Structural and Multidisciplinary Optimization

, Volume 57, Issue 3, pp 1079–1091 | Cite as

Convergence control of single loop approach for reliability-based design optimization

  • Zeng Meng
  • Dixiong Yang
  • Huanlin Zhou
  • Bo Ping Wang


For solution of reliability-based design optimization (RBDO) problems, single loop approach (SLA) shows high efficiency. Thus SLA is extensively used in RBDO. However, the iteration solution procedure by SLA is often oscillatory and non-convergent for RBDO with nonlinear performance function. This prevents the application of SLA to engineering design problems. In this paper, the chaotic single loop approach (CLSA) is proposed to achieve the convergence control of original iterative algorithm in SLA. The modification involves automated selection of the chaos control factor by solving a novel one-dimensional optimization model. Additionally, a new oscillation-checking method is constructed to detect the oscillation of iterative process of design variables. The computational capability of CLSA is demonstrated through five benchmark examples and one stiffened shell application. The comparison of numerical results indicates that CSLA is more efficient than the double loop approach and the decoupled approach. CSLA also solves the RBDO problems with highly nonlinear performance function and non-normal random variables stably.


Reliability-based design optimization Iterative computation Convergence control Chaotic single loop approach Oscillation-checking method 



Reliability-based design optimization


First order reliability method


Reliability index approach


Performance measure approach


Advanced mean value


Hybrid mean value method


Conjugate gradient analysis


Chaos control


Modified chaos control method


Sequential optimization and reliability assessment


Single loop single vector


Single loop approach


Reliable design space


Chaotic single loop approach


Standard normal space


Most probable target point


Objective function


Performance function


Design variables


Low bounds of design variables


Upper bounds of design variables


Random design variables


Mean of random design variables

\( {\boldsymbol{\upmu}}_{\mathbf{x}}^L \)

Low bounds of random variables

\( {\boldsymbol{\upmu}}_{\mathbf{x}}^U \)

Upper bounds of random variables


Standard deviation variables


Random variables


Mean of random variables


Standard deviations of random variables


Admissible failure probability


Target reliability index


Failure probability

fx , p(x, p)

joint probability density function


Normalized random variable


Chaos control factor


Involutory matrix


Response function vector


Iterative step number


Performance function


Performance function in U-space


Sensitivities of performance function with respect to random design variables x


Sensitivities of performance function with respect to random variables p



The supports of the National Natural Science Foundation of China (Grant Nos. 11602076 and 51605127), the Natural Science Foundation of Anhui Province (No. 1708085QA06) and the Fundamental Research Funds for the Central Universities of China (No JZ2016HGBZ0751) are much appreciated. The authors also thank Dr. Yuxue Pu for his comments and discussion.


  1. Aoues Y, Chateauneuf A (2010) Benchmark study of numerical methods for reliability-based design optimization. Struct Multidiscip Optim 41:277–294MathSciNetCrossRefzbMATHGoogle Scholar
  2. Chen X, Hasselman TK, Neill DJ (1997) Reliability based structural design optimization for practical applications. Paper presented at the Proceedings of the 38th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference, Kissimmee, Florida, 7–10 AprilGoogle Scholar
  3. Chen ZZ, Qiu HB, Gao L, Li PG (2013a) An optimal shifting vector approach for efficient probabilistic design. Struct Multidiscip Optim 47:905–920CrossRefGoogle Scholar
  4. Chen ZZ, Qiu HB, Gao L, Su L, Li PG (2013b) An adaptive decoupling approach for reliability-based design optimization. Comput Struct 117:58–66CrossRefGoogle Scholar
  5. Cheng GD, Xu L, Jiang L (2006) A sequential approximate programming strategy for reliability-based structural optimization. Comput Struct 84:1353–1367CrossRefGoogle Scholar
  6. Cho TM, Lee BC (2011) Reliability-based design optimization using convex linearization and sequential optimization and reliability assessment method. Struct Saf 33:42–50CrossRefGoogle Scholar
  7. Du XP, Chen W (2004) Sequential optimization and reliability assessment method for efficient probabilistic design. ASME J Mech Des 126:225–233CrossRefGoogle Scholar
  8. Du XP, Sudjianto A, Chen W (2004) An integrated framework for optimization under uncertainty using inverse reliability strategy. ASME J Mech Des 126:562–570CrossRefGoogle Scholar
  9. Ezzati G, Mammadov M, Kulkarni S (2015) A new reliability analysis method based on the conjugate gradient direction. Struct Multidiscip Optim 51:89–98MathSciNetCrossRefGoogle Scholar
  10. Kaymaz I, Marti K (2007) Reliability-based design optimization for elastoplastic mechanical structures. Comput Struct 85:615–625MathSciNetCrossRefGoogle Scholar
  11. Keshtegar B (2016) Chaotic conjugate stability transformation method for structural reliability analysis. Comput Methods Appl Mech Eng 310:866–885MathSciNetCrossRefGoogle Scholar
  12. Keshtegar B, Lee I (2016) Relaxed performance measure approach for reliability-based design optimization. Struct Multidiscip Optim 54:1439–1454MathSciNetCrossRefGoogle Scholar
  13. Kidane S, Li G, Helms J, Pang SS, Woldesenbet E (2003) Buckling load analysis of grid stiffened composite cylinders. Compos Part B 34:1–9CrossRefGoogle Scholar
  14. Lamberti L, Venkataraman S, Haftka RT, Johnson TF (2003) Preliminary design optimization of stiffened panels using approximate analysis models. Int J Numer Methods Eng 57:1351–1380CrossRefzbMATHGoogle Scholar
  15. Lee JO, Yang YS, Ruy WS (2002) A comparative study on reliability-index and target-performance-based probabilistic structural design optimization. Comput Struct 80:257–269CrossRefGoogle Scholar
  16. Lee I, Choi KK, Du L, Gorsich D (2008) Inverse analysis method using MPP-based dimension reduction for reliability-based design optimization of nonlinear and multi-dimensional systems. Comput Methods Appl Mech Eng 198:14–27CrossRefzbMATHGoogle Scholar
  17. Lee I, Choi KK, Gorsich D (2010) Sensitivity analyses of FORM-based and DRM-based performance measure approach (PMA) for reliability-based design optimization (RBDO). Int J Numer Methods Eng 82:26–46zbMATHGoogle Scholar
  18. Leriche R, Haftka RT (1993) Optimization of laminate stacking sequence for buckling load maximization by genetic algorithm. AIAA J 31:951–956CrossRefzbMATHGoogle Scholar
  19. Li G, Meng Z, Hu H (2015) An adaptive hybrid approach for reliability-based design optimization. Struct Multidiscip Optim 51:1051–1065MathSciNetCrossRefGoogle Scholar
  20. Liang J, Mourelatos ZP, Nikolaidis E (2007) A single-loop approach for system reliability-based design optimization. ASME J Mech Des 129(12):1215–1224CrossRefGoogle Scholar
  21. 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 100:773–792MathSciNetCrossRefzbMATHGoogle Scholar
  22. Meng Z, Hao P, Li G, Wang BP, Zhang K (2015a) Non-probabilistic reliability-based design optimization of stiffened shells under buckling constraint. Thin-Walled Struct 94:325–333CrossRefGoogle Scholar
  23. Meng Z, Li G, Wang BP, Hao P (2015b) A hybrid chaos control approach of the performance measure functions for reliability-based design optimization. Comput Struct 146:32–43CrossRefGoogle Scholar
  24. Meng Z, Li G, Yang DX, Zhan LC (2017) A new directional stability transformation method of chaos control for first order reliability analysis. Struct Multidiscip Optim 55:601–612MathSciNetCrossRefGoogle Scholar
  25. Pingel D, Schmelcher P, Diakonos FK (2004) Stability transformation: a tool to solve nonlinear problems. Phys Rep 400:67–148MathSciNetCrossRefGoogle Scholar
  26. Qu X, Haftka RT (2004) Reliability-based design optimization using probabilistic sufficiency factor. Struct Multidiscip Optim 27:314–325CrossRefGoogle Scholar
  27. Reddy MV, Grandhi RV (1994) Reliability based structural optimization: a simplified safety index approach. Comput Struct 53:1407–1418CrossRefzbMATHGoogle Scholar
  28. Royset J, Der Kiureghian A, Polak E (2001) Reliability-based optimal structural design by the decoupling approach. Reliab Eng Syst Saf 73:213–221CrossRefGoogle Scholar
  29. Schuëller GI, Jensen HA (2008) Computational methods in optimization considering uncertainties – an overview. Comput Methods Appl Mech Eng 198:2–13CrossRefzbMATHGoogle Scholar
  30. Shan S, Wang GG (2008) Reliable design space and complete single-loop reliability-based design optimization. Reliab Eng Syst Saf 93:1218–1230CrossRefGoogle Scholar
  31. Torii AJ, Lopez RH, Miguel LF (2016) A general RBDO decoupling approach for different reliability analysis methods. Struct Multidiscip Optim 54:317–332MathSciNetCrossRefGoogle Scholar
  32. Tu J, Choi KK, Park YH (1999) A new study on reliability-based design optimization. ASME J Mech Des 121:557–564CrossRefGoogle Scholar
  33. Valdebenito M, Schuëller G (2010) A survey on approaches for reliability-based optimization. Struct Multidiscip Optim 42:645–663MathSciNetCrossRefzbMATHGoogle Scholar
  34. Venkataraman S, Lamberti L, Haftka RT, Johnson TF (2003) Challenges in comparing numerical solutions for optimum weights of stiffened shells. J Spacecr Rocket 40:183–192CrossRefGoogle Scholar
  35. Yang DX (2010) Chaos control for numerical instability of first order reliability method. Commun Nonlinear Sci Numer Simul 15:3131–3141CrossRefzbMATHGoogle Scholar
  36. Youn BD, Choi KK, Park YH (2003) Hybrid analysis method for reliability-based design optimization. ASME J Mech Des 125:221–232CrossRefGoogle Scholar
  37. Zou T, Mahadevan S (2006) A direct decoupling approach for efficient reliability-based design optimization. Struct Multidiscip Optim 31:190–200CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.School of Civil EngineeringHefei University of TechnologyHefeiPeople’s Republic of China
  2. 2.Department of Engineering Mechanics, State Key Laboratory of Structural Analyses for Industrial EquipmentDalian University of TechnologyDalianPeople’s Republic of China
  3. 3.Department of Mechanical and Aerospace EngineeringUniversity of Texas at ArlingtonArlingtonUSA

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