Research in stability of periodic motions Quarterly progress report, 5 May - 4 Aug. 1966
Instability of periodic orbits in restricted and reduced three-body problems using mapping by fixed point method
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Solution of a few nonlinear problems in aerodynamics by the finite elements and functional least squares methods
The numerical simulation of the transonic flows of idealized fluids and of incompressible viscous fluids, by the nonlinear least squares methods is presented. The nonlinear equations, the boundary conditions, and the various constraints controlling the two types of flow are described. The standard iterative methods for solving a quasi elliptical nonlinear equation with partial derivatives are reviewed with emphasis placed on two examples: the fixed point method applied to the Gelder functional in the case of compressible subsonic flows and the Newton method used in the technique of decomposition of the lifting potential. The new abstract least squares method is discussed. It consists of substituting the nonlinear equation by a problem of minimization in a H to the minus 1 type Sobolev functional space.
Periaux, J.↗
Numerical analysis of Weyl's method for integrating boundary layer equations
A fast method for accurate numerical integration of Blasius equation is proposed. It is based on the limit interchange in Weyl's fixed point method formulated as an iterated limit process. Each inner limit represents convergence to a discrete solution. It is shown that the error in a discrete solution admits asymptotic expansion in even powers of step size. An extrapolation process is set up to operate on a sequence of discrete solutions to reach the outer limit. Finally, this method is extended to related boundary layer equations.
Najfeld, I.↗
On a class of approximate iterative processes.
Combination into unified setting of various results for approximate solution of fixed point equation, using iterative process
APPROXIMATION METHOD↗
Investigation of Turbulence in Wind Tunnels by a Study of the Flow About Cylinders
With the assistance and cooperation of the National Advisory Committee for Aeronautics the Bureau of Standards has been engaged for the past year in an investigation of turbulence in wind tunnels, especially in so far as turbulence affects the results of measurements in different wind tunnels. Two methods of making such studies are described in this report together with the results of the use in the 54-inch wind tunnel of the Bureau of Standards. The first method consists in measuring the drag of circular cylinders; the second in measuring the static pressure at some fixed point. Both methods show that the flow is not entirely free from irregularities.
Dryden, H L↗
A renormalization group analysis of two-dimensional magnetohydrodynamic turbulence
The renormalization group (RNG) method is used to study the physics of two-dimensional (2D) magnetohydrodynamic (MHD) turbulence. It is shown that, for a turbulent magnetofluid in two dimensions, no RNG transformation fixed point exists on account of the coexistence of energy transfer to small scales and mean-square magnetic flux transfer to large scales. The absence of a fixed point renders the RNG method incapable of describing the 2D MHD system. A similar conclusion is reached for 2D hydrodynamics, where enstrophy flows to small scales and energy to large scales. These analyses suggest that the applicability of the RNG method to turbulent systems is intrinsically limited, especially in the case of systems with dual-direction transfer.
Liang, Wenli Z.↗
Brightness and polarization of the zodiacal light - Results of fixed-position observations from Skylab
A method is outlined for evaluating discrete and background starlight at the Skylab wavelengths so as to derive the total brightness of zodiacal light at five sky positions. These sky positions are north celestial pole, south ecliptic pole, vernal equinox and two places near the north galactic pole. Pioneer 10/11 imaging photopolarimeters were used to periodically measure sky brightness and polarization in the blue and red at heliocentric distances beyond 1.002 AU. Mean zodiacal light, which is assumed to be also solar color in total light, is then estimated by applying the method to all fixed-point observations.
Weinberg, J. L.↗
On options for interdisciplinary analysis and design optimization
The interdisciplinary optimization of engineering systems is discussed from the standpoint of the computational alternatives available to the designer. The analysis of such systems typically requires the solution of coupled systems of nonlinear algebraic equations. The solution procedure is necessarily iterative in nature. It is shown that the system can be solved by fixed point iteration, by Newton's method, or by a combination of the two. However, the need for sensitivity analysis may affect the choice of analysis solution method. Similarly, the optimization of the system can be formulated in several ways that are discussed in the paper. It is shown that the effect of the topology of the interaction between disciplines is a key factor in the choice of analysis, sensitivity and optimization methods. Several examples are presented to illustrate the discussion.
Haftka, R. T.↗
Community Noise Assessment of Urban Air Mobility Vehicle Operations using the FAA Aviation Environmental Design Tool
In contrast to most commercial air traffic today, vehicles serving the urban air mobility (UAM) market are anticipated to operate in communities close to the public at large. The approved model for assessing environmental impact of air traffic actions in the United States, the Federal Aviation Administration’s Aviation Environmental Design Tool (AEDT), does not support analysis of such operations due to a combined lack of a UAM aircraft performance model and aircraft noise data. This paper discusses the initial development of a method to assess the acoustic impact of UAM fleet operations on the community using AEDT and demonstrates its use for representative UAM operations. In particular, methods were developed using fixed-point flight profiles and user-supplied noise data in a manner that avoids unwanted behavior in AEDT. A set of 32 routes in the Dallas-Ft. Worth area were assessed for single and multiple (fleet) operations for two concept vehicles.
urban air mobility↗
Community Noise Assessment of Urban Air Mobility Vehicle Operations using the FAA Aviation Environmental Design Tool
In contrast to most commercial air traffic today, vehicles serving the urban air mobility (UAM) market are anticipated to operate in communities close to the public at large. The approved model for assessing environmental impact of air traffic actions in the United States, the Federal Aviation Administration’s Aviation Environmental Design Tool (AEDT), does not support analysis of such operations due to a combined lack of a UAM aircraft performance model and aircraft noise data. This paper discusses the initial development of a method to assess the acoustic impact of UAM fleet operations on the community using AEDT and demonstrates its use for representative UAM operations. In particular, methods were developed using fixed-point flight profiles and user-supplied noise data in a manner that avoids unwanted behavior in AEDT. A set of 32 routes in the Dallas-Ft. Worth area were assessed for single and multiple (fleet) operations for two concept vehicles.
urban air mobility↗
Adaptive computational methods for SSME internal flow analysis
Adaptive finite element methods for the analysis of classes of problems in compressible and incompressible flow of interest in SSME (space shuttle main engine) analysis and design are described. The general objective of the adaptive methods is to improve and to quantify the quality of numerical solutions to the governing partial differential equations of fluid dynamics in two-dimensional cases. There are several different families of adaptive schemes that can be used to improve the quality of solutions in complex flow simulations. Among these are: (1) r-methods (node-redistribution or moving mesh methods) in which a fixed number of nodal points is allowed to migrate to points in the mesh where high error is detected; (2) h-methods, in which the mesh size h is automatically refined to reduce local error; and (3) p-methods, in which the local degree p of the finite element approximation is increased to reduce local error. Two of the three basic techniques have been studied in this project: an r-method for steady Euler equations in two dimensions and a p-method for transient, laminar, viscous incompressible flow. Numerical results are presented. A brief introduction to residual methods of a-posterior error estimation is also given and some pertinent conclusions of the study are listed.
Oden, J. T.↗
Efficient Implementation of an Optimal Interpolator for Large Spatial Data Sets
Scattered data interpolation is a problem of interest in numerous areas such as electronic imaging, smooth surface modeling, and computational geometry. Our motivation arises from applications in geology and mining, which often involve large scattered data sets and a demand for high accuracy. The method of choice is ordinary kriging. This is because it is a best unbiased estimator. Unfortunately, this interpolant is computationally very expensive to compute exactly. For n scattered data points, computing the value of a single interpolant involves solving a dense linear system of size roughly n x n. This is infeasible for large n. In practice, kriging is solved approximately by local approaches that are based on considering only a relatively small'number of points that lie close to the query point. There are many problems with this local approach, however. The first is that determining the proper neighborhood size is tricky, and is usually solved by ad hoc methods such as selecting a fixed number of nearest neighbors or all the points lying within a fixed radius. Such fixed neighborhood sizes may not work well for all query points, depending on local density of the point distribution. Local methods also suffer from the problem that the resulting interpolant is not continuous. Meyer showed that while kriging produces smooth continues surfaces, it has zero order continuity along its borders. Thus, at interface boundaries where the neighborhood changes, the interpolant behaves discontinuously. Therefore, it is important to consider and solve the global system for each interpolant. However, solving such large dense systems for each query point is impractical. Recently a more principled approach to approximating kriging has been proposed based on a technique called covariance tapering. The problems arise from the fact that the covariance functions that are used in kriging have global support. Our implementations combine, utilize, and enhance a number of different approaches that have been introduced in literature for solving large linear systems for interpolation of scattered data points. For very large systems, exact methods such as Gaussian elimination are impractical since they require 0(n(exp 3)) time and 0(n(exp 2)) storage. As Billings et al. suggested, we use an iterative approach. In particular, we use the SYMMLQ method, for solving the large but sparse ordinary kriging systems that result from tapering. The main technical issue that need to be overcome in our algorithmic solution is that the points' covariance matrix for kriging should be symmetric positive definite. The goal of tapering is to obtain a sparse approximate representation of the covariance matrix while maintaining its positive definiteness. Furrer et al. used tapering to obtain a sparse linear system of the form Ax = b, where A is the tapered symmetric positive definite covariance matrix. Thus, Cholesky factorization could be used to solve their linear systems. They implemented an efficient sparse Cholesky decomposition method. They also showed if these tapers are used for a limited class of covariance models, the solution of the system converges to the solution of the original system. Matrix A in the ordinary kriging system, while symmetric, is not positive definite. Thus, their approach is not applicable to the ordinary kriging system. Therefore, we use tapering only to obtain a sparse linear system. Then, we use SYMMLQ to solve the ordinary kriging system. We show that solving large kriging systems becomes practical via tapering and iterative methods, and results in lower estimation errors compared to traditional local approaches, and significant memory savings compared to the original global system. We also developed a more efficient variant of the sparse SYMMLQ method for large ordinary kriging systems. This approach adaptively finds the correct local neighborhood for each query point in the interpolation process.
Memarsadeghi, Nargess↗
Digital random-number generator
For binary digit array of N bits, use N noise sources to feed N nonlinear operators; each flip-flop in digit array is set by nonlinear operator to reflect whether amplitude of generator which feeds it is above or below mean value of generated noise. Fixed-point uniform distribution random number generation method can also be used to generate random numbers with other than uniform distribution.
Brocker, D. H.↗
Arbitrary Steady-State Solutions with the K-epsilon Model
Widely-used forms of the K-epsilon turbulence model are shown to yield arbitrary steady-state converged solutions that are highly dependent on numerical considerations such as initial conditions and solution procedure. These solutions contain pseudo-laminar regions of varying size. By applying a nullcline analysis to the equation set, it is possible to clearly demonstrate the reasons for the anomalous behavior. In summary, the degenerate solution acts as a stable fixed point under certain conditions, causing the numerical method to converge there. The analysis also suggests a methodology for preventing the anomalous behavior in steady-state computations.
Rumsey, Christopher L.↗
Reconfigurable Flight Control Designs With Application to the X-33 Vehicle
Two methods for control system reconfiguration have been investigated. The first method is a robust servomechanism control approach (optimal tracking problem) that is a generalization of the classical proportional-plus-integral control to multiple input-multiple output systems. The second method is a control-allocation approach based on a quadratic programming formulation. A globally convergent fixed-point iteration algorithm has been developed to make onboard implementation of this method feasible. These methods have been applied to reconfigurable entry flight control design for the X-33 vehicle. Examples presented demonstrate simultaneous tracking of angle-of-attack and roll angle commands during failures of the right body flap actuator. Although simulations demonstrate success of the first method in most cases, the control-allocation method appears to provide uniformly better performance in all cases.
Burken, John J.↗
Two Reconfigurable Flight-Control Design Methods: Robust Servomechanism and Control Allocation
Two methods for control system reconfiguration have been investigated. The first method is a robust servomechanism control approach (optimal tracking problem) that is a generalization of the classical proportional-plus-integral control to multiple input-multiple output systems. The second method is a control-allocation approach based on a quadratic programming formulation. A globally convergent fixed-point iteration algorithm has been developed to make onboard implementation of this method feasible. These methods have been applied to reconfigurable entry flight control design for the X-33 vehicle. Examples presented demonstrate simultaneous tracking of angle-of-attack and roll angle commands during failures of the fight body flap actuator. Although simulations demonstrate success of the first method in most cases, the control-allocation method appears to provide uniformly better performance in all cases.
Burken, John J.↗
Dynamical Chaos in the Wisdom-Holman Integrator: Origins and Solutions
We examine the nonlinear stability of the Wisdom-Holman (WH) symplectic mapping applied to the integration of perturbed, highly eccentric (e-0.9) two-body orbits. We find that the method is unstable and introduces artificial chaos into the computed trajectories for this class of problems, unless the step size chosen 1s small enough that PeriaPse is always resolved, in which case the method is generically stable. This 'radial orbit instability' persists even for weakly perturbed systems. Using the Stark problem as a fiducial test case, we investigate the dynamical origin of this instability and argue that the numerical chaos results from the overlap of step-size resonances; interestingly, for the Stark-problem many of these resonances appear to be absolutely stable. We similarly examine the robustness of several alternative integration methods: a time-regularized version of the WH mapping suggested by Mikkola; the potential-splitting (PS) method of Duncan, Levison, Lee; and two original methods incorporating approximations based on Stark motion instead of Keplerian motion. The two fixed point problem and a related, more general problem are used to conduct a comparative test of the various methods for several types of motion. Among the algorithms tested, the time-transformed WH mapping is clearly the most efficient and stable method of integrating eccentric, nearly Keplerian orbits in the absence of close encounters. For test particles subject to both high eccentricities and very close encounters, we find an enhanced version of the PS method-incorporating time regularization, force-center switching, and an improved kernel function-to be both economical and highly versatile. We conclude that Stark-based methods are of marginal utility in N-body type integrations. Additional implications for the symplectic integration of N-body systems are discussed.
Rauch, Kevin P.↗
The Windvan pulsed CO2 Doppler lidar wide-area wind sensor
Wind sensing using a Doppler lidar is achieved by sensing the Doppler content of narrow frequency laser light backscattered by the ambient atmospheric aerosols. The derived radial wind components along several directions are used to generate wind vectors, typically using the Velocity Azimuth Display (VAD) method described below. Range resolved information is obtained by range gating the continuous scattered return. For a CO2 laser (10.6 mu) the Doppler velocity scaling factor is 188 kHz/ms(exp -1). In the VAD scan method the zenith angle of the pointing direction is fixed and its azimuth is continuously varied through 2 pi. A spatially uniform wind field at a particular altitude yields a sinusoidal variation of the radial component vs. azimuth. The amplitude, phase and dc component of this sinusoid yield the horizontal wind speed, direction and vertical component of the wind respectively. In a nonuniform wind field the Fourier components of the variation yields the required information.
Lawrence, Rhidian↗