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An active learning Kriging-assisted method for reliability-based design optimization under distributional probability-box model

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Abstract

Due to lack of sufficient data and information in engineering practice, it is often difficult to obtain precise probability distributions of some uncertain variables and parameters in reliability-based design optimization (RBDO). In this paper, distributional probability-box (p-box) model is employed to quantify these uncertain variables and parameters. To reduce the computational cost in RBDO associated with expensive and time-consuming constraints, an active learning Kriging-assisted method is proposed. In this method, the sequential optimization and reliability assessment (SORA) method is extended for RBDO under distributional p-box model. Kriging metamodels are constructed to make the replacement of actual constraints. To remove unnecessary computational expense on constructing Kriging metamodels, a screening criterion is built and employed for the judgment of active constraints in RBDO. Then, an active learning function is defined to find out update samples, which are adopted for sequentially refining Kriging metamodel of each active constraint by focusing on its limit-state surface (LSS) around the most probable target point (MPTP) at the solution of SORA. Several examples, including a welded beam problem and a piezoelectric energy harvester design, are provided to test the accuracy and efficiency of the proposed active learning Kriging-assisted method.

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Abbreviations

P(⋅):

Probability of an event

g(⋅):

Constraint function

f(⋅):

Object function

d :

Vector of deterministic variables

d l :

Lower bound vector of deterministic variables

d u :

Upper bound vector of deterministic variables

X :

Vector of p-box variables

x :

Realization of X

μ x :

Vector of the nominal values of X

\( {\boldsymbol{\upmu}}_{\mathbf{x}}^l \) :

Lower bound vector of μx

\( {\boldsymbol{\upmu}}_{\mathbf{x}}^u \) :

Upper bound vector of μx

P :

Vector of p-box parameters

p :

Realization of P

μ p :

Vector of the nominal values of P

\( {\boldsymbol{\upmu}}_{\mathbf{p}}^l \) :

Lower bound vector of distribution mean of P

\( {\boldsymbol{\upmu}}_{\mathbf{p}}^u \) :

Upper bound vector of distribution mean of P

σ p :

Vector of the distribution standard deviation of P

\( {\boldsymbol{\upsigma}}_{\mathbf{p}}^u \) :

Upper bound vector of σp

Z :

Vector containing both X and P

z :

Realization of Z

μ z :

Vector of the nominal values of Z

Y :

Vector of interval distribution parameters of Z

y :

Realization of Y

z MPTP :

The most probable target point of Z

s :

Offset vector of Z

W :

Vector of random parameters with the standard normal distribution

w :

Realization of W

R l :

Lower bound of the reliability degree of a constraint

R u :

Upper bound of the reliability degree of a constraint

R t :

Target reliability degree of a constraint

β t :

Target reliability index of a constraint

\( \hat{g}\left(\cdot \right) \) :

Mean of Kriging prediction

\( {\sigma}_{\hat{g}}^2\left(\cdot \right) \) :

Variance of Kriging prediction

Φ(⋅):

Cumulative distribution function of the standard normal distribution

U(⋅) :

A learning function (Echard et al. 2011)

Fp(⋅):

An active learning function proposed in this work

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Funding

This research was supported by the National Key R&D Program of China (2018AAA0101704), the National Natural Science Foundation of China (grant numbers 51675196 and 51721092), the Natural Science Foundation of Hubei Province (grant number 2019CFA059), the Program for HUST Academic Frontier Youth Team (grant number 2017QYTD04), and the Graduate Innovation Fund of Huazhong University of Science and Technology (No. 2019YGSCXCY070).

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Authors and Affiliations

  1. State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan, 430074, China

    Jinhao Zhang, Liang Gao & Mi Xiao

  2. Department of Mechanical Engineering, University of Maryland, Baltimore County, Baltimore, MD, 21250, USA

    Soobum Lee & Amin Toghi Eshghi

Authors
  1. Jinhao Zhang
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  2. Liang Gao
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  3. Mi Xiao
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  4. Soobum Lee
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  5. Amin Toghi Eshghi
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Correspondence to Mi Xiao.

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The authors declare that they have no conflict of interest.

Replication of results

The Matlab codes of extended SORA and the proposed method are provided on https://github.com/Jinhao218/paper-for-smo.

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Responsible Editor: Pingfeng Wang

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Zhang, J., Gao, L., Xiao, M. et al. An active learning Kriging-assisted method for reliability-based design optimization under distributional probability-box model. Struct Multidisc Optim 62, 2341–2356 (2020). https://doi.org/10.1007/s00158-020-02604-5

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