to be smooth. Some of the obstacles to wider application of spectral methods were: (1) poor resolution of discontinuous solutions; (2) inefficient implementation of implicit methods; and (3) drastic geometric constraints. All of these barriers have undergone some erosion during the 1980's, particularly the latter two. As a result, the applicability and appeal of spectral methods for computational fluid dynamics has broadened considerably. The motivation for the use of spectral methods in numerical calculations stems from the attractive approximation properties of orthogonal polynomial expansions.

Zang, T. A.↗

NASA NTRS1977

On the stability of Galerkin methods for initial-boundary value problems for hyperbolic systems

The stability of approximating the solution of mixed initial-boundary value problems for hyperbolic systems by semidiscrete Galerkin methods is studied. It is shown that a particular straightforward Galerkin method yields an unstable approximation, and that this numerical instability is caused by an improper treatment of the boundary. Stable schemes are then presented, one of which differs from the unstable scheme only insofar as the treatment of the boundary is concerned. These stable schemes make use of a particular matrix which symmetrizes the differential system. It is therefore shown that the use of this matrix is crucial to the stability of the computations as well as for obtaining a priori bounds on the energy of the continuous system. This symmetrizing matrix is also related to the diagonalizing matrix for the system of hyperbolic equations and to the Liapunov matrix for the system of ordinary differential equations resulting from the application of Galerkin's method.

Gunzburger, M. D.↗

NASA NTRSDate not supplied

Derivatives Across Adaptive-Step Integration Accurately and Efficiently

The numerical integration of ordinary differential equations (ODEs) is common in trajectory optimization. Previous studies have shown that the use of adaptive-step Runge-Kutta integration schemes can introduce errors in the derivatives due to non-differentiable behavior within the integration scheme itself. The authors previously demonstrated the use of an implicit adaptation to the Runge-Kutta methods to mitigate this issue, but at a significant computational cost. In this work, the authors demonstrate an explicit adaptation to the Runge-Kutta methods to address the issue that does not incur a significant computational cost.

Kaushik Ponnapalli↗

NASA NTRS1982

Geometrically derived difference formulae for the numerical integration of trajectory problems

An initial value problem for the autonomous system of ordinary differential equations dy/dt = f(y), where y is a vector, is considered. In a number of practical applications the interest lies in obtaining the curve traced by the solution y. These applications include the computation of trajectories in mechanical problems. The term 'trajectory problem' is employed to refer to these cases. Lambert and McLeod (1979) have introduced a method involving local rotation of the axes in the y-plane for the two-dimensional case. The present investigation continues the study of difference schemes specifically derived for trajectory problems. A simple geometrical way of constructing such methods is presented, and the local accuracy of the schemes is investigated. A circularly exact, fixed-step predictor-corrector algorithm is defined, and a variable-step version of a circularly exact algorithm is presented.

Mcleod, R. J. Y.↗

NASA NTRS1982

The applicability of Lyapunov characteristic numbers in the study of the stability of satellite orbits

It is pointed out that the Lyapunov Characteristic Numbers constitute a new tool for determining stability of trajectories of dynamical systems, or, even more generally, of solutions of systems of ordinary differential equations. In contrast with the characteristic exponents, which apply only to periodic solutions, the Lyapunov Characteristic Numbers apply to arbitrary nonperiodic solutions as well. A description is presented of the numerical experiments which have been made in order to investigate the practical value of the Lyapunov Characteristic Number and the Kolmogorov Entropy for the purpose of estimating the stability of trajectories and/or numerical integration methods in celestial mechanics. It is found that the Lyapunov Characteristic Numbers are extremely useful for the classification of the solutions of nonintegrable dynamical systems, especially in order to distinguish between quasi-periodic and chaotic solutions. However, the Lyapunov Characteristics Numbers do not appear to be useful for the purpose of evaluating numerical integration methods.

Broucke, R.↗

NASA NTRS2013

High-Order Space-Time Methods for Conservation Laws

Current high-order methods such as discontinuous Galerkin and/or flux reconstruction can provide effective discretization for the spatial derivatives. Together with a time discretization, such methods result in either too small a time step size in the case of an explicit scheme or a very large system in the case of an implicit one. To tackle these problems, two new high-order space-time schemes for conservation laws are introduced: the first is explicit and the second, implicit. The explicit method here, also called the moment scheme, achieves a Courant-Friedrichs-Lewy (CFL) condition of 1 for the case of one-spatial dimension regardless of the degree of the polynomial approximation. (For standard explicit methods, if the spatial approximation is of degree p, then the time step sizes are typically proportional to 1/p(exp 2)). Fourier analyses for the one and two-dimensional cases are carried out. The property of super accuracy (or super convergence) is discussed. The implicit method is a simplified but optimal version of the discontinuous Galerkin scheme applied to time. It reduces to a collocation implicit Runge-Kutta (RK) method for ordinary differential equations (ODE) called Radau IIA. The explicit and implicit schemes are closely related since they employ the same intermediate time levels, and the former can serve as a key building block in an iterative procedure for the latter. A limiting technique for the piecewise linear scheme is also discussed. The technique can suppress oscillations near a discontinuity while preserving accuracy near extrema. Preliminary numerical results are shown

Huynh, H. T.↗

NASA NTRS1989

Numerical integration of asymptotic solutions of ordinary differential equations

Classical asymptotic analysis of ordinary differential equations derives approximate solutions that are numerically stable. However, the analysis also leads to tedious expansions in powers of the relevant parameter for a particular problem. The expansions are replaced with integrals that can be evaluated by numerical integration. The resulting numerical solutions retain the linear independence that is the main advantage of asymptotic solutions. Examples, including the Falkner-Skan equation from laminar boundary layer theory, illustrate the method of asymptotic analysis with numerical integration.

Thurston, Gaylen A.↗