A Newton iteration‐based interval analysis method for nonlinear structural systems with uncertain‐but‐bounded parameters

Abstract

In this article, the propagating effect of uncertainties in nonlinear structural systems is investigated. Via considering uncertain-but-bounded parameters as interval variables, a Newton iteration-based interval analysis method (NI-IAM) is proposed to quantify the uncertainty of the nonlinear structural response, namely predicting the upper and lower bounds of the response. In the proposed method, a nonlinear structural system with uncertain interval parameters is described as a series of interval nonlinear equations. With Taylor series expansion, an interval iteration scheme is established to successively approximate the upper and lower bounds of the response based on the Newton method. The structural response bounds in every iteration step are updated by solving linear interval increment determination equations via the Lagrangian multiplier method. Thereout, an uncertain nonlinear problem is simplified to a series of uncertain linear issues. The convergence of the proposed method is further discussed in this article. Numerical examples are provided to demonstrate the effectiveness and applicability of the proposed method by contrast with several existing methods. Moreover, the compatibility between numerical results of the perturbation-based probabilistic method and the proposed method is then studied and verified.

A novel univariate dimension‐reduction based interval finite element method for static response prediction of uncertain structures

“…Uncertainties of things are divided into three types: stochastic uncertainty, [4][5][6] fuzzy uncertainty 7,8 and interval uncertainty. [9][10][11][12] They belong to the research areas of probability theory and mathematical statistics, fuzzy mathematics and interval mathematics, respectively. Probabilistic/ stochastic models deal with uncertain engineering problems under the premise that empirical data are sufficient to determine the probability density function of uncertainty.…”

“…Uncertainties of things are divided into three types: stochastic uncertainty, 4‐6 fuzzy uncertainty 7,8 and interval uncertainty 9‐12 . They belong to the research areas of probability theory and mathematical statistics, fuzzy mathematics and interval mathematics, respectively.…”

“…With the development of the optimization theory, many methods based on optimization algorithms have been proposed for uncertainty analysis. The gradient descent method 25 and Newton iteration method 10 are commonly‐used optimization methods. Wang et al 26 proposed the adjoint sensitivity‐based Taylor expansion method (ATEM) to approximate the structural response with small uncertainty ranges, and developed an adjoint‐based optimization method for the case with large uncertainty parameters.…”

An Adaptive Subinterval Finite Element Method Based on Dynamic Sensitivity Analysis for Structures with Uncertain-but-Bounded Parameters

“…De et al [20] reduce the computational cost of structural analyses with interval uncertainty using gradient descent. Qiu et al [21] proposed a Newton iteration interval analysis method to quantify the uncertainty of nonlinear structural response. Dinh-Cong et al [22] applied the lightning attachment procedure optimization to the interval finite element analysis.…”