Numerical Instability and Convergence Control of First Order Reliability Method

摘要

The HL-RF algorithm of the first order reliability method (FORM) is widely applied to calculate the reliability index in structural reliability analysis and reliability-based design optimization. Nevertheless, its iterative solution could yield the phenomena of numerical instabilities such as periodic oscillation, bifurcation and chaos for some nonlinear problems. This paper proposes a novel method to overcome the numerical instabilities of HL-RF algorithm of FORM based on the principle of chaos control. The essential causes of chaotic dynamics for numerical instabilities including periodic oscillation and chaos of iterative solutions of FORM are exposed. Moreover, the geometrical properties of periodic oscillation of the iterative formulas derived from the FORM and performance measure approach are analyzed and compared. Finally, the stability transformation method (STM) of chaos feedback control is suggested to implement the convergence control of FORM. Several numerical examples validate the effectiveness of STM for oscillation and chaos control of FORM.