Search NASA⌕ Search

SEARCH · Search NASA

Results for “Newton Method”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 109 records · Page 6

Molecular Simulations in Astrobiology

One of the main goals of astrobiology is to understand the origin of cellular life. The most direct approach to this problem is to construct laboratory models of protocells. Such efforts, currently underway in the NASA Astrobiology Program, are accompanied by computational studies aimed at explaining self-organization of simple molecules into ordered structures that are capable of performing protocellular functions. Many of these functions, such as importing nutrients, capturing energy and responding to changes in the environment, are carried out by proteins bound to membranes. We use computer simulations to address the following questions about these proteins: (1) How do small proteins self-organize into ordered structures at water-membrane interfaces and insert into membranes? (2) How do peptides form membrane-spanning structures (e.g. channels)? (3) By what mechanisms do such structures perform their functions? The simulations are performed using the molecular dynamics method. In this method, Newton's equations of motion for each atom in the system are solved iteratively. At each time step, the forces exerted on each atom by the remaining atoms are evaluated by dividing them into two parts. Short-range forces are calculated in real space while long-range forces are evaluated in reciprocal space, using a particle-mesh algorithm which is of order O(NInN). With a time step of 2 femtoseconds, problems occurring on multi-nanosecond time scales (10(exp 6)-10(exp 8) time steps) are accessible. To address a broader range of problems, simulations need to be extended by three orders of magnitude, which requires algorithmic improvements and codes scalable to a large number of processors. Work in this direction is in progress. Two series of simulations are discussed. In one series, it is shown that nonpolar peptides, disordered in water, translocate to the nonpolar interior of the membrane and fold into helical structures (see Figure). Once in the membrane, the peptides exhibit orientational flexibility with changing conditions, which may have provided a mechanism of transmitting signals between the protocell and its environment. In another series of simulations, the mechanism by which a simple protein channel efficiently mediates proton transport across membranes was investigated. This process is a key step in cellular bioenergetics. In the channel under study, proton transport is gated by four histidines that occlude the channel pore. The simulations identify the mechanisms by which protons move through the gate.

Pohorille, Andrew↗

Molecular Simulations in Astrobiology

One of the main goals of astrobiology is to understand the origin of cellular life. In the absence of any record of the earliest ancestors of contemporary cells, protocells, the most direct way to test our understanding of their characteristics is to construct laboratory models of protocells. Such efforts, currently underway in the NASA Astrobiology Program, are accompanied by computational studies aimed at explaining self-organization of simple molecules into ordered structures and developing designs of molecules that are capable of performing protocellular functions. Many of these functions, such as importing nutrients, capturing and storing energy, and responding to changes in the environment, are carried out by proteins bound to membranes. We use computer simulations to address the following, questions about these proteins: (1) How do small proteins (peptides) organize themselves into ordered structures at water-membrane interfaces and insert into membranes? (2) How do peptides aggregate to form membrane-spannin(y structures (e.g., channels)? (3) By what mechanisms do such aggregates perform their functions? The simulations are performed using the molecular dynamics (MD) method. In this method, Newton's equations of motion for each atom in the system are solved iteratively. At each time step, the forces exerted on each atom by the remaining atoms are evaluated by dividing them into two parts. Short-range forces are calculated directly in real space while long-range forces are evaluated in reciprocal space, usually using a particle-mesh algorithm which is of order O(NlnN). Currently, a time step of 2 femtoseconds is typically used, thereby making studies of problems occurring on multi-nanosecond time scales (10(exp 6) - 10(exp 8) time steps) accessible. To address a broader range of problems, simulations need to be extended by three orders of magnitude. Such an extension requires both algorithmic improvements and codes scalable to a large number of parallel processors. Work in this direction is in progress. Two specific series of simulations that demonstrate how peptides self-organize and function in membranes are discussed. In one series of simulations, it was shown that nonpolar peptides, disordered in water, translocate to the nonpolar interior of the membrane and, simultaneously, fold into two different helical structures, which remain in equilibrium. Once in the membrane, the peptides can readily change their orientation, especially in response to local electric fields. This structural and orientational flexibility of peptides with changing conditions may have provided a mechanism of transmitting signals between the environment and the interior of the protocell. In another series of simulations, the mechanism by which a simple protein channel efficiently mediates proton transport across membranes was investigated. This process is a key step in cellular bioenergetics. In the channel under study, proton transport is gated by four histidines that occlude the channel pore. The simulations demonstrate that protons move through the gate by a "shuttle" mechanism, wherein one histidine is protonated on the extracellular side and, subsequently, the proton bound on the opposite side is released.

Pohorille, Andrew↗

Finding the Closest Surface Point

To find the closest surface point to a given point is one of the most frequently used operations in geometric computation. Traditionally, this problem has been formulated by a linear Newton method. To improve robustness for this operation, I present a different formulation, with a second order method, for the general cases, and ways to handle the degeneracy at the apex of a cone, a commonly occurring surface degeneracy.

Chou, Jin J.↗

Variable X-Ray Absorption in the Mini-BAL QSO PG 1126-041

Context. X-ray studies of AGN with powerful nuclear winds are important to constrain the physics of the inner accretion/ejection flow around SMBH, and to understand the impact of such winds on the AGN environment. Aims. Our main scientific goal is to constrain the properties of a variable outflowing absorber that is thought to be launched near the SMBH of the mini-BAL QSO PG 1126-041 using a multi-epoch observational campaign performed with XMM-Newton. Methods. We performed temporally resolved X-ray spectroscopy and simultaneous UV and X-ray photometry on the most complete set of observations and on the deepest X-ray exposure of a mini-BAL QSO to date. Results. We found complex X-ray spectral variability on time scales of both months and hours, best reproduced by means of variable massive ionized absorbers along the line of sight. As a consequence, the observed optical-to-X-ray spectral index is found to be variable with time. In the highest signal-to-noise observation we detected highly ionized X-ray absorbing material outflowing much faster (u(sub X) approx. 16 500 km/s) than the UV absorbing one (u(sub uv) approx. 5,000 km/s). This highly ionized absorber is found to be variable on very short (a few kiloseconds) time scales. Conclusions. Our findings are qualitatively consistent with line driven accretion disk winds scenarios. Our observations have opened the time-resolved X-ray spectral analysis field for mini-BAL QSOs; only with future deep studies will we be able to map the dynamics of the inner flow and understand the physics of AGN winds and their impact on the environment.

Giustini, M.↗

An Experimental and Computational Study on Soot Formation in a Coflow Jet Flame Under Microgravity and Normal Gravity

Upon the completion of the Structure and Liftoff in Combustion Experiment (SLICE) in March 2012, a comprehensive and unique set of microgravity coflow diffusion flame data was obtained. This data covers a range of conditions from weak flames near extinction to strong, highly sooting flames, and enabled the study of gravitational effects on phenomena such as liftoff, blowout and soot formation. The microgravity experiment was carried out in the Microgravity Science Glovebox (MSG) on board the International Space Station (ISS), while the normal gravity experiment was performed at Yale utilizing a copy of the flight hardware. Computational simulations of microgravity and normal gravity flames were also carried out to facilitate understanding of the experimental observations. This paper focuses on the different sooting behaviors of CH4 coflow jet flames in microgravity and normal gravity. The unique set of data serves as an excellent test case for developing more accurate computational models.Experimentally, the flame shape and size, lift-off height, and soot temperature were determined from line-of-sight flame emission images taken with a color digital camera. Soot volume fraction was determined by performing an absolute light calibration using the incandescence from a flame-heated thermocouple. Computationally, the MC-Smooth vorticity-velocity formulation was employed to describe the chemically reacting flow, and the soot evolution was modeled by the sectional aerosol equations. The governing equations and boundary conditions were discretized on an axisymmetric computational domain by finite differences, and the resulting system of fully coupled, highly nonlinear equations was solved by a damped, modified Newtons method. The microgravity sooting flames were found to have lower soot temperatures and higher volume fraction than their normal gravity counterparts. The soot distribution tends to shift from the centerline of the flame to the wings from normal gravity to microgravity.

microgravity↗

The GPM Combined Algorithm

In this paper, the operational Global Precipitation Measurement (GPM) mission combined radar-radiometer algorithm is thoroughly described. The operational combined algorithm is designed to reduce uncertainties in GPM Core Observatory precipitation estimates by effectively integrating complementary information from the GPM Dual-Frequency Precipitation Radar (DPR) and the GPM Microwave Imager (GMI) into an optimal, physically consistent precipitation product. Although similar in many respects to previously developed combined algorithms, the GPM combined algorithm has several unique features that are specifically designed to meet the GPM objectives of deriving, based on GPM Core Observatory information, accurate and physically consistent precipitation estimates from multiple spaceborne instruments, and ancillary environmental data from reanalyses. The algorithm features an optimal estimation framework based on a statistical formulation of the Gauss-Newton method, a parameterization for the nonuniform distribution of precipitation within the radar fields of view, a methodology to detect and account for multiple scattering in Ka-band DPR observations, and a statistical deconvolution technique that allows for an efficient sequential incorporation of radiometer information into DPR precipitation retrievals.

Grecu, Mircea↗

Rapid and Accurate Computation of Invariant Tori, Manifolds, and Connections Near Mean Motion Resonances in the Periodically Perturbed Planar Circular Restricted 3-Body Problem Models

When the planar circular restricted 3-body problem (RTBP) is periodically perturbed, most unstable resonant periodic orbits become invariant tori. In this study, we 1) develop a quasi-Newton method which simultaneously solves for the tori and their center, stable, and unstable directions; 2) implement continuation by both perturbation as well as rotation numbers; 3) compute Fourier-Taylor parameterizations of the stable and unstable manifolds; 4) globalize these manifolds; 5) compute homoclinic and heteroclinic connections. Our methodology improves on efficiency and accuracy compared to prior studies, and applies to a variety of periodic perturbations. We demonstrate the tools on the planar elliptic RTBP.

de la Llave, Rafael↗

An actuator extension transformation for a motion simulator and an inverse transformation applying Newton-Raphson's method

A set of equations which transform position and angular orientation of the centroid of the payload platform of a six-degree-of-freedom motion simulator into extensions of the simulator's actuators has been derived and is based on a geometrical representation of the system. An iterative scheme, Newton-Raphson's method, has been successfully used in a real time environment in the calculation of the position and angular orientation of the centroid of the payload platform when the magnitude of the actuator extensions is known. Sufficient accuracy is obtained by using only one Newton-Raphson iteration per integration step of the real time environment.

Dieudonne, J. E.↗

Efficient shallow Ritz method for 1D diffusion problems

This paper studies the shallow Ritz method for solving the one-dimensional diffusion problem. It is shown that the shallow Ritz method improves the order of approximation dramatically for non-smooth problems. To realize this optimal or nearly optimal order of the shallow Ritz approximation, we develop a damped block Newton (dBN) method that alternates between updates of the linear and non-linear parameters. Per each iteration, the linear and the non-linear parameters are updated by exact inversion and one step of a modified, damped Newton method applied to a reduced non-linear system, respectively. The computational cost of each dBN iteration is $\mathcal{O}$(n). Starting with the non-linear parameters as a uniform partition of the interval, numerical experiments show that the dBN is capable of efficiently moving mesh points to nearly optimal locations. In conclusion, to improve the efficiency of the dBN further, we propose an adaptive damped block Newton (AdBN) method by combining the dBN with the adaptive neuron enhancement (ANE) method [28].

Diffusion problems↗

Davidon-Broyden rank-one minimization methods in Hilbert space with application to optimal control problems

The Davidon-Broyden class of rank one, quasi-Newton minimization methods is extended from Euclidean spaces to infinite-dimensional, real Hilbert spaces. For several techniques of choosing the step size, conditions are found which assure convergence of the associated iterates to the location of the minimum of a positive definite quadratic functional. For those techniques, convergence is achieved without the problem of the computation of a one-dimensional minimum at each iteration. The application of this class of minimization methods for the direct computation of the solution of an optimal control problem is outlined. The performance of various members of the class are compared by solving a sample optimal control problem. Finally, the sample problem is solved by other known gradient methods, and the results are compared with those obtained with the rank one quasi-Newton methods.

Straeter, T. A.↗

Demonstration of the Dynamic Flowgraph Methodology using the Titan 2 Space Launch Vehicle Digital Flight Control System

Dynamic Flowgraph Methodology (DFM) is a new approach developed to integrate the modeling and analysis of the hardware and software components of an embedded system. The objective is to complement the traditional approaches which generally follow the philosophy of separating out the hardware and software portions of the assurance analysis. In this paper, the DFM approach is demonstrated using the Titan 2 Space Launch Vehicle Digital Flight Control System. The hardware and software portions of this embedded system are modeled in an integrated framework. In addition, the time dependent behavior and the switching logic can be captured by this DFM model. In the modeling process, it is found that constructing decision tables for software subroutines is very time consuming. A possible solution is suggested. This approach makes use of a well-known numerical method, the Newton-Raphson method, to solve the equations implemented in the subroutines in reverse. Convergence can be achieved in a few steps.

Yau, M.↗

Newton like: Minimal residual methods applied to transonic flow calculations

A computational technique for the solution of the full potential equation is presented. The method consists of outer and inner iterations. The outer iterate is based on a Newton like algorithm, and a preconditioned Minimal Residual method is used to seek an approximate solution of the system of linear equations arising at each inner iterate. The present iterative scheme is formulated so that the uncertainties and difficulties associated with many iterative techniques, namely the requirements of acceleration parameters and the treatment of additional boundary conditions for the intermediate variables, are eliminated. Numerical experiments based on the new method for transonic potential flows around the NACA 0012 airfoil at different Mach numbers and different angles of attack are presented, and these results are compared with those obtained by the Approximate Factorization technique. Extention to three dimensional flow calculations and application in finite element methods for fluid dynamics problems by the present method are also discussed. The Inexact Newton like method produces a smoother reduction in the residual norm, and the number of supersonic points and circulations are rapidly established as the number of iterations is increased.

Wong, Y. S.↗

Newton-like minimal residual methods applied to transonic flow calculations

A computational technique for the solution of the full potential equation is presented. The method consists of outer and inner iterations. The outer iterate is based on a Newton like algorithm, and a preconditioned Minimal Residual method is used to seek an approximate solution of the system of linear equations arising at each inner iterate. The present iterative scheme is formulated so that the uncertainties and difficulties associated with many iterative techniques, namely the requirements of acceleration parameters and the treatment of additional boundary conditions for the intermediate variables, are eliminated. Numerical experiments based on the new method for transonic potential flows around the NACA 0012 airfoil at different Mach numbers and different angles of attack are presented, and these results are compared with those obtained by the Approximate Factorization technique. Extention to three dimensional flow calculations and application in finite element methods for fluid dynamics problems by the present method are also discussed. The Inexact Newton like method produces a smoother reduction in the residual norm, and the number of supersonic points and circulations are rapidly established as the number of iterations is increased.

Wong, Y. S.↗

Natural convection in steady solidification - Finite element analysis of a two-phase Rayleigh-Benard problem

Galerkin finite-element approximations and Newton's method for solving free boundary problems are combined with computer-implemented techniques from nonlinear perturbation analysis to study solidification problems with natural convection in the melt. The Newton method gives rapid convergence to steady state velocity, temperature and pressure fields and melt-solid interface shapes, and forms the basis for algebraic methods for detecting multiple steady flows and assessing their stability. The power of this combination is demonstrated for a two-phase Rayleigh-Benard problem composed of melt and solid in a veritical cylinder with the thermal boundary conditions arranged so that a static melt with a flat melt-solid interface is always a solution. Multiple cellular flows bifurcating from the static state are detected and followed as Rayleigh number is varied. Changing the boundary conditions to approach those appropriate for the vertical Bridgman solidification system causes imperfections that eliminate the static state. The flow structure in the Bridgman system is related to those for the Rayleigh-Benard system by a continuous evolution of the boundary conditions.

Chang, C. J.↗

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.↗