A novel mixed uncertainty support vector machine method for structural reliability analysis


For structures with both random and fuzzy uncertainty parameters, a novel method for obtaining the membership function of reliability in the fuzziness failure criterion is presented using a Random Fuzzy Support Vector Machine based on the Particle Swarm Optimization method (PSO-RFSVM). The proposed method is used to solve the structural reliability problem with implicit limit state function in the presence of mixed variables. Under each cut level, a new distance measure between the mixed variables is presented, named the advanced Yang distance; then, the RFSVM model can be constructed using the kernel function built by the advanced Yang distance and random fuzzy mixed sampling points. Furthermore, to obtain satisfactory fitting, a PSO algorithm is used to optimize the super-parameters in RFSVM. The limit state function is subsequently approximated at the given cut level, and then, the reliability bound under the given cut level is readily obtained. The PSO-RFSVM method provides a new efficient analysis framework in the case of implicit approximation under mixed uncertainty when the classical method has excessive consumption in virtual prototype simulation. Two examples are used to demonstrate the validity and advantage of PSO-RFSVM compared to the Extreme Response Surface based on Simulated Annealing (SAERS) method and Monte Carlo simulation (MCS).

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  1. 1.

    Kiureghian, A.D., Stefano, M.D.: Efficient algorithm for second-order reliability analysis. J. Eng. Mech. 117(12), 2904–2923 (1991)

    Google Scholar 

  2. 2.

    Chiralaksanakul, A., Mahadevan, S.: First-order approximation methods in reliability-based design optimization. J. Mech. Des. 127(5), 851–857 (2005)

    Google Scholar 

  3. 3.

    Roudak, M.A., Shayanfar, M.A., Barkhordari, M.A., Karamloo, M.: A robust approximation method for nonlinear cases of structural reliability analysis. Int. J. Mech. Sci. 133, 11–20 (2017)

    Google Scholar 

  4. 4.

    Grandhi, R.V., Wang, L.P.: Higher-order failure probability calculation using nonlinear approximations. J. Comput. Meth. Appl. Mech. Eng. 168(1–4), 185–206 (1999)

    MATH  Google Scholar 

  5. 5.

    Yang, L.C., Guo, Y.L., Kong, Z.F.: On the performance evaluation of a hierarchical-structure prototype product using inconsistent prior information and limited test data. Inf. Sci. 485, 362–375 (2019)

    Google Scholar 

  6. 6.

    Melchers, R.E.: Importance sampling in structural systems. Struct. Saf. 6(1), 3–10 (1989)

    Google Scholar 

  7. 7.

    Melchers, R.E.: Radial importance sampling for structural reliability. J. Eng. Mech. 116(1), 189–203 (1990)

    Google Scholar 

  8. 8.

    Koutsourelakis, P.S., Pradlwarter, H.J., Schuëller, G.I.: Reliability of structures in high dimensions, part i: algorithms and applications. J. Eng. Mech. 19(4), 409–417 (2004)

    Google Scholar 

  9. 9.

    Li, X.K., Qiu, H.B., Chen, Z.Z., Gao, L., Shao, X.Y.: A local kriging approximation method using mpp for reliability-based design optimization. Comput. Struct. 162, 102–115 (2016)

    Google Scholar 

  10. 10.

    Alabbas, Al.-A., Michael, H.S.: Response sensitivity for geometrically nonlinear displacement-based beam-column elements. Comput. Struct. 220, 43–54 (2019)

    Google Scholar 

  11. 11.

    Jiang, C., Zhang, Q.F., Han, X., Qian, Y.H.: A non-probabilistic structural reliability analysis method based on a multidimensional parallelepiped convex model. Acta Mech. 225(2), 383–395 (2014)

    Google Scholar 

  12. 12.

    Cornelis, C., Cock, M.D., Kerre, E.: Representing reliability and hesitation in possibility theory: a general framework. Springer Press, Berlin (2004)

    Google Scholar 

  13. 13.

    Jiang, C., Lu, G.Y., Han, X., Liu, L.X.: A new reliability analysis method for uncertain structures with random and interval variables. Int. J. Mech. Mater. Des. 8(2), 169–182 (2012)

    Google Scholar 

  14. 14.

    Long, X.Y., Mao, D.L., Jiang, C., Wei, F.Y., Li, G.J.: Unified uncertainty analysis under probabilistic, evidence, fuzzy and interval uncertainties. Comput. Meth. Appl. Mech. Eng. 355(1), 1–26 (2019)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Möller, B., Graf, W., Beer, M.: Safety assessment of structures in view of fuzzy randomness. Comput. Struct. 81(15), 1567–1582 (2003)

    Google Scholar 

  16. 16.

    Huang, H.Z.: Structural reliability analysis using fuzzy sets theory. Eksploat. Niezawodn. 14(4), 284–294 (2012)

    Google Scholar 

  17. 17.

    Khaniyev, T., Baskir, M.B., Gokpinar, F., Mirzayev, F.: Statistical distributions and reliability functions with type-2 fuzzy parameters. Eksploat. Niezawodn. 21(2), 268–274 (2019)

    Google Scholar 

  18. 18.

    Bagheri, M., Miri, M., Shabakhty, N.: Fuzzy reliability analysis using a new alpha level set optimization approach based on particle swarm optimization. J. Intell. Fuzzy Syst. 30(1), 235–244 (2016)

    Google Scholar 

  19. 19.

    Penmetsa, R.C., Grandhi, R.V.: Uncertainty propagation using possibility theory and function approximations. Mech. Based Des. Struct. Mech. 31(2), 257–279 (2003)

    Google Scholar 

  20. 20.

    Wang, Z.L., Li, Y.F., Huang, H.Z., Liu, Y.: Reliability analysis of structure for fuzzy safety state. J. Intell. Autom. Soft. Comput. 18(3), 215–224 (2012)

    Google Scholar 

  21. 21.

    Adduri, P.R., Penmetsa, R.C.: Confidence bounds on component reliability in the presence of mixed uncertain variables. Int. J. Mech. Sci. 50(3), 481–489 (2008)

    MATH  Google Scholar 

  22. 22.

    Li, L.Y., Lu, Z.Z.: Interval optimization based line sampling method for fuzzy and random reliability analysis. J. Appl. Math. Model. 38(13), 3124–3135 (2014)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Wang, C., Matthies, H.G., Xu, M.H., Li, Y.L.: Hybrid reliability analysis and optimization for spacecraft structural system with random and fuzzy parameters. J. Aerosp. Sci. Technol. 77, 353–361 (2018)

    Google Scholar 

  24. 24.

    Shi, Y., Lu, Z.Z., Zhou, Y.C.: Time-dependent safety and sensitivity analysis for structure involving both random and fuzzy inputs. Struct. Multidiscip. Optim. 58(6), 2655–2675 (2018)

    MathSciNet  Google Scholar 

  25. 25.

    Ebenuwa, A.U., Tee, K.F.: Fuzzy-based optimised subset simulation for reliability analysis of engineering structures. Struct. Infrastruct. Eng. 25(3), 413–425 (2019)

    Google Scholar 

  26. 26.

    Vapnik, V.N.: The nature of statistical learning theory. Springer Press, New York (1995)

    MATH  Google Scholar 

  27. 27.

    Guerbai, Y., Chibani, Y., Hadjadji, B.: The effective use the one-class SVM classifier for handwritten signature verification based on writer-independent parameters. Pattern Recognit. 48(1), 103–113 (2015)

    Google Scholar 

  28. 28.

    Fei, C.W., Bai, G.C.: Distributed collaborative probabilistic design for turbine blade-tip radial running clearance using support vector machine of regression. Mech. Syst. Sig. Process. 49(1–2), 196–208 (2014)

    Google Scholar 

  29. 29.

    Rocco, C.M., Moreno, J.A.: Fast Monte Carlo reliability evaluation using support vector machine. Reliab. Eng. Syst. Saf. 76(3), 237–243 (2002)

    MATH  Google Scholar 

  30. 30.

    Ghiasi, R., Torkzadeh, P., Noori, M.: A machine-learning approach for structural damage detection using least square support vector machine based on a new combinational kernel function. Struct. Health. Moni. 15(3), 302–316 (2016)

    Google Scholar 

  31. 31.

    Pan, Q.J., Dias, D.: An efficient reliability method combining adaptive support vector machine and Monte Carlo simulation. Struct. Saf. 67, 85–95 (2017)

    Google Scholar 

  32. 32.

    Ju, Y.P., Parks, G., Zhang, C.H.: A bisection-sampling-based support vector regression- high-dimensional model representation metamodeling technique for high-dimensional problems. Proc. IMechE. Part C J. Mech. Eng. Sci. 231(12), 2173–2186 (2017).

  33. 33.

    Lin, C.F., Wang, S.D.: Fuzzy support vector machines. IEEE Trans. Neural Networks 13(2), 464–471 (2002)

    Google Scholar 

  34. 34.

    Jaya, T., Dheeba, J., Singh, N.A.: Detection of hard exudates in colour fundus images using fuzzy support vector machine-based expert system. J. Digit. Imaging. 28(6), 761–768 (2015)

    Google Scholar 

  35. 35.

    Zhang, Y.D., Wang, S.H., Yang, X.J., Dong, Z.C., Liu, G., Phillips, P., Yuan, T.F.: Pathological brain detection in MRI scanning by wavelet packet Tsallis entropy and fuzzy support vector machine. Springer Plus 4(1), 1–16 (2015)

    Google Scholar 

  36. 36.

    Forghani, Y., Yazdi, H., Effati, S.: An extension to fuzzy support vector data description (FSVDD*). Pattern. Anal. Appl. 15(3), 237–247 (2012)

    MathSciNet  Google Scholar 

  37. 37.

    Li, H.S., Lu, Z.Z., Yue, Z.F.: Support Vector Machine for structural reliability analysis. Appl. Math. Mech. 27(10), 1295–1303 (2006)

    MATH  Google Scholar 

  38. 38.

    Guo, Z.W., Bai, G.C.: Application of least squares support vector machine for regression to reliability analysis. Chin. J. Aeronaut. 22(2), 160–166 (2009)

    Google Scholar 

  39. 39.

    Tan, X.H., Bi, W.H., Hou, X.L., Wang, W.: Reliability analysis using radial basis function networks and support vector machines. Comput. Geotech. 38(2), 178–186 (2011)

    Google Scholar 

  40. 40.

    Khatibinia, M., Fadaee, M.J., Salajegheh, J., Salajegheh, E.: Seismic reliability assessment of RC structures including soil–structure interaction using wavelet weighted least squares support vector machine. Reliab. Eng. Syst. Saf. 110, 22–33 (2013)

    Google Scholar 

  41. 41.

    Wang, Y.H., Zhao, X.Y., Wang, B.T.: LS-SVM and Monte Carlo methods based reliability analysis for settlement of soft clayey foundation. J. Rock. Mech. Geotech. Eng. 5, 312–317 (2013)

    Google Scholar 

  42. 42.

    Alibrandi, U., Alani, A.M., Ricciardi, G.: A new sampling strategy for SVM-based response surface for structural reliability analysis. Probab. Eng. Mech. 41, 1–12 (2015)

    Google Scholar 

  43. 43.

    Jiang, Y.B., Luo, J., Liao, G.Y., Zhao, Y.L.: An efficient method for generation of uniform support vector and its application in structural failure function fitting. Struct. Saf. 54, 1–9 (2015)

    Google Scholar 

  44. 44.

    Zhao, H.B., Li, S.J., Ru, Z.L.: Adaptive reliability analysis based on a support vector machine and its application to rock engineering. J. Appl. Math. Model. 44, 508–522 (2017)

    MATH  Google Scholar 

  45. 45.

    Feng, J.W., Liu, L., Wu, D., Li, G.Y., Beer, M., Gao, W.: Dynamic reliability analysis using the extended support vector regression (X-SVR). Mech. Syst. Signal. Proc. 126, 368–391 (2019)

    Google Scholar 

  46. 46.

    Shyamal, G., Atin, R., Subrata, C.: Support vector regression based metamodeling for seismic reliability analysis of structures. J. Appl. Math. Model. 64, 584–602 (2018)

    MathSciNet  MATH  Google Scholar 

  47. 47.

    Roy, A., Manna, R., Chakraborty, S.: Support vector regression based metamodeling for structural reliability analysis. Probab. Eng. Eng. Mech. 55, 78–89 (2019)

    Google Scholar 

  48. 48.

    Zhao, W., Tao, T., Zio, E., Wang, W.B.: A novel hybrid method of parameters tuning in support vector regression for reliability prediction: particle swarm optimization combined with analytical selection. IEEE. Trans. Reliab. 65(3), 1393–1405 (2016)

    Google Scholar 

  49. 49.

    Wang, Z.Q., Wang, P.F.: A new approach for reliability analysis with time-variant performance characteristics. Reliab. Eng. Syst. Saf. 115(115), 70–81 (2013)

    Google Scholar 

  50. 50.

    Radecki, T.: Level fuzzy sets. J. Cybern. 7(3), 189–198 (1977)

    MathSciNet  MATH  Google Scholar 

  51. 51.

    Yang, M.S., Ko, C.H.: On a class of fuzzy c-numbers clustering procedures for fuzzy data. Fuzzy. Sets. Syst. 84(1), 49–60 (1996)

    MathSciNet  MATH  Google Scholar 

  52. 52.

    Claude, J.P. Bélisle.: Convergence theorems for a class of simulated annealing algorithms on ℝd. J. Appl. Probab. 29(4), 885–895 (1992).

  53. 53.

    Rahul, M., Narinder, S., Yaduvir, S.: Genetic algorithms: concepts, design for optimization of process controllers. J. Com. Inf. Sci. 4(2), 39–54 (2011)

    Google Scholar 

  54. 54.

    Sekar, V., Zhang, M.Q., Shu, C.: Inverse design of airfoil using a deep convolutional neural network. AIAA J. 57(3), 993–1003 (2019)

    Google Scholar 

  55. 55.

    You, L.F., Zhang, J.G., Du, X.S., Wu, J.: A new structural reliability analysis method in presence of mixed uncertainty variables. Chin. J. Aeronaut. (2020). https://doi.org/10.1016/j.cja.2019.12.008

    Article  Google Scholar 

  56. 56.

    Anescu, G., Ulmeanua, A.P.: A no speeds and coefficients PSO approach to reliability optimization problems. Comput. Ind. Eng. 120, 31–41 (2018)

    Google Scholar 

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This research is supported by the National Natural Science Foundation of China (No. 51675026) and Aeronautical Science Foundation of China (2018ZC74001).

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You, LF., Zhang, JG., Zhou, S. et al. A novel mixed uncertainty support vector machine method for structural reliability analysis. Acta Mech 232, 1497–1513 (2021). https://doi.org/10.1007/s00707-020-02906-1

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