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A new method to solve the structural reliability index based on homotopy analysis

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Abstract

In this paper, we introduce a homotopy analysis method into the structural reliability analysis and propose a different algorithm from the traditional Hasofer–Lind–Rackwitz–Fiessler type of iterations to solve the reliability index. Due to the powerful function of homotopy analysis in solving nonlinear equations, the method presented in this paper yields high efficiency and strong convergence of the reliability analysis. We first establish the Karush–Kuhn–Tucker (KKT) condition of the optimization problem for reliability index in the first-order reliability method. Next, we construct the corresponding combined homotopy equations for the KKT condition and apply the path-tracking algorithm to efficiently solve the equations. Finally, several numerical examples and an engineering application are provided to validate the effectiveness of the present method.

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References

  1. Hasofer, A.M., Lind, N.C.: Exact and invariant second-moment code format. J. Eng. Mech. Div. 100(1), 111–121 (1974)

    Google Scholar 

  2. Rackwitz, R., Flessler, B.: Structural reliability under combined random load sequences. Comput. Struct. 9(5), 489–494 (1978)

    Article  MATH  Google Scholar 

  3. Hohenbichler, M., Rackwitz, R.: Non-normal dependent vectors in structural safety. J. Eng. Mech. Div. 107(6), 1227–1238 (1981)

    Google Scholar 

  4. Liu, P.L., Der Kiureghian, A.: Optimization algorithms for structural reliability. Struct. Saf. 9(3), 161–177 (1991)

    Article  Google Scholar 

  5. Zhang, Y., der Kiureghian, A.: Two improved algorithms for reliability analysis. In: Rackwitz, R., et al. (eds) Reliability and Optimization of Structural Systems. Proceedings of the Sixth IFIP WG7.5 Working Conference on Reliability and Optimization of Structural Systems. Assisi, Italy (1995)

  6. Lee, J.O., Yang, Y.S., Ruy, W.S.: A comparative study on reliability-index and target-performance-based probabilistic structural design optimization. Comput. Struct. 80(3), 257–269 (2002)

    Article  Google Scholar 

  7. Yang, D., Li, G., Cheng, G.: Convergence analysis of first order reliability method using chaos theory. Comput. Struct. 84(8), 563–571 (2006)

    Article  Google Scholar 

  8. Yang, D.: Chaos control for numerical instability of first order reliability method. Commun. Nonlinear. Sci. 15(10), 3131–3141 (2010)

    Article  MATH  Google Scholar 

  9. Zhang, Z.: Computing reliability index of structure by use of Lagrange multiplier method. Eng. Mech., 11(1), 90–98 (1994) (in Chinese)

  10. Li, G., Cheng, G.D.: Performance-based seismic design of structures, method and application. Science Press, Beijing (2004). (in Chinese)

  11. Gong, J.X.: Engineering Structural Reliability Calculation Method. Dut. Press, Dalian (2003). (in Chinese)

  12. Kang, Z., Luo, Y.J.: A modified iteration algorithm for structural reliability index evaluation. Eng. Mech. 25(11), 20–26 (2008) (in Chinese)

  13. Liu, P.L., Der Kiureghian, A.: Multivariate distribution models with prescribed marginals and covariances. Probab. Eng. Mech. 1(2), 105–112 (1986)

    Article  Google Scholar 

  14. Madsen, H.O., Krenk, S., Lind, N.C.: Methods of Structural Safety, Prentice-Hall, Englewood Cliffs, NJ (1986)

  15. Kellogg, R.B., Li, T.Y., Yorke, J.: A constructive proof of the Brouwer fixed-point theorem and computational results. SIAM. J. Numer. Anal. 13(4), 473–483 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  16. Watson, L.T.: Theory of globally convergent probability-one homotopies for nonlinear programming. SIAM. J. Optim. 11(3), 761–780 (2001)

    Article  Google Scholar 

  17. Smale, S.: A convergent process of price adjustment and global Newton methods. J. Math. Econ. 3(2), 107–120 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  18. Chow, S.N., Mallet-Paret, J., Yorke, J.A.: Finding zeroes of maps: homotopy methods that are constructive with probability one. Math. Comput. 32(143), 887–899 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  19. Karmarkar, N.: A new polynomial-time algorithm for linear programming. In: Proceedings of the Sixteenth Annual ACM Symposium on Theory of Computing, pp. 302–311. ACM (1984)

  20. Allgower, E.L., Georg, K.: Numerical Continuation Methods. Springer, Berlin (1990)

    Book  MATH  Google Scholar 

  21. Liao, S.J.: An explicit, totally analytic approximate solution for Blasius’ viscous flow problems. Int. J. Nonlinear. Mech. 34(4), 759–778 (1999)

    Article  MATH  Google Scholar 

  22. Lin, Z.H., Li, Y., Yu, B.: A combined homotopy interior point method for general nonlinear programming problems. Appl. Math. Comput. 80(2), 759–778 (1996)

    MathSciNet  Google Scholar 

  23. Lin, Z.H., Yu, B., Feng, G.C.: A combined homotopy interior point method for convex nonlinear programming. Appl. Math. Comput. 84(2), 193–211 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  24. Jiang, C., Deng, S.L.: Multi-objective optimization and design considering automotive high-speed and low-speed crashworthiness, Chinese. J. Comput. Mech. 31(4), 471–479 (2014) (in Chinese)

  25. Marzougui, D., Samaha, R.R., Cui, C.: Extended Validation of the Finite Element Model for the 2010 Toyota Yaris Passenger Sedan. Working Paper, NCAC 2012-W-005 (2012)

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Authors and Affiliations

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

    Chao Jiang, Shuo Han, Meng Ji & Xu Han

Authors
  1. Chao Jiang
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  2. Shuo Han
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  3. Meng Ji
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  4. Xu Han
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Correspondence to Chao Jiang.

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Jiang, C., Han, S., Ji, M. et al. A new method to solve the structural reliability index based on homotopy analysis. Acta Mech 226, 1067–1083 (2015). https://doi.org/10.1007/s00707-014-1226-x

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  • DOI: https://doi.org/10.1007/s00707-014-1226-x

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