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NASA NTRS1976

Advances in the application of fast semidirect computational methods in transonic flow

The paper uses finite-difference algorithms called 'fast direct elliptic solvers' within an iteration scheme for the rapid solution of the equations of inviscid transonic aerodynamics. The methods are called 'direct' (or semidirect) because the entire computational field is solved at once rather than in successive traverses over the field. These semidirect iterative methods have been limited here to the investigation of two-dimensional steady inviscid flow over airfoils in subsonic free stream.

Martin, E. D.↗

NASA NTRS1992

Schroedinger's radial equation - Solution by extrapolation

A high-accuracy numerical method for the solution of a 1D Schroedinger equation that is suitable for a diatomic molecule, obtained by combining a finite-difference method with iterative extrapolation to the limit, is presently shown to have several advantages over more conventional methods. Initial guesses for the term values are obviated, and implementation of the algorithm is straightforward. The method is both less sensitive to round-off error, and faster than conventional methods for equivalent accuracy. These advantages are illustrated through the solution of Schroedinger's equation for a Morse potential function suited for HCl and a numerically derived Rydberg-Klein-Rees potential function for the X 1Sigma(+) state of CO.

Goorvitch, D.↗

NASA NTRS2003

Factors Affecting the Evolution of Hurricane Erin and the Distributions of Hydrometeors: Role of Microphysical Processes

Fine-resolution simulations of Hurricane Erin 2001 are conducted using the Penn State University/National Center for Atmospheric Research mesoscale model version 3.5 to investigate the role of thermodynamic, boundary layer and microphysical processes in Erin's growth and maintenance, and their effects on the horizontal and vertical distributions of hydrometeors. Through comparison against radar, radiometer, and dropsonde data collected during the Convection and Moisture Experiment 4, it is seen that realistic simulations of Erin are obtained provided that fine resolution simulations with detailed representations of physical processes are conducted. The principle findings of the study are as follows: 1) a new iterative condensation scheme, which limits the unphysical increase of equivalent potential temperature associated with most condensation schemes, increases the horizontal size of the hurricane, decreases its maximum rainfall rate, reduces its intensity, and makes its eye more moist; 2) in general, microphysical parameterization schemes with more categories of hydrometeors produce more intense hurricanes, larger hydrometeor mixing ratios, and more intense updrafts and downdrafts; 3) the choice of coefficients describing hydrometeor fall velocities has as big of an impact on the hurricane simulations as does choice of microphysical parameterization scheme with no clear relationship between fall velocity and hurricane intensity; and 4) in order for a tropical cyclone to adequately intensify, an advanced boundary layer scheme (e.g., Burk-Thompson scheme) must be used to represent boundary layer processes. The impacts of varying simulations on the horizontal and vertical distributions of different categories of hydrometeor species, on equivalent potential temperature, and on storm updrafts and downdrafts are examined to determine how the release of latent heat feedbacks upon the structure of Erin. In general, all simulations tend to overpredict precipitation rate and hydrometeor mixing ratios. The ramifications of these findings for quantitative precipitation forecasts (QPFs) of tropical cyclones are discussed.

McFarquhar, Greg M.↗

NASA NTRS2011

AGN Obscuration Through Dusty Infrared Dominated Flows: Radiation-Hydrodynamics Solution for the Wind - 1

We construct a radiation-hydrodynamics model for the obscuring toroidal structure in active galactic nuclei. In this model the obscuration is produced at parsec scale by a dense, dusty wind which is supported by infrared radiation pressure on dust grains. To find the distribution of radiation pressure, we numerically solve the 2D radiation transfer problem in a flux limited diffusion approximation. We iteratively couple the solution with calculations of stationary 1D models for the wind, and obtain the z-component of the velocity. Our results demonstrate that for AGN luminosities greater than 0.1 L(sub edd) external illumination can support a geometrically thick obscuration via outflows driven by infrared radiation pressure. The terminal velocity of marginally Compton-thin models (0.2 < tau(sub T) < 0.6), is comparable to or greater than the escape velocity. In Compton thick models the maximum value of the vertical component of the velocity is lower than the escape velocity, suggesting that a significant part of our torus is in the form of failed wind. The results demonstrate that obscuration via normal or failed infrared-driven winds is a viable option for the AGN torus problem and AGN unification models. Such winds can also provide an important channel for AGN feedback.

Dorodnitsyn, A.↗

NASA NTRS1999

Ames Optimized TCA Configuration

Configuration design at Ames was carried out with the SYN87-SB (single block) Euler code using a 193 x 49 x 65 C-H grid. The Euler solver is coupled to the constrained (NPSOL) and the unconstrained (QNMDIF) optimization packages. Since the single block grid is able to model only wing-body configurations, the nacelle/diverter effects were included in the optimization process by SYN87's option to superimpose the nacelle/diverter interference pressures on the wing. These interference pressures were calculated using the AIRPLANE code. AIRPLANE is an Euler solver that uses a unstructured tetrahedral mesh and is capable of computations about arbitrary complete configurations. In addition, the buoyancy effects of the nacelle/diverters were also included in the design process by imposing the pressure field obtained during the design process onto the triangulated surfaces of the nacelle/diverter mesh generated by AIRPLANE. The interference pressures and nacelle buoyancy effects are added to the final forces after each flow field calculation. Full details of the (recently enhanced) ghost nacelle capability are given in a related talk. The pseudo nacelle corrections were greatly improved during this design cycle. During the Ref H and Cycle 1 design activities, the nacelles were only translated and pitched. In the cycle 2 design effort the nacelles can translate vertically, and pitch to accommodate the changes in the lower surface geometry. The diverter heights (between their leading and trailing edges) were modified during design as the shape of the lower wing changed, with the drag of the diverter changing accordingly. Both adjoint and finite difference gradients were used during optimization. The adjoint-based gradients were found to give good direction in the design space for configurations near the starting point, but as the design approached a minimum, the finite difference gradients were found to be more accurate. Use of finite difference gradients was limited by the CPU time limit available on the Cray machines. A typical optimization run using finite difference gradients can use only 30 to 40 design variables and one optimization iteration within the 8 hour queue limit for the chosen grid size and convergence level. The efficiency afforded by the adjoint method allowed for 50-120 design variables and 5-10 optimization iterations in the 8 hour queue. Geometric perturbations to the wing and fuselage were made using the Hicks/Henne (HH) shape functions. The HH functions were distributed uniformly along the chords of the wing defining sections and lofted linearly. During single-surface design, constraints on thickness and volume at selected wing stations were imposed. Both fuselage camber and cross-sectional area distributions were permitted to change during design. The major disadvantage to the use of these functions is the inherent surface waviness produced by repeated use of such functions. Many smoothing operations were required following optimization runs to produce a configuration with reasonable smoothness. Wagner functions were also used on the wing sections but were never used on the fuselage. The Wagner functions are a family of increasingly oscillatory functions that have also been used extensively in airfoil design. The leading and trailing edge regions of the wing were designed by use of polynomial and monomial functions respectively. Twist was attempted but was abandoned because of little performance improvement available from changing the baseline twist.

Cliff, Susan E.↗

NASA NTRS2011

Iterative Strain-Gage Balance Calibration Data Analysis for Extended Independent Variable Sets

A new method was developed that makes it possible to use an extended set of independent calibration variables for an iterative analysis of wind tunnel strain gage balance calibration data. The new method permits the application of the iterative analysis method whenever the total number of balance loads and other independent calibration variables is greater than the total number of measured strain gage outputs. Iteration equations used by the iterative analysis method have the limitation that the number of independent and dependent variables must match. The new method circumvents this limitation. It simply adds a missing dependent variable to the original data set by using an additional independent variable also as an additional dependent variable. Then, the desired solution of the regression analysis problem can be obtained that fits each gage output as a function of both the original and additional independent calibration variables. The final regression coefficients can be converted to data reduction matrix coefficients because the missing dependent variables were added to the data set without changing the regression analysis result for each gage output. Therefore, the new method still supports the application of the two load iteration equation choices that the iterative method traditionally uses for the prediction of balance loads during a wind tunnel test. An example is discussed in the paper that illustrates the application of the new method to a realistic simulation of temperature dependent calibration data set of a six component balance.

Ulbrich, Norbert Manfred↗

NASA NTRS2025

Iterating on a Design – Further Developments in the Evolution of the Ballistic Limit Equations for the Mars Sample Return Project

The goal of the Mars Sample Return–Capture, Containment, and Return System Project is to retrieve samples launched from the Martian surface and return them to Earth for detailed analysis. An important part of this project is the design of the system’s micrometeoroid protection system, which protects the Earth Entry System and the collected samples during their journey back to Earth. As mission parameters and the micrometeoroid and orbital debris threat became better understood, the design of the micrometeoroid protection system evolved. A key element used in the shield development process is the ballistic limit equation, which is an equation that is used to determine whether or not a particular structural element or system will end up in a failed state as a result of a specified impact. As the design of the Earth Entry System and the micrometeoroid protection system evolved, a new set of ballistic limit equations was needed to better predict and assess the performance of developing shield system designs in anticipation of possible damage from micrometeoroid and orbital debris particle impacts. This paper provides a summary of how a set of initial BLEs were either extended or modified so that the resulting equations were better suited to new types of target configurations being considered, as well as how additional ballistic limit equations were developed where none previously existed.

William P Schonberg↗

NASA NTRS1973

Finite element stress analysis of polymers at high strains

A numerical analysis is presented for the problem of a flat rectangular rubber membrane with a circular rigid inclusion undergoing high strains due to the action of an axial load. The neo-hookean constitutive equations are introduced into the general purpose TITUS program by means of equivalent hookean constants and initial strains. The convergence is achieved after a few iterations. The method is not limited to any specific program. The results are in good agreement with those of a company sponsored photoelastic stress analysis. The theoretical and experimental deformed shapes also agree very closely with one another. For high strains it is demonstrated that using the conventional HOOKE law the stress concentration factor obtained is unreliable in the case of rubberlike material.

Durand, M.↗

NASA NTRS1991

Remote sensing of earth terrain

In remote sensing, the encountered geophysical media such as agricultural canopy, forest, snow, or ice are inhomogeneous and contain scatters in a random manner. Furthermore, weather conditions such as fog, mist, or snow cover can intervene the electromagnetic observation of the remotely sensed media. In the modelling of such media accounting for the weather effects, a multi-layer random medium model has been developed. The scattering effects of the random media are described by three-dimensional correlation functions with variances and correlation lengths corresponding to the fluctuation strengths and the physical geometry of the inhomogeneities, respectively. With proper consideration of the dyadic Green's function and its singularities, the strong fluctuation theory is used to calculate the effective permittivities which account for the modification of the wave speed and attenuation in the presence of the scatters. The distorted Born approximation is then applied to obtain the correlations of the scattered fields. From the correlation of the scattered field, calculated is the complete set of scattering coefficients for polarimetric radar observation or brightness temperature in passive radiometer applications. In the remote sensing of terrestrial ecosystems, the development of microwave remote sensing technology and the potential of SAR to measure vegetation structure and biomass have increased effort to conduct experimental and theoretical researches on the interactions between microwave and vegetation canopies. The overall objective is to develop inversion algorithms to retrieve biophysical parameters from radar data. In this perspective, theoretical models and experimental data are methodically interconnected in the following manner: Due to the complexity of the interactions involved, all theoretical models have limited domains of validity; the proposed solution is to use theoretical models, which is validated by experiments, to establish the region in which the radar response is most sensitive to the parameters of interest; theoretically simulated data will be used to generate simple invertible models over the region. For applications to the remote sensing of sea ice, the developed theoretical models need to be tested with experimental measurements. With measured ground truth such as ice thickness, temperature, salinity, and structure, input parameters to the theoretical models can be obtained to calculate the polarimetric scattering coefficients for radars or brightness temperature for radiometers and then compare theoretical results with experimental data. Validated models will play an important role in the interpretation and classification of ice in monitoring global ice cover from space borne remote sensors in the future. We present an inversion algorithm based on a recently developed inversion method referred to as the Renormalized Source-Type Integral Equation approach. The objective of this method is to overcome some of the limitations and difficulties of the iterative Born technique. It recasts the inversion, which is nonlinear in nature, in terms of the solution of a set of linear equations; however, the final inversion equation is still nonlinear. The derived inversion equation is an exact equation which sums up the iterative Neuman (or Born) series in a closed form and, thus, is a valid representation even in the case when the Born series diverges; hence, the name Renormalized Source-Type Integral Equation Approach.

Yueh, Herng-Aung↗

NASA NTRS1966

Problems Arising from Current Trends in Propulsion System Design and Guidance Schemes

Mission requirements in various NASA programs dictate the necessity for high reliability, continual and chronic increases in payload capability, and precise injection into the prescribed orbit. In some vehicles, concepts employed to meet these criteria in the areas of guidance and propulsion have been found to be somewhat in conflict. For the express purpose of increasing reliability, the primary emphasis in engine and feed design has been placed on simplification including the removal of control systems. In the guidance area considerable attention has been given to the development of methods of implementation which allow complete reshaping of the pitch program if the stage should perform at some level other than predicted. In such a guidance scheme the assumption is made that observed differences in performance levels are constant throughout flight. In the absence of propulsion control systems, however, some oscillations about a mean value representative of the shift can be expected. Analyses showed that this occurrence would severely limit the effectiveness of the Iterative Guidance Mode. The purpose of this paper is to deal with the definition, causes and cures of this problem.

McKay, George H., Jr.↗

NASA NTRS2010

Real-Time Exponential Curve Fits Using Discrete Calculus

An improved solution for curve fitting data to an exponential equation (y = Ae(exp Bt) + C) has been developed. This improvement is in four areas -- speed, stability, determinant processing time, and the removal of limits. The solution presented avoids iterative techniques and their stability errors by using three mathematical ideas: discrete calculus, a special relationship (be tween exponential curves and the Mean Value Theorem for Derivatives), and a simple linear curve fit algorithm. This method can also be applied to fitting data to the general power law equation y = Ax(exp B) + C and the general geometric growth equation y = Ak(exp Bt) + C.

Rowe, Geoffrey↗

NASA NTRS1985

Restoration of multichannel microwave radiometric images

A constrained iterative image restoration method is applied to multichannel diffraction-limited imagery. This method is based on the Gerchberg-Papoulis algorithm utilizing incomplete information and partial constraints. The procedure is described using the orthogonal projection operators which project onto two prescribed subspaces iteratively. Its properties and limitations are presented. The effect of noise was investigated and a better understanding of the performance of the algorithm with noisy data has been achieved. The restoration scheme with the selection of appropriate constraints was applied to a practical problem. The 6.6, 10.7, 18, and 21 GHz satellite images obtained by the scanning multichannel microwave radiometer (SMMR), each having different spatial resolution, were restored to a common, high resolution (that of the 37 GHz channels) to demonstrate the effectiveness of the method. Both simulated data and real data were used in this study. The restored multichannel images may be utilized to retrieve rainfall distributions.

Chin, R. T.↗

NASA NTRS1990

A superlinear interior points algorithm for engineering design optimization

We present a quasi-Newton interior points algorithm for nonlinear constrained optimization. It is based on a general approach consisting of the iterative solution in the primal and dual spaces of the equalities in Karush-Kuhn-Tucker optimality conditions. This is done in such a way to have primal and dual feasibility at each iteration, which ensures satisfaction of those optimality conditions at the limit points. This approach is very strong and efficient, since at each iteration it only requires the solution of two linear systems with the same matrix, instead of quadratic programming subproblems. It is also particularly appropriate for engineering design optimization inasmuch at each iteration a feasible design is obtained. The present algorithm uses a quasi-Newton approximation of the second derivative of the Lagrangian function in order to have superlinear asymptotic convergence. We discuss theoretical aspects of the algorithm and its computer implementation.

Herskovits, J.↗

NASA NTRS1991

Single-agent parallel window search

Parallel window search is applied to single-agent problems by having different processes simultaneously perform iterations of Iterative-Deepening-A(asterisk) (IDA-asterisk) on the same problem but with different cost thresholds. This approach is limited by the time to perform the goal iteration. To overcome this disadvantage, the authors consider node ordering. They discuss how global node ordering by minimum h among nodes with equal f = g + h values can reduce the time complexity of serial IDA-asterisk by reducing the time to perform the iterations prior to the goal iteration. Finally, the two ideas of parallel window search and node ordering are combined to eliminate the weaknesses of each approach while retaining the strengths. The resulting approach, called simply parallel window search, can be used to find a near-optimal solution quickly, improve the solution until it is optimal, and then finally guarantee optimality, depending on the amount of time available.

Powley, Curt↗

NASA NTRS1999

Structural Reliability Analysis and Optimization: Use of Approximations

This report is intended for the demonstration of function approximation concepts and their applicability in reliability analysis and design. Particularly, approximations in the calculation of the safety index, failure probability and structural optimization (modification of design variables) are developed. With this scope in mind, extensive details on probability theory are avoided. Definitions relevant to the stated objectives have been taken from standard text books. The idea of function approximations is to minimize the repetitive use of computationally intensive calculations by replacing them with simpler closed-form equations, which could be nonlinear. Typically, the approximations provide good accuracy around the points where they are constructed, and they need to be periodically updated to extend their utility. There are approximations in calculating the failure probability of a limit state function. The first one, which is most commonly discussed, is how the limit state is approximated at the design point. Most of the time this could be a first-order Taylor series expansion, also known as the First Order Reliability Method (FORM), or a second-order Taylor series expansion (paraboloid), also known as the Second Order Reliability Method (SORM). From the computational procedure point of view, this step comes after the design point identification; however, the order of approximation for the probability of failure calculation is discussed first, and it is denoted by either FORM or SORM. The other approximation of interest is how the design point, or the most probable failure point (MPP), is identified. For iteratively finding this point, again the limit state is approximated. The accuracy and efficiency of the approximations make the search process quite practical for analysis intensive approaches such as the finite element methods; therefore, the crux of this research is to develop excellent approximations for MPP identification and also different approximations including the higher-order reliability methods (HORM) for representing the failure surface. This report is divided into several parts to emphasize different segments of the structural reliability analysis and design. Broadly, it consists of mathematical foundations, methods and applications. Chapter I discusses the fundamental definitions of the probability theory, which are mostly available in standard text books. Probability density function descriptions relevant to this work are addressed. In Chapter 2, the concept and utility of function approximation are discussed for a general application in engineering analysis. Various forms of function representations and the latest developments in nonlinear adaptive approximations are presented with comparison studies. Research work accomplished in reliability analysis is presented in Chapter 3. First, the definition of safety index and most probable point of failure are introduced. Efficient ways of computing the safety index with a fewer number of iterations is emphasized. In chapter 4, the probability of failure prediction is presented using first-order, second-order and higher-order methods. System reliability methods are discussed in chapter 5. Chapter 6 presents optimization techniques for the modification and redistribution of structural sizes for improving the structural reliability. The report also contains several appendices on probability parameters.

Grandhi, Ramana V.↗

NASA NTRS1983

Restoration of multichannel microwave radiometric images

Chin, R. T.↗

NASA NTRS1986

Preconditioned conjugate residual methods for the solution of spectral equations

Conjugate residual methods for the solution of spectral equations are described. An inexact finite-difference operator is introduced as a preconditioner in the iterative procedures. Application of these techniques is limited to problems for which the symmetric part of the coefficient matrix is positive definite. Although the spectral equation is a very ill-conditioned and full matrix problem, the computational effort of the present iterative methods for solving such a system is comparable to that for the sparse matrix equations obtained from the application of either finite-difference or finite-element methods to the same problems. Numerical experiments are shown for a self-adjoint elliptic partial differential equation with Dirichlet boundary conditions, and comparison with other solution procedures for spectral equations is presented.

Wong, Y. S.↗

NASA NTRS1995

Programmable Iterative Optical Image And Data Processing

Proposed method of iterative optical image and data processing overcomes limitations imposed by loss of optical power after repeated passes through many optical elements - especially, beam splitters. Involves selective, timed combination of optical wavefront phase conjugation and amplification to regenerate images in real time to compensate for losses in optical iteration loops; timing such that amplification turned on to regenerate desired image, then turned off so as not to regenerate other, undesired images or spurious light propagating through loops from unwanted reflections.

Jackson, Deborah J.↗