Search NASA⌕ Search

SEARCH · Search NASA

Results for “nonlinear 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.

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

At least 19 records
All sourcesNASA NTRSDOE OSTI
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 NTRS1988

Reliable use of determinants to solve nonlinear structural eigenvalue problems efficiently

The analytical derivation, numerical implementation, and performance of a multiple-determinant parabolic interpolation method (MDPIM) for use in solving transcendental eigenvalue (critical buckling or undamped free vibration) problems in structural mechanics are presented. The overall bounding, eigenvalue-separation, qualified parabolic interpolation, accuracy-confirmation, and convergence-recovery stages of the MDPIM are described in detail, and the numbers of iterations required to solve sample plane-frame problems using the MDPIM are compared with those for a conventional bisection method and for the Newtonian method of Simpson (1984) in extensive tables. The MDPIM is shown to use 31 percent less computation time than bisection when accuracy of 0.0001 is required, but 62 percent less when accuracy of 10 to the -8th is required; the time savings over the Newtonian method are about 10 percent.

Williams, F. W.↗
DOE OSTI2023

Subspace recursive Fermi-operator expansion strategies for large-scale DFT eigenvalue problems on HPC architectures

Quantum mechanical calculations for material modeling using Kohn–Sham density functional theory (DFT) involve the solution of a nonlinear eigenvalue problem for N smallest eigenvector-eigenvalue pairs, with N proportional to the number of electrons in the material system. Here, these calculations are computationally demanding and have asymptotic cubic scaling complexity with the number of electrons. Large-scale matrix eigenvalue problems arising from the discretization of the Kohn–Sham DFT equations employing a systematically convergent basis traditionally rely on iterative orthogonal projection methods, which are shown to be computationally efficient and scalable on massively parallel computing architectures. However, as the size of the material system increases, these methods are known to incur dominant computational costs through the Rayleigh–Ritz projection step of the discretized Kohn–Sham Hamiltonian matrix and the subsequent subspace diagonalization of the projected matrix. This work explores the potential of polynomial expansion approaches based on recursive Fermi-operator expansion as an alternative to the subspace diagonalization of the projected Hamiltonian matrix to reduce the computational cost. Subsequently, we perform a detailed comparison of various recursive polynomial expansion approaches to the traditional approach of explicit diagonalization on both multi-node central processing unit and graphics processing unit architectures and assess their relative performance in terms of accuracy, computational efficiency, scaling behavior, and energy efficiency.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
NASA NTRS1990

Numerical simulation of spatially-evolving instability

A computational study of the spatial stability of plane Poiseuille flow is presented. The numerical scheme employs a time-splitting method to integrate the full Navier-Stokes equations using spectral collocation/finite-difference discretization on a non-staggered mesh. The eigenvalue decomposition procedure is applied for the solution of the Poisson equations using the capacitance matrix technique. The buffer domain method is incorporated for the outflow boundary conditions. The input perturbation velocities are obtained by solving the Orr-Sommerfeld equation for the nonlinear eigenvalue problem employing the companion matrix method. Computational results are compared with the linear theory for two-dimensional disturbances.

Danabasoglu, G.↗
NASA NTRS1982

Calculation of exact vibration modes for plane grillages by the dynamic stiffness method

A dynamic stiffness method is developed for the calculation of the exact modal parameters for plane grillages which consist of straight and uniform beams with coincident elastic and inertial axes. Elementary bending-torsion beam theory is utilized, and bending translation is restricted to one direction. The exact bending-torsion dynamic stiffness matrix is obtained for a straight and uniform beam element with coincident elastic and inertial axes. The element stiffness matrices are assembled using the standard procedure of the static stiffness method to form the dynamic stiffness matrix of the complete grillage. The exact natural frequencies, mode shapes, and generalized masses of the grillage are then calculated by solving a nonlinear eigenvalue problem based on the dynamic stiffness matrix. The exact modal solutions for an example grillage are calculated and compared with the approximate solutions obtained by using the finite element method.

Hallauer, W. L., Jr.↗
NASA NTRS1985

Long-time asymptotics of a system for plasma diffusion

A system of parabolic nonlinear equations that describe the diffusion of a fully collisional plasma across a strong magnetic field is discussed. It is demonstrated that the solution to this system tends to a time asymptotic state which is of space-time separable form, theta(t)f(x). Furthermore, f(x) is independent of the initial conditions and theta(t) depends slightly on the initial conditions. The rate of decay of the temporal part is governed by a nonlinear eigenvalue problem. Since the equations are considered in a bounded domain we are able to analyze the effect of boundary conditions on the evolution of the system. Additional effects on radiation, heating, and particle injection can also be accounted for. Essential differences between the behavior of a fully-coupled system and a scalar equation are observed.

Rosenau, P.↗
NASA NTRS1987

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.↗
NASA NTRS1987

Vortex breakdown simulation based on a nonlinear inviscid model

It is shown that the inviscid equations governing steady axisymmetric flow with swirl, admit solutions with closed streamlines. Results are obtained using two different numerical algorithms. The first is based on a multigrid method for nonlinear eigenvalue problems, while the second is based on a least squares formulation.

Hafez, M. M.↗
NASA NTRS1989

Finite element large-amplitude free and forced vibrations of rectangular thin composite plates

A finite element formulation is presented for determining the large-amplitude free and steady-state forced vibration response of arbitrarily laminated anisotropic composite rectangular thin plates. The nonlinear stiffness and harmonic force matrices of an arbitrarily laminated composite rectangular plate element are developed for nonlinear free and forced vibration analyses. The linearized updated-mode method with nonlinear time function approximation is employed for the solution of the system nonlinear eigenvalue equations. The amplitude-frequency relations for convergence with gridwork refinement, different boundary conditions, aspect ratios, lamination angles and number of plies are presented. The finite element results are compared with available approximate continuum solutions.

Chiang, C. K.↗
NASA NTRS1992

Buckling of imperfect, anisotropic, ring-stiffened cylinders under combined loads

The objectives of this study is to develop an analysis to predict loads of ring-stiffened anisotropic cylinders subject to axial compression, torsion and internal pressure. This structure is modeled as a branched shell. A nonlinear axisymmetric prebuckling equilibrium state is assumed which is amenable to exact solution within each branch. Axisymmetric geometric imperfections are included. Buckling displacements are represented by a Fourier series in the circumferential coordinate and the finite element method in the axial coordinate. Application of Trefftz criterion to the second variation of the total potential energy leads to a nonlinear eigenvalue problem for the buckling load and mode. Results are presented for both unstiffened and ring-stiffened cylinders in the form of buckling interaction diagrams. Imperfections can cause an unexpected buckling mode in the ring web which would not occur for the perfect structure, and pressurization diminishes the benefit of adding rings to the unstiffened shell to increase the buckling load. The implementation of the analysis methodology into a structural sizing algorithm is discussed.

Ley, Robert P.↗
NASA NTRS1993

Nonlinear vibrations of thin arbitrarily laminated composite plates subjected to harmonic excitations using DKT elements

A finite element formulation is presented for determining the large-amplitude free and steady-state forced vibration response of arbitrarily laminated anisotropic composite thin plates using the Discrete Kirchhoff Theory (DKT) triangular elements. The nonlinear stiffness and harmonic force matrices of an arbitrarily laminated composite triangular plate element are developed for nonlinear free and forced vibration analyses. The linearized updated-mode method with nonlinear time function approximation is employed for the solution of the system nonlinear eigenvalue equations. The amplitude-frequency relations for convergence with gridwork refinement, triangular plates, different boundary conditions, lamination angles, number of plies, and uniform versus concentrated loads are presented.

Chiang, C. K.↗
NASA NTRS1992

Comparison of level-1 and level-2 buckling and postbuckling solutions

The accuracy of the solutions obtained with the Level-1 computational modules of DISDECO (Delft Interactive Shell Design Code) are investigated by comparing them with solutions obtained with ANILISA, a Level-2 computational module. Whereas the Level-1 solutions are based on the use of trigonometric functions to represent the out-of-plane displacement W and involve Galerkin type approximations to reduce the governing partial differential equations to simple algebraic expressions, the Level-2 solutions employ a truncated Fourier expansion in the circumferential direction and solve the resulting ordinary differential equations forming a nonlinear eigenvalue problem by a Stodola like technique. This involves the numerical integration of the set of ordinary differential equations, which in turn makes it possible to satisfy the specified boundary conditions exactly. It is shown that in many applications the Level-1 predictions concerning the critical buckling load of the perfect structure and its imperfection sensitivity are quite accurate and reliable. For those cases where this is not true, it is shown how the pre and postprocessing facilities of DISDECO can be used efficiently to arrive at a reliable prediction. This involves always the use of the Level-2 computational module ANILISA with its rigorous prebuckling solution and its feature which makes it possible to satisfy the specified boundary conditions exactly.

Arbocz, J.↗
NASA NTRS1994

Buckling of imperfect, anisotropic, ring-stiffened cylinders under combined loads

The objective of this study is to develop an anlysis to predict buckling loads of ring-stiffened anisotropic cylinders subject to axial compression, torsion, and internal pressure. This structure is modeled as a branched shell. A nonlinear axisymmetric prebuckling equilibrium state is assumed which is amenable to an exact solution within each branch. Axisymmetric geometric imperfections are included. Buckling displacements are represented by a Fourier series in the circumferential coordination and the finite-element method in the axial coordinate. Application of the Trefftz criterion to the second variation of the total potential energy leads to a nonlinear eigenvalue problem for the buckling load and mode. Results are presented for both unstiffened and ring-stiffened cylinders in the form of buckling interaction diagrams. Imperfections can cause an unexpected buckling mode in the ring web which would not occur for the perfect structure, and pressurization diminishes the benefit of adding rings to the unstiffened shell to increase the buckling load. The implementation of the analysis methodology into a structural sizing algorithm is discussed.

Ley, Robert P.↗
NASA NTRS1994

Rapid solution of large-scale systems of equations

The analysis and design of complex aerospace structures requires the rapid solution of large systems of linear and nonlinear equations, eigenvalue extraction for buckling, vibration and flutter modes, structural optimization and design sensitivity calculation. Computers with multiple processors and vector capabilities can offer substantial computational advantages over traditional scalar computer for these analyses. These computers fall into two categories: shared memory computers and distributed memory computers. This presentation covers general-purpose, highly efficient algorithms for generation/assembly or element matrices, solution of systems of linear and nonlinear equations, eigenvalue and design sensitivity analysis and optimization. All algorithms are coded in FORTRAN for shared memory computers and many are adapted to distributed memory computers. The capability and numerical performance of these algorithms will be addressed.

Storaasli, Olaf O.↗