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

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Abstract

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.

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Abbreviations

CC:

Chaotic conjugate of FORM

CCC:

Chaotic conjugate chaos control

CHL–RF:

Conjugate HL–RF

FORM:

First-order reliability method

FSL:

Finite-step length

HL–RF:

Hasofer and Lind–Rackwitz and Fiessler method

IHL–RF:

Improved HL–RF method

LSF:

Limit state function

MCS:

Monte Carlo simulation

RHL–RF:

Relaxed HL–RF

STM:

Stability transformation method

\(g({{\mathbf {X}}})\) :

Limit state function

\(g({{\mathbf {X}}})\le 0\) :

Failure region

\(f_{X}\) :

Joint probability density function

\({{\mathbf {U}}}^{*}, {{\mathbf {X}}}^{*}\) :

Most probable point (MPP) in U-space, X-space

\({{\mathbf {U}}}_{k}^{\mathrm{CC}}\) :

Point of the chaotic conjugate (CC) formula

\({{\mathbf {X}}}\) :

Basic random variables

\(\beta \) :

Reliability index

\(P_\mathrm{f}\) :

Failure probability

\(\varPhi \) :

Standard normal cumulative distribution function

\(\xi \) :

Chaos control factor

\({\varvec{\alpha }}_{k}\) :

Negative unit normal vector

\({\varvec{\alpha }}_{k}^{C}\) :

Conjugate unit vector

\({{\mathbf {d}}}_{k}\) :

Conjugate search direction vector

\(\lambda \) :

Finite-step size

\(c_\mathrm{m}\) :

Chaotic adjusting coefficient

\(f^{C}(u_{k})\) :

Discrete conjugate dynamical map

\(\mu , \sigma \) :

Mean, standard deviation

\(\rho ({{\mathbf {J}}})\) :

Jacobian matrix

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Authors and Affiliations

  1. Department of Civil Engineering, University of Zabol, P.B. 9861335-856, Zabol, Iran

    Behrooz Keshtegar

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  1. Behrooz Keshtegar
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Keshtegar, B. Stability iterative method for structural reliability analysis using a chaotic conjugate map. Nonlinear Dyn 84, 2161–2174 (2016). https://doi.org/10.1007/s11071-016-2636-1

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