Chaos control for numerical instability of first order reliability method |
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| Institution: | 1. Center for Optimization and Reliability in Engineering (CORE), Department of Civil Engineering – UFSC, Rua Joao Pio Duarte, s/n, Florianopolis, SC CEP 88034-000, Brazil;2. Center for Optimization and Reliability in Engineering (CORE), Department of Scientific Computing – UFPB, Cidade Universitaria, Joao Pessoa, PB CEP 58051-900, Brazil;3. Institut National des Sciences Appliquées (INSA) de Rouen, 76801 Saint Etienne du Rouvray CEDEX, France;1. Department of Civil Engineering, University of Zabol, P.B. 9861335856, Zabol, Iran;2. Department of Civil Engineering, University of Sistan and Baluchestan, P.B. 9816745563-161, Zahedan, Iran |
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| Abstract: | The HL-RF algorithm of the first order reliability method (FORM) is a kind of popular iterative algorithm for solving the reliability index in structural reliability analysis and reliability-based design optimization. However, there are the phenomena of convergence failure such as periodic oscillation, bifurcation and chaos in the FORM for some nonlinear problems. This paper suggests 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 revealed. 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 proposed to implement the convergence control of FORM. Several numerical examples with explicit or implicit HL-RF iterative formulas illustrate that the STM is effective, simple and versatile, and can control the periodic oscillation, bifurcation and chaos of the FORM iterative algorithm. |
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