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NASA NTRS1985

Numerical solution of Euler's equation by perturbed functionals

A perturbed functional iteration has been developed to solve nonlinear systems. It adds at each iteration level, unique perturbation parameters to nonlinear Gauss-Seidel iterates which enhances its convergence properties. As convergence is approached these parameters are damped out. Local linearization along the diagonal has been used to compute these parameters. The method requires no computation of Jacobian or factorization of matrices. Analysis of convergence depends on properties of certain contraction-type mappings, known as D-mappings. In this article, application of this method to solve an implicit finite difference approximation of Euler's equation is studied. Some representative results for the well known shock tube problem and compressible flows in a nozzle are given.

Dey, S. K.↗

NASA NTRS2004

Refinement of the ICRF

Since the ICRF was generated in 1995, VLBI modeling and estimation, data quality: source position stability analysis, and supporting observational programs have improved markedly. There are developing and potential applications in the areas of space navigation Earth orientation monitoring and optical astrometry from space that would benefit from a refined ICRF with enhanced accuracy, stability and spatial distribution. The convergence of analysis, focused observations, and astrometric needs should drive the production of a new realization in the next few years.

Ma, Chopo↗

NASA NTRS1983

A quasi-Newton versus a homotopy method for nonlinear structural analysis

The globally convergent quasi-Newton minimization algorithm and the homotopy algorithms are discussed in detail and their effectiveness in solving certain classes of highly nonlinear problems of structural analysis is demonstrated. The application of the double dogleg strategy controls the directions and step-lengths of the quasi-Newtonian algorithm and overcomes the problem of nonpositive definite Hessians being produced during the iteration process. The algorithms are applied to a centrally loaded clamped beam, the snap-through of a shallow arch, and a shallow reticulated dome.

Kamat, M. P.↗

NASA NTRS1994

System Sensitivity Analysis Applied to the Conceptual Design of a Dual-Fuel Rocket SSTO

This paper reports the results of initial efforts to apply the System Sensitivity Analysis (SSA) optimization method to the conceptual design of a single-stage-to-orbit (SSTO) launch vehicle. SSA is an efficient, calculus-based MDO technique for generating sensitivity derivatives in a highly multidisciplinary design environment. The method has been successfully applied to conceptual aircraft design and has been proven to have advantages over traditional direct optimization methods. The method is applied to the optimization of an advanced, piloted SSTO design similar to vehicles currently being analyzed by NASA as possible replacements for the Space Shuttle. Powered by a derivative of the Russian RD-701 rocket engine, the vehicle employs a combination of hydrocarbon, hydrogen, and oxygen propellants. Three primary disciplines are included in the design - propulsion, performance, and weights & sizing. A complete, converged vehicle analysis depends on the use of three standalone conceptual analysis computer codes. Efforts to minimize vehicle dry (empty) weight are reported in this paper. The problem consists of six system-level design variables and one system-level constraint. Using SSA in a 'manual' fashion to generate gradient information, six system-level iterations were performed from each of two different starting points. The results showed a good pattern of convergence for both starting points. A discussion of the advantages and disadvantages of the method, possible areas of improvement, and future work is included.

Olds, John R.↗

NASA NTRS1973

Validity test for linear error analysis

To determine whether estimation process simulated by linear error analysis will converge, criterion has been developed based on extension of classical observability. Particular application of technique is with groups of batched navigation data where statistics of estimation errors are derived with classical minimum-variance methods.

Diamant, L. S.↗

NASA NTRS1978

Adaptive finite element analysis based on p-convergence

The results of numerical experiments are presented in which a posteriori estimators of error in strain energy were examined on the basis of a typical problem in linear elastic fracture mechanics. Two estimators were found to give close upper and lower bounds for the strain energy error. The potential significance of this is that the same estimators may provide a suitable basis for adaptive redistribution of the degrees of freedom in finite element models.

Szabo, B. A.↗

NASA NTRS1981

Prediction of subsonic aircraft flows with jet exhaust interactions

A numerical procedure to calculate the flow fields resulting from the viscous inviscid interactions that occur when a strong jet exhaust and aircraft flow field coupling exists was developed. The approach divides the interaction region into zones which are either predominantly viscous or inviscid. The flow in the inviscid zone, which surrounds most of the aircraft, is calculated using an existing potential flow code. The viscous flow zone, which encompasses the jet plume, is modeled using a parabolized Navier-Stokes code. The procedure features the coupling of the zonal solutions such that sufficient information is transferred between the zones to preserve the effects of the interactions. The zonal boundaries overlap and the boundary conditions are the information link between zones. An iteration scheme iterates the coupled analysis until convergence has been obtained.

Roberts, D. W.↗

NASA NTRS1982

Flow analysis of nozzle installations with strong airplane flow interactions

A numerical procedure has been developed to calculate the flow fields resulting from the viscous-inviscid interactions that occur when a strong jet exhaust and aircraft flow field coupling exists. The approach used in the current procedure is to divide the interaction region into zones which are either predominantly viscous or inviscid. The flow in the inviscid zone, which surrounds most of the aircraft, is calculated using an existing linearized potential flow code. The viscous flow zone, which encompasses the jet plume, is modeled using a parabolized Navier-Stokes code. The key feature of the present procedure is the coupling of the zonal solutions such that sufficient information is transferred between the zones to preserve the effects of the interactions. The zonal boundaries overlap with the boundary conditions being the information link between zones. An iteraction scheme iterates the coupled analysis until convergence has been obtained. The procedure has been successfully used for several test cases for which the computed results are presented.

Roberts, D. W.↗

NASA NTRS1987

A zonal method for modeling 3-D aircraft flow fields with jet plume effects

A coupled zonal method has been developed for modeling flows about aircraft with highly integrated advanced nozzle installations in which the exhaust plume interacts with the aircraft flow field to enhance performance. A cost effective flow analysis is achieved by dividing the flow domain into zones and coupling the zonal solutions. PAN AIR is used to model the flow in the inviscid zone which surrounds the aircraft. The plume zone flow field is solved with a PNS code. The flow in the optional internal nozzle zone is modeled using a full Navier-Stokes code. An automated iterative procedure couples the inviscid and plume zones by passing updated boundary conditions between the zones. The coupled analysis provides converged solutions for the overall flow field. The method is demonstrated for a highly integrated nonaxisymmetric vectored nozzle installed on a swept-wing aircraft.

Roberts, D. W.↗

NASA NTRS1989

Applications of a transonic wing design method

A method for designing wings and airfoils at transonic speeds using a predictor/corrector approach was developed. The procedure iterates between an aerodynamic code, which predicts the flow about a given geometry, and the design module, which compares the calculated and target pressure distributions and modifies the geometry using an algorithm that relates differences in pressure to a change in surface curvature. The modular nature of the design method makes it relatively simple to couple it to any analysis method. The iterative approach allows the design process and aerodynamic analysis to converge in parallel, significantly reducing the time required to reach a final design. Viscous and static aeroelastic effects can also be accounted for during the design or as a post-design correction. Results from several pilot design codes indicated that the method accurately reproduced pressure distributions as well as the coordinates of a given airfoil or wing by modifying an initial contour. The codes were applied to supercritical as well as conventional airfoils, forward- and aft-swept transport wings, and moderate-to-highly swept fighter wings. The design method was found to be robust and efficient, even for cases having fairly strong shocks.

Campbell, Richard L.↗

NASA NTRS2020

An Overview of the Null-Field Method. II: Convergence and Numerical Stability

In this paper we provide an analysis of the convergence and numerical stability of the null-field method with discrete sources. We show that (i) if the null-field scheme is numerically stable then we can decide whether or not convergence can be achieved; (ii) if the null-field scheme is numerically unstable then we cannot draw any conclusion about the convergence issue; and (iii) the numerical stability is closely related to the property of a tangential system of radiating discrete sources to form a Riesz basis. Our numerical analysis indicates that for prolate spheroids and localized vector spherical wave functions, the null-field scheme is numerically unstable (this system of vector functions does not form a Riesz basis), while for distributed vector spherical wave functions, the numerical instability is not so pronounced (this system of discrete sources almost possesses the property of being a Riesz basis). We also describe an analytical method for computing the surface integrals in the framework of the conventional null-field method with localized vector spherical wave functions which increases the stability of the numerical scheme.

Null-field method↗

NASA NTRS1976

Convergence of an iterative procedure for large-scale static analysis of structural components

The paper proves convergence of an iterative procedure for calculating the deflections of built-up component structures which can be represented as consisting of a dominant, relatively stiff primary structure and a less stiff secondary structure, which may be composed of one or more substructures that are not connected to one another but are all connected to the primary structure. The iteration consists in estimating the deformation of the primary structure in the absence of the secondary structure on the assumption that all mechanical loads are applied directly to the primary structure. The j-th iterate primary structure deflections at the interface are imposed on the secondary structure, and the boundary loads required to produce these deflections are computed. The cycle is completed by applying the interface reaction to the primary structure and computing its updated deflections. It is shown that the mathematical condition for convergence of this procedure is that the maximum eigenvalue of the equation relating primary-structure deflection to imposed secondary-structure deflection be less than unity, which is shown to correspond with the physical requirement that the secondary structure be more flexible at the interface boundary.

Austin, F.↗

NASA NTRS1977

Moisture convergence from a combined mesoscale moisture analysis and wind field for 24 April 1975

Precipitable water values inferred from the Vertical Temperature Profile Radiometer data of the polar orbiting NOAA-4 satellite are used in conjunction with wind-field analyses obtained from Synchronous Meteorological Satellite visible-channel data to study the moisture convergence in the boundary layer immediately preceding a storm. This combination of data simulates the information that will be available from the Visible and Infrared Spin-Scan Radiometer on board the GOES-D satellite, which is scheduled to begin operation in the 1980s. Serviceable representations of boundary layer flow are developed through analysis of the satellite infrared cumulus velocities, although the flow representations are not exactly located in the vertical.

Negri, A. J.↗

NASA NTRS2008

Deflection Analysis of the Space Shuttle External Tank Door Drive Mechanism

Upon observing an abnormal closure of the Space Shuttle s External Tank Doors (ETD), a dynamic model was created in MSC/ADAMS to conduct deflection analyses of the Door Drive Mechanism (DDM). For a similar analysis, the traditional approach would be to construct a full finite element model of the mechanism. The purpose of this paper is to describe an alternative approach that models the flexibility of the DDM using a lumped parameter approximation to capture the compliance of individual parts within the drive linkage. This approach allows for rapid construction of a dynamic model in a time-critical setting, while still retaining the appropriate equivalent stiffness of each linkage component. As a validation of these equivalent stiffnesses, finite element analysis (FEA) was used to iteratively update the model towards convergence. Following this analysis, deflections recovered from the dynamic model can be used to calculate stress and classify each component s deformation as either elastic or plastic. Based on the modeling assumptions used in this analysis and the maximum input forcing condition, two components in the DDM show a factor of safety less than or equal to 0.5. However, to accurately evaluate the induced stresses, additional mechanism rigging information would be necessary to characterize the input forcing conditions. This information would also allow for the classification of stresses as either elastic or plastic.

Tosto, Michael A.↗

NASA NTRS1984

A two-dimensional analysis of laser heat addition in converging nozzles

The two-dimensional equations of motion describing the interaction between a laser beam and a flowing gas are considered. An implicit numerical scheme is used to solve these equations for unchoked flow through a converging-diverging nozzle. Separate grids are used for the fluid dynamics and the radiation equations. The effects of beam focusing and cross-beam intensity profiles are included. The calculations are based upon real gas properties for all quantities except the gas absorptivity, which is taken as a constant. The solutions contain the expected hot central core region with cool gas near the walls. This results in steep temperature gradients in both the streamwise and cross-stream directions. The absorption zone acts as a blockage in the nozzle causing a nonuniform velocity profile at the inlet and an overall decrease in mass flow. The absorption region also forces the streamlines to move away from the axis of symmetry, although this effect is not strong.

Molvik, G. A.↗

NASA NTRS2014

Statistical Symbolic Execution with Informed Sampling

Symbolic execution techniques have been proposed recently for the probabilistic analysis of programs. These techniques seek to quantify the likelihood of reaching program events of interest, e.g., assert violations. They have many promising applications but have scalability issues due to high computational demand. To address this challenge, we propose a statistical symbolic execution technique that performs Monte Carlo sampling of the symbolic program paths and uses the obtained information for Bayesian estimation and hypothesis testing with respect to the probability of reaching the target events. To speed up the convergence of the statistical analysis, we propose Informed Sampling, an iterative symbolic execution that first explores the paths that have high statistical significance, prunes them from the state space and guides the execution towards less likely paths. The technique combines Bayesian estimation with a partial exact analysis for the pruned paths leading to provably improved convergence of the statistical analysis. We have implemented statistical symbolic execution with in- formed sampling in the Symbolic PathFinder tool. We show experimentally that the informed sampling obtains more precise results and converges faster than a purely statistical analysis and may also be more efficient than an exact symbolic analysis. When the latter does not terminate symbolic execution with informed sampling can give meaningful results under the same time and memory limits.

Reliability↗

NASA NTRS1986

Necessary and sufficient conditions for parameter convergence in adaptive control

Using Generalized Harmonic Analysis, a complete description of parameter convergence in Model Reference Adaptive Control (MRAC) is given in terms of the spectrum of the exogenous reference input signal. Roughly speaking, if the reference signal 'contains enough frequencies' then the parameter vector converges to its correct value. If not, it converges to an easily characterizable subspace in parameter space.

Boyd, Stephen↗

NASA NTRS1989

Multiple boundary condition testing error analysis

Techniques for interpreting data from multiple-boundary-condition (MBC) ground tests of large space structures are developed analytically and demonstrated. The use of MBC testing to validate structures too large to stand alone on the ground is explained; the generalized least-squares mass and stiffness curve-fitting methods typically applied to MBC test data are reviewed; and a detailed error analysis is performed. Consideration is given to sensitivity coefficients, covariance-matrix theory, the correspondence between test and analysis modes, constraints and step sizes, convergence criteria, and factor-analysis theory. Numerical results for a simple beam problem are presented in tables and briefly characterized. The improved error-updating capabilities of MBC testing are confirmed, and it is concluded that reasonably accurate results can be obtained using a diagonal covariance matrix.

Glaser, R. J.↗