Nonlinear Dynamics

, Volume 84, Issue 4, pp 2161–2174

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

• Behrooz Keshtegar
Original Paper

## 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.

## Keywords

Chaotic conjugate map First-order reliability method  Chaotic step size Chaos feedback control Logistic map

## 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

## List of symbols

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

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

$$c_\mathrm{m}$$

$$f^{C}(u_{k})$$

Discrete conjugate dynamical map

$$\mu , \sigma$$

Mean, standard deviation

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

Jacobian matrix

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