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Static response analysis of structures with interval parameters using the second-order Taylor series expansion and the DCA for QB

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Abstract

In this paper, based on the second-order Taylor series expansion and the difference of convex functions algorithm for quadratic problems with box constraints (the DCA for QB), a new method is proposed to solve the static response problem of structures with fairly large uncertainties in interval parameters. Although current methods are effective for solving the static response problem of structures with interval parameters with small uncertainties, these methods may fail to estimate the region of the static response of uncertain structures if the uncertainties in the parameters are fairly large. To resolve this problem, first, the general expression of the static response of structures in terms of structural parameters is derived based on the second-order Taylor series expansion. Then the problem of determining the bounds of the static response of uncertain structures is transformed into a series of quadratic problems with box constraints. These quadratic problems with box constraints can be solved using the DCA approach effectively. The numerical examples are given to illustrate the accuracy and the efficiency of the proposed method when comparing with other existing methods.

Graphical Abstract

Although the first-order parameter perturbation method (FM) is effective for solving the static response problem of structures with interval parameters with small uncertainties, the method may fail to estimate the region of the static response of uncertain structures if the uncertainties in the parameters are fairly large. To resolve this problem, a new approach (the proposed method:PM) is presented based on the second-order Taylor series expansion and the DCA for QB(the difference of convex functions algorithm for quadratic problems with box constraints).

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References

  1. Stefanou, G.: The stochastic finite element method: Past, present and future. Comput. Methods Appl. Mech. Eng. 198, 1031–1051 (2009)

    Article  MATH  Google Scholar 

  2. Moore, R.E.: Methods and Applications of Interval Analysis. Prentice-Hall Inc., London (1979)

  3. Alefeld, G., Herzberger, J.: Introductions to Interval Computations. Academic Press, New York (1983)

    Google Scholar 

  4. Rao, S.S., Berke, L.: Analysis of uncertain structural systems using interval analysis. AIAA J. 35, 727–735 (1997)

    Article  MATH  Google Scholar 

  5. Rohn, J., Kreinovich, V.: Computing exact componentwise bounds on solutions of linear systems with interval data is NP-hard. SIAM J. Matrix Anal. Appl. 16, 4–15 (1995)

    Article  MathSciNet  Google Scholar 

  6. Qiu, Z.P., Wang, X.J., Chen, G.Y.: Exact bounds for the static response set of structures with uncertain-but-bounded parameters. Int. J. Solids Struct. 43, 6574–6593 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Qiu, Z.P., Chen, S.H., Song, D.: The displacement bound estimation for structures with an interval description of uncertain parameters. Commun. Numer. Methods Eng. 12, 1–11 (1996)

    Article  MATH  Google Scholar 

  8. Qiu, Z.P., Elishakoff, I.: Antioptimization of structures with large uncertain-but-non-random parameters via interval analysis. Comput. Methods Appl. Mech. Eng. 152, 361–372 (1998)

    Article  MATH  Google Scholar 

  9. McWilliam, S.: Anti-optimisation of uncertain structures using interval analysis. Comput. Struct. 79, 421–430 (2001)

    Article  Google Scholar 

  10. Chen, S.H., Yang, X.W.: Interval finite element method for beam structures. Finite Elem. Anal. Des. 34, 75–78 (2000)

    Article  MATH  Google Scholar 

  11. Chen, S.H., Lian, H.D., Yang, X.W.: Interval static displacement analysis for structures with interval parameters. Int. J. Numer. Meth. Eng. 53, 393–407 (2002)

    Article  MATH  Google Scholar 

  12. Vavasis, S.A.: Nonlinear Optimization: Complexity Issues. Oxford University Press, New York (1991)

    MATH  Google Scholar 

  13. Tao, P.D., Souad, E.B.: Duality in d.c. (difference of convex functions) optimization: Subgradient methods, trends in mathematical optimization. Int. Ser. Numer. Math. 84, 276–294 (1988)

    Google Scholar 

  14. An, L.T.H.: Tao, P.D.: A branch and bound method via d.c. optimization algorithms and ellipsoidal technique for box constrained nonconvex quadratic problems. J. Global Optim. 13, 171–206 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Shi, Y.H., Eberhart, R.C.: A modified particle swarm optimizer. In: Proceeding of the IEEE Congress on Evolutionary Computation, 69–73 (1998)

  16. Svanberg, K.: The method of moving asymptotes–a new method for structural optimization. Int. J. Numer. Methods. Eng. 24, 359–373 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  17. Tao, P.D., An, L.T.H.: D.C. optimization algorithm for solving the trust region sub-problem. SIAM J. Optim. 8, 1–30 (1998)

    Article  MathSciNet  Google Scholar 

  18. Qiu, Z.P.: Comparison of static response of structures using convex models and interval analysis method. Int. J. Numer. Meth. Eng. 56, 1735–1753 (2003)

    Article  MATH  Google Scholar 

  19. Guo, X., Bai, W., Zhang, W.S.: Extreme structural response analysis of truss structures under material uncertainty via linear mixed 0–1 programming. Int. J. Numer. Meth. Eng. 76, 253–277 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grants 11002013, 11372025), the Defense Industrial Technology Development Program (Grants A0820132001, JCKY2013601B), the Aeronautical Science Foundation of China (Grant 2012ZA51010) and 111 Project (Grant B07009) for support.

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  1. Institute of Solid Mechanics, Beihang University (Beijing University of Aeronautics and Astronautics), Beijing, 100191, China

    Qi Li, Zhiping Qiu & Xudong Zhang

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  1. Qi Li
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  2. Zhiping Qiu
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  3. Xudong Zhang
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Correspondence to Qi Li.

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Li, Q., Qiu, Z. & Zhang, X. 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). https://doi.org/10.1007/s10409-015-0501-y

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  • DOI: https://doi.org/10.1007/s10409-015-0501-y

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