Abstract
A vital challenge problem of structural reliability analysis is how to estimate the small failure probability with a minimum number of model evaluations. The Adaptive Kriging combined with Importance Sampling method (AK-IS) which is developed from the adaptive Kriging combined with Monte Carlo simulation (AK-MCS) is a viable method to deal with this challenge. The aim of this paper is to reduce the number of model evaluations of the existing AK-IS algorithm. Firstly, we use a contributive weight function to divide the candidate samples of model input variables generated in AK-IS. The candidate samples are used to select the best next sample to update the Kriging model in AK-IS. Secondly, select the best next sample only in the important area obtained according to the contributive weight value to failure probability to update the Kriging model until the stopping condition is satisfied. Thirdly, use the Kriging model constructed in the important area to predict the other area and update the important area by adding the point with the maximum contributive weight value in the area except the important area ceaselessly until the probability of the accurate identification on the limit state function’s signs (positive limit state value or negative limit state value) of all the importance sampling points satisfies a criterion. Finally, the updated Kriging model is used to estimate the failure probability especially for the small failure probability. The proposed method uses a thought from local to global in order to reduce the computational cost of AK-IS and simultaneously guarantees the accuracy of estimation. A non-linear oscillator system, a roof truss structure and a planar ten-bar structure are analyzed by the proposed method. The results demonstrate the efficiency and accuracy of the proposed method in structural reliability analysis especially for small failure probability.
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References
Au SK, Beck JL (1999) A new adaptive importance sampling scheme. Struct Saf 21:135–158
Au SK, Beck JL (2002) Importance sampling in high dimensions. Struct Saf 25:139–163
Balesdent M, Morio J, Marzat J (2013) Kriging-based adaptive importance sampling algorithms for rare event estimation. Struct Saf 44:1–10
Bourinet JM, Deheeger F, Lemaire M (2011) Assessing small failure probabilities by combined subset simulation and support vector machines. Struct Saf 33(6):343–353
Der Kiureghian A, Stefano M (1991) Efficient algorithm for second-order reliability analysis. J Eng Mech ASCE 117(2):2904–2923
Echard B, Gayton N, Lemaire M (2011) AK-MCS: An active learning reliability method combining Kriging and Monte Carlo Simulation. Struct Saf 33:145–154
Echard B, Gayton N, Lemaire M, Relun N (2013) A combined importance sampling and Kriging reliability method for small failure probabilities with time-demanding numerical models. Reliab Eng Syst Saf 111:232–240
Fauriat W, Gayton N (2014) AK-SYS: An adaptation of the AK-MCS method for system reliability. Reliab Eng Syst Saf 123:137–144
Grooteman F (2008) Adaptive radial-based importance sampling method for structural reliability. Struct Saf 30:533–542
Harbitz A (1986) An efficient sampling method for probability of failure calculation. Struct Saf 3:109–115
Hasofer AM, Lind NC (1974) An exact and invariant first order reliability format. J Eng Mech ASCE 100(1):111–121
Hu Z, Mahadevan S (2016) Global sensitivity analysis-enhanced surrogate (GSAS) modeling for reliability analysis. Struct Multidiscip Optim 53:501–521
Kim SH, Na SW (1997) Response surface method using vector projected sampling points. Struct Saf 19(1):3–19
Melchers RE (1989) Importance sampling in structural system. Struct Saf 6:3–10
Papadrakakis M, Lagaros N (2001) Reliability-based structural optimization using neural networks and Monte Carlo simulation. Comput Methods Appl Mech Eng 191(32):3491–3507
Rajashekhar MR, Ellingwood BR (1993) A new look at the response surface approach for reliability analysis. Struct Saf 12(3):205–220
Rubinstein RY, Kroese DP (2016) Simulation and the Monte Carlo method. Wiley, Hokoben
Sacks J, Schiller SB, Welch WJ (1989) Design for computer experiment. Technometrics 31(1):41–47
Schueremans L, Van GD (2005) Benefit of splines and neural networks in simulation based structural reliability analysis. Struct Saf 37(3):246–261
Song H, Choi KK, Lee CI, Zhao L, Lamb D (2013) Adaptive virtual support vector machine for reliability analysis of high-dimensional problems. Struct Multidiscip Optim 47:479–491
Xu L, Cheng GD (2003) Discussion on: moment methods for structural reliability. Struct Saf 25:193–199
Yun WY, Lu ZZ, Jiang X (2017) A modified importance sampling method for structural reliability and its global reliability sensitivity analysis. Struct Multidiscip Optim. https://doi.org/10.1007/s00158-017-1832-z
Zhao YG, Ono T (1999a) A general procedure for first/s-order reliability method (FORM/SORM). Struct Saf 21(2):95–112
Zhao YG, Ono T (1999b) New approximations for SORM: part 1. J Eng Mech ASCE 125(1):79–85
Zhao YG, Ono T (2001) Moment method for structural reliability. Struct Saf 23(1):47–75
Zhao YG, Ono T (2004) On the problems of the fourth moment method. Struct Saf 26(3):343–347
Zhao YG, Lu ZH, Ono T (2006) A simple-moment method for structural reliability. J Asian Archit Build Eng 5(1):129–136
Zhao HL, Yue ZF, Liu YS, Gao ZZ, Zhang YS (2015) An efficient reliability method combining adaptive importance sampling and Kriging metamodel. Appl Math Model 39:1853–1866
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant 51,475,370, 51,775,439), the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (Grant CX201708), and the Natural Science Basic Research Plan in Shaanxi Province of China (2017JQ1007) behind (Grant CX201708).
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Appendix 1. Kriging model
Appendix 1. Kriging model
The Kriging model is a semi-parametric interpolation technique based on the statistical theory (Sacks et al. 1989) including the parametric linear regression part and the nonparametric stochastic process. For an unknown function g(x), the Kriging model is given as follows:
where B(x) = [B1(x), B2(x), …, Bp(x)]T are the base functions of vector x, β = [β1, β2, …, βp]T is the regression coefficient vector, and p denotes the number of base functions. Z(x) is a stationary Gaussian process with zero mean and covariance which can be defined as follows:
where N0 denotes the number of training points.
Define \( \boldsymbol{R}=\left[\begin{array}{ccc}R\left(\left({\boldsymbol{x}}^{(1)}\right),\left({\boldsymbol{x}}^{(2)}\right)\right)& \cdots & R\left(\left({\boldsymbol{x}}^{(1)}\right),\left({\boldsymbol{x}}^{\left({N}_0\right)}\right)\right)\\ {}\vdots & \ddots & \vdots \\ {}R\left(\left({\boldsymbol{x}}^{\left({N}_0\right)}\right),\left({\boldsymbol{x}}^{(1)}\right)\right)& \cdots & R\left(\left({\boldsymbol{x}}^{\left({N}_0\right)}\right),\left({\boldsymbol{x}}^{\left({N}_0\right)}\right)\right)\end{array}\right] \), B is a vector of B(X) and g is corresponding vector of the limit state functions calculated at each experimental points (x(i))(i = 1, 2, …, N0), the unknown β and σ2 can be estimated as follows:
Therefore, for any unknown point x, the Best Linear Unbiased Predictor of model gK(x) is shown to be a Gaussian random \( {g}_K\left(\boldsymbol{x}\right)\sim N\left({\mu}_{g_K}\left(\boldsymbol{x}\right),{\sigma}_{g_K}\left(\boldsymbol{x}\right)\right) \) where the mean and the variance are given as follows:
where \( {\boldsymbol{r}}^{\mathrm{T}}\left(\boldsymbol{x}\right)={\left[R\left(\left(\boldsymbol{x}\right),\left({\boldsymbol{x}}^{(1)}\right)\right),\dots, R\left(\left(\boldsymbol{x}\right),\left({\boldsymbol{x}}^{\left({N}_0\right)}\right)\right)\right]}^{\mathrm{T}} \).
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Yun, W., Lu, Z. & Jiang, X. An efficient reliability analysis method combining adaptive Kriging and modified importance sampling for small failure probability. Struct Multidisc Optim 58, 1383–1393 (2018). https://doi.org/10.1007/s00158-018-1975-6
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DOI: https://doi.org/10.1007/s00158-018-1975-6