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NASA NTRS1973

Direct numerical solution of three-dimensional equations containing elliptic operators.

A direct three-dimensional elliptic solver is presented for application in a wide class of numerical methods for solving partial differential equations in physics and engineering. The derived algorithm and FORTRAN code implement Buzbee, Golub and Nielson's proposed extension of Buneman's Cyclic-Reduction Poisson solver to three dimensions. Both a 'most direct' cyclic reduction and a revised method (to eliminate roundoff error difficulties) are derived. Tests on an IBM 360/67 computer, using various optional combinations of subroutines, showed significant differences in accuracy and computing time, with the optimum subroutine combination depending on mesh size.

Martin, E. D.↗

NASA NTRS1974

The modified equation approach to the stability and accuracy analysis of finite-difference methods

The stability and accuracy of finite-difference approximations to simple linear partial differential equations are analyzed by studying the modified partial differential equation. Aside from round-off error, the modified equation represents the actual partial differential equation solved when a numerical solution is computed using a finite-difference equation. The modified equation is derived by first expanding each term of a difference scheme in a Taylor series and then eliminating time derivatives higher than first order by certain algebraic manipulations. The connection between 'heuristic' stability theory based on the modified equation approach and the von Neumann (Fourier) method is established. In addition to the determination of necessary and sufficient conditions for computational stability, a truncated version of the modified equation can be used to gain insight into the nature of both dissipative and dispersive errors.

Warming, R. F.↗

NASA NTRS1974

Linear viscous stability analysis of the stratified Bickley jet

The stability of the stably stratified Bickley jet to small disturbances in a viscous, heat conducting fluid is explored by numerical integration of the governing differential equation. Curves of neutral stability are presented in the alpha-R plane and the parametric dependence of these curves upon Richardson number and Prandtl number is examined. The results show a curious bimodal behavior of the curve of neutral stability for small Richardson numbers. In addition, closed neutral curves are found for Richardson numbers in excess of the critical value for the inviscid flow. The unstable regions in the interior of these curves increase with decreasing Prandtl number. Energy balances are computed which show the relative magnitude of the Reynolds stress production, buoyancy dissipation, and viscous dissipation terms across the flow. As the thermal diffusivity becomes large compared with the kinematic viscosity, the buoyancy term is suppressed and the thermal stratification becomes irrelevant.

Gage, K. S.↗

NASA NTRS1975

Simulation of multicorrelated random processes using the FFT algorithm

A technique for the digital simulation of multicorrelated Gaussian random processes is described. This technique is based upon generating discrete frequency functions which correspond to the Fourier transform of the desired random processes, and then using the fast Fourier transform (FFT) algorithm to obtain the actual random processes. The main advantage of this method of simulation over other methods is computation time; it appears to be more than an order of magnitude faster than present methods of simulation. One of the main uses of multicorrelated simulated random processes is in solving nonlinear random vibration problems by numerical integration of the governing differential equations. The response of a nonlinear string to a distributed noise input is presented as an example.

Wittig, L. E.↗

NASA NTRS1975

A technique for accelerating iterative convergence in numerical integration, with application in transonic aerodynamics

A technique is described for the efficient numerical solution of nonlinear partial differential equations by rapid iteration. In particular, a special approach is described for applying the Aitken acceleration formula (a simple Pade approximant) for accelerating the iterative convergence. The method finds the most appropriate successive approximations, which are in a most nearly geometric sequence, for use in the Aitken formula. Simple examples are given to illustrate the use of the method. The method is then applied to the mixed elliptic-hyperbolic problem of steady, inviscid, transonic flow over an airfoil in a subsonic free stream.

Martin, E. D.↗

NASA NTRS1975

Study of viscous flow about airfoils by the integro-differential method

An integro-differential method was used for numerically solving unsteady incompressible viscous flow problems. A computer program was prepared to solve the problem of an impulsively started 9% thick symmetric Joukowski airfoil at an angle of attack of 15 deg and a Reynolds number of 1000. Some of the results obtained for this problem were discussed and compared with related work completed previously. Two numerical procedures were used, an Alternating Direction Implicit (ADI) method and a Successive Line Relaxation (SLR) method. Generally, the ADI solution agrees well with the SLR solution and with previous results are stations away from the trailing edge. At the trailing edge station, the ADI solution differs substantially from previous results, while the vorticity profiles obtained from the SLR method there are in good qualitative agreement with previous results.

Wu, J. C.↗

NASA NTRS1976

A technique for accelerating iterative convergence in numerical integration, with application in transonic aerodynamics

Martin, E. D.↗

NASA NTRS1978

Computation of three-dimensional nozzle-exhaust flow fields with the GIM code

A methodology is introduced for constructing numerical analogs of the partial differential equations of continuum mechanics. A general formulation is provided which permits classical finite element and many of the finite difference methods to be derived directly. The approach, termed the General Interpolants Method (GIM), can combined the best features of finite element and finite difference methods. A quasi-variational procedure is used to formulate the element equations, to introduce boundary conditions into the method and to provide a natural assembly sequence. A derivation is given in terms of general interpolation functions from this procedure. Example computations for transonic and supersonic flows in two and three dimensions are given to illustrate the utility of GIM. A three-dimensional nozzle-exhaust flow field is solved including interaction with the freestream and a coupled treatment of the shear layer. Potential applications of the GIM code to a variety of computational fluid dynamics problems is then discussed in terms of existing capability or by extension of the methodology.

Spradley, L. W.↗

NASA NTRS1979

Multi-level adaptive finite element methods. 1: Variation problems

A general numerical strategy for solving partial differential equations and other functional problems by cycling between coarser and finer levels of discretization is described. Optimal discretization schemes are provided together with very fast general solvers. It is described in terms of finite element discretizations of general nonlinear minimization problems. The basic processes (relaxation sweeps, fine-grid-to-coarse-grid transfers of residuals, coarse-to-fine interpolations of corrections) are directly and naturally determined by the objective functional and the sequence of approximation spaces. The natural processes, however, are not always optimal. Concrete examples are given and some new techniques are reviewed. Including the local truncation extrapolation and a multilevel procedure for inexpensively solving chains of many boundary value problems, such as those arising in the solution of time-dependent problems.

Brandt, A.↗

NASA NTRS1980

Solution of Euler's Equations of Motion and Eulerian Angles for near symmetric rigid bodies subject to constant moments

Analytic expressions are found for Euler's Equations of Motion and for the Eulerian Angles for both symmetric and near symmetric rigid bodies under the influence of arbitrary constant body-fixed torques. These solutions provide the body-fixed angular velocities and the attitude of the body, respectively, as functions of time. They are of special interest in applications to spinning spacecraft (such as the Galileo Spacecraft to be launched in 1984) because they include the effect of time-varying spin rate. Thus they can be applied to spin-up and spin-down maneuvers as well as to error analysis for thruster misalignments. The solutions are given for arbitrary initial conditions in terms of Fresnel, Sine and Cosine Integrals. Numerical integration of the governing differential equations has verified that the approximate analytic solutions are very accurate in many physical situations of interest.

Longuski, J. M.↗

NASA NTRS1981

Surface roughness effect on finite oil journal bearings

A theoretical study of the performance of finite oil journal bearings is made, considering the surface roughness effect. The total load supporting ability under such a condition derives from the hydrodynamic as well as asperity contact pressure. These two components of load are calculated separately. The average Reynolds equation for partially lubricated surfaces is used to evaluate hydrodynamic pressure. An analytical expression for average film thickness is obtained and introduced to modify the average Reynolds equation. The resulting differential equation is then solved numerically by finite difference methods for mean hydrodynamic pressure, which in turn gives the hydrodynamic load. Assuming the surface height distribution as Gaussian, the asperity contact pressure is found. The effect of surface roughness parameter, surface pattern, eccentricity ratio, and length to diameter ratio on hydrodynamic load and on side leakage is investigated. It is shown that hydrodynamic load increases with increasing surface roughness when both journal and bearing surfaces have identical roughness structures or when the journal only has a rough surface. The trend of hydrodynamic load is reversed if the journal surface is smooth and the bearing surface is rough.

Majumdar, B. C.↗

NASA NTRS1981

The interaction of helium with alpha-quartz

A detailed method of calculating the three-dimensional interaction of gas atoms with solid surfaces has been developed in order to provide a quantitative means of determining atmospheric evolution and global distribution on exospheric planetary bodies. The computational method involves accurate structural models of the surface and subsurface, and numerical solution of 60-dimensional differential equations of motion in a Monte Carlo calculation of the collision process. Preliminary results for the He-alpha-quartz system are described.

Kunc, J. A.↗

NASA NTRS1982

Postbuckling of orthotropic composite plates loaded in compression

The nonlinear large deflection equations of von Karman are written for 'specially' orthotropic plates. The equations are then manipulated to determine the parameters required to establish postbuckling behavior. It is found that only two new parameters are needed beyond those required for buckling. By assuming trigonometric functions in one direction, the plate equations are converted into ordinary nonlinear differential equations which are solved numerically using a two point boundary problem solver that makes use of Newton's method. The postbuckling behavior is obtained for simply supported and clamped, long, rectangular, orthotropic plates covering the complete range of dimensions and material properties.

Stein, M.↗

NASA NTRS1983

Numerical computation of transonic flow governed by the full-potential equation

Basic concepts associated with the numerical solution of elliptic partial differential equations are introduced, and procedures used to solve the full potential equation for transonic flow fields are discussed. Governing equations, classical relaxation schemes and concepts regarding transonic, full potential equation algorithms are covered. The equation transformation and grid generation procedures; full potential spatial differencing schemes; full potential iteration schemes, emphasizing convergence acceleration; and three dimensional applications are presented.

Holst, T. L.↗

NASA NTRS1984

A comparison of the efficiency of numerical methods for integrating chemical kinetic rate equations

The efficiency of several algorithms used for numerical integration of stiff ordinary differential equations was compared. The methods examined included two general purpose codes EPISODE and LSODE and three codes (CHEMEQ, CREK1D and GCKP84) developed specifically to integrate chemical kinetic rate equations. The codes were applied to two test problems drawn from combustion kinetics. The comparisons show that LSODE is the fastest code available for the integration of combustion kinetic rate equations. It is shown that an iterative solution of the algebraic energy conservation equation to compute the temperature can be more efficient then evaluating the temperature by integrating its time-derivative.

Radhakrishnan, K.↗

NASA NTRS1984

Computational problems on composite grids

Most currently used algorithms for the numerical solution of the partial differential equations encountered in fluid flow problems can be implemented on composite grid systems. Finite volume formulations are easier to derive on composite grids, and may in principle be derived for partial differential equations of all types. Except when using Alternating Difference Implicit-type schemes, the overlapping of grids is an alternative to the more common grid construction procedure where grid lines continue smoothly from one subregion to the next. In the solution of model problems, the correct choice of an interpolation formula has been found able to reduce errors by a factor of two.

Mastin, C. W.↗

NASA NTRS1984

A comparison of the efficiency of numerical methods for integrating chemical kinetic rate equations

A comparison of the efficiency of several algorithms recently developed for the efficient numerical integration of stiff ordinary differential equations is presented. The methods examined include two general-purpose codes EPISODE and LSODE and three codes (CHEMEQ, CREK1D, and GCKP84) developed specifically to integrate chemical kinetic rate equations. The codes are applied to two test problems drawn from combustion kinetics. The comparisons show that LSODE is the fastest code currently available for the integration of combustion kinetic rate equations. An important finding is that an iterative solution of the algebraic energy conservation equation to compute the temperature can be more efficient than evaluating the temperature by integrating its time-derivative.

Radhakrishnan, K.↗

NASA NTRS1984

Secular solution for delta-V during spin rate change maneuvers of rigid body spacecraft

Analytic expressions have been found for Euler's Equations of Motion and for the Eulerian Angles for both symmetric and near symmetric rigid bodies under the influence of arbitrary constant body-fixed torques. These solutions have been used to solve for the secular terms in the translational delta-V equations in inertial space. This secular delta-V solution is of interest in application to spinning spacecraft in that it describes the average direction of the delta-V of the spacecraft during a spin-up maneuver. Numerical integration of the governing differential equations has verified that the secular delta-V solution is valid for large time and is accurate in many physical situations including spin-up maneuvers of the Galileo spacecraft.

Klumpe, E. W.↗