Skip to main content
SpringerLink
Log in

A novel evidence theory model dealing with correlated variables and the corresponding structural reliability analysis method

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

Abstract

Evidence theory serves as a powerful tool to deal with epistemic uncertainty which widely exists in the design stages of many complex engineering systems or products. However, the traditional evidence theory model cannot handle parameter correlations that may have profound influences on the reliability analysis results. This paper is supposed to develop a novel evidence theory model with consideration of parameter correlations and its corresponding structural reliability analysis method. First, a multidimensional parallelepiped uncertainty domain which takes into account the influence of parameter correlations is constructed. Second, the corresponding joint basic probability assignments are established for each focal element in the uncertainty domain. Finally, the reliability interval composed of the belief and plausibility measures are computed. Several numerical examples are investigated to demonstrate the effectiveness of the proposed model and the corresponding reliability analysis method.

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.

Institutional subscriptions

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

Similar content being viewed by others

References

  • Agarwal H, Renaud JE, Preston EL, Padmanabhan D (2004) Uncertainty quantification using evidence theory in multidisciplinary design optimization. Reliab. Eng. Syst. Saf. 85(1–3):281–294

    Article  Google Scholar 

  • Bae HR, Grandhi RV, Canfield RA (2004a) An approximation approach for uncertainty quantification using evidence theory. Reliab. Eng. Syst. Saf. 86(3):215–225

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Basir O, Yuan X (2007) Engine fault diagnosis based on multi-sensor information fusion using Dempster–Shafer evidence theory. Inform Fusion 8(4):379–386

    Article  Google Scholar 

  • Bauer M (1997) Approximation algorithms and decision making in the Dempster-Shafer theory of evidence—an empirical study. Int J Approx Reason 17(2–3):217–237

    Article  MathSciNet  MATH  Google Scholar 

  • Ben-Haim Y, Elishakoff I (2013) Convex models of uncertainty in applied mechanics. Elsevier, Amsterdam

    MATH  Google Scholar 

  • Beynon M, Cosker D, Marshall D (2001) An expert system for multi-criteria decision making using Dempster Shafer theory. Expert Syst Appl 20(4):357–367

    Article  Google Scholar 

  • Cai KY (2012) Introduction to fuzzy reliability. Springer, New York

    Google Scholar 

  • Dempster AP (1967) Upper and lower probabilities induced by a multivalued mapping. Ann Math Stat 38(2):325–339

    Article  MathSciNet  MATH  Google Scholar 

  • Dong W, Shah HC (1987) Vertex method for computing functions of fuzzy variables. Fuzzy Sets Syst 24(1):65–78

    Article  MathSciNet  MATH  Google Scholar 

  • Du X (2006) Uncertainty analysis with probability and evidence theories. In ASME 2006 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Pennsylvania, USA

  • Du X, Hu Z (2012) First order reliability method with truncated random variables. J Mech Des 134(9):091005

    Article  Google Scholar 

  • Eldred MS, Swiler LP, Tang G (2011) Mixed aleatory-epistemic uncertainty quantification with stochastic expansions and optimization-based interval estimation. Reliab. Eng. Syst. Saf. 96(9):1092–1113

    Article  Google Scholar 

  • Elishakoff I, Haftka R, Fang J (1994) Structural design under bounded uncertainty—optimization with anti-optimizati.n. Comput Struct 53(6):1401–1405

    Article  MATH  Google Scholar 

  • Ferson S, Nelsen RB, Hajagos J, Berleant DJ, Zhang J, Tucker WT, Ginzburg LR, Oberkampf WL (2004) Dependence in probabilistic modeling, Dempster-Shafer theory, and probability bounds analysis. Report No SAND 2004–3072, Sandia National Laboratories

  • Gogu C, Qiu Y, Segonds S, Bes C (2012) Optimization based algorithms for uncertainty propagation through functions with multidimensional output within evidence theory. J Mech Des 134(10):614–620

    Article  Google Scholar 

  • Helton JC (2011) Quantification of margins and uncertainties: Conceptual and computational basis. Reliab. Eng. Syst. Saf. 96(9):976–1013

    Article  Google Scholar 

  • Helton JC, Johnson JD, Oberkampf WL (2004) An exploration of alternative approaches to the representation of uncertainty in model predictions. Reliab. Eng. Syst. Saf. 85(1–3):39–71

    Article  Google Scholar 

  • Hoffman FO, Hammonds JS (1994) Propagation of uncertainty in risk assessments: the need to distinguish between uncertainty due to lack of knowledge and uncertainty due to variability. Risk Anal 14(5):707–712

    Article  Google Scholar 

  • Huang Z, Jiang C, Zhang Z, Fang T, Han X (2017a) A decoupling approach for evidence-theory-based reliability design optimization. Struct Multidiscip O 56(3):647–661

    Article  Google Scholar 

  • Huang Z, Jiang C, Zhou Y, Zheng J, Long X (2017b) Reliability-based design optimization for problems with interval distribution parameters. Struct Multidiscip O 55(2):513–528

    Article  MathSciNet  Google Scholar 

  • Jiang C, Han X, Lu G, Liu ZZ, Bai Y (2011) Correlation analysis of non-probabilistic convex model and corresponding structural reliability technique. Comput Meth Appl M 200(33–36):2528–2546

    Article  MATH  Google Scholar 

  • Jiang C, Wang B, Li ZR, Han X, Yu DJ (2013a) An evidence-theory model considering dependence among parameters and its application in structural reliability analysis. Eng Struct 57:12–22

    Article  Google Scholar 

  • Jiang C, Zhang Q, Han X, Liu J, Hu D (2015) Multidimensional parallelepiped model—a new type of non-probabilistic convex model for structural uncertainty analysis. Int J Numer Meth Eng 103(16):31–59

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  • Kiureghian AD, Ditlevsen O (2009) Aleatory or epistemic? Does it matter? Struct Safe 31(2):105–112

    Article  Google Scholar 

  • Lee JO, Yang YS, Ruy WS (2002) A comparative study on reliability-index and target-performance-based probabilistic structural design optimization. Comput Struct 80(3–4):257–269

    Article  Google Scholar 

  • Liu YW, Moses M (1994) A sequential response surface method and its application in the reliability analysis of aircraft structural systems. Struct Safe 16(1–2):39–46

    Google Scholar 

  • Madsen HO, Krenk S, Lind NC (2006) Methods of structural safety. Dover Publications, New York

    Google Scholar 

  • Meng D, Zhang H, Huang T (2016) A concurrent reliability optimization procedure in the earlier design phases of complex engineering systems under epistemic uncertainties. Adv Mech Eng. https://doi.org/10.1177/1687814016673976

  • Morio J (2011) Non-parametric adaptive importance sampling for the probability estimation of a launcher impact position. Reliab. Eng. Syst. Saf. 96(1):178–183

    Article  Google Scholar 

  • Mourelatos ZP, Zhou J (2005) A design optimization method using evidence theory. J Mech Des 128(4):901–908

    Article  Google Scholar 

  • Ni B, Jiang C, Han X (2016) An improved multidimensional parallelepiped non-probabilistic model for structural uncertainty analysis. Appl Math Model 40(7–8):4727–4745

    Article  MathSciNet  Google Scholar 

  • Oberkampf WL, Helton JC, Sentz K (2001) Mathematical representation of uncertainty. In 19th AIAA Applied Aerodynamics Conference, CA

  • Park I, Grandhi RV (2012) Quantification of model-form and parametric uncertainty using evidence theory. Struct Safe 39:44–51

    Article  Google Scholar 

  • Qiu Z, Wang L (2016) The need for introduction of non-probabilistic interval conceptions into structural analysis and design. Sci China Phys Mech 59:114632

    Article  Google Scholar 

  • Rao KD, Kushwaha H, Verma AK, Srividya A (2007) Quantification of epistemic and aleatory uncertainties in level-1 probabilistic safety assessment studies. Reliab. Eng. Syst. Saf. 92(7):947–956

    Article  Google Scholar 

  • Rao SS, Annamdas K (2013) A comparative study of evidence theories in the modeling, analysis, and design of engineering systems. J Mech Des 135(6):061006

    Article  Google Scholar 

  • Riley ME (2015) Evidence-based quantification of uncertainties induced via simulation-based modeling. Reliab. Eng. Syst. Saf. 133:79–86

    Article  Google Scholar 

  • Shafer G (1976) A mathematical theory of evidence. Princeton University Press, New Jersey

    MATH  Google Scholar 

  • Shah H, Hosder S, Winter T (2015) Quantification of margins and mixed uncertainties using evidence theory and stochastic expansions. Reliab. Eng. Syst. Saf. 138:59–72

    Article  Google Scholar 

  • Sobczyk K, Trcebicki J (1999) Approximate probability distributions for stochastic systems: maximum entropy method. Comput Method Appl M 168(1–4):91–111

    Article  MathSciNet  MATH  Google Scholar 

  • Srivastava RK, Deb K, Tulshyan R (2013) An evolutionary algorithm based approach to design optimization using evidence theory. J Mech Des 135:081003

    Article  Google Scholar 

  • Tang H, Li D, Chen W, Xue S (2016) Uncertainty quantification using evidence theory in concrete fatigue damage prognosis. In IEEE International Conference on Prognostics and Health Management (ICPHM), Ottawa

  • Tang H, Li D, Li J, Xue S (2017) Epistemic uncertainty quantification in metal fatigue crack growth analysis using evidence theory. Int J Fatigue 99:163–174

    Article  Google Scholar 

  • Wang L, Wang X, Xia Y (2014) Hybrid reliability analysis of structures with multi-source uncertainties. Acta Mech 225:413–430

    Article  MATH  Google Scholar 

  • Wang L, Wang X, Su H, Lin G (2017) Reliability estimation of fatigue crack growth prediction via limited measured data. Int J Mech Sci 121:44–57

    Article  Google Scholar 

  • Wang Z, Huang HZ, Li Y, Pang Y, Xiao NC (2012) An approach to system reliability analysis with fuzzy random variables. Mech Mach Theory 52:35–46

    Article  Google Scholar 

  • Warren-Hicks W, Hart A (2010) Application of Uncertainty Analysis to Ecological Risks of Pesticides. CRC Press, Florida

    Google Scholar 

  • Yager R, Fedrizzi M, Kacprzyk J (1994) Advances in the Dempster-Shafer theory of evidence. John Wiley & Sons, New York

    MATH  Google Scholar 

  • Yang X, Liu Y, Gao Y (2016) Unified reliability analysis by active learning Kriging model combining with random-set based Monte Carlo simulation method. Int J Numer Meth Eng 108(11):1343–1361

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Yin S, Yu D, Yin H, Xia B (2017) A new evidence-theory-based method for response analysis of acoustic system with epistemic uncertainty by using Jacobi expansion. Comput Method Appl M 322:419–440

    Article  MathSciNet  Google Scholar 

  • Yu X, Chang KH, Choi KK (1998) Probabilistic structural durability prediction. AIAA J 36(4):628–637

    Article  Google Scholar 

  • Zadeh LA (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst 1(1):3–28

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang Z, Jiang C, Wang GG, Han X (2015) First and second order approximate reliability analysis methods using evidence theory. Reliab. Eng. Syst. Saf. 137:40–49

    Article  Google Scholar 

Download references

Acknowledgements

This work is supported by the National Natural Science Foundation of China for Distinguished Young Scholars (Grant No. 51725502), National Natural Science Foundation of China (Grant No. 51490662), National Key Research and Development Plan (Grant No. 2016YFD0701105), National Natural Science Foundation of China (Grant No. 11402296).

Author information

Authors and Affiliations

  1. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha, People’s Republic of China, 410082

    Z. Zhang, C. Jiang & X. X. Ruan

  2. Science and Technology on Integrated Logistics Support Laboratory, National University of Defense Technology, Changsha, People’s Republic of China, 410073

    F. J. Guan

Authors
  1. Z. Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. X. X. Ruan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. F. J. Guan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. Jiang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, Z., Jiang, C., Ruan, X.X. et al. A novel evidence theory model dealing with correlated variables and the corresponding structural reliability analysis method. Struct Multidisc Optim 57, 1749–1764 (2018). https://doi.org/10.1007/s00158-017-1843-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00158-017-1843-9

Keywords

Search

Navigation