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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.

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At least 289 records · Page 16
All sourcesNASA NTRSDOE OSTI
NASA NTRS1980

The finite element duct eigenvalue problem - An improved formulation with Hermitian elements and no-flow condensation

Hermitian elements are used in a finite element solution for the eigenvalue problem in lined ducts with flow. These elements give significantly greater accuracy for reduced dimensionality when compared with Lagrangian elements. Spurious mode generation associated with the Lagrangian formulation is eliminated. A dramatic improvement in the ratio of the number of reliable eigenvalues to the total number of computed eigenvalues is effected by the use of a condensation scheme based on the no-flow eigenvectors. Results are presented for two dimensional and axisymmetric ducts. In the axisymmetric case good resolution is obtained even for high order, high frequency modes by the use of continuously graded meshes.

Astley, R. J.↗
NASA NTRS1994

Multigrid techniques for nonlinear eigenvalue probems: Solutions of a nonlinear Schroedinger eigenvalue problem in 2D and 3D

This paper presents multigrid (MG) techniques for nonlinear eigenvalue problems (EP) and emphasizes an MG algorithm for a nonlinear Schrodinger EP. The algorithm overcomes the mentioned difficulties combining the following techniques: an MG projection coupled with backrotations for separation of solutions and treatment of difficulties related to clusters of close and equal eigenvalues; MG subspace continuation techniques for treatment of the nonlinearity; an MG simultaneous treatment of the eigenvectors at the same time with the nonlinearity and with the global constraints. The simultaneous MG techniques reduce the large number of self consistent iterations to only a few or one MG simultaneous iteration and keep the solutions in a right neighborhood where the algorithm converges fast.

Costiner, Sorin↗
NASA NTRS1992

A subspace iteration for eigenvector derivatives

An iterative procedure for calculating eigenvector derivatives has been developed based on the subspace iteration concept. The basic formulation is derived directly from the first variation of the resulting equations of subspace iteration for solving eigenvalue problems. Since the basic formulation conserves the convergence properties of the original subspace iteration for eigen-problems, an overrelaxation scheme similar to Bathe's approach is employed to accelerate the subspace iteration process for calculating eigenvector derivatives.

Ting, T.↗
NASA NTRS1992

On a finite-state inflow application to flap-lag-torsion damping in hover

An aerodynamic model with a coupled set of generalized dynamic wake equations and hybrid equations of motion for an elastic blade are applied here to a two-blade untwisted stiff in-plane hingeless small-scale model rotor with torsionally soft blades. Blade root offset, precone, blade droop, pitch control stiffness, and blade pitch angle are included in the model rotor. Numerical results show that 3D tip relief effects within the nonuniform steady-state inflow are significant to predict steady-state aerodynamic loads and blade deflections. Eigenvalue results confirm the importance of unsteady 3D aerodynamics in predicting lead-lag damping and frequency. Eigenvector analysis correlations reinforced qualitative and quantitative shortcomings associated with quasi-steady 2D aerodynamic theory for aeroelastic applications in hover.

De Andrade, Donizeti↗
NASA NTRS1990

An efficient design sensitivity analysis of eigenvectors

Subspace iteration has been a major advance in solving large eigen problems when only a subset of eigen-pairs is required. The essence of this method is a transformation from displacement coordinates of an n-th order eigensystem to generalized coordinates of a smaller q-th order. The eigenvalue problem is then solved in the reduced space. The method was first developed by Clint and Jennings for real symmetric systems and was then called 'simultaneous iteration'. The success of the method prompted further research along this line and there have been many improved algorithms developed. This approach has been widely used by structural engineers for extracting the most useful natural frequencies and mode shapes of large-scale dynamic systems. This paper exploits into a new direction which is in the form of iterative process for simultaneously calculating eigenvector derivatives of many eigenvectors with respect to multi-variables. The method fully uses all the available information from preceding eigenvalue solution and, thus, effectively economizes computational efforts. It iterates through two equations derived from the first variation of the two fundamental equations used in subspace iteration method. There is no expensive large matrix decomposition required and the process converges to acceptable solution in a finite number of iterations. Therefore, the procedure increases its efficiency superiority over the others as the system size or the number of interested eigenvectors become larger and larger.

Ting, T.↗
NASA NTRS1990

Dynamic substructuring by the boundary flexibility vector method of component mode synthesis

Component mode synthesis (CMS) is a method of dynamic analysis, for structures having a large number of degrees of freedom (DOF). These structures often required lengthy computer CPU time and large computer memory resources, if solved directly by the finite-element method (FEM). In CMS, the structure is divided into independent components in which the DOF are defined by a set of generalized coordinates defined by displacement shapes. The number of the generalized coordinates are much less than the original number of physical DOF, in the component. The displacement shapes are used to transform the component property matrices and any applied external loads, to a reduced system of coordinates. Reduced system property matrices are assembled, and any type of dynamic analysis is carried out in the reduced coordinate system. Any obtained results are back transformed to the original component coordinate systems. In all conventional methods of CMS, the mode shapes used for components are dynamic mode shapes, supplemented by static deflected shapes. Historically, all the dynamic mode shapes used in conventional CMS are the natural modes (eigenvectors) of components. A new method of CMS, namely the boundary flexibility vector method of CMS, is presented. The method provides for the incorporation of a set of static Ritz vectors, referred to as boundary flexibility vectors, as a replacement and/or supplement to conventional eigenvectors, as displacement shapes for components. The generation of these vectors does not require the solution of a costly eigenvalue problem, as in the case of natural modes in conventional CMS, and hence a substantial saving in CPU time can be achieved. The boundary flexibility vectors are generated from flexibility (or stiffness) properties of components. The formulation presented is for both free and fixed-interface components, and for both the free and forced vibration problems. Free and forced vibration numerical examples are presented to verify the accuracy of the method and the saving in CPU time. Compared to conventional methods of CMS, the results indicate that by using the new method, more accurate results can be obtained with a substantial saving in CPU time.

Abdallah, Ayman Ahmed↗
DOE OSTI2021

Rodeo Algorithm for Quantum Computing

We present a stochastic quantum computing algorithm that can prepare any eigenvector of a quantum Hamiltonian within a selected energy interval $\ [E-\epsilon, E+\epsilon]$. In order to reduce the spectral weight of all other eigenvectors by a suppression factor δ, the required computational effort scales as $\ O[|\log \delta|/(p \epsilon)]$, where p s the squared overlap of the initial state with the target eigenvector. The method, which we call the rodeo algorithm, uses auxiliary qubits to control the time evolution of the Hamiltonian minus some tunable parameter E . In this manner, we converge to the target eigenvector with exponential accuracy in the number of measurements. In addition to preparing eigenvectors, the method can also compute the full spectrum of the Hamiltonian. We illustrate the performance with several examples. For energy eigenvalue determination with error $\epsilon$, the computational scaling is $\ O[(\log \epsilon)^2/(p \epsilon)]$. For eigenstate preparation, the computational scaling is $\ O(\log \Delta/p)$, where $\Delta$ is the magnitude of the orthogonal component of the residual vector. The speed for eigenstate preparation is exponentially faster than that for phase estimation or adiabatic evolution.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1989

On eigensystem assignment with output feedback

A new approach for the eigenvalue assignment of linear, first-order, time-invariant systems using output feedback is developed. The approach can assign the maximum allowable number of closed-loop eigenvalues through output feedback provided that the system is fully controllable and observable, and both the input influence and output influence matrices are full rank. First, a collection of bases for the space of attainable closed-loop eigenvectors is generated using the Singular Value Decomposition or QR Decomposition techniques. Then, an algorithm based on subspace intersections is developed and used to compute the corresponding coefficients of the bases, and the required output feedback gain matrix. Moreover, the additional freedom provided by the multi-inputs and multi-outputs beyond the eigenvalue assignment is characterized for possible exploitation. A numerical example is given to demonstrate the viability of the proposed approach.

Maghami, Peiman G.↗
NASA NTRS1989

Sequential design of discrete linear quadratic regulators via optimal root-locus techniques

A sequential method employing classical root-locus techniques has been developed in order to determine the quadratic weighting matrices and discrete linear quadratic regulators of multivariable control systems. At each recursive step, an intermediate unity rank state-weighting matrix that contains some invariant eigenvectors of that open-loop matrix is assigned, and an intermediate characteristic equation of the closed-loop system containing the invariant eigenvalues is created.

Shieh, Leang S.↗
NASA NTRS1974

Development of technology for fluid-structure interaction modeling of a 1/8-scale dynamic model of shuttle External Tank (ET). Volume i: Technical report

A NASTRAN analysis of the external tank (ET) substructure of the 1/8-scale space shuttle structural dynamics model is described. The NASTRAN hydroelastic procedures were used to form a model of the liquid oxygen portion of the ET. Large computer storage requirements and running times were required unless these procedures were modified. Several possibilities were demonstrated including the substitution of the real for the complex eigenvalue routine and the use of the OMIT capability to reduce the number of fluid coordinates. A NASTRAN model of the complete ET was formed and reduced to 252 degrees of freedom. A review of the eigenvector extracted, using the unsymmetrical inverse power method, indicated that the structural OMIT's resulted in unsatisfactory modal deflections. Modifications to the NASTRAN hydroelastic capability appeared necessary to generate a successful ET model. An approach is described which, by assuming incompressibility, reduces the fluid representation to a symmetric mass matrix which can be added to the structural mass. The problem is solved using faster and more efficient eigenvalue routines. Using this approach the ET NASTRAN model was analyzed for three separate weight conditions.

Bernstein, M.↗
NASA NTRS1995

Separation analysis, a tool for analyzing multigrid algorithms

The separation of vectors by multigrid (MG) algorithms is applied to the study of convergence and to the prediction of the performance of MG algorithms. The separation operator for a two level cycle algorithm is derived. It is used to analyze the efficiency of the cycle when mixing of eigenvectors occurs. In particular cases the separation analysis reduces to Fourier type analysis. The separation operator of a two level cycle for a Schridubger eigenvalue problem, is derived and analyzed in a Fourier basis. Separation analysis gives information on how to choose performance relaxations and inter-level transfers. Separation analysis is a tool for analyzing and designing algorithms, and for optimizing their performance.

Costiner, Sorin↗
NASA NTRS1988

Multiple boundary condition test (MBCT) - Identification with mode shapes

The multiple boundary condition test (MBCT) approach is a ground test method to test a class of large flexible structures which cannot be ground tested by state-of-the-art test methods due to the adverse terrestrial environment. The ultimate objective of a ground test is considered to be the validation and update of a mathematical model of the structure. The research to date has indicated the MBCT does work on numerical simulations and on experimental laboratory hardware. To date only the eigenvalue has been used in the model correlation/update by inclusion of the information in the nonlinear terms resulting from the difference between the analytical and measured eigenvectors. This paper presents the results of utilizing additional information, namely the difference in the analytical and the test eigenvectors, in the validation and update of the mathematical model.

Kuo, C. P.↗
NASA NTRS1994

The Topology of Three-Dimensional Symmetric Tensor Fields

We study the topology of 3-D symmetric tensor fields. The goal is to represent their complex structure by a simple set of carefully chosen points and lines analogous to vector field topology. The basic constituents of tensor topology are the degenerate points, or points where eigenvalues are equal to each other. First, we introduce a new method for locating 3-D degenerate points. We then extract the topological skeletons of the eigenvector fields and use them for a compact, comprehensive description of the tensor field. Finally, we demonstrate the use of tensor field topology for the interpretation of the two-force Boussinesq problem.

Lavin, Yingmei↗
NASA NTRS1978

Eigenvalue/eigenvector assignment using output feedback

The problem of pole-assignment in a linear time-invariant multivariable system using output feedback is considered. New sufficient conditions are derived to assign an almost arbitrary set of min (n, m+r-1) distinct eigenvalues, where n, m, and r are the number of states, inputs, and outputs, respectively. The analysis also highlights the freedom in selection of closed-loop eigenvectors which can be used for response shaping.

Srinathkumar, S.↗
NASA NTRS1975

Modal coupling procedures adapted to NASTRAN analysis of the 1/8-scale shuttle structural dynamics model. Volume 1: Technical report

A dynamic substructuring analysis, utilizing the component modes technique, of the 1/8 scale space shuttle orbiter finite element model is presented. The analysis was accomplished in 3 phases, using NASTRAN RIGID FORMAT 3, with appropriate Alters, on the IBM 360-370. The orbiter was divided into 5 substructures, each of which was reduced to interface degrees of freedom and generalized normal modes. The reduced substructures were coupled to yield the first 23 symmetric free-free orbiter modes, and the eigenvectors in the original grid point degree of freedom lineup were recovered. A comparison was made with an analysis which was performed with the same model using the direct coordinate elimination approach. Eigenvalues were extracted using the inverse power method.

Zalesak, J.↗
DOE OSTI2024

Solving a class of infinite-dimensional tensor eigenvalue problems by translational invariant tensor ring approximations

Here, we examine a method for solving an infinite-dimensional tensor eigenvalue problem Hx = λx, where the infinite-dimensional symmetric matrix H exhibits a translational invariant structure. We provide a formulation of this type of problem from a numerical linear algebra point of view and describe how a power method applied to e -Ht is used to obtain an approximation to the desired eigenvector. This infinite-dimensional eigenvector is represented in a compact way by a translational invariant infinite Tensor Ring (iTR). Low rank approximation is used to keep the cost of subsequent power iterations bounded while preserving the iTR structure of the approximate eigenvector. We show how the averaged Rayleigh quotient of an iTR eigenvector approximation can be efficiently computed and introduce a projected residual to monitor its convergence. In the numerical examples, we illustrate that the norm of this projected iTR residual can also be used to automatically modify the time step to ensure accurate and rapid convergence of the power method.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2000

Low-dimensional Representation of Error Covariance

Ensemble and reduced-rank approaches to prediction and assimilation rely on low-dimensional approximations of the estimation error covariances. Here stability properties of the forecast/analysis cycle for linear, time-independent systems are used to identify factors that cause the steady-state analysis error covariance to admit a low-dimensional representation. A useful measure of forecast/analysis cycle stability is the bound matrix, a function of the dynamics, observation operator and assimilation method. Upper and lower estimates for the steady-state analysis error covariance matrix eigenvalues are derived from the bound matrix. The estimates generalize to time-dependent systems. If much of the steady-state analysis error variance is due to a few dominant modes, the leading eigenvectors of the bound matrix approximate those of the steady-state analysis error covariance matrix. The analytical results are illustrated in two numerical examples where the Kalman filter is carried to steady state. The first example uses the dynamics of a generalized advection equation exhibiting nonmodal transient growth. Failure to observe growing modes leads to increased steady-state analysis error variances. Leading eigenvectors of the steady-state analysis error covariance matrix are well approximated by leading eigenvectors of the bound matrix. The second example uses the dynamics of a damped baroclinic wave model. The leading eigenvectors of a lowest-order approximation of the bound matrix are shown to approximate well the leading eigenvectors of the steady-state analysis error covariance matrix.

Tippett, Michael K.↗
DOE OSTI2024

Analysis and Mitigation of Cascading Failures Using a Stochastic Interaction Graph with Eigen-analysis

In studies on complex network systems using graph theory, eigen-analysis is typically performed on an undirected graph model of the network. However, when analyzing cascading failures in a power system, the interactions among failures suggest the need for a directed graph beyond the topology of the power system to model directions of failure propagation. To accurately quantify failure interactions for effective mitigation strategies, this paper proposes a stochastic interaction graph model and associated eigen-analysis. Different types of modes on failure propagations are defined and characterized by the eigenvalues of a stochastic interaction matrix, whose absolute values are unity, zero, or in between. Finding and interpreting these modes helps identify the probable patterns of failure propagation, either local or widespread, and the participating components based on eigenvectors. Then, by lowering the failure probabilities of critical components highly participating in a mode of widespread failures, cascading can be mitigated. Here, the validity of the proposed stochastic interaction graph model, eigen-analysis and the resulting mitigation strategies is demonstrated using simulated cascading failure data on an NPCC 140-bus system.

24 POWER TRANSMISSION AND DISTRIBUTION↗