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NASA NTRS1995

Modified Interior Distance Functions (Theory and Methods)

In this paper we introduced and developed the theory of Modified Interior Distance Functions (MIDF's). The MIDF is a Classical Lagrangian (CL) for a constrained optimization problem which is equivalent to the initial one and can be obtained from the latter by monotone transformation both the objective function and constraints. In contrast to the Interior Distance Functions (IDF's), which played a fundamental role in Interior Point Methods (IPM's), the MIDF's are defined on an extended feasible set and along with center, have two extra tools, which control the computational process: the barrier parameter and the vector of Lagrange multipliers. The extra tools allow to attach to the MEDF's very important properties of Augmented Lagrangeans. One can consider the MIDFs as Interior Augmented Lagrangeans. It makes MIDF's similar in spirit to Modified Barrier Functions (MBF's), although there is a fundamental difference between them both in theory and methods. Based on MIDF's theory, Modified Center Methods (MCM's) have been developed and analyzed. The MCM's find an unconstrained minimizer in primal space and update the Lagrange multipliers, while both the center and the barrier parameter can be fixed or updated at each step. The MCM's convergence was investigated, and their rate of convergence was estimated. The extension of the feasible set and the special role of the Lagrange multipliers allow to develop MCM's, which produce, in case of nondegenerate constrained optimization, a primal and dual sequences that converge to the primal-dual solutions with linear rate, even when both the center and the barrier parameter are fixed. Moreover, every Lagrange multipliers update shrinks the distance to the primal dual solution by a factor 0 less than gamma less than 1 which can be made as small as one wants by choosing a fixed interior point as a 'center' and a fixed but large enough barrier parameter. The numericai realization of MCM leads to the Newton MCM (NMCM). The approximation for the primal minimizer one finds by Newton Method followed by the Lagrange multipliers update. Due to the MCM convergence, when both the center and the barrier parameter are fixed, the condition of the MDF Hessism and the neighborhood of the primal ninimizer where Newton method is 'well' defined remains stable. It contributes to both the complexity and the numerical stability of the NMCM.

Polyak, Roman A.↗

NASA NTRS1986

An optimization model for the US Air-Traffic System

A systematic approach for monitoring U.S. air traffic was developed in the context of system-wide planning and control. Towards this end, a network optimization model with nonlinear objectives was chosen as the central element in the planning/control system. The network representation was selected because: (1) it provides a comprehensive structure for depicting essential aspects of the air traffic system, (2) it can be solved efficiently for large scale problems, and (3) the design can be easily communicated to non-technical users through computer graphics. Briefly, the network planning models consider the flow of traffic through a graph as the basic structure. Nodes depict locations and time periods for either individual planes or for aggregated groups of airplanes. Arcs define variables as actual airplanes flying through space or as delays across time periods. As such, a special case of the network can be used to model the so called flow control problem. Due to the large number of interacting variables and the difficulty in subdividing the problem into relatively independent subproblems, an integrated model was designed which will depict the entire high level (above 29000 feet) jet route system for the 48 contiguous states in the U.S. As a first step in demonstrating the concept's feasibility a nonlinear risk/cost model was developed for the Indianapolis Airspace. The nonlinear network program --NLPNETG-- was employed in solving the resulting test cases. This optimization program uses the Truncated-Newton method (quadratic approximation) for determining the search direction at each iteration in the nonlinear algorithm. It was shown that aircraft could be re-routed in an optimal fashion whenever traffic congestion increased beyond an acceptable level, as measured by the nonlinear risk function.

Mulvey, J. M.↗

NASA NTRS1989

Error analysis of finite element solutions for postbuckled cylinders

A general method of error analysis and correction is investigated for the discrete finite-element results for cylindrical shell structures. The method for error analysis is an adaptation of the method of successive approximation. When applied to the equilibrium equations of shell theory, successive approximations derive an approximate continuous solution from the discrete finite-element results. The advantage of this continuous solution is that it contains continuous partial derivatives of an order higher than the basis functions of the finite-element solution. Preliminary numerical results are presented in this paper for the error analysis of finite-element results for a postbuckled stiffened cylindrical panel modeled by a general purpose shell code. Numerical results from the method have previously been reported for postbuckled stiffened plates. A procedure for correcting the continuous approximate solution by Newton's method is outlined.

Sistla, Rajaram↗

NASA NTRS1991

Rapid Design of Gravity Assist Trajectories

Several International Solar Terrestrial Physics (ISTP) missions require the design of complex gravity assisted trajectories in order to investigate the interaction of the solar wind with the Earth's magnetic field. These trajectories present a formidable trajectory design and optimization problem. The philosophy and methodology that enable an analyst to design and analyse such trajectories are discussed. The so called 'floating end point' targeting, which allows the inherently nonlinear multiple body problem to be solved with simple linear techniques, is described. The combination of floating end point targeting with analytic approximations with a Newton method targeter to achieve trajectory design goals quickly, even for the very sensitive double lunar swingby trajectories used by the ISTP missions, is demonstrated. A multiconic orbit integration scheme allows fast and accurate orbit propagation. A prototype software tool, Swingby, built for trajectory design and launch window analysis, is described.

Carrico, J.↗

NASA NTRS1991

Rapid design of gravity assist trajectories

Several International Solar-Terrestrial Physics (ISTP) missions require the design of complex gravity-assisted trajectories in order to investigate the interaction of the solar wind with the earth's magnetic field. These trajectories present a formidable trajectory design and optimization problem. This paper discusses the philosophy and methodology that enable an analyst to design and analyze such trajectories. The authors describe what is called 'floating end-point' targeting, which allows the inherently nonlinear multiple body problem to be solved with simple linear techniques. This paper demonstrates how floating end-point targeting combines analytic approximations with a Newton method targeter to achieve trajectory design goals quickly, even for the very sensitive double lunar swingby trajectories used by the ISTP missions. A Multiconic orbit integration scheme allows fast and accurate orbit propagation. This paper also describes a prototype software tool, Swingby, which has been built for trajectory design and launch window analysis.

Carrico, J.↗

NASA NTRS1992

Eigenvalue error analysis of viscously damped structures using a Ritz reduction method

The efficient solution of the eigenvalue problem that results from inserting passive dampers with variable stiffness and damping coefficients into a structure is addressed. Eigenanalysis of reduced models obtained by retaining a number of normal modes augmented with Ritz vectors corresponding to the static solutions resulting from the load patterns introduced by the dampers has been empirically shown to yield excellent approximations to the full eigenvalue problem. An analysis of this technique in the case of a single damper is presented. A priori and a posteriori error estimates are generated and tested on numerical examples. Comparison theorems with modally truncated models and a Markov parameter matching reduced-order model are derived. These theorems corroborate the heuristic that residual flexibility methods improve low-frequency approximation of the system. The analysis leads to other techniques for eigenvalue approximation. Approximate closed-form solutions are derived that include a refinement to eigenvalue derivative methods for approximation. An efficient Newton scheme is also developed. A numerical example is presented demonstrating the effectiveness of each of these methods.

Chu, Cheng-Chih↗

NASA NTRS1994

A Numerical Comparison of Barrier and Modified Barrier Methods for Large-Scale Bound-Constrained Optimization

When a classical barrier method is applied to the solution of a nonlinear programming problem with inequality constraints, the Hessian matrix of the barrier function becomes increasingly ill-conditioned as the solution is approached. As a result, it may be desirable to consider alternative numerical algorithms. We compare the performance of two methods motivated by barrier functions. The first is a stabilized form of the classical barrier method, where a numerically stable approximation to the Newton direction is used when the barrier parameter is small. The second is a modified barrier method where a barrier function is applied to a shifted form of the problem, and the resulting barrier terms are scaled by estimates of the optimal Lagrange multipliers. The condition number of the Hessian matrix of the resulting modified barrier function remains bounded as the solution to the constrained optimization problem is approached. Both of these techniques can be used in the context of a truncated-Newton method, and hence can be applied to large problems, as well as on parallel computers. In this paper, both techniques are applied to problems with bound constraints and we compare their practical behavior.

Nash, Stephen G.↗

NASA NTRS2013

Variability of a Stellar Corona on a Time Scale of Days: Evidence for Abundance Fractionation in an Emerging Coronal Active Region

Elemental abundance effects in active coronae have eluded our understanding for almost three decades, since the discovery of the first ionization potential (FIP) effect on the sun. The goal of this paper is to monitor the same coronal structures over a time interval of six days and resolve active regions on a stellar corona through rotational modulation. We report on four iso-phase X-ray spectroscopic observations of the RS CVn binary EI Eri with XMM-Newton, carried out approximately every two days, to match the rotation period of EI Eri. We present an analysis of the thermal and chemical structure of the EI Eri corona as it evolves over the six days. Although the corona is rather steady in its temperature distribution, the emission measure and FIP bias both vary and seem to be correlated. An active region, predating the beginning of the campaign, repeatedly enters into our view at the same phase as it rotates from beyond the stellar limb. As a result, the abundances tend slightly, but consistently, to increase for high FIP elements (an inverse FIP effect) with phase. We estimate the abundance increase of high FIP elements in the active region to be of about 75% over the coronal mean. This observed fractionation of elements in an active region on time scales of days provides circumstantial clues regarding the element enrichment mechanism of non-flaring stellar coronae.

Nordon, R.↗

NASA NTRS2019

Boson Star from Repulsive Light Scalars and Gravitational Waves

We study properties of boson stars consisting of ultra-light scalar dark matter with repulsive self-interactions. We investigate the origin of the maximum mass of spherically symmetric stable stars which emerges only when solving the full equations of motion in curved space-time, but not when solving the approximated Schrödinger-Newton equations. When the repulsion is weak, the backreaction of the curvature on the scalars acts as an additional source of attraction and can overcome the repulsion, resulting in a maximum star mass and compactness. We also point out that the potential in a UV completed particle physics model of light scalar dark matter is generally more complicated than the widely used φ 4 interaction. Additional interactions beyond φ 4 in the potential can dramatically change the properties of boson stars as well as modify the prospect of LIGO gravitational wave detection for binary mergers of boson stars.

Djuna Croon↗

NASA NTRS1980

Computer-aided analysis of nonlinear problems in transport phenomena

The paper describes algorithms for equilibrium and steady-state problems with coefficients in the expansions derived by the Galerkin weighted residual method and calculated from the resulting sets of nonlinear algebraic equations by the Newton-Raphson method. Initial approximations are obtained from nearby solutions by continuation techniques as parameters are varied. The Newton-Raphson technique is preferred because the Jacobian of the solution is useful for continuation, for analyzing the stability of solutions, for detecting bifurcation of solution families, and for computing asymptotic estimates of the effects on any solution of small changes in parameters, boundary conditions, and boundary shape.

Brown, R. A.↗

NASA NTRS1976

The Adams formulas for numerical integration of differential equations from 1st to 20th order

The Adams Bashforth predictor coefficients and the Adams Moulton corrector coefficients for the integration of differential equations are presented for methods of 1st to 20th order. The order of the method as presented refers to the highest order difference formula used in Newton's backward difference interpolation formula, on which the Adams method is based. The Adams method is a polynomial approximation method derived from Newton's backward difference interpolation formula. The Newton formula is derived and expanded to 20th order. The Adams predictor and corrector formulas are derived and expressed in terms of differences of the derivatives, as well as in terms of the derivatives themselves. All coefficients are given to 18 significant digits. For the difference formula only, the ratio coefficients are given to 10th order.

Kirkpatrick, J. C.↗

NASA NTRS2016

Evidence for Intermediate Polars as the Origin of the Galactic Center Hard X-Ray Emission

Recently, unresolved hard (20-40 keV) X-ray emission has been discovered within the central 10 pc of the Galaxy, possibly indicating a large population of intermediate polars (IPs). Chandra and XMM-Newton measurements in the surrounding approximately 50 pc imply a much lighter population of IPs with (M(sub WD)) approximately 0.5 solar mass. Here we use broadband NuSTAR observations of two IPs: TV Columbae, which has a fairly typical but widely varying reported mass of (M(sub WD)) approximately 0.5-1.0 solar mass, and IGR J17303-0601, with a heavy reported mass of (M(sub WD)) approximately 1.0-1.2 solar mass. We investigate how varying spectral models and observed energy ranges influences estimated white dwarf mass. Observations of the inner 10 pc can be accounted for by IPs with (M(sub WD) approximately 0.9 solar mass, consistent with that of the CV population in general and the X-ray observed field IPs in particular. The lower mass derived by Chandra and XMM-Newton appears to be an artifact of narrow energy-band fitting. To explain the (unresolved) central hard X-ray emission (CHXE) by IPs requires an X-ray (2-8 keV) luminosity function (XLF) extending down to at least 5 x 10(exp 31) per erg s. The CHXE XLF, if extended to the surrounding approximately 50 pc observed by Chandra and XMM-Newton, requires that at least approximately 20%-40% of the approximately 9000 point sources are IPs. If the XLF extends just a factor of a few lower in luminosity, then the vast majority of these sources are IPs. This is in contrast to recent observations of the Galactic ridge, where the bulk of the 2-8 keV emission is ascribed to non-magnetic CVs.

Galaxy: center↗

NASA NTRS2017

X-Ray Brightening and UV Fading of Tidal Disruption Event ASASSN-15oi

We present late-time observations by Swift and XMM-Newton of the tidal disruption event (TDE) ASASSN-15oi that reveal that the source brightened in the X-rays by a factor of approximately 10 one year after its discovery, while it faded in the UV (Ultra Violet)/optical by a factor of approximately 100. The XMM-Newton observations measure a soft X-ray blackbody component with kT(sub bb) (blackbody kiloteslas) approximating 45 electronvolts, corresponding to radiation from several gravitational radii of a central approximately 10 (sup 6) solar mass black hole. The last Swift epoch taken almost 600 days after discovery shows that the X-ray source has faded back to its levels during the UV/optical peak. The timescale of the X-ray brightening suggests that the X-ray emission could be coming from delayed accretion through a newly forming debris disk and that the prompt UV/optical emission is from the prior circularization of the disk through stream-stream collisions. The lack of spectral evolution during the X-ray brightening disfavors ionization breakout of a TDE "veiled" by obscuring material. This is the first time a TDE has been shown to have a delayed peak in soft X-rays relative to the UV/optical peak, which may be the first clear signature of the real-time assembly of a nascent accretion disk, and provides strong evidence for the origin of the UV/optical emission from circularization, as opposed to reprocessed emission of accretion radiation.

Gezari, S.↗

NASA NTRS1973

Automated procedure for design of wing structures to satisfy strength and flutter requirements

A pilot computer program was developed for the design of minimum mass wing structures under flutter, strength, and minimum gage constraints. The wing structure is idealized by finite elements, and second-order piston theory aerodynamics is used in the flutter calculation. Mathematical programing methods are used for the optimization. Computation times during the design process are reduced by three techniques. First, iterative analysis methods used to reduce significantly reanalysis times. Second, the number of design variables is kept small by not using a one-to-one correspondence between finite elements and design variables. Third, a technique for using approximate second derivatives with Newton's method for the optimization is incorporated. The program output is compared witH previous published results. It is found that some flutter characteristics, such as the flutter speed, can display discontinous dependence on the design variables (which are the thicknesses of the structural elements). It is concluded that it is undesirable to use such quantities in the formulation of the flutter constraint.

Haftka, R. T.↗

NASA NTRS1980

Uniaxial and biaxial tensioning effects on thin membrane materials

Thin laminated membranes are being considered for various surface applications on future large space structural systems. Some of the thin membranes would be stretched across or between structural members with the requirement that the membrane be maintained within specified limits of smoothness which would be dictated by the particular applications such as antenna reflector requirements. The multiaxial tensile force required to maintain the smoothness in the membrane needs to be determined for use in the structure design. Therefore, several types of thicknesses of thin membrane materials have been subjected to varied levels of uniaxial and biaxial tensile loads. During the biaxial tests, deviations of the material surface smoothness were measured by a noncontacting capacitance probe. Basic materials consisted of composites of vacuum deposited aluminum on Mylar and Kapton ranging in thickness from 0.00025 in (0.000635 cm) to 0.002 in (0.00508 cm). Some of the material was reinforced with Kevlar and Nomex scrim. The uniaxial tests determined the material elongation and tensile forces up to ultimate conditions. Biaxial tests indicated that a relatively smooth material surface could be achieved with tensile force of approximately 1 to 15 Newtons per centimeter, depending upon the material thickness and/or reinforcement.

Hinson, W. F.↗

NASA NTRS1980

Preliminary design of composite wing-box structures for global damage tolerance

A procedure is presented that incorporates the influence of potential global damage conditions into the design process for minimum-mass wing-box structures. The procedure is based on mathematical-programming optimization techniques. Material-strength, minimum-gage, and panel-buckling constraints are introduced by penalty functions, and Newton's method with approximate second derivatives of the penalty terms is used as the search algorithm to obtain minimum-mass designs. A potential global damage condition is represented by a structural model with the damaged components removed. Example minimum-mass designs are obtained that simultaneously satisfy the constraints of the damaged and undamaged configurations of both graphite-epoxy and aluminum wing-box structural models. These examples are designed with and without the influence of potential damage conditions, and results indicate that for equal mass cases the residual strength of a damaged structure is higher when the influence of potential damage is properly included in the design from the outset. Results of these examples also identify the minimum structural mass increase required to increase residual strength levels.

Starnes, J. H., Jr.↗

NASA NTRS1983

The drag of magnetically suspended wind-tunnel models with nose-cones of various shapes

This article concerns the experimental determination of optimum nose-cones (minimum drag) of a body of revolution at supersonic and hypersonic speeds by means of ONERA magnetic suspension. The study concerns two groups of models, specifically: a group whose nose-cone has a profile in the shape of X(n); the AGARD B group whose nose-cone is plotted in accordance with a given law. The results obtained for the first group are comparable to those calculated with the approximations of Cole and Newton and the experiments carried out by Kubota.

Dubois, G.↗

NASA NTRS1984

An improved computational approach for multilevel optimum design

A penalty-function algorithm employing Newton's method with approximate second derivatives (Haftka and Starnes, 1980) is developed for two-level hierarchical design optimization problems. The difficulties posed by discontinuous behavior in typical multilevel problems are explained and illustrated for the case of a three-bar truss; the algorithm is formulated; and its advantages are demonstrated in the problem of a portal framework having three beams (described by six cross-section parameters), subjected to two loading conditions, and to be constructed in six different materials for comparison. The final design parameters are listed in a table.

Haftka, R. T.↗