Skip to main content
Log in

Limited conjugate gradient method for structural reliability analysis

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

A limited conjugate gradient method is proposed to improve the robustness and efficiency of the first-order reliability method (FORM). A new search direction vector is developed for structural reliability analysis using a limited scalar factor with Armijo’s rule and sufficient descent condition, namely limited Fletcher–Reeves (LFR) method. The conjugate gradient search direction is adaptively determined based on limited scalar factor and the instabilities of FORM formula are dynamically controlled by sufficient descent condition in the proposed LFR method. The LFR method is compared with the Hasofer–Lind and Rackwitz–Fiessler (HL–RF), stability transformation method (STM), finite-step size (FSS), chaotic conjugate stability transformation method (CCSTM), and directional stability transformation method (DSTM) using five mathematical and structural problems. Results of numerical examples illustrated that LFR is more efficient than the STM, FSS, CCSTM, and DSTM reliability methods and more robust than the HL–RF for highly non-linear performance functions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Abbreviations

C :

Involutory matrix

\(c\) :

Adjusting coefficient

CCSTM:

Chaotic conjugate stability transformation method

CGM:

Conjugate gradient method

DSTM:

Directional stability transformation method

FORM:

First-order reliability method

FR:

Conjugate Fletcher–Reeves method

FSS:

Finite-step size method

HL-RF:

Hasofer and Lind–Rackwitz and Fiessler method

LFR:

Limited conjugate Fletcher–Reeves method

LSF:

Limit state function

MPP:

Most probability failure point

SORM:

Second-order reliability method

STM:

Stability transformation method

\(\varvec{d}_{k}^{{}}\) :

Conjugate search direction vector

\(g()\) :

Limit state function

\(g() \le 0\) :

Failure region

\(f_{X}\) :

Joint probability density function

\(P_{\text{f}}\) :

Failure probability

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

Basic random variables

\(\varvec{X}^{\varvec{*}}\) :

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

\(\varvec{\alpha}_{k}^{c\lambda }\) :

Conjugate sensitivity vector

\(\varvec{\alpha}^{\lambda }\) :

Finite sensitivity vector

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

Sensitivity vector

\(\beta\) :

Reliability index

\(\varepsilon\) :

Stopping criterion

\(\mu\), \(\sigma\) :

Mean standard deviation

\(\lambda_{{}}^{C}\) :

Chaotic control factor

\(\lambda\) :

Finite-step size

\(\varPhi\) :

Standard normal cumulative distribution function

\(\nabla g(\varvec{U}_{k}^{{}} )\) :

Gradient vector of the LSF at point \(\varvec{U}_{k}^{{}}\)

\(\xi\) :

Step size or chaos control factor

\(\theta_{k}\) :

Conjugate scalar at iteration k

References

  1. Rashki M, Miri M, Moghaddam MA (2012) A new efficient simulation method to approximate the probability of failure and most probable point. Struct Saf 39:22–29

    Article  Google Scholar 

  2. Hamzehkolaei NS, Miri M, Rashki M (2016) An enhanced simulation-based design method coupled with meta-heuristic search algorithm for accurate reliability-based design optimization. Eng Comput 32(3):477–495

    Article  Google Scholar 

  3. Hasfer AM, Lind NC (1974) Exact and invariant second moment code format. J Eng Mech ASCE 100:111–121

    Google Scholar 

  4. Keshtegar B, Miri M (2014) Introducing conjugate gradient optimization for modified HL-RF method. Eng Comput 31:775–790

    Article  Google Scholar 

  5. Rackwitz R, Fiessler B (1978) Structural reliability under combined random load sequences. Comput Struct 9:489–494

    Article  MATH  Google Scholar 

  6. Gong JX, Yi P (2011) A robust iterative algorithm for structural reliability analysis. Struct Multidisc Optim 43:519–527

    Article  MATH  Google Scholar 

  7. Liu PL, Kiureghian AD (1991) Optimization algorithms for structural reliability. Struct Saf 9:161–177

    Article  Google Scholar 

  8. Santosh TV, Saraf RK, Ghosh AK, Kushwaha HS (2006) Optimum step length selection rule in modified HL–RF method for structural reliability. Int J Press Vessel Pip 83:742–748

    Article  Google Scholar 

  9. Wang LP, Grandhi RV (1994) Efficient safety index calculation for structural reliability analysis. Comput Struct 52:103–111

    Article  MATH  Google Scholar 

  10. Wang LP, Grandhi RV (1996) Safety index calculation using intervening variables for structural reliability analysis. Comput Struct 59:1139–1148

    Article  MATH  Google Scholar 

  11. Yang D (2010) Chaos control for numerical instability of first order reliability method. Commun Non-linear Sci Numer Simulat 15:3131–3141

    Article  MATH  Google Scholar 

  12. Keshtegar B, Miri M (2013) An enhanced HL-RF Method for the computation of structural failure probability based on relaxed approach. Civil Eng Infrastruct 1:69–80

    Google Scholar 

  13. Keshtegar B (2016) Stability iterative method for structural reliability analysis using a chaotic conjugate map. Nonlinear Dyn 84(4):2161–2174

    Article  MathSciNet  Google Scholar 

  14. Keshtegar B, Miri M (2014) Reliability analysis of corroded pipes using conjugate HL–RF algorithm based on average shear stress yield criterion. Eng Fail Anal 46:104–117

    Article  Google Scholar 

  15. Keshtegar B (2016) Chaotic conjugate stability transformation method for structural reliability analysis. Comput Methods Appl Mech Eng 310:866–885

    Article  MathSciNet  Google Scholar 

  16. Meng Z, Li G, Yang D, Zhan L (2016) A new directional stability transformation method of chaos control for first order reliability analysis. Struct Multidiscipl Optim. doi:10.1007/s00158-016-1525-z:1-12

    Google Scholar 

  17. van den Eshof J, Sleijpen GL (2004) Accurate conjugate gradient methods for families of shifted systems. Appl Numer Math 49(1):17–37

    Article  MathSciNet  MATH  Google Scholar 

  18. An XM, Li DH, Xiao YH (2011) Sufficient descent directions in unconstrained optimization. Comput Optim Appl 48(3):515–532

    Article  MathSciNet  MATH  Google Scholar 

  19. Deng S, Wan Z (2015) A three-term conjugate gradient algorithm for large-scale unconstrained optimization problems. Appl Numer Math 92:70–81

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhang L, Zhou W, Li D (2006) Global convergence of a modified Fletcher–Reeves conjugate gradient method with Armijo-type line search. Numer Math 104:561–572

    Article  MathSciNet  MATH  Google Scholar 

  21. Sastry SP, Shontz SM (2012) Performance characterization of nonlinear optimization methods for mesh quality improvement. Eng Comput 28(3):269–286

    Article  Google Scholar 

  22. Rivaie M, Mustafa M, Ismail M, Fauzi M (2011) A comparative study of conjugate gradient coefficient for unconstrained optimization. Aus J Bas Appl Sci 5:947–951

    MATH  Google Scholar 

  23. Fletcher R, Reeves C (1964) Function minimization by conjugate gradients. J Comput 7:149–154

    Article  MathSciNet  MATH  Google Scholar 

  24. Keshtegar B, Hao P (2016) A hybrid loop approach using the sufficient descent condition for accurate, robust and efficient reliability-based design optimization. J Mech Des 138(12):121401–121411

    Article  Google Scholar 

  25. Melchers RE, Ahammed M (2004) A fast approximate method for parameter sensitivity estimation in Monte Carlo structural reliability. Comput Struct 82:55–61

    Article  Google Scholar 

  26. Echard B, Gayton N, Lemaire M (2011) AK-MCS: an active learning reliability method combining Kriging and Monte Carlo simulation. Struct Saf 33:145–154

    Article  Google Scholar 

Download references

Acknowledgements

The authors are grateful to Dr. Zhen Hu for his significant comments to improve this work.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Behrooz Keshtegar.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Keshtegar, B. Limited conjugate gradient method for structural reliability analysis. Engineering with Computers 33, 621–629 (2017). https://doi.org/10.1007/s00366-016-0493-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-016-0493-7

Keywords

Navigation