SEARCH · Search NASA

Results for “eigenvalue problems”

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.

SourceResearch connectionFrom yearThrough year

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

At least 739 records · Page 41

All sources NASA NTRS DOE OSTI

NASA NTRS1973

Program for solution of ordinary differential equations

A program for the solution of linear and nonlinear first order ordinary differential equations is described and user instructions are included. The program contains a new integration algorithm for the solution of initial value problems which is particularly efficient for the solution of differential equations with a wide range of eigenvalues. The program in its present form handles up to ten state variables, but expansion to handle up to fifty state variables is being investigated.

Sloate, H.↗

NASA NTRS1989

Elimination of spurious eigenvalues in the Chebyshev tau spectral method

Spectral methods have been used to great advantage in hydrodynamic stability calculations; the concepts are described in Orszag's seminal application of the Chebyshev tau method to the Orr-Sommerfeld equation for plane Poiseuille flow in 1971. Orszag discusses both the Chebyshev Galerkin and the Chebyshev tau methods, but presents results for the tau method, which is easier to implement than the Galerkin method. The tau method has the disadvantage that two unstable eigenvalues are produced that are artifacts of the discretization. An extremely simple modification to the Chebyshev tau method is presented which eliminates the spurious eigenvalues. First a simplified model of the Orr-Sommerfeld equation discussed by Gottlieb and Orszag was studied. Then the Chebyshev tau method is considered, which has two spurious eigenvalues, and then a modification which eliminates them is described. Finally, results for the Orr-Sommerfeld equation are considered where the modified tau method also eliminates the spurious eigenvalues. The simplicity of the modification makes it a convenient alternative to other approaches to the problem.

Mcfadden, G. B.↗

NASA NTRS1991

A discourse on sensitivity analysis for discretely-modeled structures

A descriptive review is presented of the most recent methods for performing sensitivity analysis of the structural behavior of discretely-modeled systems. The methods are generally but not exclusively aimed at finite element modeled structures. Topics included are: selections of finite difference step sizes; special consideration for finite difference sensitivity of iteratively-solved response problems; first and second derivatives of static structural response; sensitivity of stresses; nonlinear static response sensitivity; eigenvalue and eigenvector sensitivities for both distinct and repeated eigenvalues; and sensitivity of transient response for both linear and nonlinear structural response.

Adelman, Howard M.↗

DOE OSTI2018

The generalized double shift-splitting preconditioner for nonsymmetric generalized saddle point problems from the steady Navier–Stokes equations

In this paper, a generalized double shift-splitting (GDSS) preconditioner induced by a newmatrix splitting method is proposed and implemented for nonsymmetric generalized saddle point problems having a nonsymmetric positive definite (1,1)-block and a positive definite (2,2)-block. Detailed theoretical analysis of the iteration matrix is provided to show the GDSS method, which corresponds to the GDSS preconditioner, is unconditionally convergent. Additionally,a deteriorated GDSS (DGDSS) method is proposed. It is shown that, with suitable choice of parameter matrix, the DGDSS preconditioned matrix has an eigenvalue at 1 with multiplicity n,and the other m eigenvalues are of the form1−λwith|λ| < 1,independently of the Schur complement matrix related. Finally, numerical experiments arising from a model Navier–Stokes problem are provided to validate and illustrate the effectiveness of the proposed preconditioner, with which a faster convergence for Krylov subspace iteration methods can be achieved.

42 ENGINEERING↗

NASA NTRS1982

Free vibration analysis of coupled fluid-structure systems

An efficient numerical technique for the eigenvalue solution in the free vibration analysis of compressible fluid-structure coupled systems is presented. The fluid is assumed to be compressible in nature and the incompressible problem is only a special case of the present generalized algorithm. A natural frequency analysis of the structure in the absence of any fluid is achieved by a combined Sturm sequence and inverse iteration technique that computes only the required eigenvalues and vectors. A special inverse iteration scheme is then developed for the coupled system that uses the computed eigenvalues as starting iteration values for convergence. Numerical results obtained by solving a number of standard test cases indicate the pattern of root convergence corresponding to various simplifying assumptions.

Gupta, K. K.↗

DOE OSTI2025

A Scalable Interior‐Point Gauss–Newton Method for PDE‐Constrained Optimization With Bound Constraints

Here, we present a scalable approach to solve a class of partial differential equation (PDE)‐constrained optimization problems with bound constraints. This approach utilizes a robust full‐space interior‐point (IP)‐Gauss–Newton optimization method. To cope with the poorly‐conditioned IP‐Gauss–Newton saddle‐point linear systems that need to be solved approximately, once per optimization step, we propose two spectrally related preconditioners. These preconditioners leverage the limited informativeness of data in regularized PDE‐constrained optimization problems. A block Gauss–Seidel preconditioner is proposed for the GMRES‐based solution of the IP‐Gauss–Newton linear systems. It is shown, for a large‐class of PDE‐ and bound‐constrained optimization problems, that the spectrum of the block Gauss–Seidel preconditioned IP‐Gauss–Newton matrix is asymptotically independent of discretization and is not impacted by the ill‐conditioning that notoriously plagues interior‐point methods. We exploit symmetry of the IP‐Gauss–Newton linear systems and propose a regularization and log‐barrier Hessian preconditioner for the preconditioned conjugate gradient (PCG)‐based solution of the equivalent IP‐Gauss–Newton–Schur complement linear systems. The eigenvalues of the block Gauss–Seidel preconditioned IP‐Gauss–Newton matrix, that are not equal to one, are identical to the eigenvalues of the regularization and log‐barrier Hessian preconditioned Schur complement matrix. The scalability of the approach is demonstrated on two example problems. The numerical solution of these optimization problems is shown to require a discretization independent number of IP‐Gauss–Newton linear solves. Furthermore, the linear systems are solved in a discretization and IP ill‐conditioning independent number of preconditioned Krylov subspace iterations. The parallel scalability of the preconditioner, achieved via algebraic multigrid component solvers when applicable, and the aforementioned algorithmic scalability permits a parallel scalable means to compute solutions of a large class of PDE‐ and bound‐constrained problems.

PDE-constrained optimization↗

DOE OSTI2017

Proposed Semi-Analytic Benchmark for Coupled Neutronics/Thermal-Hydraulics Feedback in Multiphysics Simulations

The past decade has seen considerable advancements in multiscale and multiphysics simulations that account for nonlinear effects of feedback between different physical processes within a single calculation. One area of particular interest has been multiphysics simulations that include both neutronics and thermal-hydraulics effects to accurately model the thermal feedback effects present in nuclear reactors. During this time, many groups have successfully modeled thermal feedback effects by coupling a variety of different types of neutron transport and thermal-hydraulic solvers together. However, formal verification and validation of these coupled code systems has proven challenging due to a lack of benchmark problems with well-characterized reference solutions. One approach is to confirm gross energy balance using bulk coolant properties (outlet temperature/enthalpy) for code verification purposes. Alternatively, code-to-code comparisons of results for a predefined reference/challenge problem have been used to establish reasonability of local results (neutron flux, heating, enthalpy rise) for various implementations, for example the VERA Core Physics Progression Problems. While community solutions are valuable for assessing the relative performance and fidelity of different implementations, they are not guaranteed to provide a measure of the absolute accuracy of a given implementation. In fact, these code-to-code comparisons can give a false sense of confidence by masking the effects of errors or approximations that are common to all of the solution methods tested. This is a particular concern when many methods derive from a common approach, such as Picard iteration. To this end, an analytical benchmark would be very valuable for verifying individual coupled physics implementations, as well as for providing a formal assessment of algorithm accuracy and sensitivity to model approximations such as mesh size and number of convergence iterations. However, creation of an analytical benchmark is difficult due to the complex nature of the eigenvalue form of the neutron transport equation (which does not permit a simple analytical solution for non-trivial cases) as well as the variety of different empirical closure relationships used in thermal-hydraulic modeling. In this paper we propose a benchmark for coupled neutronics/thermal-hydraulics simulations. The proposed benchmark overcomes the challenges associated with a multiphysics benchmark problem by using a fixed-source (rather than eigenvalue) formulation of the neutron transport equation and limiting fluid properties to the single-phase regime where the density and temperature of the coolant are related to enthalpy by a known relationship, while still preserving significant nuclear/thermal feedback effects within the simplified problem. The new benchmark is easy to simulate using existing coupled physics implementations, and allows for a direct comparison with a reference solution, as provided in this summary.

42 ENGINEERING↗

DOE OSTI2017

Big Data-Based Approach to Detect, Locate, and Enhance the Stability of an Unplanned Microgrid Islanding

In this paper, a big data-based approach is proposed for the security improvement of an unplanned microgrid islanding (UMI). The proposed approach contains two major steps: the first step is big data analysis of wide-area monitoring to detect a UMI and locate it; the second step is particle swarm optimization (PSO)-based stability enhancement for the UMI. First, an optimal synchrophasor measurement device selection (OSMDS) and matching pursuit decomposition (MPD)-based spatial-temporal analysis approach is proposed to significantly reduce the volume of data while keeping appropriate information from the synchrophasor measurements. Second, a random forest-based ensemble learning approach is trained to detect the UMI. When combined with grid topology, the UMI can be located. Then the stability problem of the UMI is formulated as an optimization problem and the PSO is used to find the optimal operational parameters of the UMI. An eigenvalue-based multiobjective function is proposed, which aims to improve the damping and dynamic characteristics of the UMI. Finally, the simulation results demonstrate the effectiveness and robustness of the proposed approach.

24 POWER TRANSMISSION AND DISTRIBUTION↗

NASA NTRS1980

Numerical methods for studying anharmonic oscillator approximations to the phi super 4 sub 2 quantum field theory

This paper is an expanded version of a talk given at the 1979 T.I.C.O.M. conference. It is a self-contained introduction, for applied mathematicians and numerical analysts, to quantum mechanics and quantum field theory. It also contains a brief description of the authors' numerical approach to the problems of quantum field theory, which may best be summarized by the question; Can we compute the eigenvalues and eigenfunctions of Schrodinger operators in infinitely many variables.

Isaacson, D.↗

NASA NTRS1986

Inclusion of elastically connected members in exact buckling and frequency calculations

A standard stiffness matrix procedure which permits any combination of rigid, elastic, pinned or sliding connections of the degrees of freedom at the ends of a member to the nodes of its parent structure is described, in order to show how easily it can be extended to allow an existing algorithm to be used to ensure that no eigenvalues of the parent structure can be missed even when 'exact' member theory is used. The eigenvalues are the natural frequencies of undamped free vibration analyses or the critical load factors of buckling problems. The method preserves the exactness of the member theory and an efficient method for computer application is indicated. The theory also permits any combination of rigid, elastic, pinned or sliding connections between the freedoms of a substructure and those of its parent structure.

Williams, F. W.↗

DOE OSTI2024

Enhancing scalability and accuracy of quantum poisson solver

The Poisson equation has many applications across the broad areas of science and engineering. Most quantum algorithms for the Poisson solver presented so far either suffer from lack of accuracy and/or are limited to very small sizes of the problem and thus have no practical usage. In this regard, our previous work showed a proof-of-concept demonstration in advancing quantum Poisson solver algorithm and validated preliminary results for a simple case of 3 x 3 problem. In this work, we delve into comprehensive research details, presenting the results on up to 15 x 15 problems that include step-by-step improvements in Poisson equation solutions, scaling performance, and experimental exploration. In particular, we demonstrate the implementation of eigenvalue amplification by a factor of up to 2 8 , achieving a significant improvement in the accuracy of our quantum Poisson solver and comparing that to the exact solution. Additionally, we present success probability results, highlighting the reliability of our quantum Poisson solver. Moreover, we explore the scaling performance of our algorithm against the circuit depth and width, demonstrating how our approach scales with larger problem sizes and thus further solidifies the practicality of easy adaptation of this algorithm in real-world applications. We also discuss a multilevel strategy for how this algorithm might be further improved to explore much larger problems with greater performance. Finally, through our experiments on the IBM quantum hardware, we conclude that though overall results on the existing NISQ hardware are dominated by the error in the CNOT gates, this work opens a path to realizing a multidimensional Poisson solver on near-term quantum hardware.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1985

Multigrid method for nearly singular and slightly indefinite problems

This paper deals with nearly singular, possibly indefinite problems for which the usual multigrid solvers converge very slowly or even diverge. The main difficulty is related to some badly approximated smooth functions which correspond to eigenfunctions with nearly zero eigenvalues. A correction to the usual coarse-grid equations is derived, both in the correction scheme and in the full approximation scheme. The performance of the new algorithm using this correction is essentially as that of usual multigrid for definite problems.

Brandt, A.↗

NASA NTRS1986

Multigrid method for nearly singular and slightly indefinite problems

Brandt, A.↗

DOE OSTI2018

A low-order block preconditioner for saddle point linear systems

A preconditioner is proposed for the large and sparse linear saddle point problems, which is based on a low-order three-by-three block saddle point form. The eigenvalue distribution and an upper bound of the degree of the minimal polynomial for the preconditioned matrix are discussed. Numerical results show that the optimal convergence behavior can be achieved when the new preconditioner is used to accelerate the convergence rate of Krylov subspace methods such as GMRES.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2019

Rationally extended many-body truncated Calogero–Sutherland model

We construct a rational extension of the truncated Calogero–Sutherland model by Pittman et al. The exact solution of this rationally extended model is obtained analytically and it is shown that while the energy eigenvalues remain unchanged, however the eigenfunctions are completely different and written in terms of exceptional X{sub 1} Laguerre orthogonal polynomials. The rational model is further extended to a more general X{sub m} case by introducing m dependent interaction term. As expected, in the special case of m=0, the extended model reduces to the conventional model of Pittman et al. In the two appropriate limits, we thereby obtain rational extensions of the celebrated Calogero–Sutherland as well as Jain–Khare models. The multi-index extension of the model is also discussed.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2025

Ab-initio simulation of spin-vibronic spectra of methoxy radical

Despite the fact that experimental and theoretical work on the spectrum of methoxy has stretched from the microwave to the ultraviolet and proceeded for nearly 50 years, parts of the spectrum have remained a challenge to simulate theoretically and make reliable line-by-line assignments. The spectral complexity arises because the radical has a non-zero electron spin and significant vibronic coupling between the two elec- tronic components of the ground state due to the presence of a conical intersection. This work describes a completely ab initio effort to understand and assign the spin- vibronic levels of the X 2E state from 0 to above 3000 cm−1, a region that includes the fundamental transitions of the C-H symmetric and asymmetric stretches that have not previously been identified uniquely. A potential energy surface for methoxy was calculated at the EOM-CCSDT/ANO1 level of theory. Subsequently this potential energy surface was fit to a quartic power series expansion of all nine vibrational nor- mal coordinates (as determined at the minimum of the conical intersection) by the use of a machine-learning-based algorithm. After the addition of spin-orbit coupling, the spin-vibronic problem was solved using both the Krylov-Schur and Lanczos algorithms with the SOCJT3 software to converge eigenvalues up to 3500 cm−1 and their eigen- vectors. The latter were used, in conjunction with the calculated dipole moment and its derivatives (calculated using finite differences at EOM-CCSDT/ANO1 level), to determine spectral intensities for the spin-vibronic spectra. The calculated transition frequencies and intensities were used to simulate and assign the observed transitions of the spin-vibronic spectra of the radical. The credibility of the assignments and their significance is discussed in detail.

Sharma, Ketan [University of Florida, Gainesville,↗

NASA NTRS1978

On the append and continue features in NASTRAN

The APPEND feature is described which is applicable in the case of real eigenvalue analysis. This feature permitted the addition of new eigenvalues and eigenvectors to those already computed in a previously checkpointed run without re-executing the entire problem. The next feature was the CONTINUE feature which was applicable in the case of transient analysis of coupled equations. This feature enabled the integration of coupled equations to be continued beyond the last output time for which the solution was obtained in a previously checkpointed run (without re-executing the entire problem). These two features are illustrated by suitable examples.

Pamidi, P. R.↗

NASA NTRS1978

Kalman filter divergence and aircraft motion estimators

Kalman filters designed for many aerospace systems turn out to be unsatisfactory. The estimate errors become large compared to the errors predicted by the theory ('divergence'). One of the principal causes of this failure is that the system model contains states or modes that are undisturbed by the modeled process noise, and are neutrally stable (NS). One cure for such problems is periodic restarting of a time-varying Kalman filter. Other cures include minimum variance observers with eigenvalue constraints, added noise, pole-shifting, and destabilization. Several examples are given, including effective time-invariant estimators for the longitudinal and lateral motions of an airplane where several NS modes are undisturbed by wind gusts. An interpretation of these estimators as a 'strapdown IMU' without accelerometers, gimbaled gyros, or servos is given.

Bryson, A. E., Jr.↗