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An enhanced active learning Kriging model for evidence theory-based reliability analysis

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Abstract

Evidence theory-based reliability analysis (ETRA) is investigated in this paper. To estimate the plausibility (Pl) and belief (Bel) measures of failure probability, the active learning method based on Kriging model for ETRA (ALK-ETRA) has been proposed. However, ALK-ETRA pays too much attention to rightly predicting the signs of performance function at points throughout the uncertain space. To minimize the waste of training points, an enhanced version of ALK-ETRA (En-ALK-ETRA) is proposed in this paper. Pl or Bel is determined by the sign associated with an individual joint focal element (JFE) rather than a single point. Based on this idea, a brand-new learning function and a novel stopping condition are proposed in En-ALK-ETRA. Aided by the new learning function, the most dangerous JFE in which the lower (or upper) bound of performance function has the largest probability of wrong sign prediction is identified. The added training point is chosen from the most dangerous JFE and thus the convergence speed of learning process is accelerated. In the new stopping condition, the number of JFEs where the minimum (or maximum) of performance function with wrong sign predictions is explicitly deduced and thus the error of ETRA can be estimated. The error of ETRA is real-time monitored and thus the learning process is timely terminated without accuracy sacrifice. The performance of the proposed method is demonstrated by six case studies.

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Abbreviations

n X :

Total number of epistemic variables

n j :

Total number of focal elements of the jth (j = 1, ⋯, nX) variable

n E :

Total number of joint focal elements of all variable

N MC :

Total number of samples of interval Monte Carlo simulation

N A :

Total number of samples in each joint focal element

N T :

Total number of training points of Kriging model

\( {A}_j^i \) :

The ith (i = 1, ⋯, nj) focal element of the jth variable

\( {\tilde{N}}_F^{Bel} \) :

Total number of joint focal elements which are predicted to be in the failure region

\( {N}_F^{Bel} \) :

Total number of joint focal elements which are truly in the failure region

\( {N}_{\mathrm{wsp}}^{\mathrm{max}} \) :

Total number of joint focal elements where the sign predictions for maximum of performance function are wrong

\( {\overline{N}}_{\mathrm{wsp}}^{\mathrm{max}} \) :

Upper-bound estimation of \( {N}_{\mathrm{wsp}}^{\mathrm{max}} \)

\( {N}_{\mathrm{wsp}}^{\mathrm{min}} \) :

True number of joint focal elements where the sign predictions for the minimum of performance function are wrong

\( {\overline{N}}_{\mathrm{wsp}}^{\mathrm{min}} \) :

Upper-bound estimation of \( {N}_{\mathrm{wsp}}^{\mathrm{min}} \)

N r :

A constant in stopping condition

A k :

The kth (k = 1, 2, ⋯, nE) joint focal element

A (k) :

The kth sample of interval Monte Carlo simulation

JFE:

Joint focal element

ETRA:

Evidence theory-based reliability analysis

Bel :

Belief measure of the failure probability

Pl :

Plausibility measure of the failure probability

CPM:

Cartesian product method

IMCS:

Interval Monte Carlo simulation

MPFE:

Most probable focal element

RS:

Response surface

FARM/SARM:

First-order and second-order approximate reliability methods

DRD:

Dimension reduction decomposition

ALK:

Active learning methods based on Kriging

BPA:

Basic probability assignment

Cov:

Coefficients of variation

LHS:

Latin hypercube sampling

DoE:

Design of experiments

ERF:

Expected risk function

WSP:

Wrong sign prediction

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Funding

This work is supported by the National Natural Science Foundation of China (Grant No. 51705433), the Sichuan Science and Technology Program (Grant No. 2021YFG0178), and the Fundamental Research Funds for the Central Universities (Grant No. 2682017CX028).

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Authors and Affiliations

  1. School of Mechanical Engineering, Southwest Jiaotong University, Chengdu, 610031, People’s Republic of China

    Xufeng Yang, Liu Zeqing & Xin Cheng

Authors
  1. Xufeng Yang
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  2. Liu Zeqing
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  3. Xin Cheng
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Correspondence to Xufeng Yang.

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Replication of results

Details on the numerical implementation for the replication of the results have been provided in Section 6, with the pseudocode and the optimization parameters. The detailed procedures of the proposed method are given in Section 5. If the information provided in the paper is not enough, we sincerely welcome scientists or interested parties to contact us for further explanation. Data or implementation code is available upon request.

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Responsible Editor: Gengdong Cheng

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Yang, X., Zeqing, L. & Cheng, X. An enhanced active learning Kriging model for evidence theory-based reliability analysis. Struct Multidisc Optim 64, 2165–2181 (2021). https://doi.org/10.1007/s00158-021-02973-5

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