Structural and Multidisciplinary Optimization

, Volume 57, Issue 4, pp 1731–1747 | Cite as

Enhanced single-loop method for efficient reliability-based design optimization with complex constraints

  • Behrooz Keshtegar
  • Peng Hao


Reliability-based design optimization (RBDO) has been widely implemented for engineering design optimization when considering the uncertainty. The single loop approaches (SLA) are highly efficient but is prone to converge with inappropriate results for highly nonlinear probabilistic constraints. In this paper, a novel RBDO algorithm is proposed based on single loop approach and the enhanced chaos control method, named as enhanced single-loop method (ESM). The performance of SLA is enhanced using an adaptive inverse reliability method with limited number of iterations. The adaptive step size is computed based on a merit function which is computed using the results of the new and previous iterations. The iterations of the probabilistic constraints of RBDO models are manually controlled in the range from 1 to 10 in ESM. The efficiency and accuracy of the ESM are compared through four nonlinear RBDO problems with complex constraints, including a nonlinear mathematical problem, two engineering problems and a practical complex stiffened panel example with complex buckling constraint for aircraft design. Results illustrate that the proposed ESM is more efficient and robust than the performance measure approach and reliability index approach for RBDO problems.


Enhanced single-loop method Merit function Reliability-based design optimization Adaptive step size Complex aircraft panel 



design variables.


Young’s modulus.


objective function.


joint probability density function of the basic random variables.


limit state function.


stiffener height.

\( \tilde{\boldsymbol{n}} \)

enhanced search direction.


total number of iterations.


critical buckling load.


failure probability.

\( {\boldsymbol{s}}_k^j \)

shift vector of the j th probabilistic constraint at the k th cycle.


skin thickness.


stiffener thickness.


independent standard normal random variable.


most probable failure point.


structural weight.


random variables.


normalized steepest descent direction.


normalized sensitivities vector.


reliability index.

\( {\beta}_t^j \)

prescribed reliability index.


chaos control factor.


standard deviation.


adaptive step size.




Poisson’s ratio.


standard normal cumulative distribution function.


mean value.



This work was supported by University of Zabol under Grant No. UOZ-GR-9517-3, National Natural Science Foundation of China under Grant Nos. 11772078 and 11402049, and International Joint Research Project by University of Zabol under Grant No. IR-UOZ96-8.


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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Civil EngineeringUniversity of ZabolZabolIran
  2. 2.State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, International Center for Computational MechanicsDalian University of TechnologyDalianChina

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