Keywords: convergence control, optimization algorithms, reliability analysis, iterative failure, chaotic dynamics, stability transformation method.

As is known, both the structural optimization and reliability analysis algorithms such as the Newton method, convex approximation methods, first order reliability method (FORM) and the performance measure approach (PMA) etc., usually adopt iterative schemes. However, they could produce the numerical instabilities of iterative oscillation and chaos for some nonlinear problems. This paper presents a novel method to control the convergent failure of iterative procedures in optimization and reliability analysis algorithms from the perspective of chaotic dynamics and chaos control.

The Newton method as an ancient optimization algorithm could generate the periodic oscillation of solution for minimizing nonlinear functions. Furthermore, the convex approximation methods also yield the non-convergence of iterative oscillation for solving some structural optimization problems. In the reliability based structural optimization two approaches are proposed to deal with the probabilistic constraints, i.e. the first order reliability method and performance measure approach. Generally, the Hazofer-Lind-Rackwitz-Fiessler (HLRF) scheme and advanced mean value method are iteratively used to search the most probable point for the FORM and PMA respectively. Nevertheless, there is the phenomenon of periodic oscillation for computing some nonlinear performance functions.

Up to date, the essential causes for convergent failure of iterative algorithms in engineering system analysis have not yet explored and revealed clearly. And usually, a superficial explanation is given and the algorithm improvement is implemented based on the geometric meaning, intuition and experience [1]. In this paper, we introduce the chaos theory to analyze and control the non-convergence of iterative schemes for structural optimization and reliability analysis.

From the viewpoint of chaotic dynamics, the iterative procedure in structural optimization and reliability analysis forms a nonlinear map or discrete dynamical system. If the maximum of the absolute eigenvalues (i.e. spectral radius) of Jacobian matrix of the dynamical system at the fixed point is larger than 1, the fixed point will lose its stability, then the periodic oscillation and even chaos occurs. Fortunately, the chaos control methods can catch the unstable fixed point embedded in the chaotic attractor or periodic orbit of the nonlinear dynamical system through implementing the target guidance and position. As for a kind of chaos feedback control method, the stability transformation method can stabilize the all unknown and unstable fixed points involved in the periodic or chaotic orbits of a discrete dynamical system with the advantages of convenient implementation and easy comprehension.

Some numerical examples of structural optimization and reliability analysis demonstrate that the convergence failure can be overcome and the stable convergence solutions of dynamical system can be obtained by the stability transformation method. For convergence control of iterative algorithms in engineering systems the stability transformation method is simple, general and effective.

1

D.X. Yang, G. Li, G.D. Cheng, "Convergence analysis of first order reliability method using chaos theory", Computers and Structures, 84(8-9), 563-571, 2006. doi:10.1016/j.compstruc.2005.11.009

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