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At least 865 records · Page 48

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NASA NTRS1995

An approach to the development of numerical algorithms for first order linear hyperbolic systems in multiple space dimensions: The constant coefficient case

Two methods for developing high order single step explicit algorithms on symmetric stencils with data on only one time level are presented. Examples are given for the convection and linearized Euler equations with up to the eighth order accuracy in both space and time in one space dimension, and up to the sixth in two space dimensions. The method of characteristics is generalized to nondiagonalizable hyperbolic systems by using exact local polynominal solutions of the system, and the resulting exact propagator methods automatically incorporate the correct multidimensional wave propagation dynamics. Multivariate Taylor or Cauchy-Kowaleskaya expansions are also used to develop algorithms. Both of these methods can be applied to obtain algorithms of arbitrarily high order for hyperbolic systems in multiple space dimensions. Cross derivatives are included in the local approximations used to develop the algorithms in this paper in order to obtain high order accuracy, and improved isotropy and stability. Efficiency in meeting global error bounds is an important criterion for evaluating algorithms, and the higher order algorithms are shown to be up to several orders of magnitude more efficient even though they are more complex. Stable high order boundary conditions for the linearized Euler equations are developed in one space dimension, and demonstrated in two space dimensions.

Goodrich, John W.↗

NASA NTRS1986

Excitation of nearly steady finite-amplitude barotropic waves

An exact nonlinear stationary solution is obtained for barotropic waves in a beta-plane channel and it is shown that it can be excited under a range of initial conditions. Results show that a finite-amplitude wave in a constant shear flow, given an initial phase tilt against the shear and a sufficient initial amplitude, interacts with the mean flow to produce a nearly steady state close to the exact stationary solution. This equilibration process involves nonlinear transients; in particular, as the flow equilibrates, the emergence of critical levels is accompanied by the neutralization of local mean vorticity gradients at these levels, thus allowing the solution to attain a nonsingular modal structure.

Hou, A. Y.↗

NASA NTRS1987

Exact finite element method analysis of viscoelastic tapered structures to transient loads

A general method is presented for determining the dynamic torsional/axial response of linear structures composed of either tapered bars or shafts to transient excitations. The method consists of formulating and solving the dynamic problem in the Laplace transform domain by the finite element method and obtaining the response by a numerical inversion of the transformed solution. The derivation of the torsional and axial stiffness matrices is based on the exact solution of the transformed governing equation of motion, and it consequently leads to the exact solution of the problem. The solution permits treatment of the most practical cases of linear tapered bars and shafts, and employs modeling of structures with only one element per member which reduces the number of degrees of freedom involved. The effects of external viscous or internal viscoelastic damping are also taken into account.

Spyrakos, Constantine Chris↗

NASA NTRS1987

Validation of the SURE Program, phase 1

Presented are the results of the first phase in the validation of the SURE (Semi-Markov Unreliability Range Evaluator) program. The SURE program gives lower and upper bounds on the death-state probabilities of a semi-Markov model. With these bounds, the reliability of a semi-Markov model of a fault-tolerant computer system can be analyzed. For the first phase in the validation, fifteen semi-Markov models were solved analytically for the exact death-state probabilities and these solutions compared to the corresponding bounds given by SURE. In every case, the SURE bounds covered the exact solution. The bounds, however, had a tendency to separate in cases where the recovery rate was slow or the fault arrival rate was fast.

Dotson, Kelly J.↗

NASA NTRS1974

Analytical solution of the optimal three dimensional reentry problem using Chapman's exact equations

This paper presents the general solution for the optimal three dimensional aerodynamic control of a lifting vehicle entering a planetary atmosphere. A set of dimensionless variables is introduced, and the resulting exact equations of motion have the distinctive advantage that they are completely free of the physical characteristics of the vehicle. Furthermore, a general lift-drag polar is used to define the aerodynamic control. Hence, the results obtained apply to any type of vehicle of arbitrary weight, dimensions and shape, having an arbitrary polar and entering any planetary atmosphere.

Vinh, N. X.↗

NASA NTRS1987

Experience in distributed parameter modeling of the Spacecraft Control Laboratory Experiment (SCOLE) structure

The Spacecraft Control Laboratory Experiment (SCOLE) configuration is used to compare exact and approximate solutions of the partial differential equations which define its structural dynamics. The need for a proof model for evaluating competing control laws demands that solutions be generated which not only exhibit accurate modal characteristics, but precise static deflections as well. Because precise pointing is required, the motion of the end bodies of the Shuttle-attached antenna must be known with great accuracy. Modal models are attractive because of their stable solutions but require hundreds of modes to obtain a static deflection accuracy of only one percent. Although proportional damping in bending agrees well with experimental results using the SCOLE experimental apparatus, modes which involve both torsion and bending differ significantly from proportional damping. A lumped mass model is used to generate exact static deflections, but only approximate modal characteristics. Asymptotic solutions to the distributed parameter system approximate very accurately the modal characteristics at high mode numbers. Ways are examined for refining the approximate solutions by applying a first-order variation and by employing singular perturbation techniques which are usually limited to ordinary differential equations. The most accurate solutions of the distributed parameter model of SCOLE are obtained by combining exact and asymptotic solutions.

Taylor, L. W.↗

NASA NTRS2010

Tuning Parameters in Heuristics by Using Design of Experiments Methods

With the growing complexity of today's large scale problems, it has become more difficult to find optimal solutions by using exact mathematical methods. The need to find near-optimal solutions in an acceptable time frame requires heuristic approaches. In many cases, however, most heuristics have several parameters that need to be "tuned" before they can reach good results. The problem then turns into "finding best parameter setting" for the heuristics to solve the problems efficiently and timely. One-Factor-At-a-Time (OFAT) approach for parameter tuning neglects the interactions between parameters. Design of Experiments (DOE) tools can be instead employed to tune the parameters more effectively. In this paper, we seek the best parameter setting for a Genetic Algorithm (GA) to solve the single machine total weighted tardiness problem in which n jobs must be scheduled on a single machine without preemption, and the objective is to minimize the total weighted tardiness. Benchmark instances for the problem are available in the literature. To fine tune the GA parameters in the most efficient way, we compare multiple DOE models including 2-level (2k ) full factorial design, orthogonal array design, central composite design, D-optimal design and signal-to-noise (SIN) ratios. In each DOE method, a mathematical model is created using regression analysis, and solved to obtain the best parameter setting. After verification runs using the tuned parameter setting, the preliminary results for optimal solutions of multiple instances were found efficiently.

Arin, Arif↗

NASA NTRS1994

Numerical simulation of receptivity and transition in a boundary layer on a flat plate with a suction hole

A Navier-Stokes solver has been modified to allow for unsteady boundary conditions to be imposed at the inflow boundary. Because the streamwise derivatives are discretized with three-point compact differences, there is no need for special treatment of the points adjacent to the inflow boundary, provided that all necessary derivatives are properly specified at the boundary points. The unsteady boundary conditions can be obtained from a previous computation, from a solution of the parabolized stability equations (PSE), or from an analytical exact or approximate solution of the Navier-Stokes equations, e.g. an asymptotic expansion solution to model a sound wave interacting with the boundary layer.

Fasel, Hermann F.↗

NASA NTRS1994

Matching Inviscid/Boundary Layer Flowfields

New boundary-layer equations are developed and the solutions match all the boundary-layer properties, except for the normal velocity, exactly with the corresponding inviscid properties. The numerical procedure solves tridiagonal matrices at each marching station. As part of the solution an inviscid transpiration velocity at the surface is calculated from the boundary-layer solution. This transpiration velocity could be used as a boundary condition to calculate a new inviscid solution. If the inviscid/boundary layer solutions are iterated, then the normal velocity from the boundary layer solution will match the inviscid values exactly. Solutions are calculated for shear flows over a flat plate. Results from the present method compared well with Navier-Stokes solutions for incompressible constant shear flow and sinusoidal shear flow. Iterations of the inviscid/boundary layer solutions were not necessary. Compressible flow of an exponential shear at Mach 4 was calculated for cold, adiabatic, and hot walls. It was found that inviscid shear flows have a significant effect on skin friction and heating rates. The present method should yield accurate boundary-layer solutions at Reynolds numbers lower than the traditional boundary-layer equations.

DeJarnette, Fred R.↗

NASA NTRS1997

Thermal Radiation Effects Analyzed in Translucent Composite and Thermal Barrier Coating

Ceramic parts and coatings are needed to withstand high temperatures in advanced aircraft engines. In hot environments, such as in the combustion chambers of these engines, infrared and visible radiation can penetrate into some ceramics and heat them internally. The internal temperatures depend on radiative effects combined with heat conduction, and on convection and radiation at the material boundaries. Since engine temperatures are high, radiant emission can be large from within translucent parts and coatings, and this must be included in the analysis. Transient and steady-state behavior are both important. During a transient, radiant penetration provides more rapid internal heating than conduction alone, and the temperature distributions are usually considerably different than for steady-state conditions; this can produce transient thermal stresses. Analytical and numerical methods are being used at the NASA Lewis Research Center to predict transient temperatures and heat flows in translucent materials. A transient analysis was done for a composite of two translucent layers. The layer refractive indices were larger than 1, producing internal reflections at the boundaries and at the internal interface. In addition, steady-state results were computed for a two-layer composite with one layer opaque. The results were used to assess the importance of internal radiation in a zirconia thermal barrier coating on a cooled metal wall. Since the radiative transfer equations are rather complex, especially when internal scattering is included, approximate methods are being investigated that might be convenient for computer design programs. An approximate two-flux method was used and verified by comparisons with solutions that used exact radiative transfer relations. The two-flux equations include scattering without increasing solution difficulty.

Siegel, Robert↗

NASA NTRS1989

A useful approximation for the flat surface impulse response

The flat surface impulse response (FSIR) is a very useful quantity in computing the mean return power for near-nadir-oriented short-pulse radar altimeters. However, for very small antenna beamwidths and relatively large pointing angles, previous analytical descriptions become very difficult to compute accurately. An asymptotic approximation is developed to overcome these computational problems. Since accuracy is of key importance, a condition is developed under which this solution is within 2 percent of the exact answer. The asymptotic solution is shown to be in functional agreement with a conventional clutter power result and gives a 1.25-dB correction to this formula to account properly for the antenna-pattern variation over the illuminated area.

Brown, Gary S.↗

NASA NTRS1992

Practical aspects of spatially high accurate methods

The computational qualities of high order spatially accurate methods for the finite volume solution of the Euler equations are presented. Two dimensional essentially non-oscillatory (ENO), k-exact, and 'dimension by dimension' ENO reconstruction operators are discussed and compared in terms of reconstruction and solution accuracy, computational cost and oscillatory behavior in supersonic flows with shocks. Inherent steady state convergence difficulties are demonstrated for adaptive stencil algorithms. An exact solution to the heat equation is used to determine reconstruction error, and the computational intensity is reflected in operation counts. Standard MUSCL differencing is included for comparison. Numerical experiments presented include the Ringleb flow for numerical accuracy and a shock reflection problem. A vortex-shock interaction demonstrates the ability of the ENO scheme to excel in simulating unsteady high-frequency flow physics.

Godfrey, Andrew G.↗

NASA NTRS1987

Particle paths and phase plane for time-dependent similarity solutions of the one-dimensional Vlasov-Maxwell equations

The phase trajectories of particles in a plasma described by the one-dimensional Vlasov-Maxwell equations are determined qualitatively, analyzing exact general similarity solutions for the cases of temporally damped and growing (sinusoidal or localized) electric fields. The results of numerical integration in both untransformed and Lie-group point-transformed coordinates are presented in extensive graphs and characterized in detail. The implications of the present analysis for the stability of BGK equilibria are explored, and the existence of nonlinear solutions arbitrarily close to and significantly different from the BGK solutions is demonstrated.

Roberts, Dana Aaron↗

NASA NTRS1975

Unsteady transonic aerodynamics - An aeronautics challenge

The paper presents a review of the historical development in unsteady transonic aerodynamics, along with the foundations and accomplishments of several approaches to solve the equations of unsteady transonic flow. The discussion covers the linearized unsteady flow theory, numerical solution of the exact equations for an inviscid compressible gas, nonlinear small disturbance theory of transonic flow and linearization of the unsteady component about the nonlinear solution for the steady state, local linearization solution for unsteady transonic flow, unsteady transonic flow theory for slender wings and bodies, and three-dimensional unsteady transonic flows. The relation between the calculated results and experiment is examined. It is shown that the newly emerging numerical methods are capable of solving the nonlinear equations for two-dimensional flow and can be extended to three-dimensional flows.

Spreiter, J. R.↗

NASA NTRS1986

A rapid and accurate method for calculation of stratospheric photolysis rates with molecular scattering

A method for calculating the photodissociation rates needed for photochemical modeling of the stratosphere, which includes the effects of molecular scattering, is described. The procedure is based on Sokolov's method of averaging functional correction. The radiation model and approximations used to calculate the radiation field are examined. The approximated diffuse fields and photolysis rates are compared with exact data. It is observed that the approximate solutions differ from the exact result by 10 percent or less at altitudes above 15 km; the photolysis rates differ from the exact rates by less than 5 percent for altitudes above 10 km and all zenith angles, and by less than 1 percent for altitudes above 15 km.

Boughner, Robert E.↗

NASA NTRS1981

An iterative finite element-integral technique for predicting sound radiation from turbofan inlets

A new iterative solution technique for predicting the sound field radiated from a turbofan inlet is presented. The sound field is divided into two regions; the sound field within the inlet which is computed using the finite element method and the radiation field outside the inlet which is calculated using an integral solution technique. A 'unified' solution is obtained by matching the finite element and integral solutions at the interface between the interior and exterior regions. The applicability of the iterative technique is demonstrated by considering several simple cases for which exact or 'classical' solutions for the sound field are available. These examples show that the proper solution is obtained within five iterations. The overall accuracy of the method is demonstrated by comparison with experimental data.

Horowitz, S. J.↗

NASA NTRS1987

Guidance and control strategies for aerospace vehicles

A simplified method of matched asymptotic expansions was developed where the common part in composite solution is generated as a polynomial in stretched variable instead of actually evaluating the same from the outer solution. This methodology was applied to the solution of the exact equations for three dimensional atmospheric entry problems. Compared to previous works, the present simplified methodology yields explicit analytical expressions for various components of the composite solution without resorting to any type of transcendental equations to be solved only by numerical methods. The optimal control problem arising in the noncoplanar orbital transfer employing aeroassist was also addressed.

Hibey, Joseph L.↗

NASA NTRS1982

Validation of zero-order feedback strategies for medium range air-to-air interception in a horizontal plane

A zero order feedback solution of a variable speed interception game between two aircraft in the horizontal plane, obtained by using the method of forced singular perturbation (FSP), is compared with the exact open loop solution. The comparison indicates that for initial distances of separation larger than eight turning radii of the evader, the accuracy of the feedback approximation is better than one percent. The result validates the zero order FSP approximation for medium range air combat analysis.

Shinar, J.↗