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NASA NTRS1981

Optimality criteria solution strategies in multiple constraint design optimization

Procedures and solution strategies are described to solve the conventional structural optimization problem using the Lagrange multiplier technique. The multipliers, obtained through solution of an auxiliary nonlinear optimization problem, lead to optimality criteria to determine the design variables. It is shown that this procedure is essentially equivalent to an alternative formulation using a dual method Lagrangian function objective. Although mathematical formulations are straight-forward, successful applications and computational efficiency depend upon execution procedure strategies. Strategies examined, with application examples, include selection of active constraints, move limits, line search procedures, and side constraint boundaries.

Levy, R.↗

NASA NTRS1974

Microscopic plasma Hamiltonian

A Hamiltonian for the microscopic plasma model is derived from the Low Lagrangian after the dual roles of the generalized variables are taken into account. The resulting Hamilton equations are shown to agree with the Euler-Lagrange equations of the Low Lagrangian.

Peng, Y.-K. M.↗

NASA NTRS2002

Progress in the Simulation of Steady and Time-Dependent Flows with 3D Parallel Unstructured Cartesian Methods

The proposed paper will present recent extensions in the development of an efficient Euler solver for adaptively-refined Cartesian meshes with embedded boundaries. The paper will focus on extensions of the basic method to include solution adaptation, time-dependent flow simulation, and arbitrary rigid domain motion. The parallel multilevel method makes use of on-the-fly parallel domain decomposition to achieve extremely good scalability on large numbers of processors, and is coupled with an automatic coarse mesh generation algorithm for efficient processing by a multigrid smoother. Numerical results are presented demonstrating parallel speed-ups of up to 435 on 512 processors. Solution-based adaptation may be keyed off truncation error estimates using tau-extrapolation or a variety of feature detection based refinement parameters. The multigrid method is extended to for time-dependent flows through the use of a dual-time approach. The extension to rigid domain motion uses an Arbitrary Lagrangian-Eulerlarian (ALE) formulation, and results will be presented for a variety of two- and three-dimensional example problems with both simple and complex geometry.

Aftosmis, M. J.↗

NASA NTRS1998

A Posteriori Bounds for Linear-Functional Outputs of Crouzeix-Raviart Finite Element Discretizations of the Incompressible Stokes Problem

We present a finite element technique for the efficient generation of lower and upper bounds to outputs which are linear functionals of the solutions to the incompressible Stokes equations in two space dimensions; the finite element discretization is effected by Crouzeix-Raviart elements, the discontinuous pressure approximation of which is central to our approach. The bounds are based upon the construction of an augmented Lagrangian: the objective is a quadratic "energy" reformulation of the desired output; the constraints are the finite element equilibrium equations (including the incompressibility constraint), and the intersubdomain continuity conditions on velocity. Appeal to the dual max-min problem for appropriately chosen candidate Lagrange multipliers then yields inexpensive bounds for the output associated with a fine-mesh discretization; the Lagrange multipliers are generated by exploiting an associated coarse-mesh approximation. In addition to the requisite coarse-mesh calculations, the bound technique requires solution only of local subdomain Stokes problems on the fine-mesh. The method is illustrated for the Stokes equations, in which the outputs of interest are the flowrate past, and the lift force on, a body immersed in a channel.

Patera, Anthony T.↗

NASA NTRS1989

Quantum disorder, duality, and fractional statistics in 2 + 1 dimensions

A low-energy equivalence between two apparently unrelated Lagrangians with fractional statistics is reported. Exploiting this equivalence, it is possible to study the quantum disordered phase of the nonlinear sigma model with Hopf term. It is found that the quasi-particles in the disordered phase also have fractional statistics. There appears to be a dual relationship between the ordered and disordered phases.

Wen, X. G.↗

NASA NTRS1989

Dynamic analysis of a 6 DOF CKCM robot end-effector for dual-arm telerobot systems

A dynamical analysis is performed for a six DOF robot end-effector built to study telerobotic service and maintenance of NASA hardwares in space. The design of the end-effector is based on the concept of closed-kinematic chain mechanism capable of performing precise motion in a small workspace. After presenting a closed-form solution for the inverse kinematic problem, the Lagrangian approach is used to derive a set of equations of motion for the end-effector where the generalized coordinates are selected to be the Cartesian coordinates. A computer simulation study shows that the centrifugal and Coriolis terms can be neglected for slow motion.

Nguyen, Charles C.↗

NASA NTRS2013

An Investigation of the Kinematic and Microphysical Control of Lightning Rate, Extent and NOx Production using DC3 Observations and the NASA Lightning Nitrogen Oxides Model (LNOM)

The Deep Convective Clouds and Chemistry (DC3) experiment seeks to quantify the relationship between storm physics, lightning characteristics and the production of nitrogen oxides via lightning (LNOx). The focus of this study is to investigate the kinematic and microphysical control of lightning properties, particularly those that may govern LNOx production, such as flash rate, type and extent across Alabama during DC3. Prior studies have demonstrated that lightning flash rate and type is correlated to kinematic and microphysical properties in the mixed‐phase region of thunderstorms such as updraft volume and graupel mass. More study is required to generalize these relationships in a wide variety of storm modes and meteorological conditions. Less is known about the co‐evolving relationship between storm physics, morphology and three‐dimensional flash extent, despite its importance for LNOx production. To address this conceptual gap, the NASA Lightning Nitrogen Oxides Model (LNOM) is applied to North Alabama Lightning Mapping Array (NALMA) and Vaisala National Lightning Detection Network(TM) (NLDN) observations following ordinary convective cells through their lifecycle. LNOM provides estimates of flash rate, flash type, channel length distributions, lightning segment altitude distributions (SADs) and lightning NOx production profiles. For this study, LNOM is applied in a Lagrangian sense to multicell thunderstorms over Northern Alabama on two days during DC3 (21 May and 11 June 2012) in which aircraft observations of NOx are available for comparison. The LNOM lightning characteristics and LNOX production estimates are compared to the evolution of updraft and precipitation properties inferred from dual‐Doppler and polarimetric radar analyses applied to observations from a nearby radar network, including the UAH Advanced Radar for Meteorological and Operational Research (ARMOR). Given complex multicell evolution, particular attention is paid to storm morphology, cell mergers and possible dynamical, microphysical and electrical interaction of individual cells when testing various hypotheses.

Carey, Lawrence↗

NASA NTRS2013

An Investigation of the Kinematic and Microphysical Control of Lightning Rate, Extent and NOX Production using DC3 Observations and the NASA Lightning Nitrogen Oxides Model (LNOM)

The Deep Convective Clouds and Chemistry (DC3) experiment seeks to quantify the relationship between storm physics, lightning characteristics and the production of nitrogen oxides via lightning (LNOx). The focus of this study is to investigate the kinematic and microphysical control of lightning properties, particularly those that may govern LNOx production, such as flash rate, type and extent across Alabama during DC3. Prior studies have demonstrated that lightning flash rate and type is correlated to kinematic and microphysical properties in the mixed‐phase region of thunderstorms such as updraft volume and graupel mass. More study is required to generalize these relationships in a wide variety of storm modes and meteorological conditions. Less is known about the co‐evolving relationship between storm physics, morphology and three‐dimensional flash extent, despite its importance for LNOx production. To address this conceptual gap, the NASA Lightning Nitrogen Oxides Model (LNOM) is applied to North Alabama Lightning Mapping Array (NALMA) and Vaisala National Lightning Detection NetworkTM (NLDN) observations following ordinary convective cells through their lifecycle. LNOM provides estimates of flash rate, flash type, channel length distributions, lightning segment altitude distributions (SADs) and lightning NOx production profiles. For this study, LNOM is applied in a Lagrangian sense to multicell thunderstorms over Northern Alabama on two days during DC3 (21 May and 11 June 2012) in which aircraft observations of NOx are available for comparison. The LNOM lightning characteristics and LNOX production estimates are compared to the evolution of updraft and precipitation properties inferred from dual‐Doppler and polarimetric radar analyses applied to observations from a nearby radar network, including the UAH Advanced Radar for Meteorological and Operational Research (ARMOR). Given complex multicell evolution, particular attention is paid to storm morphology, cell mergers and possible dynamical, microphysical and electrical interaction of individual cells when testing various hypotheses.

Carey, Lawrence↗

NASA NTRS2014

The Kinematic and Microphysical Control of Lightning Rate, Extent and NOX Production

The Deep Convective Clouds and Chemistry (DC3) experiment seeks to quantify the relationship between storm physics, lightning characteristics and the production of nitrogen oxides via lightning (LNOx). The focus of this study is to investigate the kinematic and microphysical control of lightning properties, particularly those that may govern LNOx production, such as flash rate, type and extent across Alabama during DC3. Prior studies have demonstrated that lightning flash rate and type is correlated to kinematic and microphysical properties in the mixed-phase region of thunderstorms such as updraft volume and graupel mass. More study is required to generalize these relationships in a wide variety of storm modes and meteorological conditions. Less is known about the co-evolving relationship between storm physics, morphology and three-dimensional flash extent, despite its importance for LNOx production. To address this conceptual gap, the NASA Lightning Nitrogen Oxides Model (LNOM) is applied to North Alabama Lightning Mapping Array (NALMA) and Vaisala National Lightning Detection Network(TM) (NLDN) observations following ordinary convective cells through their lifecycle. LNOM provides estimates of flash rate, flash type, channel length distributions, lightning segment altitude distributions (SADs) and lightning NOx production profiles. For this study, LNOM is applied in a Lagrangian sense to multicell thunderstorms over Northern Alabama on two days during DC3 (21 May and 11 June 2012) in which aircraft observations of NOx are available for comparison. The LNOM lightning characteristics and LNOX production estimates are compared to the evolution of updraft and precipitation properties inferred from dual-Doppler and polarimetric radar analyses applied to observations from a nearby radar network, including the UAH Advanced Radar for Meteorological and Operational Research (ARMOR). Given complex multicell evolution, particular attention is paid to storm morphology, cell mergers and possible dynamical, microphysical and electrical interaction of individual cells when testing various hypotheses.

Carey, Lawrence↗

NASA NTRS2012

The Kinematic and Microphysical Control of Storm Integrated Lightning Flash Extent

The objective of this preliminary study is to investigate the kinematic and microphysical control of lightning properties, particularly those that may govern the production of nitrogen oxides (NOx) in thunderstorms, such as flash rate, type and extent. The mixed-phase region is where the noninductive charging (NIC) process is thought to generate most storm electrification during rebounding collisions between ice particles in the presence of supercooled water. As a result, prior radar-based studies have demonstrated that lightning flash rate is well correlated to kinematic and microphysical properties in the mixed-phase region of thunderstorms such as updraft volume, graupel mass, or ice mass flux. There is also some evidence that lightning type is associated with the convective state. Intracloud (IC) lightning tends to dominate during the updraft accumulation of precipitation ice mass while cloud-to-ground (CG) lightning is more numerous during the downdraft-driven descent of radar echo associated with graupel and hail. More study is required to generalize these relationships, especially regarding lightning type, in a wide variety of storm modes and meteorological conditions. Less is known about the co-evolving relationship between storm kinematics, microphysics, morphology and three-dimensional flash extent, despite its importance for lightning NOx production. To address this conceptual gap, the NASA MSFC Lightning Nitrogen Oxides Model (LNOM) is applied to North Alabama Lightning Mapping Array (NALMA) and Vaisala National Lightning Detection NetworkTM (NLDN) observations following ordinary convective cells through their lifecycle. LNOM provides estimates of flash type, channel length distributions, lightning segment altitude distributions (SADs) and lightning NOx production profiles. For this study, LNOM is applied in a Lagrangian sense to well isolated convective cells on 3 April 2007 (single cell and multi-cell hailstorm, non-severe multicell) and 6 July 2007 (non-severe multi-cell) over Northern Alabama. The LNOM lightning characteristics are compared to the evolution of updraft and precipitation properties inferred from dual-Doppler and polarimetric radar analyses applied to observations from a nearby Doppler radar network, including the UA Huntsville Advanced Radar for Meteorological and Operational Research (ARMOR, C-band, polarimetric). The LNOM estimated SAD and lightning NOx production profiles are placed in the context of radar derived profiles of vertical motion, precipitation types and amounts. Finally, these analyses are used to determine if storm integrated flash channel extent is as well correlated to volumetric updraft and precipitation ice characteristics in the mixed phase region as flash rate for these individual convective cells.

Carey, Lawrence↗

NASA NTRS1994

Wind Prelaunch Mission Operations Report (MOR)

The National Aeronautics and Space Administration (NASA) Wind mission is the first mission of the Global Geospace Science (GGS) initiative. The Wind laboratory will study the properties of particles and waves in the region between the Earth and the Sun. Using the Moon s gravity to save fuel, dual lunar swing-by orbits enable the spacecraft to sample regions close to and far from the Earth. During the three year mission, Wind will pass through the bow shock of Earth's magnetosphere to begin a thorough investigation of the solar wind. Mission objectives require spacecraft measurements in two orbits: lunar swing- by ellipses out to distances of 250 Earth radii (RE) and a small orbit around the Lagrangian point L-l that remains between the Earth and the Sun. Wind will be placed into an initial orbit for approximately 2 years. It will then be maneuvered into a transition orbit and ultimately into a halo orbit at the Earth-Sun L-l point where it will operate for the remainder of its lifetime. The Wind satellite development was managed by NASA's Goddard Space Flight Center with the Martin Marietta Corporation, Astro-Space Division serving as the prime contractor. Overall programmatic direction was provided by NASA Headquarters, Office of Space Science. The spacecraft will be launched under a launch service contract with the McDonnell Douglas Corporation on a Delta II Expendable Launch Vehicle (ELV) within a November l-l4, 1994 launch window. The Wind spacecraft carries six U.S. instruments, one French instrument, and the first Russian instrument ever to fly on an American satellite. The Wind and Polar missions are the two components of the GGS Program. Wind is also the second mission of the International Solar Terrestrial Physics (ISTP) Program. The first ISTP mission, Geotail, is a joint project of the Institute of Space and Astronautical Science of Japan and NASA which launched in 1992. The Wind mission is planned to overlap Geotail by six months and Polar by one year. The Wind and Polar missions, together with the Geotail mission (launched on July 24, 1992) and supporting equatorial measurements, will provide simultaneous data to enable the study of solar wind input to the magnetosphere and key elements of the magnetospheric response: ring current energy storage, geomagnetic tail energy storage, and ionospheric energy input.

Source record↗

NASA NTRS1989

Grid generation and inviscid flow computation about aircraft geometries

Grid generation and Euler flow about fighter aircraft are described. A fighter aircraft geometry is specified by an area ruled fuselage with an internal duct, cranked delta wing or strake/wing combinations, canard and/or horizontal tail surfaces, and vertical tail surfaces. The initial step before grid generation and flow computation is the determination of a suitable grid topology. The external grid topology that has been applied is called a dual-block topology which is a patched C (exp 1) continuous multiple-block system where inner blocks cover the highly-swept part of a cranked wing or strake, rearward inner-part of the wing, and tail components. Outer-blocks cover the remainder of the fuselage, outer-part of the wing, canards and extend to the far field boundaries. The grid generation is based on transfinite interpolation with Lagrangian blending functions. This procedure has been applied to the Langley experimental fighter configuration and a modified F-18 configuration. Supersonic flow between Mach 1.3 and 2.5 and angles of attack between 0 degrees and 10 degrees have been computed with associated Euler solvers based on the finite-volume approach. When coupling geometric details such as boundary layer diverter regions, duct regions with inlets and outlets, or slots with the general external grid, imposing C (exp 1) continuity can be extremely tedious. The approach taken here is to patch blocks together at common interfaces where there is no grid continuity, but enforce conservation in the finite-volume solution. The key to this technique is how to obtain the information required for a conservative interface. The Ramshaw technique which automates the computation of proportional areas of two overlapping grids on a planar surface and is suitable for coding was used. Researchers generated internal duct grids for the Langley experimental fighter configuration independent of the external grid topology, with a conservative interface at the inlet and outlet.

Smith, Robert E.↗

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 NTRS1996

The Newton Modified Barrier Method for QP Problems

The Modified Barrier Functions (MBF) have elements of both Classical Lagrangians (CL) and Classical Barrier Functions (CBF). The MBF methods find an unconstrained minimizer of some smooth barrier function in primal space and then update the Lagrange multipliers, while the barrier parameter either remains fixed or can be updated at each step. The numerical realization of the MBF method leads to the Newton MBF method, where the primal minimizer is found by using Newton's method. This minimizer is then used to update the Lagrange multipliers. In this paper, we examine the Newton MBF method for the Quadratic Programming (QP) problem. It will be shown that under standard second-order optimality conditions, there is a ball around the primal solution and a cut cone in the dual space such that for a set of Lagrange multipliers in this cut cone, the method converges quadratically to the primal minimizer from any point in the aforementioned ball, and continues to do so after each Lagrange multiplier update. The Lagrange multipliers remain within the cut cone and converge linearly to their optimal values. Any point in this ball will be called a "hot start". Starting at such a "hot start", at most Omicron(1n 1n epsilon(exp -1)) Newton steps are sufficient to perform the primal minimization which is necessary for the Lagrange multiplier update. Here, epsilon > 0 is the desired accuracy. Because of the linear convergence of the Lagrange multipliers, this means that only Omicron(1n epsilon(exp -1))omicron(ln 1n epsilon(exp-1)) Newton steps are required to reach an epsilon-approximation to the solution from any "hot start". In order to reach the "hot start", one has to perform Omicron(square root(m) 1n C) Newton steps, where m characterizes the size of the problem and C > 0 is the condition number of the QP problem. This condition number will be characterized explicitly in terms of key parameters of the QP problem, which in turn depend on the input data and the size of the problem.

Melman, A.↗

NASA NTRS2003

Geostationary Spectrograph (GeoSpec) for Earth and Atmospheric Science Applications

GeoSpec will support several possible future mission concepts in the Atmospheric Sciences and in Land and Ocean Sciences by measurement of both chemically linked atmospheric trace gas concentrations and profiles of important molecules such as OS, N02, CH20 and SO2 and at the same time coastal and ocean pollution events, tidal effects, and the origin and evolution of aerosol plumes. The instrument design concept we will describe is a dual spectrograph covering the WMS wavelength region of 310- 481 nm and the VIS/NIR wavelength region of 500-900 nm. A third channel in the short- wave infrared (SWIR) region between 2.2 p and 2.4 pn for total column measurements of CO, CH4, and N20 will also be described. The goal is to design a system capable of making moderate spatial resolution (750 meters at nadir) hyperspectral measurements (0.2 to 1.2 nm resolution) from a geostationary orbit. This would enable studies of time- varying pollution and coastal change processes with a temporal resolution of 5 minutes on a regional scale to 1 hour on a continental scale. Technological advances in the design and fabrication of convex holographic gratings and large format, high dynamic range PIN/CMOS detectors at the focal plane will be exploited. By simply changing the focal length of the front-end telescope GeoSpec can accommodate different orbital altitudes, including low Earth orbit, the Sun-side Lagrangian point L1, and/or different spatial resolutions.

Janz, Scott J.↗

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.↗