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Results for “generalized eigenvalues”

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 91 records · Page 5
All sourcesNASA NTRSDOE OSTI
NASA NTRS1990

Multistep matrix integrators for real-time simulation

An explicit linear multistep matrix-integration technique is presented for vector systems of ODEs which employs the stability region placement approach to permit the time-step to be chosen independently of system eigenvalues. Closed-form solutions for the general p-step method and the case where the system matrix has zero eigenvalues are given. It is shown that system mode shapes are preserved over the integration process, and that the technique remains applicable to systems with eigenvalues at their origin without need for computing a matrix inversion.

De Abreu-Garcia, J. A.↗
DOE OSTI2025

Theory and numerics of subspace approximation of eigenvalue problems

Large-scale eigenvalue problems arise in various fields of science and engineering and demand computationally efficient solutions. In this study, we investigate the subspace approximation for parametric linear eigenvalue problems, aiming to mitigate the computational burden associated with high-fidelity systems. Furthermore, we provide general error estimates under non-simple eigenvalue conditions, establishing some theoretical foundations for understanding the convergence behavior of subspace approximations. Numerical examples, including problems with one-dimensional to three-dimensional spatial domain and one-dimensional to two-dimensional parameter domain, are presented to demonstrate the efficacy of reduced basis method in handling parametric variations in boundary conditions and coefficient fields to achieve significant computational savings while maintaining high accuracy, making them promising tools for practical applications in large-scale eigenvalue computations.

Eigenvalue problems↗
NASA NTRS1992

Conservative-variable average states for equilibrium gas multi-dimensional fluxes

Modern split component evaluations of the flux vector Jacobians are thoroughly analyzed for equilibrium-gas average-state determinations. It is shown that all such derivations satisfy a fundamental eigenvalue consistency theorem. A conservative-variable average state is then developed for arbitrary equilibrium-gas equations of state and curvilinear-coordinate fluxes. Original expressions for eigenvalues, sound speed, Mach number, and eigenvectors are then determined for a general average Jacobian, and it is shown that the average eigenvalues, Mach number, and eigenvectors may not coincide with their classical pointwise counterparts. A general equilibrium-gas equation of state is then discussed for conservative-variable computational fluid dynamics (CFD) Euler formulations. The associated derivations lead to unique compatibility relations that constrain the pressure Jacobian derivatives. Thereafter, alternative forms for the pressure variation and average sound speed are developed in terms of two average pressure Jacobian derivatives. Significantly, no additional degree of freedom exists in the determination of these two average partial derivatives of pressure. Therefore, they are simultaneously computed exactly without any auxiliary relation, hence without any geometric solution projection or arbitrary scale factors. Several alternative formulations are then compared and key differences highlighted with emphasis on the determination of the pressure variation and average sound speed. The relevant underlying assumptions are identified, including some subtle approximations that are inherently employed in published average-state procedures. Finally, a representative test case is discussed for which an intrinsically exact average state is determined. This exact state is then compared with the predictions of recent methods, and their inherent approximations are appropriately quantified.

Iannelli, G. S.↗
NASA NTRS1997

Frictionless Contact of Multilayered Composite Half Planes Containing Layers With Complex Eigenvalues

A previously developed local-global stiffness matrix methodology for the response of a composite half plane, arbitrarily layered with isotropic, orthotropic or monoclinic plies, to indentation by a rigid parabolic punch is further extended to accommodate the presence of layers with complex eigenvalues (e.g., honeycomb or piezoelectric layers). First, a generalized plane deformation solution for the displacement field in an orthotropic layer or half plane characterized by complex eigenvalues is obtained using Fourier transforms. A local stiffness matrix in the transform domain is subsequently constructed for this class of layers and half planes, which is then assembled into a global stiffness matrix for the entire multilayered half plane by enforcing continuity conditions along the interfaces. Application of the mixed boundary condition on the top surface of the half plane indented by a rigid punch results in an integral equation for the unknown pressure in the contact region. The integral possesses a divergent kernel which is decomposed into Cauchy-type and regular parts using the asymptotic properties of the local stiffness matrix and a relationship between Fourier and finite Hilbert transform of the contact pressure. The solution of the resulting singular integral equation is obtained using a collocation technique based on the properties of orthogonal polynomials developed by Erdogan and Gupta. Examples are presented that illustrate the important influence of low transverse properties of layers with complex eigenvalues, such as those exhibited by honeycomb, on the load versus contact length response and contact pressure distributions for half planes containing typical composite materials.

Zhang, Wang↗
NASA NTRS1984

Squeeze-film dampers for turbomachinery stabilization

A technique for investigating the stability and damping present in centrally preloaded radially symmetric multi-mass flexible rotor bearing systems is presented. In general, one needs to find the eigenvalues of the linearized perturbation equations, though zero frequency stability maps may be found by solving as many simultaneous non-linear equations as there are dampers; and in the case of a single damper, such maps may be found directly, regardless of the number of degrees of freedom. The technique is illustrated for a simple symmetric four degree of freedom flexible rotor with an unpressurized damper. This example shows that whereas zero frequency stability maps are likely to prove to be a simple way to delineate multiple solution possibilities, they do not provide full stability information. Further, particularly for low bearing parameters, the introduction of an unpressurized squeeze film damper may promote instability in an otherwise stable system.

Mclean, L. J.↗
NASA NTRS1990

Implicit application of polynomial filters in a k-step Arnoldi method

The Arnoldi process is a well known technique for approximating a few eigenvalues and corresponding eigenvectors of a general square matrix. Numerical difficulties such as loss of orthogonality and assessment of the numerical quality of the approximations as well as a potential for unbounded growth in storage have limited the applicability of the method. These issues are addressed by fixing the number of steps in the Arnoldi process at a prescribed value k and then treating the residual vector as a function of the initial Arnoldi vector. This starting vector is then updated through an iterative scheme that is designed to force convergence of the residual to zero. The iterative scheme is shown to be a truncation of the standard implicitly shifted QR-iteration for dense problems and it avoids the need to explicitly restart the Arnoldi sequence. The main emphasis of this paper is on the derivation and analysis of this scheme. However, there are obvious ways to exploit parallelism through the matrix-vector operations that comprise the majority of the work in the algorithm. Preliminary computational results are given for a few problems on some parallel and vector computers.

Sorensen, D. C.↗
NASA NTRS2006

Rapid Aeroelastic Analysis of Blade Flutter in Turbomachines

The LINFLUX-AE computer code predicts flutter and forced responses of blades and vanes in turbomachines under subsonic, transonic, and supersonic flow conditions. The code solves the Euler equations of unsteady flow in a blade passage under the assumption that the blades vibrate harmonically at small amplitudes. The steady-state nonlinear Euler equations are solved by a separate program, then equations for unsteady flow components are obtained through linearization around the steady-state solution. A structural-dynamics analysis (see figure) is performed to determine the frequencies and mode shapes of blade vibrations, a preprocessor interpolates mode shapes from the structural-dynamics mesh onto the LINFLUX computational-fluid-dynamics mesh, and an interface code is used to convert the steady-state flow solution to a form required by LINFLUX. Then LINFLUX solves the linearized equations in the frequency domain to calculate the unsteady aerodynamic pressure distribution for a given vibration mode, frequency, and interblade phase angle. A post-processor uses the unsteady pressures to calculate generalized aerodynamic forces, response amplitudes, and eigenvalues (which determine the flutter frequency and damping). In comparison with the TURBO-AE aeroelastic-analysis code, which solves the equations in the time domain, LINFLUX-AE is 6 to 7 times faster.

Trudell, J. J.↗
NASA NTRS1987

Survey of methods for calculating sensitivity of general eigenproblems

A survey of methods for sensitivity analysis of the algebraic eigenvalue problem for non-Hermitian matrices is presented. In addition, a modification of one method based on a better normalizing condition is proposed. Methods are classified as Direct or Adjoint and are evaluated for efficiency. Operation counts are presented in terms of matrix size, number of design variables and number of eigenvalues and eigenvectors of interest. The effect of the sparsity of the matrix and its derivatives is also considered, and typical solution times are given. General guidelines are established for the selection of the most efficient method.

Murthy, Durbha V.↗
NASA NTRS1995

Aeroelastic stability of wind turbine blade/aileron systems

Aeroelastic stability analyses have been performed for the MOD-5A blade/aileron system. Various configurations having different aileron torsional stiffness, mass unbalance, and control system damping have been investigated. The analysis was conducted using a code recently developed by the General Electric Company - AILSTAB. The code extracts eigenvalues for a three degree of freedom system, consisting of: (1) a blade flapwise mode; (2) a blade torsional mode; and (3) an aileron torsional mode. Mode shapes are supplied as input and the aileron can be specified over an arbitrary length of the blade span. Quasi-steady aerodynamic strip theory is used to compute aerodynamic derivatives of the wing-aileron combination as a function of spanwise position. Equations of motion are summarized herein. The program provides rotating blade stability boundaries for torsional divergence, classical flutter (bending/torsion) and wing/aileron flutter. It has been checked out against fixed-wing results published by Theodorsen and Garrick. The MOD-5A system is stable with respect to divergence and classical flutter for all practical rotor speeds. Aileron torsional stiffness must exceed a minimum critical value to prevent aileron flutter. The nominal control system stiffness greatly exceeds this minimum during normal operation. The basic system, however, is unstable for the case of a free (or floating) aileron. The instability can be removed either by the addition of torsional damping or mass-balancing the ailerons. The MOD-5A design was performed by the General Electric Company, Advanced Energy Program Department under Contract DEN3-153 with NASA Lewis Research Center and sponsored by the Department of Energy.

Strain, J. C.↗
NASA NTRS2008

Sparse Regression as a Sparse Eigenvalue Problem

We extend the l0-norm "subspectral" algorithms for sparse-LDA [5] and sparse-PCA [6] to general quadratic costs such as MSE in linear (kernel) regression. The resulting "Sparse Least Squares" (SLS) problem is also NP-hard, by way of its equivalence to a rank-1 sparse eigenvalue problem (e.g., binary sparse-LDA [7]). Specifically, for a general quadratic cost we use a highly-efficient technique for direct eigenvalue computation using partitioned matrix inverses which leads to dramatic x103 speed-ups over standard eigenvalue decomposition. This increased efficiency mitigates the O(n4) scaling behaviour that up to now has limited the previous algorithms' utility for high-dimensional learning problems. Moreover, the new computation prioritizes the role of the less-myopic backward elimination stage which becomes more efficient than forward selection. Similarly, branch-and-bound search for Exact Sparse Least Squares (ESLS) also benefits from partitioned matrix inverse techniques. Our Greedy Sparse Least Squares (GSLS) generalizes Natarajan's algorithm [9] also known as Order-Recursive Matching Pursuit (ORMP). Specifically, the forward half of GSLS is exactly equivalent to ORMP but more efficient. By including the backward pass, which only doubles the computation, we can achieve lower MSE than ORMP. Experimental comparisons to the state-of-the-art LARS algorithm [3] show forward-GSLS is faster, more accurate and more flexible in terms of choice of regularization

Exact Sparse Least Squares (ESLS)↗
NASA NTRS1989

Quotient-difference type generalizations of the power method and their analysis

The recursion relations that were proposed by W. F. Ford and A. Sidi (Appl. Numer. Math, 4 (1988), pp. 477-489) for implementing vector extrapolation methods are used for devising generalizations of the power method for linear operators. These generalizations are shown to produce approximations to largest eigenvalues of a linear operator under certain conditions. They are similar in form to the quotient-difference algorithm and share similar convergence properties with the latter. These convergence properties also resemble those obtained for the basic LR and QR algorithms. Finally, it is shown that the convergence rate produced by one fo these generalizations is twice as fast for normal operators as it is for nonnormal operators.

Sidi, Avram↗
NASA NTRS1981

A study of the modal truncation error in the component mode analysis of a dual-rotor system

In the component mode synthesis method, the equation of motion in the generalized coordinates is built upon the undamped eigenvalue data of the component structures. Error is inevitable when truncated modes are used. In this paper, two modal truncation schemes were evaluated with regard to the critical speed, stability, and unbalance response of a two-spool gas turbine engine. The numbers of modes required to yield acceptable accuracy in these cases were determined. Guidelines for modal truncation were derived from these results.

Li, D. F.↗
NASA NTRS1985

Optimization of cascade blade mistuning. II - Global optimum and numerical optimization

The values of the mistuning which yield the most stable eigenvectors are analytically determined, using the simplified equations of motion which were developed in Part I of this work. It is shown that random mistunings, if large enough, may lead to the maximal stability, whereas the alternate mistunings cannot. The problem of obtaining maximum stability for minimal mistuning is formulated, based on numerical optimization techniques. Several local minima are obtained using different starting mistuning vectors. The starting vectors which lead to the global minimum are identified. It is analytically shown that all minima appear in multiplicities which are equal to the number of compressor blades. The effect of mistuning on the flutter speed is studied using both an optimum mistuning vector and an alternate mistuning vector. Effects of mistunings in elastic axis locations are shown to have a negligible effect on the eigenvalues. Finally, it is shown that any general two-dimensional bending-torsion system can be reduced to an equivalent uncoupled torsional system.

Nissim, E.↗
NASA NTRS1988

Experiences in applying optimization techniques to configurations for the Control Of Flexible Structures (COFS) Program

Optimization procedures are developed to systematically provide closely-spaced vibration frequencies. A general-purpose finite-element program for eigenvalue and sensitivity analyses is combined with formal mathematical programming techniques. Results are presented for three studies. The first study uses a simple model to obtain a design with two pairs of closely-spaced frequencies. Two formulations are developed: an objective function-based formulation and constraint-based formulation for the frequency spacing. It is found that conflicting goals are handled better by a constraint-based formulation. The second study uses a detailed model to obtain a design with one pair of closely-spaced frequencies while satisfying requirements on local member frequencies and manufacturing tolerances. Two formulations are developed. Both the constraint-based and the objective function-based formulations perform reasonably well and converge to the same results. However, no feasible design solution exists which satisfies all design requirements for the choices of design variables and the upper and lower design variable values used. More design freedom is needed to achieve a fully satisfactory design. The third study is part of a redesign activity in which a detailed model is used. The use of optimization in this activity allows investigation of numerous options (such as number of bays, material, minimum diagonal wall thicknesses) in a relatively short time. The procedure provides data for judgments on the effects of different options on the design.

Walsh, Joanne L.↗
NASA NTRS1989

Experiences in applying optimization techniques to configurations for the Control of Flexible Structures (COFS) program

Optimization procedures are developed to systematically provide closely-spaced vibration frequencies. A general purpose finite-element program for eigenvalue and sensitivity analyses is combined with formal mathematical programming techniques. Results are presented for three studies. The first study uses a simple model to obtain a design with two pairs of closely-spaced frequencies. Two formulations are developed: an objective function-based formulation and constraint-based formulation for the frequency spacing. It is found that conflicting goals are handled better by a constraint-based formulation. The second study uses a detailed model to obtain a design with one pair of closely-spaced frequencies while satisfying requirements on local member frequencies and manufacturing tolerances. Two formulations are developed. Both the constraint-based and the objective function-based formulations perform reasonably well and converge to the same results. However, no feasible design solution exists which satisfies all design requirements for the choices of design variables and the upper and lower design variable values used. More design freedom is needed to achieve a fully satisfactory design. The third study is part of a redesign activity in which a detailed model is used.

Walsh, Joanne L.↗
NASA NTRS1990

Eigensensitivity analysis for space structures

The main thrust of this paper is to study eigensensitivity analysis techniques and their applications to design optimization of space structures. It is found that in general, space structures show four types of eigenvalue problems most of which are related to repeated eigenvalues. This paper first gives a detailed derivation of eigensensitivity equations. Later, numerical examples are provided to verify the derived sensitivity equations as well as to demonstrate the applicability of those sensitivity equations in an iterative design environment.

Hou, Jean W.↗
NASA NTRS1990

A Shifted Block Lanczos Algorithm 1: The Block Recurrence

In this paper we describe a block Lanczos algorithm that is used as the key building block of a software package for the extraction of eigenvalues and eigenvectors of large sparse symmetric generalized eigenproblems. The software package comprises: a version of the block Lanczos algorithm specialized for spectrally transformed eigenproblems; an adaptive strategy for choosing shifts, and efficient codes for factoring large sparse symmetric indefinite matrices. This paper describes the algorithmic details of our block Lanczos recurrence. This uses a novel combination of block generalizations of several features that have only been investigated independently in the past. In particular new forms of partial reorthogonalization, selective reorthogonalization and local reorthogonalization are used, as is a new algorithm for obtaining the M-orthogonal factorization of a matrix. The heuristic shifting strategy, the integration with sparse linear equation solvers and numerical experience with the code are described in a companion paper.

Grimes, Roger G.↗
NASA NTRS1998

Nonlinear Analysis of the Space Shuttle Superlightweight LO2 Tank: Behavior Under 3g End-of-Flight Loads - Part 2

Results of linear bifurcation and nonlinear analyses of the Space Shuttle super lightweight (SLWT) external liquid-oxygen (LO2) tank are presented for an important end-of-flight loading condition. These results illustrate an important type of response mode for thin-walled shells, that are subjected to combined mechanical and thermal loads, that may be encountered in the design of other liquid-fuel launch vehicles. Linear bifurcation analyses are presented that predict several nearly equal eigenvalues that correspond to local buckling modes in the aft dome of the LO2 tank. In contrast, the nonlinear response phenomenon is shown to consist of a short-wavelength bending deformation in the aft elliptical dome of the LO2 tank that grows in amplitude in a stable manner with increasing load. Imperfection sensitivity analyses are presented that show that the presence of several nearly equal eigenvalues does not lead to a premature general instability mode for the aft dome. For the linear bifurcation and nonlinear analyses, the results show that accurate predictions of the response of the shell generally require a large-scale, high fidelity finite-element model. Results are also presented that show that the SLWT LO2 tank can support loads in excess of approximately 1.9 times the values of the operational loads considered.

Nemeth, Michael P.↗