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NASA NTRS1992

An extended Lagrangian method

A unique formulation of describing fluid motion is presented. The method, referred to as 'extended Lagrangian method', is interesting from both theoretical and numerical points of view. The formulation offers accuracy in numerical solution by avoiding numerical diffusion resulting from mixing of fluxes in the Eulerian description. Meanwhile, it also avoids the inaccuracy incurred due to geometry and variable interpolations used by the previous Lagrangian methods. Unlike the Lagrangian method previously imposed which is valid only for supersonic flows, the present method is general and capable of treating subsonic flows as well as supersonic flows. The method proposed in this paper is robust and stable. It automatically adapts to flow features without resorting to clustering, thereby maintaining rather uniform grid spacing throughout and large time step. Moreover, the method is shown to resolve multi-dimensional discontinuities with a high level of accuracy, similar to that found in one-dimensional problems.

Liou, Meng-Sing↗

NASA NTRS1993

An extended Lagrangian method

A unique formulation of describing fluid motion is presented. The method, referred to as 'extended Lagrangian method', is interesting from both theoretical and numerical points of view. The formulation offers accuracy in numerical solution by avoiding numerical diffusion resulting from mixing of fluxes in the Eulerian description. Meanwhile, it also avoids the inaccuracy incurred due to geometry and variable interpolations used by the previous Lagrangian methods. The present method is general and capable of treating subsonic flows as well as supersonic flows. The method proposed in this paper is robust and stable. It automatically adapts to flow features without resorting to clustering, thereby maintaining rather uniform grid spacing throughout and large time step. Moreover, the method is shown to resolve multidimensional discontinuities with a high level of accuracy, similar to that found in 1D problems.

Liou, Meng-Sing↗

NASA NTRS1988

The discrete correlation function - A new method for analyzing unevenly sampled variability data

A method for measuring correlation functions without interpolating in the temporal domain is proposed which provides an assumption-free representation of the correlation measured in the data and allows meaningful error estimates. Physical interpretation of the cross-correlation function of two series believed to be related by a convolution is shown to require knowledge of the input function's fluctuation power spectrum. Application of the method to two systems reveals no correlation for the optical data of Akn 120, but a strong correlation for the UV data of NGC 4151, placing bounds of between 1.2 and 20 light days on the size of the line-emitting region.

Edelson, R. A.↗

NASA NTRS1975

Sequential and simultaneous SLAR block adjustment

Two sequential methods of planimetric SLAR (Side Looking Airborne Radar) block adjustment, with and without splines, and three simultaneous methods based on the principles of least squares are evaluated. A limited experiment with simulated SLAR images indicates that sequential block formation with splines followed by external interpolative adjustment is superior to the simultaneous methods such as planimetric block adjustment with similarity transformations. The use of the sequential block formation is recommended, since it represents an inexpensive tool for satisfactory point determination from SLAR images.

Leberl, F.↗

NASA NTRS2002

Computing Aerodynamic Performance of a 2D Iced Airfoil: Blocking Topology and Grid Generation

The ice accrued on airfoils can have enormously complicated shapes with multiple protruded horns and feathers. In this paper, several blocking topologies are proposed and evaluated on their ability to produce high-quality structured multi-block grid systems. A transition layer grid is introduced to ensure that jaggedness on the ice-surface geometry do not to propagate into the domain. This is important for grid-generation methods based on hyperbolic PDEs (Partial Differential Equations) and algebraic transfinite interpolation. A 'thick' wrap-around grid is introduced to ensure that grid lines clustered next to solid walls do not propagate as streaks of tightly packed grid lines into the interior of the domain along block boundaries. For ice shapes that are not too complicated, a method is presented for generating high-quality single-block grids. To demonstrate the usefulness of the methods developed, grids and CFD solutions were generated for two iced airfoils: the NLF0414 airfoil with and without the 623-ice shape and the B575/767 airfoil with and without the 145m-ice shape. To validate the computations, the computed lift coefficients as a function of angle of attack were compared with available experimental data. The ice shapes and the blocking topologies were prepared by NASA Glenn's SmaggIce software. The grid systems were generated by using a four-boundary method based on Hermite interpolation with controls on clustering, orthogonality next to walls, and C continuity across block boundaries. The flow was modeled by the ensemble-averaged compressible Navier-Stokes equations, closed by the shear-stress transport turbulence model in which the integration is to the wall. All solutions were generated by using the NPARC WIND code.

Chi, X.↗

NASA NTRS1989

Bilinear modeling and nonlinear estimation

New methods are illustrated for online nonlinear estimation applied to the lateral deflection of an elastic beam on board measurements of angular rates and angular accelerations. The development of the filter equations, together with practical issues of their numerical solution as developed from global linearization by nonlinear output injection are contrasted with the usual method of the extended Kalman filter (EKF). It is shown how nonlinear estimation due to gyroscopic coupling can be implemented as an adaptive covariance filter using off-the-shelf Kalman filter algorithms. The effect of the global linearization by nonlinear output injection is to introduce a change of coordinates in which only the process noise covariance is to be updated in online implementation. This is in contrast to the computational approach which arises in EKF methods arising by local linearization with respect to the current conditional mean. Processing refinements for nonlinear estimation based on optimal, nonlinear interpolation between observations are also highlighted. In these methods the extrapolation of the process dynamics between measurement updates is obtained by replacing a transition matrix with an operator spline that is optimized off-line from responses to selected test inputs.

Dwyer, Thomas A. W., III↗

NASA NTRS1985

Minimal norm constrained interpolation

In computational fluid dynamics and in CAD/CAM, a physical boundary is usually known only discreetly and most often must be approximated. An acceptable approximation preserves the salient features of the data such as convexity and concavity. In this dissertation, a smooth interpolant which is locally concave where the data are concave and is locally convex where the data are convex is described. The interpolant is found by posing and solving a minimization problem whose solution is a piecewise cubic polynomial. The problem is solved indirectly by using the Peano Kernal theorem to recast it into an equivalent minimization problem having the second derivative of the interpolant as the solution. This approach leads to the solution of a nonlinear system of equations. It is shown that Newton's method is an exceptionally attractive and efficient method for solving the nonlinear system of equations. Examples of shape-preserving interpolants, as well as convergence results obtained by using Newton's method are also shown. A FORTRAN program to compute these interpolants is listed. The problem of computing the interpolant of minimal norm from a convex cone in a normal dual space is also discussed. An extension of de Boor's work on minimal norm unconstrained interpolation is presented.

Irvine, L. D.↗

NASA NTRS1976

Techniques for correcting transmission errors in video adaptive delta modulation channels

We investigated the effects of channel errors on an adaptive delta modulator used to encode video signals. The investigation revealed that channel errors caused a permanent shift in the dc level of the delta modulator's estimate. Errors in the step size, on the other hand, were transitory and had no noticeable effect on the received pictures. We then presented three error correction schemes to minimize the visibility of the channel errors in the received picture. The first scheme required the transmitter to periodically send the correct dc level of the estimate to the receiver. The second method employed a leaky integrator, and the third method used line-to-line interpolation

Scheinberg, N.↗

NASA NTRS1979

Application of the method of integral relations to unsteady fluid flow problems with shocks

A mixed method using both the method of integral relations and the finite difference technique is developed for the solution of unsteady flow problems. The integral relations method is based upon a chosen interpolating function dependent only on the time domain. The resulting local semidiscrete finite element equations obtained are assembled into a global form. The spatial derivatives at the nodes are replaced by finite difference operators and the discretized nonlinear algebraic system is solved by an iterative scheme. Solutions are obtained for the one-dimensional gasdynamics equation, the one-dimensional wave equation and Burger's model of turbulence. Agreement with other numerical and analytical solutions is excellent for the gasdynamics problem and satisfactory for the Burger equation in cases of small viscous effects.

Wadia, A. R.↗

NASA NTRS1984

Evaluation of glass model precipitation forecasts for North America during SOP-1

A study has been made of the accuracy of the GLAS fourth order forecast model's precipitation forecasts over North America during the first Special Observing Period (SOP-1) of FGGE and of the impact of the FGGE special observing systems on these forecasts. Fourteen 120 h predictions were generated using the coarse 4 deg latitude by 5 deg longitude version of the GLAS model from both the FGGE and NOSAT assimilation cycles. These forecasts were then verified against a detailed set of precipitation observations. Separate verifications were performed for precipitation accumulations or = .01 in., or = .1 in., and or = 1 in. occuring in 6, 12 and 24 time periods. In addition, three different methods of verification were applied. In the first method, model precipitation forecasts are verified against the greatest precipitation amount observed within a 4 deg latitude by 5 deg longitude gridbox centered on each gridpoint. For the prediction at a gridpoint to be verified only one observation within the gridbox is required. In the second method, a weighted mean of the closest observations to the gridpoint is used, provided that at least two observations on opposite sides of a gridpoint are available. For the third method, the model precipitation forecasts were interpolated to the observation locations.

Atlas, R.↗

NASA NTRS1991

Aerodynamic parameter studies and sensitivity analysis for rotor blades in axial flight

The analytical capability is offered for aerodynamic parametric studies and sensitivity analyses of rotary wings in axial flight by using a 3-D undistorted wake model in curved lifting line theory. The governing equations are solved by both the Multhopp Interpolation technique and the Vortex Lattice method. The singularity from the bound vortices is eliminated through the Hadamard's finite part concept. Good numerical agreement between both analytical methods and finite differences methods are found. Parametric studies were made to assess the effects of several shape variables on aerodynamic loads. It is found, e.g., that a rotor blade with out-of-plane and inplane curvature can theoretically increase lift in the inboard and outboard regions respectively without introducing an additional induced drag.

Chiu, Y. Danny↗

NASA NTRS1991

Aerodynamic parametric studies and sensitivity analysis for rotor blades in axial flight

The analytical capability is offered for aerodynamic parametric studies and sensitivity analyses of rotary wings in axial flight by using a 3D undistorted wake model in curved lifting line theory. The governing equations are solved by both the Multhopp Interpolation technique and the Vortex Lattice method. The singularity from the bound vortices is eliminated through the Hadamard's finite part concept. Good numerical agreement between both analytical methods and finite differences methods are found. Parametric studies were made to assess the effects of several shape variables on aerodynamic loads. It is found, e.g., that a rotor blade with out-of-plane and inplane curvature can theoretically increase lift in the inboard and outboard regions respectively without introducing an additional induced drag.

Chiu, Y. D.↗

NASA NTRS1993

Pyroxene equilibration temperatures in metamorphosed ordinary chondrites

Ordinary chondrites are divided into petrographic types based on observed mineralogical and textural properties consistent with progressive thermal metamorphism from low grade (type 3) to high (type 7). Regardless of the exact cause of the metamorphism, higher-type chondrites should retain information concerning peak temperatures reached and for what duration. Using the two-pyroxene geothermometer of Lindsley, we have calculated the equilibration temperatures for 26H, L and LL type 5 and 6 ordinary chondrites, to investigate the relative peak temperatures and equilibration-states reached by these various meteorite classes. The Lindsley thermometer relies on a detailed accounting of non-quadrilateral components in pyroxenes, whose recalculated compositions are then plotted onto an empirically-derived polythermal diagram from which temperatures can be interpolated. The reported uncertainty of this method is plus or minus 50 C; in addition, close spacing of isotherms on the graph (particularly for orthopyroxene compositions) increase this uncertainty. We have parameterized the Lindsley polythermal quadrilateral for 1 atm pressure (less than 2 bar), and interpreted recalculated coordinates directly in terms of 25 C temperature intervals. Meteorites selected for this study include both relatively shocked and unshoked specimens; heavily weathered or visibly brecciated specimens were avoided. Temperatures were calculated from orthopyroxene (opx) and clinopyroxene (cpx) analyses within one relative percent of ideal sums and stoichiometry. Histograms summarizing the calculated temperatures for type 5 and 6 ordinary chondrites are shown.

Harvey, R. P.↗

NASA NTRS1994

Direct numerical simulation of incompressible axisymmetric flows

In the present work, we propose to conduct direct numerical simulations (DNS) of incompressible turbulent axisymmetric jets and wakes. The objectives of the study are to understand the fundamental behavior of axisymmetric jets and wakes, which are perhaps the most technologically relevant free shear flows (e.g. combuster injectors, propulsion jet). Among the data to be generated are various statistical quantities of importance in turbulence modeling, like the mean velocity, turbulent stresses, and all the terms in the Reynolds-stress balance equations. In addition, we will be interested in the evolution of large-scale structures that are common in free shear flow. The axisymmetric jet or wake is also a good problem in which to try the newly developed b-spline numerical method. Using b-splines as interpolating functions in the non-periodic direction offers many advantages. B-splines have local support, which leads to sparse matrices that can be efficiently stored and solved. Also, they offer spectral-like accuracy that are C(exp O-1) continuous, where O is the order of the spline used; this means that derivatives of the velocity such as the vorticity are smoothly and accurately represented. For purposes of validation against existing results, the present code will also be able to simulate internal flows (ones that require a no-slip boundary condition). Implementation of no-slip boundary condition is trivial in the context of the b-splines.

Loulou, Patrick↗

NASA NTRS1995

Optimal guidance law development for an advanced launch system

The objective of this research effort was to develop a real-time guidance approach for launch vehicles ascent to orbit injection. Various analytical approaches combined with a variety of model order and model complexity reduction have been investigated. Singular perturbation methods were first attempted and found to be unsatisfactory. The second approach based on regular perturbation analysis was subsequently investigated. It also fails because the aerodynamic effects (ignored in the zero order solution) are too large to be treated as perturbations. Therefore, the study demonstrates that perturbation methods alone (both regular and singular perturbations) are inadequate for use in developing a guidance algorithm for the atmospheric flight phase of a launch vehicle. During a second phase of the research effort, a hybrid analytic/numerical approach was developed and evaluated. The approach combines the numerical methods of collocation and the analytical method of regular perturbations. The concept of choosing intelligent interpolating functions is also introduced. Regular perturbation analysis allows the use of a crude representation for the collocation solution, and intelligent interpolating functions further reduce the number of elements without sacrificing the approximation accuracy. As a result, the combined method forms a powerful tool for solving real-time optimal control problems. Details of the approach are illustrated in a fourth order nonlinear example. The hybrid approach is then applied to the launch vehicle problem. The collocation solution is derived from a bilinear tangent steering law, and results in a guidance solution for the entire flight regime that includes both atmospheric and exoatmospheric flight phases.

Calise, Anthony J.↗

NASA NTRS2011

Method and system for non-linear motion estimation

A method and system for extrapolating and interpolating a visual signal including determining a first motion vector between a first pixel position in a first image to a second pixel position in a second image, determining a second motion vector between the second pixel position in the second image and a third pixel position in a third image, determining a third motion vector between one of the first pixel position in the first image and the second pixel position in the second image, and the second pixel position in the second image and the third pixel position in the third image using a non-linear model, determining a position of the fourth pixel in a fourth image based upon the third motion vector.

Lu, Ligang↗

NASA NTRS1998

Generation of an Aerothermal Data Base for the X33 Spacecraft

The X-33 experimental program is a cooperative program between industry and NASA, managed by Lockheed-Martin Skunk Works to develop an experimental vehicle to demonstrate new technologies for a single-stage-to-orbit, fully reusable launch vehicle (RLV). One of the new technologies to be demonstrated is an advanced Thermal Protection System (TPS) being designed by BF Goodrich (formerly Rohr, Inc.) with support from NASA. The calculation of an aerothermal database is crucial to identifying the critical design environment data for the TPS. The NASA Ames X-33 team has generated such a database using Computational Fluid Dynamics (CFD) analyses, engineering analysis methods and various programs to compare and interpolate the results from the CFD and the engineering analyses. This database, along with a program used to query the database, is used extensively by several X-33 team members to help them in designing the X-33. This paper will describe the methods used to generate this database, the program used to query the database, and will show some of the aerothermal analysis results for the X-33 aircraft.

Roberts, Cathy↗

NASA NTRS1991

Quasi-static shape estimation and control of adaptive truss structures

Methods for estimating the deformation of adaptive truss structures are proposed which employ internal displacement sensors to measure changes in the length of selected truss members. Based on the measured data from the instrumented truss member, the total truss deformation pattern can be estimated through direct interpolation. To verify the validity of the methods presented here, numerical simulations are carried out for simple plane trusses, a beam truss, and a tetrahedral truss.

Kuwao, Fumihiro↗