Skip to main content
SpringerLink
Log in

A new directional stability transformation method of chaos control for first order reliability analysis

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

Abstract

The HL-RF iterative algorithm of the first order reliability method (FORM) is popularly applied to evaluate reliability index in structural reliability analysis and reliability-based design optimization. However, it sometimes suffers from non-convergence problems, such as bifurcation, periodic oscillation, and chaos for nonlinear limit state functions. This paper derives the formulation of the Lyapunov exponents for the HL-RF iterative algorithm in order to identify these complicated numerical instability phenomena of discrete chaotic dynamic systems. Moreover, the essential cause of low efficiency for the stability transform method (STM) of convergence control of FORM is revealed. Then, a novel method, directional stability transformation method (DSTM), is proposed to reduce the number of function evaluations of original STM as a chaos feedback control approach. The efficiency and convergence of different reliability evaluation methods, including the HL-RF algorithm, STM and DSTM, are analyzed and compared by several numerical examples. It is indicated that the proposed DSTM method is versatile, efficient and robust, and the bifurcation, periodic oscillation, and chaos of FORM is controlled effectively.

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

Abbreviations

FORM:

First order reliability method

STM:

Stability transfor `mation method

DSTM:

Directional stability transformation method

SORM:

Second order reliability method

MPP:

Most probable point

PMA:

Performance measure approach

U-space:

Standard normal space

SR1:

Symmetric rank-one

LE:

Lyapunov exponent

X :

Random variable vector

U :

Normalized random variable vector

U * :

Most probable point

d :

Search direction

β :

Reliability index

μ :

Mean vector

σ :

Standard deviation vector

g(X):

Performance function

∇g(·):

Sensitivities of performance function

J :

Jacobian matrix

H :

Hessian matrix

P f :

Failure probability

Ф(·):

Cumulative distribution function

f(·):

Iterative function vector

C :

Involutory matrix

λ :

Control factor

qr[.]:

QR-factorization process

ñ(U k):

Radial direction

n ⊥(U k):

Circumferential direction

References

  • Armiyoon AR, Wu CQ (2015) A novel method to identify boundaries of basins of attraction in a dynamical system using Lyapunov exponents and Monte Carlo techniques. Nonlinear Dyn 79:275–293

    Article  MathSciNet  Google Scholar 

  • Bae HR, Grandhi RV, Canfield RA (2006) Sensitivity analysis of structural response uncertainty propagation using evidence theory. Struct Multidiscip Optim 31:270–279

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Danca MF (2015) Lyapunov exponents of a class of piecewise continuous systems of fractional order. Nonlinear Dyn 81:227–237

    Article  MathSciNet  MATH  Google Scholar 

  • Du XP, Chen W (2004) Sequential optimization and reliability assessment method for efficient probabilistic design. J Mech Des 126:225–233

    Article  Google Scholar 

  • Gu L, Yang RJ, Tho CH, Makowski M, Faruque O, Li Y (2001) Optimization and robustness for crashworthiness of side impact. Int J Veh Des 26:348–360

    Article  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Hubertus F, Udwadia FE, Proskurowski W (1997) An efficient QR based method for the computation of Lyapunov exponents. Phys D 101:1–16

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Koduru S, Haukaas T (2010) Feasibility of FORM in finite element reliability analysis. Struct Saf 32:145–153

    Article  Google Scholar 

  • Lee I, Noh Y, Yoo D (2012) A novel second-order reliability method (SORM) using noncentral or generalized Chi-squared distributions. J Mech Des 134:100912

    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

  • Lim J, Lee B, Lee I (2014) Second-order reliability method-based inverse reliability analysis using Hessian update for accurate and efficient reliability-based design optimization. Int J Numer Methods Eng 100:773–792

    Article  MathSciNet  MATH  Google Scholar 

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

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

    Article  Google Scholar 

  • Petkov BH, Vitale V, Mazzola M, Lanconelli C, Lupi A (2015) Chaotic behaviour of the short-term variations in ozone column observed in Arctic. Commun Nonlinear Sci Numer Simul 26:238–249

    Article  Google Scholar 

  • Pingel D, Schmelcher P, Diakonos FK (2001) Detecting unstable periodic orbits in chaotic continuous-time dynamical systems. Phys Rev E 64:026214

    Article  Google Scholar 

  • Pingel D, Schmelcher P, Diakonos FK (2004) Stability transformation: a tool to solve nonlinear problems. Phys Rep 400:67–148

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Reddy MV, Grandhi RV (1994) Reliability based structural optimization: a simplified safety index approach. Comput Struct 53:1407–1418

    Article  MATH  Google Scholar 

  • Santos SR, Matioli LC, Beck AT (2012) New optimization algorithms for structural reliability analysis. CMES Comput Model Eng 83:23–55

    MathSciNet  Google Scholar 

  • Schmelcher P, Diakonos FK (1998) General approach to the localization of unstable periodic orbits in chaotic dynamical systems. Phys Rev E 57:2739–2746

    Article  Google Scholar 

  • ur Rehman S, Langelaar M (2015) Efficient global robust optimization of unconstrained problems affected by parametric uncertainties. Struct Multidiscip Optim 52:319–336

    Article  MathSciNet  Google Scholar 

  • Wen GL, Xu DL (2005) Nonlinear observer control for full-state projective synchronization in chaotic continuous-time systems. Chaos Solitons Fractals 26:71–77

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Yang DX, Li G, Cheng GD (2006) Convergence analysis of first order reliability method using chaos theory. Comput Struct 84:563–571

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Youn BD, Xi Z, Wang P (2008) Eigenvector dimension reduction (EDR) method for sensitivity-free probability analysis. Struct Multidiscip Optim 37:13–28

    Article  MathSciNet  MATH  Google Scholar 

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

  • Zhang J, Du X (2010) A second-order reliability method with first-order efficiency. J Mech Des 132(10):101006

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The supports of the Fundamental Research Funds for the Central Universities of China (No JZ2016HGBZ0751) and the National Natural Science Foundation of China (Grant Nos. 90815023 and 51021140006) are greatly appreciated. The authors also thank Professor Gengdong Cheng and Dr. Hao Hu for their comments and discussion.

Author information

Authors and Affiliations

  1. School of Civil Engineering, Hefei University of Technology, Hefei, 230009, People’s Republic of China

    Zeng Meng

  2. Department of Engineering Mechanics, State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, 116024, People’s Republic of China

    Gang Li, Dixiong Yang & Lichao Zhan

Authors
  1. Zeng Meng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dixiong Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Lichao Zhan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zeng Meng.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Meng, Z., Li, G., Yang, D. et al. A new directional stability transformation method of chaos control for first order reliability analysis. Struct Multidisc Optim 55, 601–612 (2017). https://doi.org/10.1007/s00158-016-1525-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00158-016-1525-z

Keywords

Search

Navigation