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At least 631 records · Page 35
All sourcesNASA NTRSDOE OSTI
NASA NTRS1949

Response of a Rotating Propeller to Aerodynamic Excitation

The flexural vibration of a rotating propeller blade with clamped shank is analyzed with the object of presenting, in matrix form, equations for the elastic bending moments in forced vibration resulting from aerodynamic forces applied at a fixed multiple of rotational speed. Matrix equations are also derived which define the critical speeds end mode shapes for any excitation order and the relation between critical speed and blade angle. Reference is given to standard works on the numerical solution of matrix equations of the forms derived. The use of a segmented blade as an approximation to a continuous blade provides a simple means for obtaining the matrix solution from the integral equation of equilibrium, so that, in the numerical application of the method presented, the several matrix arrays of the basic physical characteristics of the propeller blade are of simple form, end their simplicity is preserved until, with the solution in sight, numerical manipulations well-known in matrix algebra yield the desired critical speeds and mode shapes frame which the vibration at any operating condition may be synthesized. A close correspondence between the familiar Stodola method and the matrix method is pointed out, indicating that any features of novelty are characteristic not of the analytical procedure but only of the abbreviation, condensation, and efficient organization of the numerical procedure made possible by the use of classical matrix theory.

Arnoldi, Walter E.↗
DOE OSTI2023

Sparse adaptive basis set methods for solution of the time dependent Schrodinger equation

Scalable numerical solutions to the time dependent Schrodinger equation remain an outstanding goal in theoretical chemistry. Here we present a method which utilises recent breakthroughs in signal processing to consistently adapt a dictionary of basis functions to the dynamics of the system. Further, we show that for two low-dimensional model problems the size of the basis set does not grow quickly with time and appears only weakly dependent on dimensionality. The generality of this finding remains to be seen. The method primarily uses energies and gradients of the potential, opening the possibility for its use in on-the-fly ab initio quantum wavepacket dynamics.

74 ATOMIC AND MOLECULAR PHYSICS↗
NASA NTRS1982

Analysis of a finite difference grid

Some means of assessing the suitability of a mesh network for a finite difference calculation are investigated in this study. This has been done by a study of the nonlinear truncation errors of the scheme. It turns out that the mesh can not be properly assessed a priori. The effect of the mesh on the numerical solution depends on several factors including the mesh itself, the numerical algorithm, and the solution. Several recommendations are made with regard to generating the mesh and to assessing its suitability for a particular numerical calculation.

Klopfer, G. H.↗
DOE OSTI2020

High Order Implicit Residual-Based Spatial Discretization Error Estimation for S N Neutron Transport

This work demonstrates our novel residual source spatial discretization error estimator (LeR/TEAD) for a DGFEM-1 discretization and assesses it along with two contemporary estimators, Ragusa and Wang's h -refinement estimator (RW) and Duo, Azmy, and Zikatanov's explicit residual-based estimator (DAZ), on a suite of Method of Manufactured Solutions (MMS) 2D problems and three realistic problem geometries. LeR/TE-AD is attractive because it directly estimates the local error in the angular flux, as opposed to a mere indicator of the error's behavior, on the same mesh and method order as the original numerical solution, thus typically being less computationally intensive than a refinement-based method. On the MMS suite, LeR/TE-AD consistently displayed a reduced performance versus its DGFEM-0 results in terms of accuracy and precision metrics, though it was not typically grossly inaccurate. This is attributed to the irregularities in the true solution across singular characteristics limiting the local accuracy of the numerical flux solution, leading to poor derivative approximations used in the residual approximations. The error transport problem then spreads the error in the residual to nearby cells, causing a greater degree of imprecision that did not afflict DAZ or RW. In testing the estimators on realistic problem geometries, however, LeR/TE-AD fared better. In practice, the true error is much larger in non-idealized geometries like in MMS, and a superlinear true solution means that RW and DAZ are not beneficially biased for DGFEM-1 error estimation. LeR/TE-AD was typically first or second in accuracy, primarily competing with RW, but the latter usually consumed 2-4 times the computational time as LeR/TE-AD, and requires a solution with four times as many unknowns. Furthermore, RW and LeR/TE-AD can be used to compute direct estimates of the error in any quantity of interest that is based on the angular ux solution, such as the fission rate density in a fuel pin, whereas DAZ requires a heuristic extension due to its norm-based nature.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

An adaptive Hessian approximated stochastic gradient MCMC method

Bayesian approaches have been successfully integrated into training deep neural networks. One popular family is stochastic gradient Markov chain Monte Carlo methods (SG-MCMC), which have gained increasing interest due to their ability to handle large datasets and the potential to avoid overfitting. Although standard SG-MCMC methods have shown great performance in a variety of problems, they may be inefficient when the random variables in the target posterior densities have scale differences or are highly correlated. Here, we present an adaptive Hessian approximated stochastic gradient MCMC method to incorporate local geometric information while sampling from the posterior. The idea is to apply stochastic approximation (SA) to sequentially update a preconditioning matrix at each iteration. The preconditioner possesses second-order information and can guide the random walk of a sampler efficiently. Instead of computing and saving the full Hessian of the log posterior, we use limited memory of the samples and their stochastic gradients to approximate the inverse Hessian-vector multiplication in the updating formula. Moreover, by smoothly optimizing the preconditioning matrix via SA, our proposed algorithm can asymptotically converge to the target distribution with a controllable bias under mild conditions. To reduce the training and testing computational burden, we adopt a magnitude-based weight pruning method to enforce the sparsity of the network. Our method is user-friendly and demonstrates better learning results compared to standard SG-MCMC updating rules. The approximation of inverse Hessian alleviates storage and computational complexities for large dimensional models. Numerical experiments are performed on several problems, including sampling from 2D correlated distribution, synthetic regression problems, and learning the numerical solutions of heterogeneous elliptic PDE. The numerical results demonstrate great improvement in both the convergence rate and accuracy.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Average-atom model with Siegert states

In plasmas, electronic states can be well-localized bound states or itinerant free states, or something in between. In self-consistent treatments of plasma electronic structure such as the average-atom model, all states must be accurately resolved in order to achieve a converged numerical solution. Furthermore, this is a challenging numerical and algorithmic problem in large part due to the continuum of free states which is relatively expensive and difficult to resolve accurately. Siegert states are an appealing alternative. They form a complete eigenbasis with a purely discrete spectrum while still being equivalent to a representation in terms of the usual bound states and free states. However, many of their properties are unintuitive, and it is not obvious that they are suitable for self-consistent plasma electronic structure calculations. Here it is demonstrated that Siegert states can be used to accurately solve an average-atom model and offer advantages over the traditional finite-difference approach, including a concrete physical picture of pressure ionization and continuum resonances.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS1977

Numerical analysis of free vibrations of damped rotating structures

This paper is concerned with the efficient numerical solution of damped and undamped free vibration problems of rotating structures. While structural discretization is achieved by the finite element method, the associated eigenproblem solution is effected by a combined Sturm sequence and inverse iteration technique that enables the computation of a few required roots only without having to compute any other. For structures of complex configurations, a modal synthesis technique is also presented, which is based on appropriate combinations of eigenproblem solution of various structural components. Such numerical procedures are general in nature, which fully exploit matrix sparsity inherent in finite element discretizations, and prove to be most efficient for the vibration analysis of any damped rotating structure, such as rotating machineries, helicopter and turbine blades, spinning space stations, among others.

Gupta, K. K.↗
NASA NTRS1985

A multiple-scales model of the shock-cell structure of imperfectly expanded supersonic jets

The present investigation is concerned with the development of an analytical model of the quasi-periodic shock-cell structure of an imperfectly expanded supersonic jet. The investigation represents a part of a program to develop a mathematical theory of broadband shock-associated noise of supersonic jets. Tam and Tanna (1982) have suggested that this type of noise is generated by the weak interaction between the quasi-periodic shock cells and the downstream-propagating large turbulence structures in the mixing layer of the jet. In the model developed in this paper, the effect of turbulence in the mixing layer of the jet is simulated by the addition of turbulent eddy-viscosity terms to the momentum equation. Attention is given to the mean-flow profile and the numerical solution, and a comparison of the numerical results with experimental data.

Tam, C. K. W.↗
NASA NTRS1982

Comparison of starting values for iterative solutions to a universal Kepler's equation

General starting values for the iterative numerical solution of a universal Kepler's equation for position in a conic orbit at a specified time are investigated. Three starting values based on recent refinements of previously obtained bounds on the solution are derived and tested numerically. Of these, a simple starting value based on a cubic approximation to Kepler's equation provides the most rapid convergence using both first and second order Newton algorithms. The performance of the starting values are compared with similar studies which used the restricted case of elliptical orbits with the initial epoch at periapse.

Bergam, M. J.↗
NASA NTRS1974

Solutions for laminar boundary layers with separation and reattachment

Numerical solutions of the laminar, incompressible boundary-layer equations are presented for flows involving separation and reattachment. Regular solutions are obtained with an inverse approach in which either the displacement thickness or the skin friction is specified, and the pressure is deduced from the solution. A stream function/vorticity formulation of the boundary-layer equations is used to eliminate the unknown pressure. Solutions of the resulting finite-difference equations, in which the flow direction is taken into account, are obtained by a successive column iteration scheme. Results are compared with Klineberg and Steger's (1974) separated boundary-layer calculations, and with Briley's (1971) solution of the Navier-Stokes equations for a separated region.

Carter, J. E.↗
NASA NTRS1986

Turbulent solutions of the Navier-Stokes equations

Analytical and numerical approaches to the mechanics of turbulent flow are examined in a general introduction and illustrated with graphs. Topics discussed include the averaged and unaveraged basic equations, numerical solutions and methods, homogeneous fluctuations and turbulence with no mean flow, uniformly sheared fluctuations and turbulence, inhomogeneous fluctuations and turbulence in a developing shear layer, and steady-state homogeneous turbulence with a spatially periodic body force.

Deissler, R. G.↗
NASA NTRS1997

Reentry-F Flowfield Solutions at 80,000 ft.

Three equilibrium-air numerical solutions are presented for the Reentry-F flight-test vehicle at Mach 20, 80,000 Ft. conditions, including turbulent flow predictions. The three solutions are from a thin-layer Navier-Stokes code, coupled thin-layer and parabolized Navier-Stokes codes, and an approximate viscous shock-layer code. Boundary-layer and shock-layer profiles are presented and compared between the three solutions, revealing close agreement between the three solution methods. Notable exceptions to the close agreement, with 7-10 percent discrepancies, occur in the density profiles at the boundary-layer edge, in the boundary-layer velocity profiles, and in the shock-layer profiles in regions influenced by the nose bluntness.

Wood, William A.↗
NASA NTRS1995

Solutions of the benchmark problems by the dispersion-relation-preserving scheme

The 7-point stencil Dispersion-Relation-Preserving scheme of Tam and Webb is used to solve all the six categories of the CAA benchmark problems. The purpose is to show that the scheme is capable of solving linear, as well as nonlinear aeroacoustics problems accurately. Nonlinearities, inevitably, lead to the generation of spurious short wave length numerical waves. Often, these spurious waves would overwhelm the entire numerical solution. In this work, the spurious waves are removed by the addition of artificial selective damping terms to the discretized equations. Category 3 problems are for testing radiation and outflow boundary conditions. In solving these problems, the radiation and outflow boundary conditions of Tam and Webb are used. These conditions are derived from the asymptotic solutions of the linearized Euler equations. Category 4 problems involved solid walls. Here, the wall boundary conditions for high-order schemes of Tam and Dong are employed. These conditions require the use of one ghost value per boundary point per physical boundary condition. In the second problem of this category, the governing equations, when written in cylindrical coordinates, are singular along the axis of the radial coordinate. The proper boundary conditions at the axis are derived by applying the limiting process of r approaches 0 to the governing equations. The Category 5 problem deals with the numerical noise issue. In the present approach, the time-independent mean flow solution is computed first. Once the residual drops to the machine noise level, the incident sound wave is turned on gradually. The solution is marched in time until a time-periodic state is reached. No exact solution is known for the Category 6 problem. Because of this, the problem is formulated in two totally different ways, first as a scattering problem then as a direct simulation problem. There is good agreement between the two numerical solutions. This offers confidence in the computed results. Both formulations are solved as initial value problems. As such, no Kutta condition is required at the trailing edge of the airfoil.

Tam, Christopher K. W.↗
NASA NTRS2013

Separation-Compliant, Optimal Routing and Control of Scheduled Arrivals in a Terminal Airspace

We address the problem of navigating a set (fleet) of aircraft in an aerial route network so as to bring each aircraft to its destination at a specified time and with minimal distance separation assured between all aircraft at all times. The speed range, initial position, required destination, and required time of arrival at destination for each aircraft are assumed provided. Each aircraft's movement is governed by a controlled differential equation (state equation). The problem consists in choosing for each aircraft a path in the route network and a control strategy so as to meet the constraints and reach the destination at the required time. The main contribution of the paper is a model that allows to recast this problem as a decoupled collection of problems in classical optimal control and is easily generalized to the case when inertia cannot be neglected. Some qualitative insight into solution behavior is obtained using the Pontryagin Maximum Principle. Sample numerical solutions are computed using a numerical optimal control solver. The proposed model is first step toward increasing the fidelity of continuous time control models of air traffic in a terminal airspace. The Pontryagin Maximum Principle implies the polygonal shape of those portions of the state trajectories away from those states in which one or more aircraft pair are at minimal separation. The model also confirms the intuition that, the narrower the allowed speed ranges of the aircraft, the smaller the space of optimal solutions, and that an instance of the optimal control problem may not have a solution at all (i.e., no control strategy that meets the separation requirement and other constraints).

transportation network↗