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Hybrid perturbation-Polynomial Chaos approaches to the random algebraic eigenvalue problem

机译：随机代数特征值问题的混合摄动-多项式混沌方法

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摘要

The analysis of structures is affected by uncertainty in the structure's material properties, geometric parameters, boundary conditions and applied loads. These uncertainties can be modelled by random variables and random fields. Amongst the various problems affected by uncertainty, the random eigenvalue problem is specially important when analyzing the dynamic behavior or the buckling of a structure. The methods that stand out in dealing with the random eigenvalue problem are the perturbation method and methods based on Monte Carlo Simulation. In the past few years, methods based on Polynomial Chaos (PC) have been developed for this problem, where each eigenvalue and eigenvector are represented by a PC expansion. In this paper four variants of a method hybridizing perturbation and PC expansion approaches are proposed and compared. The methods use Rayleigh quotient, the power method, the inverse power method and the eigenvalue equation. PC expansions of eigenvalues and eigenvectors are obtained with the proposed methods. The new methods are applied to the problem of an Euler Bernoulli beam and a thin plate with stochastic properties.
机译：结构的分析受到结构材料特性，几何参数，边界条件和施加载荷的不确定性的影响。这些不确定性可以通过随机变量和随机字段来建模。在受不确定性影响的各种问题中，随机特征值问题在分析结构的动态行为或屈曲时特别重要。在处理随机特征值问题上脱颖而出的方法是摄动方法和基于蒙特卡洛模拟的方法。在过去的几年中，针对此问题开发了基于多项式混沌（PC）的方法，其中每个特征值和特征向量都由PC展开表示。在本文中，提出并比较了摄动与PC扩展方法相混合的方法的四个变体。这些方法使用瑞利商，幂方法，逆幂方法和特征值方程。利用所提出的方法获得了特征值和特征向量的PC展开。这些新方法被用于求解Euler Bernoulli梁和具有随机特性的薄板的问题。

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