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NASA NTRS1985

Flutter of swept fan blades

The effect of sweep on fan blade flutter is studied by applying the analytical methods developed for aeroelastic analysis of advance turboprops. Two methods are used. The first method utilizes an approximate structural model in which the blade is represented by a swept, nonuniform beam. The second method utilizes a finite element technique to conduct modal flutter analysis. For both methods the unsteady aerodynamic loads are calculated using two dimensional cascade theories which are modified to account for sweep. An advanced fan stage is analyzed with 0, 15 and 30 degrees of sweep. It is shown that sweep has a beneficial effect on predominantly torsional flutter and a detrimental effect on predominantly bending flutter. This detrimental effect is shown to be significantly destabilizing for 30 degrees of sweep.

Kielb, R. E.↗

NASA NTRS2001

Engine With Regression and Neural Network Approximators Designed

At the NASA Glenn Research Center, the NASA engine performance program (NEPP, ref. 1) and the design optimization testbed COMETBOARDS (ref. 2) with regression and neural network analysis-approximators have been coupled to obtain a preliminary engine design methodology. The solution to a high-bypass-ratio subsonic waverotor-topped turbofan engine, which is shown in the preceding figure, was obtained by the simulation depicted in the following figure. This engine is made of 16 components mounted on two shafts with 21 flow stations. The engine is designed for a flight envelope with 47 operating points. The design optimization utilized both neural network and regression approximations, along with the cascade strategy (ref. 3). The cascade used three algorithms in sequence: the method of feasible directions, the sequence of unconstrained minimizations technique, and sequential quadratic programming. The normalized optimum thrusts obtained by the three methods are shown in the following figure: the cascade algorithm with regression approximation is represented by a triangle, a circle is shown for the neural network solution, and a solid line indicates original NEPP results. The solutions obtained from both approximate methods lie within one standard deviation of the benchmark solution for each operating point. The simulation improved the maximum thrust by 5 percent. The performance of the linear regression and neural network methods as alternate engine analyzers was found to be satisfactory for the analysis and operation optimization of air-breathing propulsion engines (ref. 4).

Patnaik, Surya N.↗

NASA NTRS1990

An adaptive grid algorithm for one-dimensional nonlinear equations

Richards' equation, which models the flow of liquid through unsaturated porous media, is highly nonlinear and difficult to solve. Step gradients in the field variables require the use of fine grids and small time step sizes. The numerical instabilities caused by the nonlinearities often require the use of iterative methods such as Picard or Newton interation. These difficulties result in large CPU requirements in solving Richards equation. With this in mind, adaptive and multigrid methods are investigated for use with nonlinear equations such as Richards' equation. Attention is focused on one-dimensional transient problems. To investigate the use of multigrid and adaptive grid methods, a series of problems are studied. First, a multigrid program is developed and used to solve an ordinary differential equation, demonstrating the efficiency with which low and high frequency errors are smoothed out. The multigrid algorithm and an adaptive grid algorithm is used to solve one-dimensional transient partial differential equations, such as the diffusive and convective-diffusion equations. The performance of these programs are compared to that of the Gauss-Seidel and tridiagonal methods. The adaptive and multigrid schemes outperformed the Gauss-Seidel algorithm, but were not as fast as the tridiagonal method. The adaptive grid scheme solved the problems slightly faster than the multigrid method. To solve nonlinear problems, Picard iterations are introduced into the adaptive grid and tridiagonal methods. Burgers' equation is used as a test problem for the two algorithms. Both methods obtain solutions of comparable accuracy for similar time increments. For the Burgers' equation, the adaptive grid method finds the solution approximately three times faster than the tridiagonal method. Finally, both schemes are used to solve the water content formulation of the Richards' equation. For this problem, the adaptive grid method obtains a more accurate solution in fewer work units and less computation time than required by the tridiagonal method. The performance of the adaptive grid method tends to degrade as the solution process proceeds in time, but still remains faster than the tridiagonal scheme.

Gutierrez, William E.↗

NASA NTRS1993

Program Determines Minimum-State Approximations To Unsteady Aerodynamic Forces

MIST implements minimum-state method for determining rational approximations of matrix of aerodynamic force coefficients. Accepts complex tabular data representing generalized unsteady aerodynamic forces over set of reduced frequencies. Determines approximations to tabular data in Laplace domain by use of rational functions. Provides capability to select coefficients of denominator in rational approximations, to constrain approximation selectably without increasing size of problem, and determines and emphasizes criticial frequency ranges in determining approximations. Written in ANSI FORTRAN 77.

Karpel, Mordechay↗

NASA NTRS1993

Structural design utilizing updated, approximate sensitivity derivatives

A method to improve the computational efficiency of structural optimization algorithms is investigated. In this method, the calculations of 'exact' sensitivity derivatives of constraint functions are performed only at selected iterations during the optimization process. The sensitivity derivatives utilized within other iterations are approximate derivatives which are calculated using an inexpensive derivative update formula. Optimization results are presented for an analytic optimization problem (i.e., one having simple polynomial expressions for the objective and constraint functions) and for two structural optimization problems. The structural optimization results indicate that up to a factor of three improvement in computation time is possible when using the updated sensitivity derivatives.

Scotti, Stephen J.↗

NASA NTRS1993

Comparison of cross sections from the quasi-classical trajectory method and the j(z)-conserving centrifugal sudden approximation with accurate quantum results for an atom-rigid nonlinear polyatomic collision

We report the results of a series of calculations of state-to-state integral cross sections for collisions between O and nonvibrating H2O in the gas phase on a model nonreactive potential energy surface. The dynamical methods used include converged quantum mechanical scattering calculations, the j(z) conserving centrifugal sudden (j(z)-CCS) approximation, and quasi-classical trajectory (QCT) calculations. We consider three total energies 0.001, 0.002, and 0.005 E(h) and the nine initial states with rotational angular momentum less than or equal to 2 (h/2 pi). The j(z)-CCS approximation gives good results, while the QCT method can be quite unreliable for transitions to specific rotational sublevels. However, the QCT cross sections summed over final sublevels and averaged over initial sublevels are in better agreement with the quantum results.

Schwenke, David W.↗

NASA NTRS1978

Method for calculating convective heat-transfer coefficients over turbine vane surfaces

A method for calculating laminar, transitional, and turbulent convective heat-transfer coefficients for turbine vane surfaces is described. An approximate integral solution method produced results in good agreement with a finite-difference solution. Comparisons between the two are presented. The integral solution results agreed well with the finite-difference solution results in the laminar and turbulent regions. Differences in calculating the start of transition produced a later starting point for the approximate integral solution's transitional flow regime.

Gauntner, D. J.↗

NASA NTRS1992

Application of the variational-asymptotical method to composite plates

A method is developed for the 3D analysis of laminated plate deformation which is an extension of a variational-asymptotical method by Atilgan and Hodges (1991). Both methods are based on the treatment of plate deformation by splitting the 3D analysis into linear through-the-thickness analysis and 2D plate analysis. Whereas the first technique tackles transverse shear deformation in the second asymptotical approximation, the present method simplifies its treatment and restricts it to the first approximation. Both analytical techniques are applied to the linear cylindrical bending problem, and the strain and stress distributions are derived and compared with those of the exact solution. The present theory provides more accurate results than those of the classical laminated-plate theory for the transverse displacement of 2-, 3-, and 4-layer cross-ply laminated plates. The method can give reliable estimates of the in-plane strain and displacement distributions.

Hodges, Dewey H.↗

NASA NTRS1974

Digital redesign of continuous systems by matching of states at multiple sampling periods

This paper presents a method of digital approximation of a continuous-data system by the matching of states at multiple sampling periods. Two alternate methods are presented; one method samples the states every sampling period, and alters the forward and feedback gains, so as to match the states of the two systems at the end of N sampling periods. The second method requires the sampling of the states every N sampling periods, with the change of gains at each sampling period. An illustrative example using the one-axis dynamics of the Skylab Satellite is presented.

Yackel, R. A.↗

NASA NTRS1989

Globally convergent techniques in nonlinear Newton-Krylov

Some convergence theory is presented for nonlinear Krylov subspace methods. The basic idea of these methods is to use variants of Newton's iteration in conjunction with a Krylov subspace method for solving the Jacobian linear systems. These methods are variants of inexact Newton methods where the approximate Newton direction is taken from a subspace of small dimensions. The main focus is to analyze these methods when they are combined with global strategies such as linesearch techniques and model trust region algorithms. Most of the convergence results are formulated for projection onto general subspaces rather than just Krylov subspaces.

Brown, Peter N.↗

NASA NTRS2011

Comparison of Direct Solar Energy to Resistance Heating for Carbothermal Reduction of Regolith

A comparison of two methods of delivering thermal energy to regolith for the carbo thermal reduction process has been performed. The comparison concludes that electrical resistance heating is superior to direct solar energy via solar concentrators for the following reasons: (1) the resistance heating method can process approximately 12 times as much regolith using the same amount of thermal energy as the direct solar energy method because of superior thermal insulation; (2) the resistance heating method is more adaptable to nearer-term robotic exploration precursor missions because it does not require a solar concentrator system; (3) crucible-based methods are more easily adapted to separation of iron metal and glass by-products than direct solar energy because the melt can be poured directly after processing instead of being remelted; and (4) even with projected improvements in the mass of solar concentrators, projected photovoltaic system masses are expected to be even lower.

Muscatello, Anthony C.↗

NASA NTRS2013

A Floating Node Method for the Modelling of Discontinuities Within a Finite Element

This paper focuses on the accurate numerical representation of complex networks of evolving discontinuities in solids, with particular emphasis on cracks. The limitation of the standard finite element method (FEM) in approximating discontinuous solutions has motivated the development of re-meshing, smeared crack models, the eXtended Finite Element Method (XFEM) and the Phantom Node Method (PNM). We propose a new method which has some similarities to the PNM, but crucially: (i) does not introduce an error on the crack geometry when mapping to natural coordinates; (ii) does not require numerical integration over only part of a domain; (iii) can incorporate weak discontinuities and cohesive cracks more readily; (iv) is ideally suited for the representation of multiple and complex networks of (weak, strong and cohesive) discontinuities; (v) leads to the same solution as a finite element mesh where the discontinuity is represented explicitly; and (vi) is conceptually simpler than the PNM.

Pinho, Silvestre T.↗

NASA NTRS1988

Research and applications in aeroservoelasticity at the NASA Langley Research Center

A review of analytical methods used to analyze flexible vehicles with active controls is presented. Methods used to approximate and correct unsteady aerodynamic forces used in the analysis are discussed. Recent advances in the application of optimal methods to digital control law synthesis are presented. The use of active controls in an integrated design process is also discussed. Finally, the results of recent wind-tunnel studies aimed at demonstrating active control concepts and validating the analytical methods are presented.

Abel, Irving↗

NASA NTRS1976

Subsonic finite elements for wing body combinations

Capabilities, limitations, and applications of various theories for the prediction of wing-body aerodynamics are reviewed. The methods range from approximate planar representations applicable in preliminary design to surface singularity approaches applicable in the later stages of detail design. The available methods for three-dimensional configurations are limited as inviscid solutions with viscous effects included on an empirical or strip basis.

Thomas, J. L.↗

NASA NTRS1992

Algorithmic improvements for simulator motion drive

Contemporary simulator motion drive algorithms typically are designed in an analog (continuous) environment, but are implemented in a digital (discrete) environment. The intended continuous system, specified as frequency domain (Laplace transform) transfer functions, may not be represented properly by the algorithms used for digital implementation. The motion drive software in use with the Vertical Motion Simulator at Ames Research Center was investigated recently; the original algorithms (Euler) were changed to a state transition method. Comparison of the frequency responses of the original and new implementations showed that the state transition method more closely approximates the desired analog responses. In addition, test pilots who evaluated both implementations preferred the motions generated with the state transition method over those generated with the Euler integration method.

Laforce, Soren↗

NASA NTRS1994

A unifying comparison of nearly scatter free transport models

Gombosi et al. (1993) recently derived a modified telegrapher's equation for charged particle transport under the influence of isotropic scattering. This equation obeys causality and disallows upstream diffusion for particles with random velocities smaller than the bulk flow velocity. The acausal diffusion equation was obtained to lowest order in the expansion of smallness prameters. The paper by Gombosi et al. (1993) prompted responses from Pauls et al. (1993) and Earl (1993). This paper is written to explain the differences between the methods, assumptions, and results of Gombosi et al. (1993), Pauls et al. (1993), and Earl (1993) and presents a new method of obtaining approximate solutions. It is shown that the assumptions used by Gombosi et al. (1993) and Pauls et al. (1993) are physically equivalent. In our solution method, the solution of the modified telegrapher's equation is obtained as the casual limit of solutions accurate to second order in the smallness parameter expansion. In order to investigate the coherent velocity, we have also developed `wavenumber eigenfunctions' which account for all the pitch angle dependence in our Boltzmann equation. Using truncation, Earl (1993) obtains approximations for the wavenumber dependence of the lowest two frequency modes, which correspond to two of the wavenumber eigenmodes. We find that a consequence of including only two wavenumber eigenmodes is that one obtains solutions which disobey causality at sufficiently short times. Furthermore, the coherent velocity of the two eigenmodes is strongly dependent on wavenumber and approaches the particle velocity in the limit of large wavenumber for both isotropic and anisotropic scattering processes. We conclude that Earl's (1993) solutions and solutions obtained using the new solution method implicitly assume weak acausality and reasonable behavior in the temporal regime, t less than 4 tau. The solutions are not strictly consistent with the behavior of the lowest two frequency modes but have similar behavior in the regime of low wavenumber.

Schwadron, N. A.↗

NASA NTRS2014

Interpolation Method Needed for Numerical Uncertainty

Using Computational Fluid Dynamics (CFD) to predict a flow field is an approximation to the exact problem and uncertainties exist. There is a method to approximate the errors in CFD via Richardson's Extrapolation. This method is based off of progressive grid refinement. To estimate the errors, the analyst must interpolate between at least three grids. This paper describes a study to find an appropriate interpolation scheme that can be used in Richardson's extrapolation or other uncertainty method to approximate errors.

Groves, Curtis E.↗

NASA NTRS1987

A well-posed optimal spectral element approximation for the Stokes problem

A method is proposed for the spectral element simulation of incompressible flow. This method constitutes in a well-posed optimal approximation of the steady Stokes problem with no spurious modes in the pressure. The resulting method is analyzed, and numerical results are presented for a model problem.

Maday, Y.↗