Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Jacobi 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 145 records · Page 8
All sourcesNASA NTRSDOE OSTI
NASA NTRS2017

Reachability Subspace Exploration Using Continuation Methods

Reachability manifold computation suffers from the curse of dimensionality and for large state spaces is computationally intractable. This paper examines the use of continuation methods to address this issue by formulating the reachability sub-space manifold calculation into a number of initial valued problems. As a result of computing the reachability manifold for a subspace of interest, an exponential improvement in computational cost occurs. This concept is applied to a position subspace reachability problem of a spacecraft in a Keplerian orbit under maximum thrust constraints. Future work includes a comparison of the proposed method with computing reachability manifolds using viscosity solutions of the Hamilton Jacobi Bellman partial differential equation.

orbital mechanics↗
DOE OSTI2015

Improved methods for Feynman path integral calculations and their application to calculate converged vibrational–rotational partition functions, free energies, enthalpies, entropies, and heat capacities for methane

We present an improved version of our “path-by-path” enhanced same path extrapolation scheme for Feynman path integral (FPI) calculations that permits rapid convergence with discretization errors ranging from O(P{sup −6}) to O(P{sup −12}), where P is the number of path discretization points. We also present two extensions of our importance sampling and stratified sampling schemes for calculating vibrational–rotational partition functions by the FPI method. The first is the use of importance functions for dihedral angles between sets of generalized Jacobi coordinate vectors. The second is an extension of our stratification scheme to allow some strata to be defined based only on coordinate information while other strata are defined based on both the geometry and the energy of the centroid of the Feynman path. These enhanced methods are applied to calculate converged partition functions by FPI methods, and these results are compared to ones obtained earlier by vibrational configuration interaction (VCI) calculations, both calculations being for the Jordan–Gilbert potential energy surface. The earlier VCI calculations are found to agree well (within ∼1.5%) with the new benchmarks. The FPI partition functions presented here are estimated to be converged to within a 2σ statistical uncertainty of between 0.04% and 0.07% for the given potential energy surface for temperatures in the range 300–3000 K and are the most accurately converged partition functions for a given potential energy surface for any molecule with five or more atoms. We also tabulate free energies, enthalpies, entropies, and heat capacities.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2015

AETHER: A simulation platform for inductively coupled plasma

An in-house code is developed to simulate the inductively coupled plasma (ICP). The model comprises the fluid, electromagnetic and transformer submodels. Fluid equations are solved to evaluate the plasma flow parameters, including the plasma and neutral densities, ion and neutral velocities, electron flux, electron temperature, and electric potential. The model relies on the ambipolar approximation and offers the evaluation of plasma parameters without solving the sheath region. The electromagnetic model handles the calculation of the electric and magnetic fields using the magnetic vector potential. The transformer model captures the effect of the matching circuit utilized in laboratory experiments for RF power deposition. The continuity and momentum equations are solved using finite volume method. The energy, electric potential, and magnetic vector potential equations are solved using finite difference method. The resulting linear systems of equations are solved with iterative solvers including Jacobi and GMRES. The code is written using the C++ programming language, it works in parallel and has graphical user interface. The model is applied to study ICP characteristics of a plasma confined within a cylindrical chamber with dielectric walls for two different power deposition cases. The results obtained from the developed model are verified using the plasma module of COMSOL Multiphysics. The model is also applied to a plasma source configuration, and it is demonstrated that there is an overall increase in the plasma potential when current is extracted from ICP with a biased wall electrode.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2018

Performance of preconditioned iterative solvers in MFiX–Trilinos for fluidized beds

MFiX, a general-purpose Fortran-based suite, simulates the complex flow in fluidized bed applications via BiCGStab and GMRES methods along with plane relaxation preconditioners. Trilinos, an object-oriented framework, contains various first- and second-generation Krylov subspace solvers and preconditioners. We developed a framework to integrate MFiX with Trilinos as MFiX does not possess advanced linear methods. The framework allows MFiX to access advanced linear solvers and preconditioners in Trilinos. The integrated solver is called MFiX–Trilinos, here after. In the present work, we study the performance of variants of GMRES and CGS methods in MFiX–Trilinos and BiCGStab and GMRES solvers in MFiX for a 3D gas–solid fluidized bed problem. Two right preconditioners employed along with various solvers in MFiX–Trilinos are Jacobi and smoothed aggregation. The flow from MFiX–Trilinos is validated against the same from MFiX for BiCGStab and GMRES methods. And, the effect of the preconditioning on the iterative solvers in MFiX–Trilinos is also analyzed. In addition, the effect of left and right smoothed aggregation preconditioning on the solvers is studied. Finally, the performance of the first- and second-generation solver stacks in MFiX–Trilinos is studied as well for two different problem sizes.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1988

Multigrid method for stability problems

The problem of calculating the stability of steady state solutions of differential equations is treated. Leading eigenvalues (i.e., having maximal real part) of large matrices that arise from discretization are to be calculated. An efficient multigrid method for solving these problems is presented. The method begins by obtaining an initial approximation for the dominant subspace on a coarse level using a damped Jacobi relaxation. This proceeds until enough accuracy for the dominant subspace has been obtained. The resulting grid functions are then used as an initial approximation for appropriate eigenvalue problems. These problems are being solved first on coarse levels, followed by refinement until a desired accuracy for the eigenvalues has been achieved. The method employs local relaxation on all levels together with a global change on the coarsest level only, which is designed to separate the different eigenfunctions as well as to update their corresponding eigenvalues. Coarsening is done using the FAS formulation in a non-standard way in which the right hand side of the coarse grid equations involves unknown parameters to be solved for on the coarse grid. This in particular leads to a new multigrid method for calculating the eigenvalues of symmetric problems. Numerical experiments with a model problem demonstrate the effectiveness of the method proposed. Using an FMG algorithm a solution to the level of discretization errors is obtained in just a few work units (less than 10), where a work unit is the work involved in one Jacobi relization on the finest level.

Taasan, Shlomo↗
NASA NTRS1991

Photometric geodesy of main-belt asteroids. IV - An updated analysis of lightcurves for poles, periods, and shapes

The Drummond et al. (1988) analysis of main-belt asteroids is presently extended, using three independent methods to derive poles, periods, phase functions, and triaxial ellipsoid shapes from lightcurve maxima and minima. This group of 26 asteroids is also reinvestigated with a view to the distributions of triaxial shapes and obliquity distributions. Poles weakly tend to avoid asteroid orbital planes; a rough-smooth dichotomization appears to be justified by the persistence of two solar phase angle-amplitude relations. Seven of the objects may be Jacobi ellipsoids if axial ratios are slightly exaggerated by a systematic effect of the analytical method employed.

Drummond, J. D.↗
DOE OSTI2016

Measuring the diffraction properties of an imaging quartz(211) crystal

A dual goniometer X-ray system was used to measure the reflectivity curve for a spherically bent quartz(211) crystal. An analysis of the dual goniometer instrument response function for the rocking curve width measurement was developed and tested against the actual measurements. The rocking curve was measured at 4510.8 eV using the Ti Kα1 characteristic spectral line. The crystal is the dispersion element for a high resolution spectrometer used for plasma studies. It was expected to have a very narrow rocking curve width. The analysis showed that we could measure the upper bound for the rocking curve width of the Qz(211) crystal. The upper bound was 58 μrad giving a lower bound for the instrument resolving power E/ΔE = 34 000. Greatly improved insight into the dual goniometer operation and its limitations was achieved.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2021

Fermi-Löwdin-orbital self-interaction correction using the optimized-effective-potential method within the Krieger-Li-Iafrate approximation

Perdew-Zunger self-interaction correction (PZ-SIC) offers a route to remove self interaction errors on an orbital-by-orbital basis. A recent formulation of PZ-SIC by Pederson, Ruzsinszky and Perdew proposes restricting the unitary transformation to localized orbitals called Fermi-Löwdin orbitals. This formulation, called the FLOSIC method, simplifies PZ-SIC calculations and was implemented self-consistently using a Jacobi-like (FLOSIC-Jacobi) iteration scheme. In this work we implement the FLOSIC approach using the Krieger-Li-Iafrate (KLI) approximation to the optimized effective potential (OEP). We compare the results of present FLOSIC-KLI approach with FLOSIC-Jacobi scheme for atomic energies, atomization energies, ionization energies, barrier heights, polarizability of chains of hydrogen molecules etc. to validate the FLOSIC-KLI approach. The FLOSICKLI approach, which is within the realm of Kohn-Sham theory, predicts smaller energy gaps between frontier orbitals due to the lowering of eigenvalues of the lowest unoccupied orbitals. Results show that atomic energies, atomization energies, ionization energy as an absolute of highest occupied orbital eigenvalue, and polarizability of chains of hydrogen molecules between the two methods agree within 2%. Lastly the FLOSIC-KLI approach is used to determine the vertical ionization energies of water clusters.

74 ATOMIC AND MOLECULAR PHYSICS↗
NASA NTRS1982

A new numerical method for the simulation of three dimensional flow in a pipe

A new numerical technique for simulating three dimensional, unsteady, incompressible pipe flows is presented and its utility and accuracy is shown. Each vector function in the expansion of the velocity field is divergence free and satisfies the boundary conditions for viscous flow. Some of the benefits of the expansion technique are that pressure is eliminated from the dynamics, only two unknowns per mesh point are required, implicit treatment of the viscous terms is provided at no extra computational cost, and no fractional time steps are required. The method uses spectral expansions: Fourier series in the azimuthal and streamwise directions, and Jacobi polynominals in the radial direction.

Leonard, A.↗
NASA NTRS1982

A new numerical method for the simulation of three-dimensional flow in a pipe

A new numerical technique for simulating three dimensional, unsteady, incompressible pipe flows is presented and its utility and accuracy is shown. Each vector function in the expansion of the velocity field is divergence free and satisfies the boundary conditions for viscous flow. Some of the benefits of the expansion technique are that pressure is eliminated from the dynamics, only two unknowns per mesh point are required, implicit treatment of the viscous terms is provided at no extra computational cost, and no fractional time steps are required. The method uses spectral expansions: Fourier series in the azimuthal and streamwise directions, and Jacobi polynominals in the radial direction. Previously announced in STAR as N82-31644

Leonard, A.↗
NASA NTRS2002

Efficient High Order Central Schemes for Multi-Dimensional Hamilton-Jacobi Equations: Talk Slides

This viewgraph presentation presents information on the attempt to produce high-order, efficient, central methods that scale well to high dimension. The central philosophy is that the equations should evolve to the point where the data is smooth. This is accomplished by a cyclic pattern of reconstruction, evolution, and re-projection. One dimensional and two dimensional representational methods are detailed, as well.

Bryson, Steve↗
DOE OSTI2012

Modified semi-classical methods for nonlinear quantum oscillations problems

We develop a modified semi-classical approach to the approximate solution of Schroedinger's equation for certain nonlinear quantum oscillations problems. In our approach, at lowest order, the Hamilton-Jacobi equation of the conventional semi-classical formalism is replaced by an inverted-potential-vanishing-energy variant thereof. With suitable smoothness, convexity and coercivity properties imposed on its potential energy function, we prove, using methods drawn from the calculus of variations together with the (Banach space) implicit function theorem, the existence of a global, smooth 'fundamental solution' to this equation. Higher order quantum corrections thereto, for both ground and excited states, can then be computed through the integration of associated systems of linear transport equations, derived from Schroedinger's equation, and formal expansions for the corresponding energy eigenvalues obtained therefrom by imposing the natural demand for smoothness on the (successively computed) quantum corrections to the eigenfunctions. For the special case of linear oscillators our expansions naturally truncate, reproducing the well-known exact solutions for the energy eigenfunctions and eigenvalues. As an explicit application of our methods to computable nonlinear problems, we calculate a number of terms in the corresponding expansions for the one-dimensional anharmonic oscillators of quartic, sectic, octic, and dectic types and compare the results obtained with those of conventional Rayleigh/Schroedinger perturbation theory. To the orders considered (and, conjecturally, to all orders) our eigenvalue expansions agree with those of Rayleigh/Schroedinger theory whereas our wave functions more accurately capture the more-rapid-than-gaussian decay known to hold for the exact solutions to these problems. For the quartic oscillator in particular our results strongly suggest that both the ground state energy eigenvalue expansion and its associated wave function expansion are Borel summable to yield natural candidates for the actual exact ground state solution and its energy. Our techniques for proving the existence of the crucial 'fundamental solution' to the relevant (inverted-potential-vanishing-energy) Hamilton-Jacobi equation have the important property of admitting interesting infinite dimensional generalizations. In a project paralleling the present one we shall show how this basic construction can be carried out for the Yang-Mills equations in Minkowski spacetime.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2011

Exact solitary solution of Schamel equation in plasmas with negative ions

A theoretical investigation is carried out for understanding the properties of the solitary solution in plasmas with negative ions. Schamel equation for a plasma consisting of electron, positive ions and negative ions has been derived by using the reductive perturbation method. The effects of negative ions and the density on the properties of the solitary solution is discussed. We make use of the extended mapping method and auxiliary equation to obtain the solution of Schamel equation. This solution includes the Jacobi elliptic function solutions, hyperbolic function solutions, rational solutions, and periodic wave solutions. Furthermore, we show that the incorporate negative ion effects in the nonlinear propagation of ion acoustic waves that are controlled by trapped electrons and the results of the solitary solution in plasmas with negative ions model are discussed.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

Applying the Euclidean-signature semi-classical method to the quantum Taub models with a cosmological constant and aligned electromagnetic field

We prove the existence of a countably infinite number of “excited” states for the Lorentzian-signature Taub–Wheeler–DeWitt (WDW) equation when a cosmological constant is present using the Euclidean-signature semi-classical method. We also find a “ground” state solution when both an aligned electromagnetic field and cosmological constant are present; as a result, conjecture that the Euclidean-signature semi classical method can be used to prove the existence of a countably “infinite” number of “excited” states when the two aforementioned matter sources are present. Afterward, we prove the existence of asymptotic solutions to the vacuum Taub–WDW equation using the “no boundary” and “wormhole” solutions of the Taub Euclidean-signature Hamilton–Jacobi equation and compare their mathematical properties. We then discuss the fascinating qualitative properties of the wave functions we have computed. By utilizing the Euclidean-signature semi-classical method in the above manner, we further show its ability to prove the existence of solutions to Lorentzian-signature equations without having to invoke a Wick rotation. This feature of not needing to apply a Wick rotation makes this method potentially very useful for tackling a variety of problems in bosonic relativistic field theory and quantum gravity

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2017

Five-loop four-point integrand of N = 8 supergravity as a generalized double copy

Here, we use the recently developed generalized double-copy procedure to construct an integrand for the five-loop four-point amplitude of N = 8 supergravity. This construction starts from a naive double copy of the previously computed corresponding amplitude of N = 4 super-Yang-Mills theory. This is then systematically modified by adding contact terms generated in the context of the method of maximal unitarity cuts. For the simpler generalized cuts, whose corresponding contact terms tend to be the most complicated, we derive a set of formulas relating the contact contributions to the violations of the dual Jacobi identities in the relevant gauge-theory amplitudes. For more complex generalized unitarity cuts, which tend to have simpler contact terms associated with them, we use the method of maximal cuts more directly. The five-loop four-point integrand is a crucial ingredient towards future studies of ultraviolet properties of N = 8 supergravity at five loops and beyond. We further propose a nontrivial check of the consistency of the integrand, based on modern approaches for integrating over the loop momenta in the ultraviolet region.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
NASA NTRS1985

Towards developing robust algorithms for solving partial differential equations on MIMD machines

Methods for efficient computation of numerical algorithms on a wide variety of MIMD machines are proposed. These techniques reorganize the data dependency patterns to improve the processor utilization. The model problem finds the time-accurate solution to a parabolic partial differential equation discretized in space and implicitly marched forward in time. The algorithms are extensions of Jacobi and SOR. The extensions consist of iterating over a window of several timesteps, allowing efficient overlap of computation with communication. The methods increase the degree to which work can be performed while data are communicated between processors. The effect of the window size and of domain partitioning on the system performance is examined both by implementing the algorithm on a simulated multiprocessor system.

Saltz, J. H.↗
NASA NTRS1988

Towards developing robust algorithms for solving partial differential equations on MIMD machines

Methods for efficient computation of numerical algorithms on a wide variety of MIMD machines are proposed. These techniques reorganize the data dependency patterns to improve the processor utilization. The model problem finds the time-accurate solution to a parabolic partial differential equation discretized in space and implicitly marched forward in time. The algorithms are extensions of Jacobi and SOR. The extensions consist of iterating over a window of several timesteps, allowing efficient overlap of computation with communication. The methods increase the degree to which work can be performed while data are communicated between processors. The effect of the window size and of domain partitioning on the system performance is examined both by implementing the algorithm on a simulated multiprocessor system.

Saltz, Joel H.↗