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All sourcesNASA NTRSDOE OSTI
NASA NTRS1992

Non-LTE radiative transfer with lambda-acceleration - Convergence properties using exact full and diagonal lambda-operators

We investigate the convergence properties of Lambda-acceleration methods for non-LTE radiative transfer problems in planar and spherical geometry. Matrix elements of the 'exact' A-operator are used to accelerate convergence to a solution in which both the radiative transfer and atomic rate equations are simultaneously satisfied. Convergence properties of two-level and multilevel atomic systems are investigated for methods using: (1) the complete Lambda-operator, and (2) the diagonal of the Lambda-operator. We find that the convergence properties for the method utilizing the complete Lambda-operator are significantly better than those of the diagonal Lambda-operator method, often reducing the number of iterations needed for convergence by a factor of between two and seven. However, the overall computational time required for large scale calculations - that is, those with many atomic levels and spatial zones - is typically a factor of a few larger for the complete Lambda-operator method, suggesting that the approach should be best applied to problems in which convergence is especially difficult.

Macfarlane, J. J.↗
NASA NTRS1990

On domains of convergence in optimization problems

Numerical optimization algorithms require the knowledge of an initial set of design variables. Starting from an initial design x(sup 0), improved solutions are obtained by updating the design iteratively in a way prescribed by the particular algorithm used. If the algorithm is successful, convergence is achieved to a local optimal solution. Let A denote the iterative procedure that characterizes a typical optimization algorithm, applied to the problem: Find x belonging to R(sup n) that maximizes f(x) subject to x belonging to Omega contained in R(sup n). We are interested in problems with several local maxima (x(sub j))(sup *), j=1, ..., m, in the feasible design space Omega. In general, convergence of the algorithm A to a specific solution (x(sub j))(sup *) is determined by the choice of initial design x(sup 0). The domain of convergence D(sub j) of A associated with a local maximum (x(sub j))(sup *) is a subset of initial designs x(sup 0) in Omega such that the sequence (x(sup k)), k=0,1,2,... defined by x(sup k+1) = A(x(sup k)), k=0,1,... converges to (x(sub j))(sup *). The set D(sub j) is also called the basin of attraction of (x(sub j))(sup *). Cayley first proposed the problem of finding the basin of attraction for Newton's method in 1897. It has been shown that the basin of attraction for Newton's method exhibits chaotic behavior in problems with polynomial objective. This implies that there may be regions in the feasible design space where arbitrarily close starting points will converge to different local optimal solutions. Furthermore, the boundaries of the domains of convergence may have a very complex, even fractal structure. In this paper we show that even simple structural optimization problems solved using standard gradient based (first order) algorithms exhibit similar features.

Diaz, Alejandro R.↗
NASA NTRS2000

Convergent properties of vestibular-related brain stem neurons in the gerbil

Three classes of vestibular-related neurons were found in and near the prepositus and medial vestibular nuclei of alert or decerebrate gerbils, those responding to: horizontal translational motion, horizontal head rotation, or both. Their distribution ratios were 1:2:2, respectively. Many cells responsive to translational motion exhibited spatiotemporal characteristics with both response gain and phase varying as a function of the stimulus vector angle. Rotationally sensitive neurons were distributed as Type I, II, or III responses (sensitive to ipsilateral, contralateral, or both directions, respectively) in the ratios of 4:6:1. Four tested factors shaped the response dynamics of the sampled neurons: canal-otolith convergence, oculomotor-related activity, rotational Type (I or II), and the phase of the maximum response. Type I nonconvergent cells displayed increasing gains with increasing rotational stimulus frequency (0.1-2.0 Hz, 60 degrees /s), whereas Type II neurons with convergent inputs had response gains that markedly decreased with increasing translational stimulus frequency (0.25-2.0 Hz, +/-0.1 g). Type I convergent and Type II nonconvergent neurons exhibited essentially flat gains across the stimulus frequency range. Oculomotor-related activity was noted in 30% of the cells across all functional types, appearing as burst/pause discharge patterns related to the fast phase of nystagmus during head rotation. Oculomotor-related activity was correlated with enhanced dynamic range compared with the same category that had no oculomotor-related response. Finally, responses that were in-phase with head velocity during rotation exhibited greater gains with stimulus frequency increments than neurons with out-of-phase responses. In contrast, for translational motion, neurons out of phase with head acceleration exhibited low-pass characteristics, whereas in-phase neurons did not. Data from decerebrate preparations revealed that although similar response types could be detected, the sampled cells generally had lower background discharge rates, on average one-third lower response gains, and convergent properties that differed from those found in the alert animals. On the basis of the dynamic response of identified cell types, we propose a pair of models in which inhibitory input from vestibular-related neurons converges on oculomotor neurons with excitatory inputs from the vestibular nuclei. Simple signal convergence and combinations of different types of vestibular labyrinth information can enrich the dynamic characteristics of the rotational and translational vestibuloocular responses.

Non-NASA Center↗
NASA NTRSDate not supplied

Deep Neural Network Based Convergence Classification for Computational Fluid Dynamics

A supervised deep learning approach is coupled with heuristic convergence criteria to construct a classification model for detecting the completion (convergence) of computational fluid dynamics (CFD) simulations. Heuristic convergence criteria alone are not always sufficient and more complex decisions are often left to a human analyst. The proposed approach leverages heuristic convergence criteria as well as two deep neural network (DNN) models, one binary and one multi-class, to improve the efficiency and consistency of convergence classification across a wide range of flight regimes. The DNN models presented are each trained on a subset of ascent aerodynamic CFD simulations for NASA’s Space Launch System and were produced using NASA’s unstructured Navier-Stokes solver FUN3D. Individual solutions are analyzed intermittently and are classified as sufficiently converged, further iterations required, or switch from steady Reynolds Averaged Navier-Stokes (RANS) to unsteady RANS CFD based on the iterative histories of four aerodynamic coefficients. The implemented classification model is shown to produce solutions that closely correlate to solutions produced by a human analyst. This work lays groundwork for expanding the capabilities of DNNs for automating and improving more of the CFD process.

SLS↗
NASA NTRS1991

The convergence rate of approximate solutions for nonlinear scalar conservation laws

The convergence rate is discussed of approximate solutions for the nonlinear scalar conservation law. The linear convergence theory is extended into a weak regime. The extension is based on the usual two ingredients of stability and consistency. On the one hand, the counterexamples show that one must strengthen the linearized L(sup 2)-stability requirement. It is assumed that the approximate solutions are Lip(sup +)-stable in the sense that they satisfy a one-sided Lipschitz condition, in agreement with Oleinik's E-condition for the entropy solution. On the other hand, the lack of smoothness requires to weaken the consistency requirement, which is measured in the Lip'-(semi)norm. It is proved for Lip(sup +)-stable approximate solutions, that their Lip'convergence rate to the entropy solution is of the same order as their Lip'-consistency. The Lip'-convergence rate is then converted into stronger L(sup p) convergence rate estimates.

Nessyahu, Haim↗
NASA NTRS1992

On the convergence of the coupled-wave approach for lamellar diffraction gratings

Among the many existing rigorous methods for analyzing diffraction of electromagnetic waves by diffraction gratings, the coupled-wave approach stands out because of its versatility and simplicity. It can be applied to volume gratings and surface relief gratings, and its numerical implementation is much simpler than others. In addition, its predictions were experimentally validated in several cases. These facts explain the popularity of the coupled-wave approach among many optical engineers in the field of diffractive optics. However, a comprehensive analysis of the convergence of the model predictions has never been presented, although several authors have recently reported convergence difficulties with the model when it is used for metallic gratings in TM polarization. Herein, three points are made: (1) in the TM case, the coupled-wave approach converges much slower than the modal approach of Botten et al; (2) the slow convergence is caused by the use of Fourier expansions for the permittivity and the fields in the grating region; and (3) is manifested by the slow convergence of the eigenvalues and the associated modal fields. The reader is assumed to be familiar with the mathematical formulations of the coupled-wave approach and the modal approach.

Li, Lifeng↗
NASA NTRS1995

Convergence properties of finite-difference hydrodynamics schemes in the presence of shocks

We investigate the asymptotic convergence of finite-difference schemes for the Euler equations when the limiting solution contains shocks. The Lax-Wendroff theorem guarantees that certain conservative schemes converge to correct, physically valid solutions. We focus on two one-dimensional operator-split schemes with explicit artificial-viscosity terms. One, an internal-energy scheme, does not satisfy the assumptions of Lax-Wendroff; the other, a conservative total-energy scheme, does. With viscous lengths chosen proportional to the grid size, we find that both schemes converge to their zero-grid-size limits at the theoretically expected rate, but only the conversative scheme converges toward correct solutions of the inviscid fluid equations. We show that the difference in their behaviors results directly from the presence of shocks in the limiting solution. Empirically, we find that when the viscous lenghts tend toward zero more slowly than the grid size, however the nonconservative scheme also converges toward correct solutions. We characterize the asymptotic behavior of the total-energy scheme in a particular problem in which a shock forms. As the grid is refined, a Cauchy error approaches the expected rate of change slowly. We show that the changes in the artificial viscosity alter the diffusion of small-amplitude waves. The differences associated with such waves make the dominant contribution to the Cauchy error. We formulate an analytic model to relate the rate of approach to the effect of varying diffusion in waves and find quantitative agreement with our numerical results.

Kimoto, Paul A.↗
NASA NTRS2012

Influence of Primary Gage Sensitivities on the Convergence of Balance Load Iterations

The connection between the convergence of wind tunnel balance load iterations and the existence of the primary gage sensitivities of a balance is discussed. First, basic elements of two load iteration equations that the iterative method uses in combination with results of a calibration data analysis for the prediction of balance loads are reviewed. Then, the connection between the primary gage sensitivities, the load format, the gage output format, and the convergence characteristics of the load iteration equation choices is investigated. A new criterion is also introduced that may be used to objectively determine if the primary gage sensitivity of a balance gage exists. Then, it is shown that both load iteration equations will converge as long as a suitable regression model is used for the analysis of the balance calibration data, the combined influence of non linear terms of the regression model is very small, and the primary gage sensitivities of all balance gages exist. The last requirement is fulfilled, e.g., if force balance calibration data is analyzed in force balance format. Finally, it is demonstrated that only one of the two load iteration equation choices, i.e., the iteration equation used by the primary load iteration method, converges if one or more primary gage sensitivities are missing. This situation may occur, e.g., if force balance calibration data is analyzed in direct read format using the original gage outputs. Data from the calibration of a six component force balance is used to illustrate the connection between the convergence of the load iteration equation choices and the existence of the primary gage sensitivities.

Ulbrich, Norbert Manfred↗
NASA NTRS2015

An Investigation of Transonic Resonance in a Mach 2.2 Round Convergent-Divergent Nozzle

Hot-wire and acoustic measurements were taken for a round convergent nozzle and a round convergent-divergent (C-D) nozzle at a jet Mach number of 0.61. The C-D nozzle had a design Mach number of 2.2. Compared to the convergent nozzle jet flow, the Mach 2.2 nozzle jet flow produced excess broadband noise (EBBN). It also produced a transonic resonance tone at 1200 Herz. Computational simulations were performed for both nozzle flows. A steady Reynolds-Averaged Navier-Stokes simulation was performed for the convergent nozzle jet flow. For the Mach 2.2 nozzle flow, a steady RANS simulation, an unsteady RANS (URANS) simulation, and an unsteady Detached Eddy Simulation (DES) were performed. The RANS simulation of the convergent nozzle showed good agreement with the hot-wire velocity and turbulence measurements, though the decay of the potential core was over-predicted. The RANS simulation of the Mach 2.2 nozzle showed poor agreement with the experimental data, and more closely resembled an ideally-expanded jet. The URANS simulation also showed qualitative agreement with the hot-wire data, but predicted a transonic resonance at 1145 Herz. The DES showed good agreement with the hot-wire velocity and turbulence data. The DES also produced a transonic tone at 1135 Herz. The DES solution showed that the destabilization of the shock-induced separation region inside the nozzle produced increased levels of turbulence intensity. This is likely the source of the EBBN.

DES↗
NASA NTRS2015

Grid Convergence for Turbulent Flows(Invited)

A detailed grid convergence study has been conducted to establish accurate reference solutions corresponding to the one-equation linear eddy-viscosity Spalart-Allmaras turbulence model for two dimensional turbulent flows around the NACA 0012 airfoil and a flat plate. The study involved three widely used codes, CFL3D (NASA), FUN3D (NASA), and TAU (DLR), and families of uniformly refined structured grids that differ in the grid density patterns. Solutions computed by different codes on different grid families appear to converge to the same continuous limit, but exhibit different convergence characteristics. The grid resolution in the vicinity of geometric singularities, such as a sharp trailing edge, is found to be the major factor affecting accuracy and convergence of discrete solutions, more prominent than differences in discretization schemes and/or grid elements. The results reported for these relatively simple turbulent flows demonstrate that CFL3D, FUN3D, and TAU solutions are very accurate on the finest grids used in the study, but even those grids are not sufficient to conclusively establish an asymptotic convergence order.

Diskin, Boris↗
NASA NTRS2018

Implementation and Use of a Convergence Test for Iterative Wind Tunnel Balance Load Predictions

A modified and extended version of a convergence test for wind tunnel strain-gage balance load iterations was implemented. The test uses an upper bound of the Lipschitz constant to assess convergence characteristics of balance load predictions if the Iterative Method is applied. Convergence is expected within the use envelope of the balance whenever this upper bound is less than the threshold of one. It is explained in great detail how the convergence test can be applied to the two load iteration equation types that are currently being used in the aerospace testing community. In addition, the application of the test to balances with bi-directional output characteristics is discussed. It is also shown how numerical differentiation can be used in order to obtain partial derivatives that are needed for the calculation of the Lipschitz constant. Finally, machine calibration data of NASA's MC60E six-component force balance is selected to demonstrate both implementation and use of the convergence test.

calibration analysis↗
NASA NTRS1972

Convergence of oscillator spectral estimators for counted-frequency measurements.

A common intermediary connecting frequency-noise calibration or testing of an oscillator to useful applications is the spectral density of the frequency-deviating process. In attempting to turn test data into predicts of performance characteristics, one is naturally led to estimation of statistical values by sample-mean and sample-variance techniques. However, sample means and sample variances themselves are statistical quantities that do not necessarily converge (in the mean-square sense) to actual ensemble-average means and variances, except perhaps for excessively large sample sizes. This is especially true for the flicker noise component of oscillators. This article shows, for the various types of noises found in oscillators, how sample averages converge (or do not converge) to their statistical counterparts. The convergence rate is shown to be the same for all oscillators of a given spectral type.

Tausworthe, R. C.↗
NASA NTRS1973

Rate of convergence of a class of methods of feasible directions.

This paper deals with the rate of convergence of four methods of feasible directions: the Zoutendijk procedures 1 and 2 and two modifications of these procedures due to the authors. It is shown that of these methods, the two due to the authors converge linearly under convexity assumptions, that the Zoutendijk procedure 2 converges sublinearly under these assumptions, and that the Zoutendijk procedure 1 converges linearly provided the solution of the problem is a vertex of the constraint set.

Pironneau, O.↗
NASA NTRS1975

On a numerical sufficiency test for monotonic convergence of finite element models

Finite element analyses characterized by monotonic convergence include the discipline for meaningful measurements of convergence rate and consequently economical extrapolation. Few proposers of element models guarantee monotonic convergence for their elements. Thus, a need exists for an automatic test to classify available element models. This paper describes such a test - a test can be performed using a digital computer to guarantee that a particular element model imbues monotonicity. It describes the test and its basis. It examines seven element models for a rectangular membrane to illustrate the value of the tests. Besides confirming results already known, the application yields new data. It 'proves' monotonicity for two improved models, defines the range of element proportions for which another element can be guaranteed to exhibit monotonicity, and suggests that another element is deficient. In the special case of absolutely convergent membrane displacement models, proof of monotonicity is a necessary and sufficient condition to insure that upper bound estimates of strain energy are developed. Accordingly, the test furnishes a proof of bound solutions independently of requirements on displacement continuity the element basis may or may not satisfy.

Melosh, R. J.↗
NASA NTRS1979

An observational test of the quasi-geostrophic relation between eddy momentum flux convergence and eddy fluxes of potential vorticity and potential temperature

The quasi-geostrophic relation between the fluxes of momentum, potential vorticity and potential temperature is tested with atmospheric data by computing the convergence of momentum flux as a residual of the potential temperature and potential vorticity flux and comparing it to the momentum flux convergence computed directly. It is shown that in the troposphere between 18 and 74 deg N the observed momentum flux convergence differs from the quasi-geostrophic convergence by 25-60% depending on tropospheric level and season.

Heck, W. J.↗
NASA NTRS1980

Moisture convergence using satellite-derived wind fields - A severe local storm case study

Five-minute interval 1-km resolution SMS visible channel data were used to derive low-level wind fields by tracking small cumulus clouds on NASA's Atmospheric and Oceanographic Information Processing System. The satellite-derived wind fields were combined with surface mixing ratios to derive horizontal moisture convergence in the prestorm environment of April 24, 1975. Storms began developing in an area extending from southwest Oklahoma to eastern Tennessee 2 h subsequent to the time of the derived fields. The maximum moisture convergence was computed to be 0.0022 g/kg per sec and areas of low-level convergence of moisture were in general indicative of regions of severe storm genesis. The resultant moisture convergence fields derived from two wind sets 20 min apart were spatially consistent and reflected the mesoscale forcing of ensuing storm development. Results are discussed with regard to possible limitations in quantifying the relationship between low-level flow and between low-level flow and satellite-derived cumulus motion in an antecedent storm environment.

Negri, A. J.↗
NASA NTRS1982

Topics in global convergence of density estimates

The problem of estimating a density f on R sup d from a sample Xz(1),...,X(n) of independent identically distributed random vectors is critically examined, and some recent results in the field are reviewed. The following statements are qualified: (1) For any sequence of density estimates f(n), any arbitrary slow rate of convergence to 0 is possible for E(integral/f(n)-fl); (2) In theoretical comparisons of density estimates, integral/f(n)-f/ should be used and not integral/f(n)-f/sup p, p 1; and (3) For most reasonable nonparametric density estimates, either there is convergence of integral/f(n)-f/ (and then the convergence is in the strongest possible sense for all f), or there is no convergence (even in the weakest possible sense for a single f). There is no intermediate situation.

Devroye, L.↗
NASA NTRS1982

Formal convergence characteristics of elliptically constrained incremental Newton-Raphson algorithms

Various aspects of the convergence, uniqueness, and existence properties associated with solutions generated via the elliptically constrained incremental Newton-Raphson (ECINR) algorithm are analyzed. Several theorems are developed, and the formal behavior of the elliptically constrained scheme developed by Padovan (1981) is discussed in detail. Consideration is given to global and local rates of convergence, to the determination of the occurrence of safety zones wherein the algorithm yields inherently convergent results, to formal limitations on the class of functions which the scheme can be applied to solve, and to single and multidimensional formalisms on existence uniqueness and convergence. Special attention is given to functions whose Jacobian matrix exhibit positive, negative, semi and indefinite properties. Several significant advantages of ECINR over the classical INR are mentioned.

Padovan, J.↗