Skip to main content
SpringerLink
Log in

Nonparametric uncertainty representation method with different insufficient data from two sources

  • RESEARCH PAPER
  • Published:
Structural and Multidisciplinary Optimization Aims and scope Submit manuscript

Abstract

The uncertainty information of design variables is included in the available representation data, and there are differences among representation data from different sources. Therefore, the paper proposes a nonparametric uncertainty representation method of design variables with different insufficient data from two sources. The Gaussian interpolation model for sparse sampling points and/or sparse sampling intervals from a single source is constructed through maximizing the logarithmic likelihood estimation function of insufficient data. The weight ratios of probability density values at sampling points are optimized through minimizing the total deviation of the fusion model, and the fusion Gaussian model is constructed based on the weight sum of the optimum probability density values of sampling points for Source 1 and Source 2. The methodology is extended to five different fusion conditions, which contain the fusion of uncertain distribution parameters, the fusion of insufficient data and interval data, etc. Five application examples are illustrated to verify the effectiveness of the proposed methodology.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  • An H, Zhou L, Sun H (2016) Structural hybrid reliability index and its convergent solving method based on random-fuzzy-interval reliability model. Advances in Mechanical Engineering 8(8):1–13

    Article  Google Scholar 

  • Bae H-R, Grandhi RV, Canfield RA (2004) Epistemic uncertainty quantification techniques including evidence theory for large-scale structures. Comput Struct 82(13–14):1101–1112

    Article  Google Scholar 

  • Balu AS, Rao BN (2014) Efficient assessment of structural reliability in presence of random and fuzzy uncertainties. J Mech Des 136(5):051008

    Article  Google Scholar 

  • Chakraborty S, Chatterjee T, Chowdhury R, Adhikari S (2017) A surrogate based multi-fidelity approach for robust design optimization. Appl Math Model 47:726–744

    Article  MathSciNet  Google Scholar 

  • Chen S, Chen W, Lee S (2009) Level set based robust shape and topology optimization under random field uncertainties. Struct Multidiscip Optim 41(4):507–524

    Article  MathSciNet  Google Scholar 

  • Cheng J, Liu Z, Wu Z, Tang M, Tan J (2016) Direct optimization of uncertain structures based on degree of interval constraint violation. Comput Struct 164:83–94

    Article  Google Scholar 

  • Cheng J, Liu Z, Tang M, Tan J (2017) Robust optimization of uncertain structures based on normalized violation degree of interval constraint. Comput Struct 182:41–54

    Article  Google Scholar 

  • Choi CK, Yoo HH (2016) Stochastic inverse method to identify parameter random fields in a structure. Struct Multidiscip Optim 54(6):1557-1571

    Article  MathSciNet  Google Scholar 

  • 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 

  • Fuh C-F, Jea R, Su J-S (2014) Fuzzy system reliability analysis based on level (λ,1) interval-valued fuzzy numbers. Inf Sci 272:185–197

    Article  MathSciNet  Google Scholar 

  • Hu Z, Nannapaneni S, Mahadevan S (2017) Efficient kriging surrogate modeling approach for system reliability analysis. Artificial Intelligence for Engineering Design, Analysis and Manufacturing 31(02):143–160

    Article  Google Scholar 

  • Jiang C, Han X, Lu GY, Liu J, Zhang Z, Bai YC (2011) Correlation analysis of non-probabilistic convex model and corresponding structural reliability technique. Comput Methods Appl Mech Eng 200(33–36):2528–2546

    Article  Google Scholar 

  • Jiang C, Zhang Z, Han X, Liu J (2013a) A novel evidence-theory-based reliability analysis method for structures with epistemic uncertainty. Comput Struct 129:1–12

    Article  Google Scholar 

  • Jiang C, Zhang QF, Han X, Qian YH (2013b) A non-probabilistic structural reliability analysis method based on a multidimensional parallelepiped convex model. Acta Mech 225(2):383–395

    Article  Google Scholar 

  • Kang Z, Zhang W (2016) Construction and application of an ellipsoidal convex model using a semi-definite programming formulation from measured data. Comput Methods Appl Mech Eng 300:461–489

    Article  MathSciNet  Google Scholar 

  • Keshtegar B, Meng Z (2017) A hybrid relaxed first-order reliability method for efficient structural reliability analysis. Struct Saf 66:84–93

    Article  Google Scholar 

  • Liu X, Yin L, Hu L, Zhang Z (2017) An efficient reliability analysis approach for structure based on probability and probability box models. Struct Multidiscip Optim

  • Luo Y, Xing J, Niu Y, Li M, Kang Z (2017) Wrinkle-free design of thin membrane structures using stress-based topology optimization. Journal of the Mechanics and Physics of Solids 102:277–293

    Article  MathSciNet  Google Scholar 

  • 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 

  • Oberkampf WL, Helton JC, Joslyn CA, Wojtkiewicz SF, Ferson S (2004) Challenge problems: uncertainty in system response given uncertain parameters. Reliab Eng Syst Saf 85(1–3):11–19

    Article  Google Scholar 

  • Park J (2018) Simultaneous estimation based on empirical likelihood and general maximum likelihood estimation. Computational Statistics & Data Analysis 117:19–31

    Article  MathSciNet  Google Scholar 

  • Pei Z (2015) Intuitionistic fuzzy variables: concepts and applications in decision making. Expert Syst Appl 42(22):9033–9045

    Article  Google Scholar 

  • Peng X, Li J, Jiang S (2017a) Unified uncertainty representation and quantification based on insufficient input data. Struct Multidiscip Optim 56:1305–1317. https://doi.org/10.1007/s00158-017-1722-4

    Article  Google Scholar 

  • Peng X, Wu T, Li J, Jiang S, Qiu C, Yi B (2017b) Hybrid reliability analysis with uncertain statistical variables, sparse variables and interval variables. Eng Optim:1–17

  • Sakia RM (1992) The box-cox transformation technique: a review. The Statistician 41:169–178

    Article  Google Scholar 

  • Sankararaman S, Mahadevan S (2011) Likelihood-based representation of epistemic uncertainty due to sparse point data and/or interval data. Reliab Eng Syst Saf 96(7):814–824

    Article  Google Scholar 

  • Sankararaman S, Mahadevan S (2012) Likelihood-based approach to multidisciplinary analysis under uncertainty. J Mech Des 134(3):031008

    Article  Google Scholar 

  • Sankararaman S, Mahadevan S (2013) Distribution type uncertainty due to sparse and imprecise data. Mech Syst Signal Process 37(1–2):182–198

    Article  Google Scholar 

  • Shah H, Hosder S, Koziel S, Tesfahunegn YA, Leifsson L (2015) Multi-fidelity robust aerodynamic design optimization under mixed uncertainty. Aerosp Sci Technol 45:17–29

    Article  Google Scholar 

  • TaoYR, Cao L, Huang ZHH (2017) A novel evidence-based fuzzy reliability analysis method for structures. Struct Multidiscip Optim 55(4):1237-1249.

    Article  MathSciNet  Google Scholar 

  • Wang L, Wang X, Wu D, Xu M, Qiu Z (2018). Structural optimization oriented time-dependent reliability methodology under static and dynamic uncertainties. Struct Multidiscip Optim 57(4):1533-1551

    Article  Google Scholar 

  • Wang L, Wang X, Xia Y (2013) Hybrid reliability analysis of structures with multi-source uncertainties. Acta Mech 225(2):413–430

    Article  Google Scholar 

  • Wang X, Wang L, Qiu Z (2014) A feasible implementation procedure for interval analysis method from measurement data. Appl Math Model 38(9–10):2377–2397

    Article  MathSciNet  Google Scholar 

  • Wang L, Liu D, Yang Y, Wang X, Qiu Z (2017a) A novel method of non-probabilistic reliability-based topology optimization corresponding to continuum structures with unknown but bounded uncertainties. Comput Methods Appl Mech Eng 326:573–595

    Article  MathSciNet  Google Scholar 

  • Wang L, Pan R, Wang X, Fan W, Xuan J (2017b) A Bayesian reliability evaluation method with different types of data from multiple sources. Reliab Eng Syst Saf 167:128–135

    Article  Google Scholar 

  • Wang L, Wang X, Wu D, Xu M, Qiu Z (2017c) Structural optimization oriented time-dependent reliability methodology under static and dynamic uncertainties. Struct Multidiscip Optim 57:1533–1551. https://doi.org/10.1007/s00158-017-1824-z

    Article  Google Scholar 

  • Wu J, Luo Z, Li H, Zhang N (2017) A new hybrid uncertainty optimization method for structures using orthogonal series expansion. Appl Math Model 45:474–490

    Article  MathSciNet  Google Scholar 

  • Xi Z, Youn BD, Jung BC, Yoon JT (2014) Random field modeling with insufficient field data for probability analysis and design. Struct Multidiscip Optim 51(3):599–611

    Article  Google Scholar 

  • Xie S, Pan B, Du X (2017) High dimensional model representation for hybrid reliability analysis with dependent interval variables constrained within ellipsoids." Struct Multidiscip Optim

  • Yao W, Chen X, Ouyang Q, van Tooren M (2013) A reliability-based multidisciplinary design optimization procedure based on combined probability and evidence theory. Struct Multidiscip Optim 48(2):339–354

    Article  MathSciNet  Google Scholar 

  • Youn BD, Jung BC, Xi Z, Kim SB, Lee WR (2011) A hierarchical framework for statistical model calibration in engineering product development. Comput Methods Appl Mech Eng 200(13–16):1421–1431

    Article  Google Scholar 

  • Zaman K, Dey PR (2017) Likelihood-based representation of epistemic uncertainty and its application in robustness-based design optimization. Struct Multidiscip Optim

  • Zaman K, Rangavajhala S, McDonald MP, Mahadevan S (2011) A probabilistic approach for representation of interval uncertainty. Reliab Eng Syst Saf 96(1):117–130

    Article  Google Scholar 

  • Zhang J, Shields MD (2018) On the quantification and efficient propagation of imprecise probabilities resulting from small datasets. Mech Syst Signal Process 98:465–483

    Article  Google Scholar 

  • Zhang J, Yong D (2017) A method to determine basic probability assignment in the open world and its application in data fusion and classification. Appl Intell 46:934–951

    Article  Google Scholar 

Download references

Funding

This work was supported by the National Natural Science Foundation of China under grant [numbers 51505421, 51490663, U1610112]; the Zhejiang Provincial Natural Science Foundation of China under grant [number: LY18E050020]; the National Key Research and Development Program of China under grant [number: 2017YFB0603704].

Author information

Authors and Affiliations

  1. Key Laboratory of E&M, Zhejiang University of Technology, Hangzhou, 310014, China

    Xiang Peng, Xiaoqing Xu, Jiquan Li & Shaofei Jiang

  2. State Key of CAD&CG, Zhejiang University, Hangzhou, 310027, China

    Xiang Peng, Zhenyu Liu & Chan Qiu

Authors
  1. Xiang Peng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhenyu Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaoqing Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jiquan Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Chan Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Shaofei Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shaofei Jiang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Peng, X., Liu, Z., Xu, X. et al. Nonparametric uncertainty representation method with different insufficient data from two sources. Struct Multidisc Optim 58, 1947–1960 (2018). https://doi.org/10.1007/s00158-018-2003-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00158-018-2003-6

Keywords

Search

Navigation