Skip to main content

Advertisement

SpringerLink
Log in

A novel first–order reliability method based on performance measure approach for highly nonlinear problems

  • RESEARCH PAPER
  • Published:
Structural and Multidisciplinary Optimization Aims and scope Submit manuscript

Abstract

The first–order reliability method (FORM) is widely used in structural reliability analysis for its simplicity and efficiency. It can be solved by gradient–based algorithms, on which the nonlinear degree of performance functions may have a great influence. On the other hand, evolutionary algorithms could achieve convergence solutions even for highly nonlinear performance function, usually with expensive computational cost. To overcome their drawbacks, we propose a new reliability analysis method called PMA–IACC to search the most probable failure point (MPP). As the inverse reliability analysis and reliability analysis are reversible each other, the performance measure approach (PMA) of the inverse reliability analysis could be used for the reliability analysis based on the multi–objective optimization theory. To enhance the efficiency and robustness of the inverse reliability, the improved adaptive chaos control (IACC) method is proposed and then it is integrated into the PMA reliability analysis strategy. Five illustrative examples, including three two–dimensional problems and two multi–dimensional problems, demonstrate the outstanding efficiency and robustness of the PMA–IACC over other prevalent approaches.

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

Access this article

Log in via an institution

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
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  • Bai Q (2010) Analysis of particle swarm optimization algorithm. Comput Inf Sci 3(1):180–184

    Google Scholar 

  • Basudhar A, Missoum S (2009) A sampling–based approach for probabilistic design with random fields. Comput Methods Appl Mech Eng 198(47–48):3647–3655

    Article  MATH  Google Scholar 

  • Cheng J (2007) Hybrid genetic algorithms for structural reliability analysis. Comput Struct 85(19–20):1524–1533

    Article  Google Scholar 

  • Ditlevsen O (1981) Principle of normal tail approximation. J Eng Mech Div 107(6):1191–1208

    Google Scholar 

  • Elegbede C (2005) Structural reliability assessment based on particles swarm optimization. Struct Saf 27(2):171–186

    Article  Google Scholar 

  • Elishakoff I (2000) Uncertain buckling: its past, present and future. Int J Solids Struct 37(46–47):6869–6889

    Article  MATH  Google Scholar 

  • Ezzati G, Mammadov M, Kulkarni S (2015) A new reliability analysis method based on the conjugate gradient direction. Struct Multidiscip Optim 51(1):89–98

    Article  MathSciNet  Google Scholar 

  • Hao P, Wang YT, Liu C, Wang B, Wu H (2017a) A novel non-probabilistic reliability-based design optimization algorithm using enhanced chaos control method. Comput Meth Appl Mech Eng 318: 572–593

  • Hao P, Wang YT, Liu XX, Wang B, Li G, Wang LP (2017b) An efficient adaptive-loop method for non-probabilistic reliability-based design optimization. Comput Meth Appl Mech Eng 324: 689−711

  • Hasofer AM, Lind NC (1974) Exact and invariant second−moment code format. J Eng Mech Div 100(1): 111−121

  • Hu Z, Du XP (2015) First order reliability method for time–variant problems using series expansions. Struct Multidiscip Optim 51(1):1–21

    Article  MathSciNet  Google Scholar 

  • Jiang C, Han S, Ji M et al (2015) A new method to solve the structural reliability index based on homotopy analysis. Acta Mech 226(4):1067–1083

    Article  MathSciNet  Google Scholar 

  • Kennedy J, Eberhart RC (1995) Particle swarm optimization. In: Proceedings of IEEE international conference on neural networks, Perth, Australia 4, pp 1942–1948

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Keshtegar B, Lee I (2016) Relaxed performance measure approach for reliability-based design optimization. Struct Multidiscip Optim 54(6):1439–1454

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  • Kiureghian AD, Stefano MD (1991) Efficient algorithm for second–order reliability analysis. J Eng Mech 117(12):2904–2923

    Article  Google Scholar 

  • Lopez RH, Torii AJ, Miguel LFF, Souza Cursi JE (2015) Overcoming the drawbacks of the FORM using a full characterization method. Struct Saf 54:57–63

  • Lee JO, Yang YS, Ruy WS (2002) A comparative study on reliability-index and target-performance-based probabilistic structural design optimization. Comput Struct 80(3–4):257–269

    Article  Google Scholar 

  • Lee I, Choi KK, Du L, Gorsich D (2008a) Inverse analysis method using MPP-based dimension reduction for reliability-based design optimization of nonlinear and multi-dimensional systems. Comput Methods Appl Mech Eng 198(1):14–27

    Article  MATH  Google Scholar 

  • Lee I, Choi KK, Gorsich D (2008b) System reliability-based design optimization using MPP-based dimension reduction method. Struct Multidiscip Optim 41(6):823–839

    Article  MATH  Google Scholar 

  • Li G, Meng Z, Hu H (2015) An adaptive hybrid approach for reliability-based design optimization. Struct Multidiscip Optim 51(5):1051–1065

    Article  MathSciNet  Google Scholar 

  • Liu PL, Kiureghian AD (1991) Optimization algorithms for structural reliability. Struct Saf 9(3):161–177

    Article  Google Scholar 

  • Lopez RH, Torii AJ, Miguel LFF, Souza Cursi JE (2015) Overcoming the drawbacks of the FORM using a full characterization method. Struct Saf 54:57–63

  • Marler RT, Arora JS (2004) Survey of multi–objective optimization methods for engineering. Struct Multidiscip Optim 26(6):369–395

    Article  MathSciNet  MATH  Google Scholar 

  • Meng Z, Li G, Wang BP et al (2015) A hybrid chaos control approach of the performance measure functions for reliability–based design optimization. Comput Struct 146:32–43

    Article  Google Scholar 

  • Meng Z, Li G, Yang DX et al (2017a) A new directional stability transformation method of chaos control for first order reliability analysis. Struct Multidiscip Optim 55(2):601–612

    Article  MathSciNet  Google Scholar 

  • Meng Z, Yang DX, Zhou HL, Wang BP (2017b) Convergence control of single loop approach for reliability-based design optimization. Struct Multidiscip Optim. doi:10.1007/s00158-017-1796-z

  • Meng Z, Yang DX, Zhou HL, Yu B (2017c) An accurate and efficient reliability-based design optimization using the second order reliability method and improved stability transformation method. Eng Optim. doi:10.1080/0305215x.2017.1351962

  • Mezura-Montes E, Coello CAC (2011) Constraint-handling in nature-inspired numerical optimization: past, present and future. Swarm Evol Comput 1(4):173–194

    Article  Google Scholar 

  • Nataf A (1962) Détermination des distributions de probabilités dont les marges sont données. C R Hebd SeÂances Acad Sci 255:42–43

    MATH  Google Scholar 

  • Nickabadi A, Ebadzadeh MM, Safabakhsh R (2011) A novel particle swarm optimization algorithm with adaptive inertia weight. Appl Soft Comput 11(4):3658–3670

    Article  Google Scholar 

  • Parsopoulos KE, Vrahatis MN (2002) Recent approaches to global optimization problems through particle swarm optimization. Nat Comput 1(2–3):235–306

    Article  MathSciNet  MATH  Google Scholar 

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

  • 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 

  • Rosenblatt M (1952) Remarks on a multivariate transformation. Ann Math Stat 23:470–472

    Article  MathSciNet  MATH  Google Scholar 

  • Santos SR, Matioli LC, Beck AT (2012) New optimization algorithms for structural reliability analysis. Comput Model Eng Sci 83(1):23–56

    MathSciNet  MATH  Google Scholar 

  • Santosh TV, Saraf RK, Ghosh AK et al (2006) Optimum step length selection rule in modified HL–RF method for structural reliability. Int J Press Vessel Pip 83(10):742–748

    Article  Google Scholar 

  • Shi Y, Eherhart RC (1998) Parameter selection in particle swarm optimization. In: Proceedings of International conference on evolutionary programming. Springer, Berlin, Heidelberg, pp 591–600

  • Shin J, Lee I (2015) Reliability analysis and reliability–based design optimization of roadway horizontal curves using a first–order reliability method. Eng Optim 47(5):622–641

    Article  Google Scholar 

  • Tu J, Choi KK (1999) A new study on reliability–based design optimization. J Mech Des 121(4):557–564

    Article  Google Scholar 

  • Verhaeghe W, Elishakoff I (2013) Reliability based bridging of the gap between system’s safety factors associated with different failure modes. Eng Struct 49:606–614

    Article  Google Scholar 

  • Wang LP, Grandhi RV (1994) Efficient safety index calculation for structural reliability analysis. Comput Struct 52(1):103–111

    Article  MATH  Google Scholar 

  • Wang LP, Grandhi RV (1996) Safety index calculation using intervening variables for structural reliability analysis. Comput Struct 59(6):1139–1148

    Article  MATH  Google Scholar 

  • Wu YT, Millwater HR, Cruse TA (1990) Advanced probabilistic structural analysis method for implicit performance functions. AIAA J 28(9):1663–1669

    Article  Google Scholar 

  • Yang DX (2010) Chaos control for numerical instability of first order reliability method. Commun Nonlinear Sci Numer Simul 15(10):3131–3141

    Article  MATH  Google Scholar 

  • Yang DX, Yi P (2009) Chaos control of performance measure approach for evaluation of probabilistic constraints. Struct Multidiscip Optim 38(1):83–92

    Article  Google Scholar 

  • Yi P, Zhu Z (2016) Step length adjustment iterative algorithm for inverse reliability analysis. Struct Multidiscip Optim 54(4):999–1009

    Article  MathSciNet  Google Scholar 

  • Youn BD, Choi KK (2004) A new response surface methodology for reliability–based design optimization. Comput Struct 82(2–3):241–256

    Article  Google Scholar 

  • Youn BD, Choi KK, Park YH (2003) Hybrid analysis method for reliability-based design optimization. J Mech Des 125(2):221–232

    Article  Google Scholar 

  • Youn BD, Choi KK, Yang RJ et al (2004) Reliability-based design optimization for crashworthiness of vehicle side impact. Struct Multidiscip Optim 26(3):272–283

    Article  Google Scholar 

  • Youn BD, Choi KK, Du L (2005a) Adaptive probability analysis using an enhanced hybrid mean value method. Struct Multidiscip Optim 29(2):134–148

    Article  Google Scholar 

  • Youn BD, Choi KK, Du L (2005b) Enriched performance measure approach for reliability-based design optimization. AIAA J 43(4):874–884

    Article  Google Scholar 

  • Zhan ZH, Zhang J, Li Y, Chung HS (2009) Adaptive particle swarm optimization. IEEE Trans Syst Man Cybern B Cybern 39(6):1362–1381

    Article  Google Scholar 

  • Zhang Y, Kiureghian AD (1995) Two improved algorithms for reliability analysis. In: Reliability and optimization of structural systems. Springer, Boston, pp 297–304

  • Zhao YG, Jiang JR (1995) A structural reliability analysis method based on genetic algorithm. Earthq Eng Eng Vib 15(3):48–58

    Google Scholar 

  • Zhou CC, Lu ZZ, Zhang F et al (2015) An adaptive reliability method combining relevance vector machine and importance sampling. Struct Multidiscip Optim 52(5):945–957

    Article  MathSciNet  Google Scholar 

  • Zou DX, Gao LQ, Wu JH et al (2010) A novel global harmony search algorithm for reliability problems. Comput Ind Eng 58(2):307–316

    Article  Google Scholar 

Download references

Acknowledgments

The support of the National Basic Research Program of China (Grant Nos. 2014CB046506 and 2014CB046803) and the National Natural Science Foundation of China of 11372061 is greatly appreciated. The authors also thank Professor Gengdong Cheng, Professor Dixiong Yang for their comments and discussion to guarantee the quality of this paper.

Author information

Authors and Affiliations

  1. Department of Engineering Mechanics, State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, 116024, China

    Gang Li & Bin Li

  2. School of Civil Engineering, Yancheng Institute of Technology, Yancheng, 224051, China

    Hao Hu

Authors
  1. Gang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hao Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gang Li.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, G., Li, B. & Hu, H. A novel first–order reliability method based on performance measure approach for highly nonlinear problems. Struct Multidisc Optim 57, 1593–1610 (2018). https://doi.org/10.1007/s00158-017-1830-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00158-017-1830-1

Keywords

Search

Navigation