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Structural and Multidisciplinary Optimization

, Volume 59, Issue 2, pp 439–460 | Cite as

A reliability-based optimization method using sequential surrogate model and Monte Carlo simulation

  • Xu Li
  • Chunlin GongEmail author
  • Liangxian Gu
  • Zhao Jing
  • Hai Fang
  • Ruichao Gao
Research Paper
First Online:
  • 200 Downloads

Abstract

This paper presents a sequential surrogate model method for reliability-based optimization (SSRBO), which aims to reduce the number of the expensive black-box function calls in reliability-based optimization. The proposed method consists of three key steps. First, the initial samples are selected to construct radial basis function surrogate models for the objective and constraint functions, respectively. Second, by solving a series of special optimization problems in terms of the surrogate models, local samples are identified and added in the vicinity of the current optimal point to refine the surrogate models. Third, by solving the optimization problem with the shifted constraints, the current optimal point is obtained. Then, at the current optimal point, the Monte Carlo simulation based on the surrogate models is carried out to obtain the cumulative distribution functions (CDFs) of the constraints. The CDFs and target reliabilities are used to update the offsets of the constraints for the next iteration. Therefore, the original problem is decomposed to serial cheap surrogate-based deterministic problems and Monte Carlo simulations. Several examples are adopted to verify SSRBO. The results show that the number of the expensive black-box function calls is reduced exponentially without losing of precision compared to the alternative methods, which illustrates the efficiency and accuracy of the proposed method.

Keywords

Expensive black box function Radial basis function Sequential sampling Reliability-based optimization Monte Carlo simulation 

Nomenclature

X

Vector of random variables

x

Mean value of X

xL, xU

Lower and upper bounds of x

U

Vector of random variables in standard normal space

u

Mean value of U

m

Number of variables

p

Number of constraint functions

ε

Difference vector between X and x

R

Vector of constraint reliabilities

J(·)

Objective function

g(·)

Vector of constraint functions

\( \widehat{\left(\cdotp \right)} \)

Value of surrogate models

(·)i

The ith component of a vector

E(·)

Expectation of a random variable

P{·}

Probability of a random variable

CDF

Cumulative distribution function

Fε(·)

Vectorized CDF for ε

FU(·)

Vectorized CDF for U

βi

The ith reliability index of the constraint functions

ϕ(·)

CDF of the standard normal distribution

RBO

Reliability-based optimization

SSRBO

Sequential surrogate reliability-based optimization

MCS

Monte Carlo simulation

LSF

Limit state function

MPP

Most probable point

RBF

Radial basis function

AMA

Approximate moment approach

RIA

Reliability index approach

PMA

Performance measure approach

SORA

Sequential optimization and reliability assessment

SLSV

Single loop single variable

ASORA

Advanced sequential optimization and reliability assessment

SLA

Single-loop approach

AHA

Adaptive hybrid approach

AH_SLM

Adaptive hybrid single-loop method

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Notes

Acknowledgments

The authors also thank Dr. Xueyu Li for the helpful work to improve the study.

Funding information

The research is supported by the Fundamental Research Funds for the Central Universities (No. G2016KY0302) and the National Natural Science Foundation of China (No. 11572134).

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of AstronauticsNorthwestern Polytechnical UniversityXi’anPeople’s Republic of China
  2. 2.Department of MechanicsHuazhong University of Science and TechnologyWuhanPeople’s Republic of China

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