An ellipsoidal Newton’s iteration method of nonlinear structural systems with uncertain-but-bounded parameters

Abstract

This paper presents an ellipsoidal Newton’s iteration method for predicting the response of nonlinear structural systems with uncertain-but-bounded parameters. In the study, the uncertainty in parameters is expressed in terms of an ellipsoid set in an appropriate vector space and the bounds for the solution set of the nonlinear equations are aiming to be calculated effectively. In the framework of the convex set theory and Taylor series expansion, the ellipsoidal Newton’s iteration scheme is established. Two various models of the scheme depending on the different models of quantifying the region of the iterative solution in iterative calculation are discussed. The bounds of the solution are updated iteratively by using the maximum and minimum values of the solution increment, which can be obtained by solving the optimization problem. The convergence of the scheme is proved and the general procedure for its implementation is also presented. Three numerical examples are employed to illustrate the feasibility and accuracy of the proposed method in evaluating the bounds of nonlinear structural systems with uncertain-but-bounded parameters in comparison with the Monte-Carlo Simulation and the point-based iteration method.