Chaos control of performance measure approach for evaluation of probabilistic constraints

Performance measure approach (PMA) is a recently proposed method for evaluation of probabilistic constraints in reliability-based design optimization of structure. The advanced mean-value (AMV) method is well suitable for PMA due to its simplicity and efficiency. However, when the AMV iterative scheme is applied to search for the minimum performance target point for some nonlinear performance functions, the iterative sequences could fall into the periodic oscillation and even chaos. In the present paper, the phenomena of numerical instabilities of AMV iterative solutions are illustrated firstly. And the chaotic dynamics analysis on the iterative procedure of AMV method is performed. Then, the stability transformation method of chaos feedback control is suggested for the convergence control of AMV procedure in the parameter interval in which the iterative scheme fails. Numerical results of several nonlinear performance functions demonstrate that the control of periodic oscillation, bifurcation and chaos for AMV iterative procedure is achieved, and the stable convergence solutions are obtained.