Skip to main content
Log in

Interval assessments of identified parameters for uncertain structures

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

This paper investigates a kind of inverse problem for assessing the uncertainties of identified parameters with uncertainties in structural parameters and limited experimental data. The uncertainty is described by the interval model in which only the bounds of uncertain parameters are required. Directly solving this kind of inverse problem involves a double-loop problem where the outer-loop is interval analysis and the inner-loop is deterministic optimization, which requires a large number of calculations. To efficiently evaluate the effect of interval parameters on the identified parameters, a novel method based on the dimension-reduction method and adaptive collocation strategy is proposed. First, the interval inverse problem is transformed into an inverse-propagation problem, and the dimension-reduction interval method is adopted to transform the interval inverse-propagation problem into several one-dimensional interval inverse-propagation problems. Then, an adaptive collocation strategy is proposed to efficiently estimate the lower and upper bounds of identified parameters. Therefore, the double-loop problem can be transformed into several deterministic inverse problems, and the efficiency of solving the uncertain inverse problem is dramatically improved. Two numerical examples and an engineering application are applied to demonstrate the feasibility and efficiency of the proposed method.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Similar content being viewed by others

References

  1. Engl HW, Ramlau R (2000) Regularization of inverse problems. Kluwer Academic Publishers

    Google Scholar 

  2. Tarantola A (2005) Inverse problem theory and methods for model parameter estimation, vol xii. Society for Industrial & Applied Mathematics, Philadelphia, p 342

    Book  MATH  Google Scholar 

  3. Beck A, Teboulle M (2009) A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J Imag Sci 2(1):183–202

    Article  MathSciNet  MATH  Google Scholar 

  4. Beck JL, Au SK (2002) Bayesian updating of structural models and reliability using markov chain monte carlo simulation. J Eng Mech 128(4):380–391

    Google Scholar 

  5. Sohn H, Law KH (2015) Bayesian probabilistic damage detection of a reinforced-concrete bridge column. Earthq Eng Struct Dynam 29(8):1131–1152

    Article  Google Scholar 

  6. Liu J, Hu Y, Xu C, Jiang C, Han X (2016) Probability assessments of identified parameters for stochastic structures using point estimation method. Reliab Eng Syst Saf 156:51–58. https://doi.org/10.1016/j.ress.2016.07.021

    Article  Google Scholar 

  7. Fonseca JR, Friswell MI, Mottershead JE, Lees AW (2005) Uncertainty identification by the maximum likelihood method. J Sound Vib 288(3):587–599. https://doi.org/10.1016/j.jsv.2005.07.006

    Article  Google Scholar 

  8. Liu H, Tang L, Lin P (2017) Maximum likelihood estimation of model uncertainty in predicting soil nail loads using default and modified FHWA simplified methods. Math Probl Eng 2017:14. https://doi.org/10.1155/2017/7901918

    Article  Google Scholar 

  9. Tang L, Lin P (2018) Estimation of ultimate bond strength for soil nails in clayey soils using maximum likelihood method AU - Liu, Huifen. In: Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards 12 (3):190–202. doi:https://doi.org/10.1080/17499518.2017.1422525

  10. Janda T, Šejnoha M, Šejnoha J (2018) Applying Bayesian approach to predict deformations during tunnel construction. Int J Numer Anal Meth Geomech 42(15):1765–1784. https://doi.org/10.1002/nag.2810

    Article  Google Scholar 

  11. Ma C, Li X, Notarnicola C, Wang S, Wang W (2017) Uncertainty quantification of soil moisture estimations based on a bayesian probabilistic inversion. IEEE Trans Geosci Remote Sens 55(6):3194–3207. https://doi.org/10.1109/TGRS.2017.2664078

    Article  Google Scholar 

  12. Wang J, Zabaras N (2004) A Bayesian inference approach to the inverse heat conduction problem. Int J Heat Mass Transf 47(17):3927–3941. https://doi.org/10.1016/j.ijheatmasstransfer.2004.02.028

    Article  MATH  Google Scholar 

  13. Cividini A, Maier G, Nappi A (1983) Parameter estimation of a static geotechnical model using a Bayes’ approach. Int J Rock Mech Min Sci Geomech Abstracts 20(5):215–226. https://doi.org/10.1016/0148-9062(83)90002-5

    Article  Google Scholar 

  14. Zhang W, Han X, Liu J, Tan ZH (2011) A combined sensitive matrix method and maximum likelihood method for uncertainty inverse problems. Comput Mater Continua 26(3):201–225

    Google Scholar 

  15. Yang M, Zhang D, Han X (2020) Enriched single-loop approach for reliability-based design optimization of complex nonlinear problems. Eng Comput. https://doi.org/10.1007/s00366-020-01198-2

    Article  Google Scholar 

  16. Yang M, Zhang D, Han X (2020) New efficient and robust method for structural reliability analysis and its application in reliability-based design optimization. Comput Methods Appl Mech Eng 366:113018. https://doi.org/10.1016/j.cma.2020.113018

    Article  MathSciNet  MATH  Google Scholar 

  17. Xiao N-C, Zhan H, Yuan K (2020) A new reliability method for small failure probability problems by combining the adaptive importance sampling and surrogate models. Comput Methods Appl Mech Eng 372:113336. https://doi.org/10.1016/j.cma.2020.113336

    Article  MathSciNet  MATH  Google Scholar 

  18. Xiao N-C, Zuo MJ, Zhou C (2018) A new adaptive sequential sampling method to construct surrogate models for efficient reliability analysis. Reliab Eng Syst Saf 169:330–338. https://doi.org/10.1016/j.ress.2017.09.008

    Article  Google Scholar 

  19. Zhang D, Zhang N, Ye N, Fang J, Han X (2020) Hybrid learning algorithm of radial basis function networks for reliability analysis. IEEE Trans Reliabi. https://doi.org/10.1109/TR.2020.3001232

    Article  Google Scholar 

  20. Wu J, Zhang D, Jiang C, Han X, Li Q (2021) On reliability analysis method through rotational sparse grid nodes. Mech Syst Signal Process 147:107106. https://doi.org/10.1016/j.ymssp.2020.107106

    Article  Google Scholar 

  21. Yang M, Zhang D, Cheng C, Han X (2021) Reliability-based design optimization for RV reducer with experimental constraint. Struct Multidiscip Optim. https://doi.org/10.1007/s00158-020-02781-3

    Article  Google Scholar 

  22. Moore RE (1966) Interval analysis. Prentice-Hall, Englewood Cliffs

    MATH  Google Scholar 

  23. Klir GJ, Yuan B (1995) Fuzzy sets and fuzzy logic: theory and applications. Prentice-Hall, Inc.

    MATH  Google Scholar 

  24. Ben-Haim Y (1995) A non-probabilistic measure of reliability of linear systems based on expansion of convex models. Struct Saf 17(2):91–109. https://doi.org/10.1016/0167-4730(95)00004-N

    Article  Google Scholar 

  25. Lü H, Yu D (2014) Brake squeal reduction of vehicle disc brake system with interval parameters by uncertain optimization. J Sound Vib 333(26):7313–7325. https://doi.org/10.1016/j.jsv.2014.08.027

    Article  Google Scholar 

  26. Xia B, Lü H, Yu D, Jiang C (2015) Reliability-based design optimization of structural systems under hybrid probabilistic and interval model. Comput Struct 160:126–134. https://doi.org/10.1016/j.compstruc.2015.08.009

    Article  Google Scholar 

  27. Moore R (1979) Method and application of interval analysis, vol 2. Siam

  28. Rao SS, Berke L (1997) Analysis of uncertain structural systems using interval analysis. AIAA J 35(4):727–735. https://doi.org/10.2514/2.164

    Article  MATH  Google Scholar 

  29. Queipo NV, Haftka RT, Wei S, Goel T, Vaidyanathan R, Tucker PK (2005) Surrogate-based analysis and optimization. Prog Aerosp Sci 41(1):1–28

    Article  Google Scholar 

  30. Forrester AIJ, Sóbester A, Keane AJ (2008) Engineering design via surrogate modelling: a practical guide. Wiley, Chichester

  31. Moore RE, Moore RE (1979) Methods and apolications of interval analysis. Society for Industrial & Applied Mathematics, Philadelphia

    Book  Google Scholar 

  32. Jaulin L, Kieffer M, Didrit O (2001) Applied interval analysis. Springer, Berlin, p 5

    Book  MATH  Google Scholar 

  33. Mottershead JE, Friswell MI (1993) Model updating in structural dynamics: a survey. J Sound Vib 167(2):347–375

    Article  MATH  Google Scholar 

  34. Hemez FM, Doebling SW (2001) Review and Assessment of Model Updating for Non-Linear Transient Dynamics. Mech Syst Signal Process 15(1):45–74

    Article  Google Scholar 

  35. Teughels A, Maeck J, De Roeck G (2002) Damage assessment by FE model updating using damage functions. Comput Struct 80(25):1869–1879. https://doi.org/10.1016/S0045-7949(02)00217-1

    Article  Google Scholar 

  36. Fang SE, Zhang QH, Ren WX (2015) An interval model updating strategy using interval response surface models. Mech Syst Signal Process 60–61:909–927

    Article  Google Scholar 

  37. Deng Z, Guo Z, Zhang X (2017) Interval model updating using perturbation method and radial basis function neural networks. Mech Syst Signal Process 84:699–716

    Article  Google Scholar 

  38. Jiang C, Liu GR, Han X (2008) A novel method for uncertainty inverse problems and application to material characterization of composites. Exp Mech 48(4):539–548

    Article  Google Scholar 

  39. Liu J, Han X, Jiang C, Ning HM, Bai YC (2011) Dynamic load identification for uncertain structures based on interval analysis and regularization method. Int J Comput Methods 08(4):667–683

    Article  MathSciNet  MATH  Google Scholar 

  40. Feng X, Zhuo K, Wu J, Godara V, Zhang Y (2016) A new interval inverse analysis method and its application in vehicle suspension design. SAE Int J Mater Manf 9(2):315–320. https://doi.org/10.4271/2016-01-0277

    Article  Google Scholar 

  41. Liu J, Cai H, Jiang C, Han X, Zhang Z (2018) An interval inverse method based on high dimensional model representation and affine arithmetic. Appl Math Model 63:732–743. https://doi.org/10.1016/j.apm.2018.07.009

    Article  MathSciNet  MATH  Google Scholar 

  42. Golub G, Hansen P, O’Leary D (1999) Tikhonov Regularization and Total Least Squares. SIAM J Matrix Anal Appl 21(1):185–194. https://doi.org/10.1137/S0895479897326432

    Article  MathSciNet  MATH  Google Scholar 

  43. Engl HW, Hanke M, Neubauer A (1996) Regularization of inverse problems, vol 375. Springer Science & Business Media

    Book  MATH  Google Scholar 

  44. Rahman S, Xu H (2010) A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics. Probab Eng Mech 65(13):2292–2292

    Google Scholar 

  45. Ma X, Zabaras N (2010) An adaptive high-dimensional stochastic model representation technique for the solution of stochastic partial differential equations. J Comput Phys 229(10):3884–3915

    Article  MathSciNet  MATH  Google Scholar 

  46. Li G, Zhang K (2011) A combined reliability analysis approach with dimension reduction method and maximum entropy method. Struct Multidiscip Optim 43(1):121–134

    Article  MATH  Google Scholar 

  47. Huang X, Zhang Y (2013) Reliability–sensitivity analysis using dimension reduction methods and saddlepoint approximations. Int J Numer Meth Eng 93(8):857–886. https://doi.org/10.1002/nme.4412

    Article  MathSciNet  MATH  Google Scholar 

  48. Lee G, Yook S, Kang K, Choi DH (2012) Reliability-based design optimization using an enhanced dimension reduction method with variable sampling points. Int J Precis Eng Manuf 13(9):1609–1618

    Article  Google Scholar 

  49. Ren X, Yadav V, Rahman S (2016) Reliability-based design optimization by adaptive-sparse polynomial dimensional decomposition. Springer-Verlag New York, Inc.

    Book  MATH  Google Scholar 

  50. Chen SH, Ma L, Meng GW, Guo R (2009) An efficient method for evaluating the natural frequencies of structures with uncertain-but-bounded parameters. Comput Struct 87(9):582–590. https://doi.org/10.1016/j.compstruc.2009.02.009

    Article  Google Scholar 

  51. Xu M, Du J, Wang C, Li Y (2017) A dimension-wise analysis method for the structural-acoustic system with interval parameters. J Sound Vib 394:418–433

    Article  Google Scholar 

  52. Tang JC, Fu CM (2017) A dimension-reduction interval analysis method for uncertain problems. CMES-Comput Model Eng Sci 113(3):239–259

    Google Scholar 

  53. Bucher CG (1988) Adaptive sampling—an iterative fast Monte Carlo procedure. Struct Saf 5(2):119–126. https://doi.org/10.1016/0167-4730(88)90020-3

    Article  Google Scholar 

  54. Mori Y, Ellingwood BR (1993) Time-dependent system reliability analysis by adaptive importance sampling. Struct Saf 12(1):59–73. https://doi.org/10.1016/0167-4730(93)90018-V

    Article  Google Scholar 

  55. Bollapragada R, Byrd R, Nocedal J (2018) Adaptive Sampling Strategies for Stochastic Optimization. SIAM J Optim 28(4):3312–3343. https://doi.org/10.1137/17m1154679

    Article  MathSciNet  MATH  Google Scholar 

  56. Au SK, Beck JL (1999) A new adaptive importance sampling scheme for reliability calculations. Struct Saf 21(2):135–158. https://doi.org/10.1016/S0167-4730(99)00014-4

    Article  Google Scholar 

  57. Goldberg DE (1990) Genetic algorithms in search. Optim Mach Learn xiii(7):2104–2116

    Google Scholar 

  58. Kumar V (2002) Introduction to parallel computing. Addison-Wesley Longman Publishing Co., Inc.

    Google Scholar 

  59. Phillips JC, Braun R, Wang W, Gumbart J, Tajkhorshid E, Villa E, Chipot C, Skeel RD, Kalé L, Schulten K (2005) Scalable molecular dynamics with NAMD. J Comput Chem 26(16):1781–1802. https://doi.org/10.1002/jcc.20289

    Article  Google Scholar 

  60. Coelho PG, Cardoso JB, Fernandes PR, Rodrigues HC (2011) Parallel computing techniques applied to the simultaneous design of structure and material. Adv Eng Softw 42(5):219–227. https://doi.org/10.1016/j.advengsoft.2010.10.003

    Article  MATH  Google Scholar 

  61. Gao W, Kemao Q (2012) Parallel computing in experimental mechanics and optical measurement: A review. Opt Lasers Eng 50(4):608–617. https://doi.org/10.1016/j.optlaseng.2011.06.020

    Article  Google Scholar 

  62. Rubinstein RY (2008) Simulation and the Monte Carlo Method. Wiley

    MATH  Google Scholar 

  63. Wu T-J, Sepulveda A (1998) The weighted average information criterion for order selection in time series and regression models. Statist Probab Lett 39(1):1–10. https://doi.org/10.1016/S0167-7152(98)00003-0

    Article  MATH  Google Scholar 

  64. Wu T-J, Chen P, Yan Y (2013) The weighted average information criterion for multivariate regression model selection. Signal Process 93(1):49–55. https://doi.org/10.1016/j.sigpro.2012.06.017

    Article  Google Scholar 

Download references

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No. 51905257); the Natural Science Foundation of Hunan Province (Grant No: 2020JJ6075); the Outstanding Youth Foundation of Hunan Education Department (Grant No: 18B301); the Natural Science Foundation of Hebei Province (Grant No. A2019202171), the Changsha Municipal Natural Science Foundation (Grant No: kq2014050) and the Open Fund of State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body (Grant No. 31915004).

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to Jiachang Tang or Chenji Mi.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary file1 (M 2 kb)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tang, J., Cao, L., Mi, C. et al. Interval assessments of identified parameters for uncertain structures. Engineering with Computers 38 (Suppl 4), 2905–2917 (2022). https://doi.org/10.1007/s00366-021-01432-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-021-01432-5

Keywords

Navigation