Stability iterative method for structural reliability analysis using a chaotic conjugate map

In this paper, an iterative mathematical formula is developed to control instability solutions of first-order reliability method (FORM) using chaotic conjugate map. A nonlinear discrete map is proposed using a conjugate line search and a chaotic step size to search the most probability point. The chaotic step size is adjusted based on a finite-step size using Armijo line search and logistic map. A chaotic control factor is established using stability condition based on the results of the new and previous iterations, namely conjugate chaos control (CCC) method. The unstable solutions (i.e. periodic and chaos) of FORM without control are investigated using several nonlinear mathematical and structural/mechanical problems. The nonlinear conjugate map of FORM is accurately yielded to stable results. The CCC can improve the convergence speed of the FORM for concave and convex nonlinear problems. The CCC is more robust than the FORM algorithm without control and is more efficient than the other modified version algorithms of FORM.

[1]  A. M. Hasofer,et al.  Exact and Invariant Second-Moment Code Format , 1974 .

[2]  R. Rackwitz,et al.  Structural reliability under combined random load sequences , 1978 .

[3]  A. Kiureghian,et al.  Optimization algorithms for structural reliability , 1991 .

[4]  Peter Schmelcher,et al.  Detecting Unstable Periodic Orbits of Chaotic Dynamical Systems , 1997 .

[5]  D. Pingel,et al.  Stability transformation: a tool to solve nonlinear problems , 2004 .

[6]  T. V. Santosh,et al.  Optimum step length selection rule in modified HL-RF method for structural reliability , 2006 .

[7]  Yan-Gang Zhao,et al.  Fourth-Moment Standardization for Structural Reliability Assessment , 2007 .

[8]  Mohammad Saleh Tavazoei,et al.  Comparison of different one-dimensional maps as chaotic search pattern in chaos optimization algorithms , 2007, Appl. Math. Comput..

[9]  Dixiong Yang,et al.  Chaos control of performance measure approach for evaluation of probabilistic constraints , 2009 .

[10]  Aria Alasty,et al.  Stabilizing unstable fixed points of discrete chaotic systems via quasi-sliding mode method , 2009 .

[11]  Dixiong Yang.  Chaos control for numerical instability of first order reliability method , 2010 .

[12]  Dixiong Yang,et al.  Numerical instabilities and convergence control for convex approximation methods , 2010 .

[13]  Jinxin Gong,et al.  A robust iterative algorithm for structural reliability analysis , 2011 .

[14]  Hongwei Liu,et al.  A variant spectral-type FR conjugate gradient method and its global convergence , 2011, Appl. Math. Comput..

[15]  Zhen Hu,et al.  First Order Reliability Method With Truncated Random Variables , 2012 .

[16]  L. C. Matioli,et al.  New Optimization Algorithms for Structural Reliability Analysis , 2012 .

[17]  Jorge E. Hurtado,et al.  The encounter of interval and probabilistic approaches to structural reliability at the design point , 2012 .

[18]  Mahmoud Miri,et al.  An Enhanced HL-RF Method for the Computation of Structural Failure Probability Based On Relaxed Approach , 2013 .

[19]  B. Keshtegar,et al.  Reliability analysis of corroded pipes using conjugate HL–RF algorithm based on average shear stress yield criterion , 2014 .

[20]  Orcan Alpar,et al.  Analysis of a new simple one dimensional chaotic map , 2014 .

[21]  D. Baleanu,et al.  Discrete fractional logistic map and its chaos , 2014 .

[22]  B. Keshtegar,et al.  Introducing conjugate gradient optimization for modified HL-RF method , 2014 .