Nonlinear Dynamics

, Volume 84, Issue 4, pp 2161–2174 | Cite as

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

  • Behrooz KeshtegarEmail author
Original Paper


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.


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



Chaotic conjugate of FORM


Chaotic conjugate chaos control


Conjugate HL–RF


First-order reliability method


Finite-step length


Hasofer and Lind–Rackwitz and Fiessler method


Improved HL–RF method


Limit state function


Monte Carlo simulation


Relaxed HL–RF


Stability transformation method

List of symbols

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

Limit state function

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

Failure region


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


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


Chaotic adjusting coefficient


Discrete conjugate dynamical map

\(\mu , \sigma \)

Mean, standard deviation

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

Jacobian matrix


  1. 1.
    Du, X., Hu, Z.: First order reliability method with truncated random variables. J. Mech. Des. 134(9), 0910051–0910058 (2012)CrossRefGoogle Scholar
  2. 2.
    Santosh, T.V., Saraf, R.K., Ghosh, A.K., Kushwaha, H.S.: Optimum step length selection rule in modified HL-RF method for structural reliability. Int. J. Press Vessels Piping 83, 742–748 (2006)CrossRefGoogle Scholar
  3. 3.
    Hasofer, A.M., Lind, N.C.: Exact and invariant second-moment code format. J. Eng. Mech. Div. ASCE 100(1), 111–121 (1974)Google Scholar
  4. 4.
    Rackwitz, R., Fiessler, B.: Structural reliability under combined random load sequences. Comput Struct. 9, 489–494 (1978)CrossRefzbMATHGoogle Scholar
  5. 5.
    Keshtegar, B., Miri, M.: An enhanced HL–RF method for the computation of structural failure probability based on relaxed approach. Civ. Eng. Infrastruct. 1(1), 69–80 (2013)Google Scholar
  6. 6.
    Yang, D.: Chaos control for numerical instability of first order reliability method. Commun. Nonlinear Sci. Numer. Simulat. 5, 3131–3141 (2010)CrossRefzbMATHGoogle Scholar
  7. 7.
    Liu, P.L., Kiureghian, A.D.: Optimization algorithms for structural reliability. Struct. Safe 9, 161–177 (1991)CrossRefGoogle Scholar
  8. 8.
    Santos, S.R., Matioli, L.C., Beck, A.T.: New optimization algorithms for structural reliability analysis. Comput. Model. Eng. Sci. 83(1), 23–56 (2012)MathSciNetGoogle Scholar
  9. 9.
    Gong, J.X., Yi, P.: A robust iterative algorithm for structural reliability analysis. Struct. Multidisc. Optim. 43, 519–527 (2011)CrossRefzbMATHGoogle Scholar
  10. 10.
    Keshtegar, B., Miri, M.: Introducing conjugate gradient optimization for modified HL–RF Method. Eng. Comput. 31(4), 775–790 (2014)CrossRefGoogle Scholar
  11. 11.
    Keshtegar, B., Miri, M.: Reliability analysis of corroded pipes using conjugate HL–RF algorithm based on average shear stress yield criterion. Eng. Fail. Anal. 46(1), 104–117 (2014)CrossRefGoogle Scholar
  12. 12.
    Schmelcher, P., Diakonos, F.K.: Detecting unstable periodic orbits of chaotic dynamical systems. Phys. Rev. Lett. 78(25), 4733–4736 (1997)CrossRefGoogle Scholar
  13. 13.
    Salarieh, H., Kashani, S.M.M., Vossoughi, G., Alasty, A.: Stabilizing unstable fixed points of discrete chaotic systems via quasi-sliding mode method. Commun. Nonlinear Sci. Numer. Simul. 14, 839–849 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Pingel, D., Schmelcher, P., Diakonos, F.K.: Stability transformation: a tool to solve nonlinear problems. Phys. Rep. 400, 67–148 (2004)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Yang, D.X., Yi, P.: Numerical instabilities and convergence control for convex approximation methods. Nonlinear Dyn. 61, 605–622 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Yang, D.X., Yi, P.: Chaos control of performance measure approach for evaluation of probabilistic constraints. Struct. Multidiscip. Optim. 38, 83–92 (2009)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Lu, A., Liu, H., Zheng, X., Cong, W.: A variant spectral-type FR conjugate gradient method and its global convergence. Appl. Math. Comput. 217, 5547–5552 (2011)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Tavazoei, M.S., Haeri, M.: Comparison of different one-dimensional maps as chaotic search pattern in chaos optimization algorithms. Appl Math Comput. 187, 1076–1085 (2007)Google Scholar
  19. 19.
    Alpar, O.: Analysis of a new simple one-dimensional chaotic map. Nonlinear Dyn. 78, 771–778 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Wu, G.C., Baleanu, D.: Discrete fractional logistic map and its chaos. Nonlinear Dyn. 75, 283–287 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Zhao, Y.G., Lu, Z.H.: Fourth-moment standardization for structural reliability assessment. J. Struct. Eng. 133(7), 916–924 (2007)CrossRefGoogle Scholar
  22. 22.
    Hurtado, J.E., Alvarez, D.A.: The encounter of interval and probabilistic approaches to structural reliability at the design point. Comput. Methods Appl. Mech. Eng. 225–228, 74–94 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Civil EngineeringUniversity of ZabolZabolIran

Personalised recommendations