SEARCH · Engineering Papers

Results for “EIGENVALUE”

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 55 records · Page 3

All sources NASA NTRS DOE OSTI

NASA NTRS1974

Solution of an eigenvalue problem for the Laplace operator on a spherical surface

Methods for obtaining approximate solutions for the fundamental eigenvalue of the Laplace-Beltrami operator (also referred to as the membrane eigenvalue problem for the vibration equation) on the unit spherical surface are developed. Two specific types of spherical surface domains are considered: (1) the interior of a spherical triangle, i.e., the region bounded by arcs of three great circles, and (2) the exterior of a great circle arc extending for less than pi radians on the sphere (a spherical surface with a slit). In both cases, zero boundary conditions are imposed. In order to solve the resulting second-order elliptic partial differential equations in two independent variables, a finite difference approximation is derived. The symmetric (generally five-point) finite difference equations that develop are written in matrix form and then solved by the iterative method of point successive overrelaxation. Upon convergence of this iterative method, the fundamental eigenvalue is approximated by iteration utilizing the power method as applied to the finite Rayleigh quotient.

Walden, H.↗

NASA NTRS1974

Derivatives of eigenvalues and eigenvectors for a general matrix

Expressions are obtained for the derivatives of the eigenvalues and eigenvectors which are expressions of only one left-hand and one right-hand eigenvector. The approach described makes use of a Choleski decomposition or some other decomposition method. The method may be extended to find any order of derivative of the eigenvalue and eigenvector. The expressions obtained for finding the derivatives of eigenvalues and eigenvectors for nonself-adjoint systems may be applied to self-adjoint systems.

Rudisill, C. S.↗

NASA NTRS1974

Error analysis of householder transformations as applied to the standard and generalized eigenvalue problems

Backward error analyses of the application of Householder transformations to both the standard and the generalized eigenvalue problems are presented. The analysis for the standard eigenvalue problem determines the error from the application of an exact similarity transformation, and the analysis for the generalized eigenvalue problem determines the error from the application of an exact equivalence transformation. Bounds for the norms of the resulting perturbation matrices are presented and compared with existing bounds when known.

Ward, R. C.↗

NASA NTRS1977

Complex eigenvalue extraction in NASTRAN by the tridiagonal reduction (FEER) method

An extension of the Tridiagonal Reduction (FEER) method to complex eigenvalue analysis in NASTRAN is described. As in the case of real eigenvalue analysis, the eigensolutions closest to a selected point in the eigenspectrum are extracted from a reduced, symmetric, tridiagonal eigenmatrix whose order is much lower than that of the full size problem. The reduction process is effected automatically, and thus avoids the arbitrary lumping of masses and other physical quantities at selected grid points. The statement of the algebraic eigenvalue problem admits mass, damping and stiffness matrices which are unrestricted in character, i.e., they may be real, complex, symmetric or unsymmetric, singular or non-singular.

Newman, M.↗

NASA NTRS1979

A finite element formulation of the eigenvalue problem in lined ducts with flow

A finite element method is used to formulate the eigenvalue problem for a lined duct with flow. Either two dimensional or axially symmetric ducts with sheared flows can be studied, although the examples cited are two dimensional. The primitive variables of velocity and pressure are used with quadratic shape functions in each element. Results indicate that a useful level of accuracy can be achieved with a modest number of elements. Spurious eigenvalues, easily identified by obviously inconsistent eigenvectors, occur in certain instances. If the number of elements is not too small, these spurious modes are among the higher order eigenvalues of otherwise questionable accuracy. The possibility of using higher order elements which have slope continuity is proposed for future investigations to eliminate the spurious results.

Astley, R. J.↗

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 NTRS1981

Eigenvalues of the Rayleigh-Benard and Marangoni problems

The eigenvalues of the linear Benard-Marangoni stability problem are discussed. Pearson and Nield boundary conditions, which correspond to a rigid, isothermal lower boundary and a stress-free conducting upper boundary are considered. It is shown that although a critical value of the Marangoni number can be determined, the number is not, strictly speaking, an eigenvalue and cannot be used as an eigenvalue parameter for the determination of an eigenvector set.

Rosenblat, S.↗

NASA NTRS1983

Complex eigenvalues for the stability of Couette flow

The eigenvalue problem for the linear stability of Couette flow between rotating concentric cylinders to axisymmetric disturbances is considered. It is shown by numerical calculations and by formal perturbation methods that when the outer cylinder is at rest there exist complex eigenvalues corresponding to oscillatory damped disturbances. The structure of the first few eigenvalues in the spectrum is discussed. The results do not contradict the principle of exchange of stabilities, namely, for a fixed axial wavenumber the first mode to become unstable as the speed of the inner cylinder is increased is nonoscillatory as the stability boundary is crossed.

Diprima, R. C.↗

NASA NTRS1984

Differential eigenvalue problems in which the parameter appears nonlinearly

Several methods are examined for determining the eigenvalues of a system of equations in which the parameter appears nonlinearly. The equations are the result of the discretization of differential eigenvalue problems using a finite Chebyshev series. Two global methods are considered which determine the spectrum of eigenvalues without an initial estimate. A local iteration scheme with cubic convergence is presented. Calculations are performed for a model second order differential problem and the Orr-Sommerfeld problem for plane Poiseuille flow.

Bridges, T. J.↗

NASA NTRS1985

A method to stabilize linear systems using eigenvalue gradient information

Formal optimization methods and eigenvalue gradient information are used to develop a stabilizing control law for a closed loop linear system that is initially unstable. The method was originally formulated by using direct, constrained optimization methods with the constraints being the real parts of the eigenvalues. However, because of problems in trying to achieve stabilizing control laws, the problem was reformulated to be solved differently. The method described uses the Davidon-Fletcher-Powell minimization technique to solve an indirect, constrained minimization problem in which the performance index is the Kreisselmeier-Steinhauser function of the real parts of all the eigenvalues. The method is applied successfully to solve two different problems: the determination of a fourth-order control law stabilizes a single-input single-output active flutter suppression system and the determination of a second-order control law for a multi-input multi-output lateral-directional flight control system. Various sets of design variables and initial starting points were chosen to show the robustness of the method.

Wieseman, C. D.↗

NASA NTRS1987

Approximations to eigenvalues of modified general matrices

The reanalysis of non-self-adjoint dynamic models is computationally very expensive in design optimization applications. This paper describes several approximations that can be applied to eigenvalues of non-hermitian matrices to reduce that computational cost. Approximations based on eigenvalue derivatives, generalized Rayleigh quotient and the trace theorem are presented and their accuracy and computational cost are estimated. The accuracy and cost estimates are verified by applying the approximations to random matrices and matrices arising in flutter analysis of compressor blades. Recommendations are made for selection of the best approximation when the derivatives are available and when they are not. In particular, it is concluded that the quadratic approximation for eigenvalues should never be used as higher order approximations are always more accurate as well as more efficient.

Murthy, Durbha V.↗

NASA NTRS1988

Eigenvalue computation of large symmetric tridiagonal matrices on concurrent processors

Symmetric tridiagonal eigenvalue problems may arise indirectly in structural dynamic analysis. An algorithm for eigenvalue computation of large symmetric tridiagonal matrices on concurrent processors to meet the challenge of the new emerging computer hardware technology is presented. A standard bisection method in conjunction with Sylvester's Theorem is chosen to be converted into a parallel N-section algorithm. This parallel algorithm takes advantage of the multi-processor environment by carrying out N (number of processors) triangular factorizations of chosen shifted matrices in all processors concurrently and by minimizing communication between processors. The algorithm is designed for local-memory concurrent processors, i.e. message passing type processors. The efficiency and speed-up are given in terms of problem and machine parameters. The algorithm is very efficient when both the number of processors and the number of eigenvalues to be extracted are much smaller than the order of the tridiagonal matrix.

Chang, H. Y.↗

NASA NTRS1988

Generalized Eigenvalues for pairs on heritian matrices

A study was made of certain special cases of a generalized eigenvalue problem. Let A and B be nxn matrics. One may construct a certain polynomial, P(A,B, lambda) which specializes to the characteristic polynomial of B when A equals I. In particular, when B is hermitian, that characteristic polynomial, P(I,B, lambda) has real roots, and one can ask: are the roots of P(A,B, lambda) real when B is hermitian. We consider the case where A is positive definite and show that when N equals 3, the roots are indeed real. The basic tools needed in the proof are Shur's theorem on majorization for eigenvalues of hermitian matrices and the interlacing theorem for the eigenvalues of a positive definite hermitian matrix and one of its principal (n-1)x(n-1) minors. The method of proof first reduces the general problem to one where the diagonal of B has a certain structure: either diag (B) = diag (1,1,1) or diag (1,1,-1), or else the 2 x 2 principal minors of B are all 1. According as B has one of these three structures, we use an appropriate method to replace A by a positive diagonal matrix. Since it can be easily verified that P(D,B, lambda) has real roots, the result follows. For other configurations of B, a scaling and a continuity argument are used to prove the result in general.

Rublein, George↗

NASA NTRS1989

Accurate calculation of control-augmented structural eigenvalue sensitivities using reduced-order models

A method is presented for generating mode shapes for model order reduction in a way that leads to accurate calculation of eigenvalue derivatives and eigenvalues for a class of control augmented structures. The method is based on treating degrees of freedom where control forces act or masses are changed in a manner analogous to that used for boundary degrees of freedom in component mode synthesis. It is especially suited for structures controlled by a small number of actuators and/or tuned by a small number of concentrated masses whose positions are predetermined. A control augmented multispan beam with closely spaced natural frequencies is used for numerical experimentation. A comparison with reduced-order eigenvalue sensitivity calculations based on the normal modes of the structure shows that the method presented produces significant improvements in accuracy.

Livne, Eli↗

NASA NTRS1990

The kink instability in infinite cylindrical flux tubes - Eigenvalues for power-law twist profiles

Simple, accurate methods of calculating ideal MHD instability eigenvalues for infinitely long cylindrical tubes with twist function T(r) are developed. The results show that the most rapidly growing and energetic instabilities occur in the Gold-Hoyle v = 0 field, with the instability progressively weakening with increasing v. However, the maximum force eigenvalue is always small, so that even in the Gold-Hoyle case only a small proportion of the available magnetic energy can be released in the linear phase. The results also confirm that the linear pinch is remarkably weak yet relatively resistant to line-tying. It is shown that the weakness of the force eigenvalue implies that the influence of uniform gas pressure on stability is negligible. Implications for the energy-release mechanism in solar flares are discussed.

Craig, I. J. D.↗

NASA NTRS1991

Eigenvalue sensitivity analysis of planar frames with variable joint and support locations

Two sensitivity equations are derived in this study based upon the continuum approach for eigenvalue sensitivity analysis of planar frame structures with variable joint and support locations. A variational form of an eigenvalue equation is first derived in which all of the quantities are expressed in the local coordinate system attached to each member. Material derivative of this variational equation is then sought to account for changes in member's length and orientation resulting form the perturbation of joint and support locations. Finally, eigenvalue sensitivity equations are formulated in either domain quantities (by the domain method) or boundary quantities (by the boundary method). It is concluded that the sensitivity equation derived by the boundary method is more efficient in computation but less accurate than that of the domain method. Nevertheless, both of them in terms of computational efficiency are superior to the conventional direct differentiation method and the finite difference method.

Chuang, Ching H.↗

NASA NTRS1992

The eigenvalue spectrum of the Rayleigh equation for a plane shear layer

The eigenvalue spectrum of the Rayleigh equation is examined using three different solution techniques. In particular, a simple second-order finite difference scheme and two spectral methods, the Chebyshev tau and Chebyshev collocation methods, are used to discretize the equation. All of the approximation methods are shown to be capable of predicting the discrete spectrum as well as the continuous spectrum associated with the critical point singularity for the Rayleigh equation. The global eigenvalue methods considered here provide an efficient way of obtaining either an approximation to the complete eigenvalue spectrum or initial guesses for a local shooting procedure for the discrete part of the spectrum.

Liou, William W.-W.↗

NASA NTRS1993

Eigenvalue routines in NASTRAN: A comparison with the Block Lanczos method

The NASA STRuctural ANalysis (NASTRAN) program is one of the most extensively used engineering applications software in the world. It contains a wealth of matrix operations and numerical solution techniques, and they were used to construct efficient eigenvalue routines. The purpose of this paper is to examine the current eigenvalue routines in NASTRAN and to make efficiency comparisons with a more recent implementation of the Block Lanczos algorithm by Boeing Computer Services (BCS). This eigenvalue routine is now available in the BCS mathematics library as well as in several commercial versions of NASTRAN. In addition, CRAY maintains a modified version of this routine on their network. Several example problems, with a varying number of degrees of freedom, were selected primarily for efficiency bench-marking. Accuracy is not an issue, because they all gave comparable results. The Block Lanczos algorithm was found to be extremely efficient, in particular, for very large size problems.

Tischler, V. A.↗