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An efficient hierarchical fuzzy simulation method for estimating failure possibility

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Abstract

The failure possibility (FP) can reasonably measure the safety degree of the structure under fuzzy uncertainty, and the estimation of FP can be transformed into searching the point with the maximum joint membership function (MF) of fuzzy input vector in the failure domain (also known as fuzzy design point). In the current fuzzy simulation (FS) method, the fuzzy design point is searched in the maximum value region of the fuzzy input vector corresponding to the lowest membership level which is equal to 0 and the computational efficiency is low. In this paper, an efficient hierarchical fuzzy simulation (HFS) method is proposed for estimating FP. In the proposed method, by the nature that the fuzzy design point is a failure point with the maximum joint MF, the fuzzy design point is first searched in the smaller value region of input vector corresponding to the larger membership level, and the membership level is automatically reduced layer by layer to expand the search region until the failure points appear. Compared with the traditional FS method, the proposed HFS method not only guarantees the search accuracy of the fuzzy design point, but also reduces the total search region; thus the computational efficiency is improved. In addition, an adaptive Kriging model is also embedded in the search process of HFS. Since the adaptively updated Kriging model is used to replace the real performance function for recognizing the state of the simulated sample points during the search process, the strategy of combining the Kriging model with HFS method can further improve the search efficiency of the fuzzy design point. The results of examples show that the proposed HFS method is reasonable and efficient.

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Abbreviations

FP:

Failure possibility

FS:

Fuzzy simulation

HFS:

Hierarchical fuzzy simulation

MF:

Membership function

AK:

Adaptive Kriging

RE:

Relative error

\({\varvec{X}}\) :

Fuzzy input vector, and \({\varvec{X}} = {x_{1} ,x_{2} ,...,x_{n} }^{{\text{T}}}\)

\({\varvec{x}}\) :

Realization of \({\varvec{X}}\), and \({\varvec{x}} = {x_{1} ,x_{2} ,...,x_{n} }^{{\text{T}}}\)

\(F\) :

Failure domain, and \(F = \{ g({\varvec{x}}) \le 0\}\)

\(g({\varvec{x}})\) :

Performance function

\(\pi_{f}\) :

Failure possibility

\(N_{\alpha }\) :

The number of layers

\(\rho_{{\varvec{X}}} ( \cdot )\) :

Joint MF of \({\varvec{X}}\)

\(\rho_{{X_{i} }} ( \cdot )\) :

MF of \(X_{i}\)

\(\rho_{{X_{i} }}^{ - 1} ( \cdot )\) :

Inverse function of MF \(\rho_{{X_{i} }} ( \cdot )\)

\({\text{Poss}}\{ \cdot \}\) :

Possibility operator

\(N_{{{\text{call}}}}\) :

The number of calls to performance function

\(\alpha\) :

Membership level

\({\text{Sup}}\{ \cdot \}\) :

Supremum operator

\({\varvec{x}}\user2{(}\alpha \user2{)}\) :

Membership interval vector

\({\varvec{x}}^{L} \user2{(}\alpha \user2{)}\) :

Lower bound of membership interval

\({\varvec{x}}^{U} \user2{(}\alpha \user2{)}\) :

Upper bound of membership interval

\(\gamma\) :

Scale factor and \(\gamma < 1\)

\(g_{K} {(}{\varvec{x}}{)}\) :

Kriging surrogate model of performance function

\(S_{{\varvec{x}}}^{(k)} \,\) :

Candidate sample pool of fuzzy input corresponding to \(\alpha_{k}\) membership level

\(I_{{\text{F}}} ( \cdot )\) :

Indicator function of failure domain

\({\varvec{x}}^{ * }\) :

Fuzzy design point

\(U\user2{(} \cdot \user2{)}\) :

U-learning function

\(\mu_{{g_{K} }} ({\varvec{x}})\) :

Predicted mean of \(g_{K} {(}{\varvec{x}}{)}\)

\(\sigma_{{g_{K} }} ({\varvec{x}})\) :

Prediction standard deviation of \(g_{K} {(}{\varvec{x}}{)}\)

\(T\) :

The training sample set

\({\varvec{x}}_{{{\text{new}}}}\) :

New training sample point

\(N_{{\text{T}}}\) :

The number of training sample points

\(S_{{\varvec{x}}}\) :

Candidate sample pool of fuzzy input corresponding to 0 membership level

\(i\) :

\(i = 1,2,...,N_{k}\); \(N_{k}\) Is the number of samples in the \(k\)-th layer

\(k\) :

\(k = 1,2,...,N_{\alpha }\); \(N_{\alpha }\) Is the number of layers

References

  1. Wang L, Zhang XB, Li GJ, Lu ZZ (2022) Credibility distribution function based global and regional sensitivity analysis under fuzzy uncertainty. Eng Comput 38(3):1349–1362

    Article  Google Scholar 

  2. Cheng MY, Prayogo D (2017) A novel fuzzy adaptive teaching learning-based optimization (FATLBO) for solving structural optimization problems. Eng Comput 33:55–69

    Article  Google Scholar 

  3. Wang C, Qiu ZP, Xu MH, Li YL (2017) Novel reliability-based optimization method for thermal structure with hybrid random, interval and fuzzy parameters. Appl Math Model 47:573–586

    Article  MathSciNet  MATH  Google Scholar 

  4. Nahmias S (1978) Fuzzy variables. Fuzzy Sets Syst 1:79–110

    Article  MathSciNet  MATH  Google Scholar 

  5. Wang C, Matthies HG (2019) Epistemic uncertainty-based reliability analysis for engineering system with hybrid evidence and fuzzy variables. Comput Methods Appl Mech Eng 355:438–455

    Article  MathSciNet  MATH  Google Scholar 

  6. Yu SW (2010) Construction of a fuzzy membership function based on interval number. J Shandong Univ 40:32–35

    Google Scholar 

  7. Cremona C, Gao Y (1997) The possibilistic reliability theory: theoretical aspects and applications. Struct Saf 19(2):173–201

    Article  Google Scholar 

  8. Zadeh LA (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst 1:2–28

    Article  MathSciNet  Google Scholar 

  9. Tzvieli A (1990) Possibility theory: An approach to computerized processing of uncertainty. J Am Soc Inf Sci 41(2):153–154

    Article  Google Scholar 

  10. Dubois D, Prade H (1992) When upper probabilities are possibility measures. Fuzzy Sets Syst 49(1):65–74

    Article  MathSciNet  MATH  Google Scholar 

  11. Hurtado JE, Alvarez DA, Ramirez J (2012) Fuzzy structural analysis based on fundamental reliability concepts. Comput Struct 112–113:183–192

    Article  Google Scholar 

  12. Wang C, Qiu ZP, Xu MH, Qiu HC (2017) Novel fuzzy reliability analysis for heat transfer system based on interval ranking method. Int J Therm Sci 116:234–241

    Article  Google Scholar 

  13. Guo SX, Lu ZZ (2003) Procedure for computing the possibility and fuzzy probability of failure of structures. Appl Math Mech 24:338–343

    Article  MATH  Google Scholar 

  14. Mourelatos ZP, Zhou J (2005) Reliability estimation and design with insufficient data based on possibility theory. AIAA J 43(8):1696–1705

    Article  Google Scholar 

  15. Du L, Choi KK, Youn BD (2006) Inverse possibility analysis method for possibility-based design optimization. AIAA J 44(11):2682–2690

    Article  Google Scholar 

  16. Jia BX, Lu ZZ (2018) Root finding method of failure credibility for fuzzy safety analysis. Struct Multidiscip Optim 58(5):1917–1934

    Article  MathSciNet  Google Scholar 

  17. Möller B, Graf W, Beer M (2000) Fuzzy structural analysis using α-level optimization. Comput Mech 26(6):547–565

    Article  MATH  Google Scholar 

  18. Feng KX, Lu ZZ, Chao P (2019) Safety life analysis under required failure credibility constraint for unsteady thermal structure with fuzzy input parameters. Struct Multidiscip Optim 59(1):43–59

    Article  MathSciNet  Google Scholar 

  19. Liu B (2006) A survey of credibility theory. Fuzzy Optim Decis Making 5(4):387–408

    Article  MathSciNet  MATH  Google Scholar 

  20. Bououden S, Chadli M, Karimi HR (2015) An ant colony optimization-based fuzzy predictive control approach for nonlinear processes. Inf Sci 299:143–158

    Article  MathSciNet  MATH  Google Scholar 

  21. Bououden S, Chadli M, Allouani F, Filali S (2013) A new approach for fuzzy predictive adaptive controller design using particle swarm optimization algorithm. Int J Innov Comput Inf Control 9:3741–3758

    Google Scholar 

  22. Yang IT, Hsieh YH (2013) Reliability-based design optimization with cooperation between support vector machine and particle swarm optimization. Eng Comput 29(2):151–163

    Article  Google Scholar 

  23. Burke JV, Han SP (1989) A robust sequential quadratic programming method. Math Program 43(1–3):277–303

    Article  MathSciNet  MATH  Google Scholar 

  24. Weile DS, Michielssen E (1997) Genetic algorithm optimization applied to electromagnetics: a review. IEEE Trans Antennas Propag 45(3):343–353

    Article  Google Scholar 

  25. Liu B (2007) Uncertainty theory, 2nd edn. Springer Publishing Company Incorporated, New York

    MATH  Google Scholar 

  26. Echard B, Gayton N, Lemaire M (2011) AK-MCS: an active learning reliability method combining Kriging and Monte Carlo Simulation. Struct Saf 33(2):145–154

    Article  Google Scholar 

  27. Yang XF, Liu ZQ, Cheng X (2021) An enhanced active learning Kriging model for evidence theory-based reliability analysis. Struct Multidiscip Optim 64(4):2165–2181

    Article  MathSciNet  Google Scholar 

  28. Yang XF, Cheng X, Liu ZQ, Wang T (2021) An adaptive method fusing the kriging model and multimodal importance sampling for profust reliability analysis. Eng Optim, pp 1–17

  29. Ling CY, Lu ZZ, Feng KX (2019) An efficient method combining adaptive Kriging and fuzzy simulation for estimating failure credibility. Aerosp Sci Technol 92:620–634

    Article  Google Scholar 

  30. Zhai Z, Li H, Wang X (2022) An adaptive sampling method for Kriging surrogate model with multiple outputs. Eng Comput 38(1) :277–295

  31. Yang XF, Cheng X, Liu ZQ, et al (2021) A novel active learning method for profust reliability analysis based on the Kriging model. Eng Comput, pp 1–14. https://doi.org/10.1007/s00366-021-01447-y

  32. Toal D (2015) A study into the potential of GPUs for the efficient construction and evaluation of Kriging models. Eng Comput 32(3):1–28

    Google Scholar 

  33. Lophaven SN, Nielsen HB, Søndergaard J (2002) Aspects of the matlab toolbox DACE. Technical Report, Informatics and Mathematical Modeling, Technical University of Denmark, DTU

  34. Yang XF, Wang T, Li JC, Zhang C (2019) Bounds approximation of limit state surface based on active learning Kriging model with truncated candidate region for random-interval hybrid reliability analysis. Int J Numer Methods Eng 121(7):1345–1366

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant no. NSFC 52075442), National Science and Technology Major Project (2017-IV-0009-0046) and Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (CX2022018).

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  1. School of Aeronautics, Northwestern Polytechnical University, Xi’an, 710072, Shaanxi, China

    Xia Jiang, Zhenzhou Lu & Kaixuan Feng

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  1. Xia Jiang
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Jiang, X., Lu, Z. & Feng, K. An efficient hierarchical fuzzy simulation method for estimating failure possibility. Engineering with Computers 39, 3085–3097 (2023). https://doi.org/10.1007/s00366-022-01692-9

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