Solution of stochastic eigenvalue problem by improved stochastic inverse power method (I-SIPM)

Abstract

Eigenvalue analysis is an important problem in a variety of fields. In structural mechanics in the field of naval architecture and ocean engineering, eigenvalue problems commonly appear in the context of, e.g. vibrations and buckling. In eigenvalue analysis, the physical characteristics are often considered as deterministic, such as mass, geometries, stiffness in the structures. However, in many practical cases, they are not deterministic. Such uncertainties may cause serious problems because the influence of the uncertainties is in general unknown. In order to solve the stochastic eigenvalue problem, in this article, we have proposed two methods. Firstly, the improved stochastic inverse power method (I-SIPM) based on response surface methodology is proposed. The method is different with previous stochastic inverse power method. The minimum eigenvalue and eigenvector of stochastic eigenvalue problems can be evaluated by using the proposed method. Secondly, the stochastic Wielandt deflation method (SWDM) is proposed which can realize to evaluate i th (i>1) eigenvalues and eigenvectors of stochastic eigenvalue problems. This is very important for solving natural mode and buckling mode analysis problem. Next, two example problems are investigated to show the validity of two new methods compared with a Monte-Carlo simulation, i.e. the vibration problem of a discrete 2-DOF system and the buckling problem of a continuous beam. Finally, the uncertainty estimation for the dynamic damper problem is discussed by using proposed method. The probability of the natural frequency falling into the range to be avoided is shown when the dynamic damper has a stochastic mass and stiffness. Keywords Improved stochastic inverse power method (I-SIPM)  Stochastic eigenvalue problem  Polynomial chaos expansion (PCE)  Stochastic Wielandt deflation method (SWDM)

Inexact Methods for Symmetric Stochastic Eigenvalue Problems

“…Meidani and Ghanem [22,23] formulated stochastic subspace iteration using a stochastic version of the modified Gram-Schmidt algorithm. Sousedík and Elman [33] introduced stochastic inverse subspace iteration by combining the two techniques, they showed that deflation of the mean matrix can be used to compute expansions of the interior eigenvalues, and they also showed that the stochastic Rayleigh quotient alone provides a good approximation of an eigenvalue expansion; see also [3,4,27] for closely related methods. The authors of [23,33] used a quadrature-based normalization of eigenvectors.…”