Rounding error bounds by perturbation- condition analysis
Rounding error bounds by perturbation condition analysis, and conditioning for eigenvalue problem
ERROR↗
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.
Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.
Instability analysis of torispherical pressure vessel heads with triangular thin-shell finite elements
The elastic instability of an internally-pressurized cylindrical tank with a torispherical head is investigated using a triangular, doubly curved, thin-shell finite element. The formulation of the finite element, which is based upon cubic displacement functions and a modified principle of potential energy, is first described. Then, the element is verified by comparing numerical results for the linear, stable analysis to alternative solutions for the same problem. The subsequent instability investigation includes the solution of the linearized problem of equilibrium bifurcation, that is, of the classical eigenvalue problem, and a general nonlinear analysis, based on tracing the nonlinear load-displacement path. The critical pressure, obtained with use of the general nonlinear analysis, agrees closely with an experimental result as well as with a numerical solution stemming from an axisymmetric formulation.
Kanodia, V. L.↗
Free-Vibration Analysis of Structures
Improved numerical procedure more than twice as fast as previous methods. Unified numerical algorithm efficiently solves free-vibration problems of stationary or spinning structures with or without viscous or structural damping. Algorithm used to solve static problems involving multiple loads and to solve quadratic matrix eigenvalue problems associated with finite-dynamic-element structural discretization.
Gupta, K. K.↗
An efficient procedure for cascade aeroelastic stability determination using nonlinear, time-marching aerodynamic solvers
A numerical eigenvalue problem formulation and a practical calculation procedure for exact eigenvalues and corresponding eigenvectors are developed and applied to a nonlinear, two-dimensional, time-marching full potential solver for cascade aeroelastic stability analysis. This procedure is based on the Lanczos recursive method and it directly calculates stability information about a nonlinear steady state. It is compared to conventional approaches in the frequency and time domains developed earlier and is found to be 100-10.000 times more computationally efficient. Eigenvalue constellations and the flutter results for flow through a cascade SR5 propfan airfoil are presented.
Mahajan, Aparajit J.↗
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.↗
A parallel solution for the symmetric Eigenproblem
A completely parallel algorithm for the symmetric eigenproblem AX = Lambda BX is outlined. The algorithm is parallel in the sense that the numerical operations do not occur in a fixed sequence. Therefore, a large number of operations can be programmed to be performed concurrently on a computer with multiple central processing units. The standard symmetric eigenvalue problem AX = Lambda X has the property that the n eigenvalues of the principal submatrix of A of order n are separated by the (n-1) eignvalues of the principal submatrix of order (n-1). The separation property delineated n intervals containing one eigenvalue. Each eigenvalue and corresponding eigenvector can be computed independently. The n eigenproblem calculations can be divided among multiple processing units.
Thurston, Gaylen A.↗
An eigenvalue method for solving transient heat conduction problems
The eigenvalue method, which has been used by researchers in structure mechanics, is applied to problems in heat conduction. Its formulation is decribed in terms of an examination of transient heat conduction in a square slab. Taking advantage of the availability of the exact solution, we compare the accuracy and other numerical properties of the eigenvalue method with those of existing numerical schemes. The comparsion shows that, overall, the eigenvalue method appears to be fairly attractive. Furthermore, only a few dominant eigenvalues and their corresponding eigenvectors need to be computed and retained to yield reasonably high accuracy. Greater savings are attained in the computation time for a transient problem with long time duration and a large computational domain.
Shih, T. M.↗
Predicting the stability of a compressible periodic parallel jet flow
It is known that mixing enhancement in compressible free shear layer flows with high convective Mach numbers is difficult. One design strategy to get around this is to use multiple nozzles. Extrapolating this design concept in a one dimensional manner, one arrives at an array of parallel rectangular nozzles where the smaller dimension is omega and the longer dimension, b, is taken to be infinite. In this paper, the feasibility of predicting the stability of this type of compressible periodic parallel jet flow is discussed. The problem is treated using Floquet-Bloch theory. Numerical solutions to this eigenvalue problem are presented. For the case presented, the interjet spacing, s, was selected so that s/omega =2.23. Typical plots of the eigenvalue and stability curves are presented. Results obtained for a range of convective Mach numbers from 3 to 5 show growth rates omega(sub i)=kc(sub i)/2 range from 0.25 to 0.29. These results indicate that coherent two-dimensional structures can occur without difficulty in multiple parallel periodic jet nozzles and that shear layer mixing should occur with this type of nozzle design.
Miles, Jeffrey H.↗
Three-dimensional baroclinic instability of a Hadley cell for small Richardson number
A three-dimensional, linear stability analysis of a baroclinic flow for Richardson number, Ri, of order unity is presented. The model considered is a thin horizontal, rotating fluid layer which is subjected to horizontal and vertical temperature gradients. The basic state is a Hadley cell which is a solution of the complete set of governing, nonlinear equations and contains both Ekman and thermal boundary layers adjacent to the rigid boundaries; it is given in a closed form. The stability analysis is also based on the complete set of equations; and perturbation possessing zonal, meridional, and vertical structures were considered. Numerical methods were developed for the stability problem which results in a stiff, eighth-order, ordinary differential eigenvalue problem. The previous work on three-dimensional baroclinic instability for small Ri was extended to a more realistic model involving the Prandtl number, sigma, and the Ekman number, E, and to finite growth rates and a wider range of the zonal wavenumber.
Antar, B. N.↗
Eigensensitivity analysis for space structures with applications in design optimization
An approximate structural eigenvalue/vector analysis technique which uses eigensensitivities in a truncated Taylor series expression is presented. It is shown that this technique can provide the computational efficiency urgently needed for large scale control-structure optimization problems. In addition, a unified formulation and solution approach for calculating eigenvalue and eigenvector derivatives of the real symmetric structural eigenvalue problem is presented.
Kenny, Sean P.↗
Stability of streamwise vortices
A brief overview of some theoretical and computational studies of the stability of streamwise vortices is given. The local induction model and classical hydrodynamic vortex stability theories are discussed in some detail. The importance of the three-dimensionality of the mean velocity profile to the results of stability calculations is discussed briefly. The mean velocity profile is provided by employing the similarity solution of Donaldson and Sullivan. The global method of Bridges and Morris was chosen for the spatial stability calculations for the nonlinear eigenvalue problem. In order to test the numerical method, a second order accurate central difference scheme was used to obtain the coefficient matrices. It was shown that a second order finite difference method lacks the required accuracy for global eigenvalue calculations. Finally the problem was formulated using spectral methods and a truncated Chebyshev series.
Khorrami, M. K.↗
Investigating a hybrid perturbation-Galerkin technique using computer algebra
A two-step hybrid perturbation-Galerkin method is presented for the solution of a variety of differential equations type problems which involve a scalar parameter. The resulting (approximate) solution has the form of a sum where each term consists of the product of two functions. The first function is a function of the independent field variable(s) x, and the second is a function of the parameter lambda. In step one the functions of x are determined by forming a perturbation expansion in lambda. In step two the functions of lambda are determined through the use of the classical Bubnov-Gelerkin method. The resulting hybrid method has the potential of overcoming some of the drawbacks of the perturbation and Bubnov-Galerkin methods applied separately, while combining some of the good features of each. In particular, the results can be useful well beyond the radius of convergence associated with the perturbation expansion. The hybrid method is applied with the aid of computer algebra to a simple two-point boundary value problem where the radius of convergence is finite and to a quantum eigenvalue problem where the radius of convergence is zero. For both problems the hybrid method apparently converges for an infinite range of the parameter lambda. The results obtained from the hybrid method are compared with approximate solutions obtained by other methods, and the applicability of the hybrid method to broader problem areas is discussed.
Andersen, Carl M.↗
On the design derivatives of eigenvalues and eigenvectors for distributed parameter systems
In this paper, analytic expressions are obtained for the design derivatives of eigenvalues and eigenfunctions of self-adjoint linear distributed parameter systems. Explicit treatment of boundary conditions is avoided by casting the eigenvalue equation into integral form. Results are expressed in terms of the linear operators defining the eigenvalue problem, and are therefore quite general. Sufficiency conditions appropriate to structural optimization of eigenvalues are obtained.
Reiss, R.↗
Stability investigations of airfoil flow by global analysis
As the result of global, non-parallel flow stability analysis the single value of the disturbance growth-rate and respective frequency is obtained. This complex value characterizes the stability of the whole flow configuration and is not referred to any particular flow pattern. The global analysis assures that all the flow elements (wake, boundary and shear layer) are taken into account. The physical phenomena connected with the wake instability are properly reproduced by the global analysis. This enhances the investigations of instability of any 2-D flows, including ones in which the boundary layer instability effects are known to be of dominating importance. Assuming fully 2-D disturbance form, the global linear stability problem is formulated. The system of partial differential equations is solved for the eigenvalues and eigenvectors. The equations, written in the pure stream function formulation, are discretized via FDM using a curvilinear coordinate system. The complex eigenvalues and corresponding eigenvectors are evaluated by an iterative method. The investigations performed for various Reynolds numbers emphasize that the wake instability develops into the Karman vortex street. This phenomenon is shown to be connected with the first mode obtained from the non-parallel flow stability analysis. The higher modes are reflecting different physical phenomena as for example Tollmien-Schlichting waves, originating in the boundary layer and having the tendency to emerge as instabilities for the growing Reynolds number. The investigations are carried out for a circular cylinder, oblong ellipsis and airfoil. It is shown that the onset of the wake instability, the waves in the boundary layer, the shear layer instability are different solutions of the same eigenvalue problem, formulated using the non-parallel theory. The analysis offers large potential possibilities as the generalization of methods used till now for the stability analysis.
Morzynski, Marek↗
What is H sub 0/ques/
Zero order Hamiltonian in perturbation theory for eigenvalue problems of atom in magnetic field
EIGENVALUE↗
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.↗
Towards Formalization of Advanced Linear Algebra with Applications to Dynamical Systems using PVS
Linear Algebra is essential for numerous aerospace problems of interest. Formal reasoning about hybrid systems that contain variables modeled by differential equations rely on concepts from Linear Algebra such as eigenvalues, matrix decompositions, and matrix valued functions. For example, the long-term dynamics of a system of differential equations depend on the stability/instability of its equilibrium points, which often reduces to an eigenvalue problem. This talk will embark on a quest to formalize theorems and results about eigenvalues and eigenvectors using PVS. We shall start our journey with 2 x 2 complex matrices, where we will apply our PVS code to a simple example of a dynamical system. Since it can be difficult or impossible to give simple expressions of eigenvalues for larger matrices (i.e. 5 x 5 or higher), we then move towards specifying the power method for verified computation of eigenvalue approximations in PVS. This effort requires development of multivariate complex arithmetic. At the end of the day, having such additions to the PVS NASA libraries will help move towards the use of formal methods to verify concepts of control theory and system level verification.
Linear Algebra↗
Sensitivity analysis for large-scale problems
The development of efficient techniques for calculating sensitivity derivatives is studied. The objective is to present a computational procedure for calculating sensitivity derivatives as part of performing structural reanalysis for large-scale problems. The scope is limited to framed type structures. Both linear static analysis and free-vibration eigenvalue problems are considered.
Noor, Ahmed K.↗