Acta Mechanica Sinica

, Volume 34, Issue 4, pp 653–666 | Cite as

Structural eigenvalue analysis under the constraint of a fuzzy convex set model

  • Wencai SunEmail author
  • Zichun Yang
  • Guobing Chen
Research Paper


In small-sample problems, determining and controlling the errors of ordinary rigid convex set models are difficult. Therefore, a new uncertainty model called the fuzzy convex set (FCS) model is built and investigated in detail. An approach was developed to analyze the fuzzy properties of the structural eigenvalues with FCS constraints. Through this method, the approximate possibility distribution of the structural eigenvalue can be obtained. Furthermore, based on the symmetric F-programming theory, the conditional maximum and minimum values for the structural eigenvalue are presented, which can serve as non-fuzzy quantitative indicators for fuzzy problems. A practical application is provided to demonstrate the practicability and effectiveness of the proposed methods.


Structural eigenvalue Fuzzy Convex set Conditional extreme Symmetric F-programming 



This work was supported by the National Natural Science Foundation of China (Grant 51509254).


  1. 1.
    Atluri, S.N.: Methods of Computer Modeling in Engineering & the Sciences. Tech Science Press, Palmdale (2005)Google Scholar
  2. 2.
    Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach, Revised edn. Dover, New York (2003)zbMATHGoogle Scholar
  3. 3.
    Liu, W.K., Ted, B., Mani, A.: Random field finite elements. Int. J. Numer. Methods Eng. 23, 1831–1845 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Zhou, X.Y., Gosling, P.D., Ullah, Z., et al.: Stochastic multi-scale finite element based reliability analysis for laminated composite structures. Appl. Math. Model. 45, 457–473 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Balomenos, G.P., Pandey, M.D.: Probabilistic finite element investigation of prestressing loss in nuclear containment wall segments. Nucl. Eng. Des. 311, 50–59 (2017)CrossRefGoogle Scholar
  6. 6.
    Coombs, D.J., Rullkoetter, P.J., Laz, P.J.: Efficient probabilistic finite element analysis of a lumbar motion segment. J. Biomech. 61, 65–74 (2017)CrossRefGoogle Scholar
  7. 7.
    Elishakoff, I.: Possible limitations of probabilistic methods in engineering. Appl. Mech. Rev. 53, 19–36 (2000)CrossRefGoogle Scholar
  8. 8.
    Moens, D., Vandepitte, D.: A survey of non-probabilistic uncertainty treatment in finite element analysis. Comput. Methods Appl. Mech. Eng. 194, 1527–1555 (2005)CrossRefzbMATHGoogle Scholar
  9. 9.
    Muhanna, R., Mullen, R.: Formulation of fuzzy finite-element methods for solid mechanics problems. Comput. Aided Civ. Infrastruct. Eng. 14, 107–117 (1999)CrossRefGoogle Scholar
  10. 10.
    Moens, D., Vandepitte, D.: Fuzzy finite element method for frequency response function analysis of uncertain structures. AIAA J. 40, 126–136 (2002)CrossRefGoogle Scholar
  11. 11.
    Haddad Khodaparast, H., Govers, Y., Dayyani, I., et al.: Fuzzy finite element model updating of the DLR AIRMOD test structure. Appl. Math. Model. 52, 512–526 (2017)CrossRefGoogle Scholar
  12. 12.
    Wang, X.J., Wang, L.: Uncertainty quantification and propagation analysis of structures based on measurement data. Math. Comput. Model. 54, 2725–2735 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Wang, X.J., Wang, L., Qiu, Z.P.: A feasible implementation procedure for interval analysis method from measurement data. Appl. Math. Model. 38, 2377–2397 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Guo, X., Bai, W., Zhang, W.: Confidence extremal structural response analysis of truss structures under static load uncertainty via SDP relaxation. Comput. Struct. 87, 246–253 (2009)CrossRefGoogle Scholar
  15. 15.
    Guo, X., Bai, W., Zhang, W.: Extreme structural response analysis of truss structures under material uncertainty via linear mixed 0–1 programming. Int. J. Numer. Methods Eng. 76, 253–277 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Jiang, C., Fu, C.M., Ni, B.Y., et al.: Interval arithmetic operations for uncertainty analysis with correlated interval variables. Acta Mech. Sin. 32, 743–752 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Li, Q., Qiu, Z.P., Zhang, X.D.: Static response analysis of structures with interval parameters using the second-order Taylor series expansion and the DCA for QB. Acta Mech. Sin. 31, 845–854 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Sofi, A., Romeo, E.: A novel interval finite element method based on the improved interval analysis. Comput. Methods Appl. Mech. Eng. 311, 671–697 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Faes, M., Cerneels, J., Vandepitte, D., et al.: Identification and quantification of multivariate interval uncertainty in finite element models. Comput. Methods Appl. Mech. Eng. 315, 896–920 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Li, Q., Qiu, Z.P., Zhang, X.D.: Eigenvalue analysis of structures with interval parameters using the second-order Taylor series expansion and the DCA for QB. Appl. Math. Model. 49, 680–690 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Sofi, A., Muscolino, G., Elishakoff, I.: Natural frequencies of structures with interval parameters. J. Sound Vib. 347, 79–95 (2015)CrossRefzbMATHGoogle Scholar
  22. 22.
    Wang, L., Wang, X.J., Xia, Y.: Hybrid reliability analysis of structures with multi-sources uncertainties. Acta Mech. 225, 413–430 (2014)CrossRefzbMATHGoogle Scholar
  23. 23.
    Guo, X., Bai, W., Zhang, W., et al.: Confidence structural robust design and optimization under stiffness and load uncertainties. Comput. Methods Appl. Mech. Eng. 198, 3378–3399 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sun, W.C., Yang, Z.C., Li, K.F.: Non-deterministic fatigue life analysis using convex set models. Sci. China (Phys. Mech. Astron.) 56, 765–774 (2013)CrossRefGoogle Scholar
  25. 25.
    Yang, Z.C., Sun, W.C.: A set-based method for structural eigenvalue analysis using Kriging model and PSO algorithm. Comput. Model. Eng. Sci. 92, 193–212 (2013)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Zadeh, L.A.: Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst. 1, 3–28 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Yang, L.B., Gao, Y.Y.: The Principle and Application of Fuzzy Mathematics. Press of South China University of Technology, Guangzhou (2005) (in Chinese)Google Scholar
  28. 28.
    Li, Y.H., Huang, H.Z., Liu, Z.H.: Convex model in robust reliability analysis of structure. J. Basic Sci. Eng. 12, 383–391 (2004) (in Chinese)Google Scholar
  29. 29.
    Li, K.F.: Study on the non-probabilistic reliability methods for structures based on info-gap theory. Dissertation for doctoral degree. Naval University of Engineering, Wuhan (2012) (in Chinese)Google Scholar
  30. 30.
    Hansen, E.: Interval forms of Newton’s method. Computing 20, 153–163 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Qiu, Z.P.: Convex Method Based on Non-probabilistic Set-Theory and Its Application. National Defense Industry Press, Beijing (2005)Google Scholar
  32. 32.
    Atluri, S.N.: The Meshless Method (MLPG) for Domain & BIE Discretizations. Tech Science Press, Forsyth (2003)zbMATHGoogle Scholar
  33. 33.
    Hosseini, S.M., Shahabian, F., Sladek, J., et al.: Stochastic meshless local Petrov–Galerkin (MLPG) method for thermo-elastic wave propagation analysis in functionally graded thick hollow cylinders. Comput. Model. Eng. Sci. 71, 39–66 (2011)MathSciNetzbMATHGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Power EngineeringNaval University of EngineeringWuhanChina

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