A new stochastic residual error based homotopy approach for stability analysis of structures with large fluctuation of random parameters

Abstract

The large fluctuation of uncertain parameters introduces a great challenge in the stability analysis of structures. To address this problem, a novel stochastic residual error based homotopy method is proposed in this article. This new method used the concept of homotopy to reconstruct a new governing equation for stochastic elastic buckling analysis, and the closed‐form solutions of the isolated buckling eigenvalues and eigenvectors are obtained by the stochastic homotopy analysis method. On this basis, a pth order origin moment of the stochastic residual error with respect to the elastic buckling equation is defined. Then, the optimal form of the homotopy series can be determined automatically by minimizing the pth order origin moment, which overcomes the disadvantage of highly relying on sample values of the existing homotopy stochastic finite element method. Moreover, the proposed method is developed to deal with the stochastic closely spaced buckling eigenvalue problem. Three mathematical examples and three buckling eigenvalue examples, including a variable cross‐section column, a 7‐story frame, and a Kiewitt single‐layer latticed spherical shell, are performed to illustrate the accuracy and effectiveness of the proposed method by comparing with the existing methods when dealing with large fluctuation of random parameters.

An efficient Bayesian method with intrusive homotopy surrogate model for stochastic model updating

“…𝐘 • is the deterministic coefficient of eigenvalues or eigenvectors. The detailed derivation can be found in Zhang et al (2023). When 𝐘(𝜃; ℎ) represents the vector composed of all eigenvalues, each element of the structural frequency vector 𝐅(𝜽) equals to the square root of the corresponding element of 𝐘(𝜃; ℎ) divided by 2𝜋.…”

“…The detailed derivation can be found in Zhang et al. (2023). When $${{\bf Y}}(\theta ;h)$$ represents the vector composed of all eigenvalues, each element of the structural frequency vector $${{\bf F}}( {{\bm \theta }} )$$ equals to the square root of the corresponding element of $${{\bf Y}}(\theta ;h)$$ divided by $$2\pi $$.…”

“…Zhou et al, 2013), interval analysis methods (Haag et al, 2010;Khodaparast et al, 2011Khodaparast et al, , 2017, stochastic FE methods (B. Huang & Chen, 2019;Wu et al, 2023;Zhang et al, 2023), Bayesian methods (Beck & Katafygiotis, 1998;Ching et al, 2006;Yuen et al, 2019), and so on.…”

“…In recent years, numerous stochastic model updating methods emerged, which propels the uncertainty quantification of structural update parameters. These methods mainly include direct Monte Carlo simulations (MCS; Mares et al., 2006; Schuëller et al., 2009), perturbation methods (Hua et al., 2008; Khodaparast et al., 2008; Y. Xia et al., 2002), response surface method (RSM) and stochastic RSM (SRSM; Bao & Wang, 2015; Rui et al., 2013; L. Zhou et al., 2013), interval analysis methods (Haag et al., 2010; Khodaparast et al., 2011, 2017), stochastic FE methods (B. Huang & Chen, 2019; Wu et al., 2023; Zhang et al., 2023), Bayesian methods (Beck & Katafygiotis, 1998; Ching et al., 2006; Yuen et al., 2019), and so on.…”

“…To address these problems, a novel intrusive homotopy surrogate model (HSM) is proposed to combine with the MCMC sampling to update structural models. This new surrogate model comes from the explicit solution of a stochastic eigenvalue equation about update parameters, which is obtained by solving the equation using the homotopy stochastic FE method (B. Huang et al., 2018; Zhang & Huang, 2019; Zhang et al., 2023). This homotopy stochastic FE method inherits the advantages, which can accurately, efficiently, and explicitly deal with the nonlinear problems with changeable parameters, of the homotopy analysis method (Liao, 2010).…”